Algebraic Generalizations of Discrete Grs: A Path to Combinatorial Gr Theory Through One-Relator Products

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ALGEBRAIC GENERALIZATIONS OF DISCRETE GROUPS

PURE

AND APPLIED

MATHEMATICS

A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl J. Taft Rutgers University New Brunswick, New Jersey

EDITORIAL M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology

Zuhair Nashed University of Delaware Newark, Delaware

BOARD Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

Gian-Carlo Rota Massachusetts Institute of Technology

Marvin Marcus University of California, Santa Barbara

David L. Russell Virginia Polytechnic Institute and State University

W. S. Massey Yale University

Walter Schempp Universitgit Siegen

Mark Teply University of Wisconsin, Milwaukee

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS 1. K. Yano,Integral Formulas in Riemannian Geometry (1970) 2. S. Kobayashi,HyperbolicManifoldsandHolomorphic Mappings (1970) 3. V. S. Vladimirov,Equationsof Mathematical Physics(A. Jeffrey, ed.; A. Littlewood, trans.) (1970) 4. B. N. Pshenichnyi,Necessary Conditionsfor an Extremum (L. Neustadt,translation ed.; K. Makowski, trans.) (1971) 5. L. Nadciet aL, FunctionalAnalysisandValuationTheory(1971) 6. S. S. Passman, Infinite GroupRings(1971) Theory.Part A: OrdinaryRepresentation Theory. 7. L. Dornhoff,GroupRepresentation Part B: ModularRepresentation Theory(1971,1972) 8. W.BoothbyandG. L. Weiss,eds., Symmetric Spaces (1972) 9. Y. Matsushima, DifferentiableManifolds (E. T. Kobayashi, trans.) (1972) 10. L. E. Ward,Jr., Topology (1972) Cohomological Methods in GroupTheory(1972) 11. A. Babakhanian, 12. R. Gilmer,MultiplicativeIdeal Theory (1972) andthe WienerIntegral (1973) 13. J. Yeh,StochasticProcesses 14. J. Barros-Neto, Introductionto the Theory of Distributions(1973) 15. R. Larsen,FunctionalAnalysis(1973) Bundles(1973) 16. K. YanoandS. Ishihara, TangentandCotangent 17. C. ProcesL Ringswith Polynomial Identities (1973) 18. R. Hermann, Geometry,Physics, andSystems (1973) 19. N.R. Wallach, HarmonicAnalysis on Homogeneous Spaces(1973) Introductionto the Theoryof FormalGroups (1973) 20. J. Dieudonn~, 21. /. Vaisman, Cohomology andDifferential Forms(1973) 22. B.-Y. Chen,Geometry of Submanifolds (1973) Finite Dimensional MultilinearAlgebra(in twoparts) (1973,1975) 23. M.Marcus, 24. R. Larsen,Banach Algebras(1973) 25. R. O. KujalaandA. L. Vitter, eds., ValueDistributionTheory:Part A; Part B: Deficit andBezoutEstimatesby WilhelmStoll (1973) 26. K.B. Stolarsky, AlgebraicNumbers andDiophantineApproximation (1974) 27. A.R. Magid,TheSeparableGalois Theoryof Commutative Rings(1974) Finite Ringswith Identity (1974) 28. B.R.McDonald, 29. J. Satake, LinearAlgebra (S. Kohet al., trans.) (1975) 30. J. S. Golan,Localizationof Noncommutative Rings(1975) 31. G. Klambauer, Mathematical Analysis(1975) 32. M.K. Agoston,AlgebraicTopology (1976) 33. K.R. Goodearl,Ring Theory(1976) 34. L.E. Mansfield,LinearAlgebrawith Geometric Applications(1976) 35. N.J. Pullman,MatrixTheoryandIts Applications(1976) 36. B.R. McDonald, Geometric AlgebraOverLocal Rings(1976) 37. C. W.Groetsch,Generalized Inversesof LinearOperators (1977) andJ. L. Gersting,AbstractAlgebra(1977) 38. J. E. Kuczkowski 39. C. O. ChfistensonandW.L. Voxman, Aspectsof Topology(1977) 40. M. Nagata,Field Theory(1977) Theory(1977) 41. R.L. Long,AlgebraicNumber 42. W.F.Pfeffer, Integrals andMeasures (1977) 43. R.L. Wheeden andA.Zygmund, Measureand Integral (1977) Variable(1978) 44. J.H. Curtiss, Introductionto Functionsof a Complex 45. K. Hrbacek andT. Jech,Introductionto SetTheory(1978) 46. W.S. Massey,Homology and Cohomology Theory(1978) 47. M. Marcus,Introductionto Modern Algebra(1978) 48. E. C. Young,VectorandTensorAnalysis(1978) 49. S.B. Nadler,Jr., Hyperspaces of Sets(1978) 50. S.K. Segal,Topicsin GroupKings(1978) 51. A. C. M. vanRooij, Non-Archimedean FunctionalAnalysis(1978) 52. L. CorwinandR. Szczarba,Calculusin VectorSpaces(1979) 53. C. Sadosky, Interpolationof Operators andSingularIntegrals(1979) (1980) 54. J. Cronin,Differential Equations 55. C. W.Groetsch,Elements of ApplicableFunctionalAnalysis(1980)

56. 57. 58. 59. 60. 61. 62.

L Vaisman,Foundations of Three-Dimensional EuclideanGeometry (1980) H.I. Freedan,DeterministicMathematical Modelsin PopulationEcology(1980) S.B. Chae,Lebesgue Integration (1980) C. S. Reeset al., TheoryandApplicationsof FourierAnalysis(1981) L. Nachbin, Introductionto Functional Analysis(R. M.Aron,trans.) (1981) G. OrzechandM.Orzech,PlaneAlgebraicCurves(1981) R. Johnsonbaugh andW. E. Pfaffenberger, Foundationsof MathematicalAnalysis (1981) 63. W.L. Voxman andR. H. Goetschel,Advanced Calculus(1981) 64. L. J. CorwinandR. H. Szczarba, MultivariableCalculus(1982) Theory (1981) 65. V.I. Istr4tescu,Introductionto LinearOperator 66. R.D.Jarvinen,Finite andInfinite Dimensional LinearSpaces (1981) andP. E. Ehrlich, GlobalLorentzianGeometry (1981) 67. J. K. Beem 68. D. L. Arrnacost,TheStructureof LocallyCompact AbelianGroups (1981) Noether:A Tribute (1981) 69. J. W.BrewerandM. K. Smith, eds., Emmy 70. K. H. Kim,Boolean Matrix TheoryandApplications(1982) 71. T.W.Wieting, TheMathematical Theoryof ChromaticPlaneOrnaments (1982) 72. D.B.Gauld, Differential Topology (1982) 73. R.L. Faber,Foundations of EuclideanandNon-Euclidean Geometry (1983) Matrices(1983) 74. M. Carmeli,Statistical TheoryandRandom 75. J.H. Carruthet al., TheTheoryof TopologicalSemigroups (1983) andRelativity Theory(1983) 76. R.L. Faber,Differential Geometry 77. S. Barnett, Polynomials andLinearControlSystems (1983) 78. G. Karpilovsky,Commutative GroupAlgebras(1983) 79. F. VanOystaeyen andA.Verschoren, Relative Invariants of Rings(1983) 80. I. Vaisman, A First Coursein Differential Geometry (1984) 81, G. W.Swan,Applicationsof OptimalControlTheoryin Biomedicine (1984) 82. T. PetdeandJ. D. Randall,Transformation Groups onManifolds(1984) 83. K. GoebelandS. Reich, UniformConvexity,HyperbolicGeometry, andNonexpansive Mappings (1984) 84. T. AlbuandC. N#st#sescu, RelativeFinitenessin Module Theory(1984) 85. K. Hrbacek andT. Jech, Introductionto Set Theory:Second Edition (1984) 86. F. VanOystaeyen andA.Verschoren, Relative Invariants of Rings(1964) 87. B.R. McDonald, Linear AlgebraOverCommutative Rings(1984) 88, M. Namba, Geometry of Projective AlgebraicCurves(1984) 89. G.F. Webb,Theoryof NonlinearAge-Dependent PopulationDynamics (1985) et aL, Tablesof Dominant WeightMultiplicities for Representations of 90, M. R. Bremner SimpleLie Algebras(1985) 91. A. E. Fekete,RealLinearAlgebra(1985) 92. S.B. Chae,Holomorphy andCalculus in Normed Spaces(1985) 93. A.J. Jerri, Introductionto IntegralEquations with Applications (1985) 94. G. Karpilovsky,ProjectiveRepresentations of Finite Groups (1985) 95. L. NadciandE. Beckenstein, TopologicalVectorSpaces (1985) 96. J. Weeks,TheShapeof Space(1985) 97. P. R. Gribik andK. O. Kortanek,ExtremalMethods of OperationsResearch (1985) 98. J.-A. ChaoandW.A. Woyczynski, eds., Probability TheoryandHarmonic Analysis (1986) 99. G. D. Crownet aL, AbstractAlgebra(1986) 100. J. H. Carruthet aL, TheTheoryof TopologicalSemigroups, Volume 2 (1986) 101. R. S. DoranandV. A. Belfi, Characterizations of C*-AIgebras (1986) 102. M. W.Jeter, Mathematical Programming (1986) 103. M. Altman, A Unified Theoryof Nonlinear Operatorand Evolution Equationswith Applications(1986) 104. A. Verschoren, RelativeInvariantsof Sheaves (1987) 105. R.A. UsmanL AppliedLinear Algebra(1987) 106. P. BlassandJ. Lang,Zariski Surfaces andDifferential Equations in Characteristicp > 0 (1987) et aL, StructuredHereditarySystems (1987) 107. J.A. Reneke 108. H. Busemann andB. B. ’Phadke,Spaceswith DistinguishedGeodesics (1987) 109. R. HaRe, Invertibility andSingularityfor Bounded LinearOperators (1988) 110. G. S. Laddeet aL, Oscillation Theoryof Differential Equations with DeviatingArguments(1987) 111. L. Dudkinet aL, Iterative Aggregation Theory(1987) (1987) 112. T. Okubo,Differential Geometry 113. D.L. StanclandM. L. Stancl, RealAnalysiswith Point-SetTopology (1987)

114.T.C.Gard,Introductionto StochasticDifferential Equations (1988) 115. S. S. Abhyankar, Enumerative Combinatorics of YoungTableaux(1988) 116. H. StradeandR. Famsteiner, ModularLie AlgebrasandTheir Representations (1988) 117. J.A. Huckaba, Commutative Ringswith ZeroDivisors (1988) 118. W.D.Wallis, CombinatorialDesigns(1988) 119. WWi~slaw,TopologicalFields (1988) 120. G. Karpilovsky,Field Theory(1988) 121. S. Caenepeel and F. VanOystaeyen,BrauerGroupsand the Cohomology of Graded Rings(1989) 122. W.Kozlowski,ModularFunctionSpaces(1988) 123. E. Lowen-Colebunders, FunctionClassesof Cauchy ContinuousMaps(1989) 124. M. Pavel, Fundamentals of PatternRecognition(1989) 125. V. Lakshmikantham et al., Stability Analysisof NonlinearSystems (1989) 126. R. Sivaramakfishnan, TheClassicalTheoryof ArithmeticFunctions(1989) 127. N.A.Watson, ParabolicEquations on anInfinite Strip (1989) 128. K.J. Hastings,Introductionto the Mathematics of Operations Research (1989) 129. B. Fine, AlgebraicTheoryof the BianchiGroups (1989) 130. D.N.Dikranjanet aL, TopologicalGroups (1989) 131. J. C. Morgan II, Point Set Theory(1990) 132. P. BilerandA.Witkowski,Problems in Mathematical Analysis(1990) 133. H.J. Sussmann, NonlinearControllability andOptimalControl(1990) 134.J.-P. Florenset aL, Elements of Bayesian Statistics (1990) 135. N Shell, TopologicalFields andNearValuations(1990) 136. B. F. Doolin andC. F. Martin, Introduction to Differential Geometry for Engineers (1990) 137.S. S. Holland,Jr., AppliedAnalysisby the Hilber~Space Method (1990) 138. J. Okninski,Semigroup Algebras(1990) 139. K. Zhu,OperatorTheoryin FunctionSpaces (1990) 140. G. B Pdce,AnIntroductionto Multicomplex Spaces andFunctions(1991) 141. R. B. Darst, Introductionto LinearProgramming (1991) 142. P.L. Sachdev, NonlinearOrdinaryDifferential Equations andTheir Applications(1991) 143. T. Husain,OrthogonalSchauder Bases(1991) 144. J. Foran,Fundamentals of RealAnalysis(1991) 145. W.C. Brown,MatricesandVectorSpaces(1991) 146. M.M.RaoandZ. D. Ren,Theoryof Orlicz Spaces (1991 147. J. S. GolanandT. Head,Modules andthe Structuresof Rings(1991) 148.C. Small,Arithmeticof Finite Fields(1991) 149. K. Yang,Complex Algebraic Geometry (1991) 150. D. G. Hoffmanet aL, CodingTheory(1991) 151. M.O.Gonz~lez,Classical Complex Analysis(1992) 152. M. O. Gonz~lez,Complex Analysis (1992) 153. L. W.Baggett,FunctionalAnalysis(1992) 154. M. Sniedovich,DynamicProgramming (1992) 155. R. P. Agarwal,DifferenceEquations andInequalities (1992) 156.C. Brezinski,Biorthogonality andIts Applicationsto Numerical Analysis(1992) 157.C. Swartz,AnIntroductionto FunctionalAnalysis(1992) 158. S. B. Nadler,Jr., Continuum Theory(1992) 159. M.A.AI-Gwaiz,Theoryof Distributions(1992) 160. E. Perry, Geometry: AxiomaticDevelopments with ProblemSolving(1992) 161. E. Castillo andM. R. Ruiz-Cobo,FunctionalEquationsandModellingin Scienceand Engineering (1992) 162. A. J. Jerri, Integral andDiscreteTransforms with ApplicationsandError Analysis (1992) 163.A. CharlieretaL, Tensors andthe Clifford Algebra(1992) 164. P. BilerandT. Nadzieja,Problems andExamples in Differential Equations (1992) 165. E. Hansen, GlobalOptimizationUsingInterval Analysis(1992) 166. S. Guerre-Delabr~re, Classical Sequences in Banach Spaces(1992) 167. Y.C. Wong,IntroductoryTheoryof TopologicalVectorSpaces(1992) 168. S. H. KulkamiandB. V. Limaye,RealFunctionAlgebras(1992) 169. W. C. Brown,MatdcesOverCommutative Rings (1993) 170. J. LoustauandM. Dillon, Linear Geometry with Computer Graphics(1993) 171. W.V. Petryshyn,Approximation-Solvability of NonlinearFunctionalandDifferential Equations(1993) 172. E.C. Young,VectorandTensorAnalysis: Second Edition (1993) 173. T.A. Bick, ElementaryBoundary ValueProblems(1993)

174. M. Pavel,Fundamentals of PatternRecognition:Second Edition (1993) 175. S. A. Albevedo et aL, Noncommutative Distributions (1993) 176. W.Fulks, Complex Variables (1993) 177. M.M.Rao,ConditionalMeasures andApplications (1993) 178. A. Janicki andA. Weron,SimulationandChaotic Behaviorof c(-Stable Stochastic Processes(1994) 179. P. Neittaanm~ki andD. Tiba, OptimalControlof NonlinearParabolicSystems (1994) 180. J. Cronin,Differential Equations: IntroductionandQualitativeTheory,Second Edition (1994) 181. S. Heikkila andV. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) 182. X. Mao,Exponential Stability of StochasticDifferential Equations (1994) 183. B. S. Thomson, Symmetric Propertiesof Real Functions(1994) Analysis(1994) 184. J.E. Rubio,OptimizationandNonstandard 185. J.L. Bueso et al., Compatibility,Stability, andSheaves (1995) Systems (1995) 186. A. N MichelandK. Wang,Qualitative Theoryof Dynamical 187. M.R.Darnel,Theoryof Lattice-Ordered Groups(1995) 188. Z. NaniewiczandP. D. Panagiotopoulos,MathematicalTheoryof Hemivadational InequalitiesandApplications(1995) 189. L.J. CorwinandR.H. Szczarba,Calculusin VectorSpaces:Second Edition (1995) (1995) 190. L.H. Erbeet aL, OscillationTheoryfor FunctionalDifferential Equations 191. S. Agaianet aL, BinaryPolynomial Transforms andNonlinearDigital Filters (1995) Estimations for Operation-Valued FunctionsandApplications(1995) 192. M.I. Gil’, Norm 193. P.A.Grillet, Semigroups: AnIntroductionto the StructureTheory(1995) NonlinearWaveEquations(1996) 194. S. Kichenassamy, 195. V.F. Krotov, GlobalMethods in OptimalControlTheory(1996) Identities (1996) 196. K. I. BeidaretaL, RingswithGeneralized 197. V. I. Amautov et aL, Introduction to the Theoryof TopologicalRingsandModules (1996) 198. G. Sierksma,Linear andInteger Programming (1996) 199. R. Lasser,Introductionto FourierSeries(1996) 200. V. Sima,Algorithms for Linear-Quadratic Optimization (1996) 201. D. Redmond, NumberTheory(1996) et al., GlobalLorentzianGeometry: Second Edition (1996) 202. J.K. Beem 203. M. Fontanaet aL, PrOferDomains (1997) 204. H. Tanabe, Functional AnalyticMethods for Partial Differential Equations (1997) 205. C. Q. Zhang,Integer FlowsandCycleCoversof Graphs(1997) 206. E. SpiegelandC. J. O’Donnell,IncidenceAlgebras (1997) Geometry of Feedback andOptimalControl (1998) 207. B. JakubczykandW. Respondek, 208. T. W.Haynes et aL, Fundamentals of Domination in Graphs(1998) et aL, Domination in Graphs:Advanced Topics(1998) 209. T. W.Haynes 210. L. A. D’Alotto et al., A Unified SignalAlgebraApproach to Two-Dimensional Parallel Digital SignalProcessing (1998) 211. F. Halter-Koch,Ideal Systems (1998) Theory(1998) 212. N.K.Govil et al., Approximation 213. R. Cross,MultivaluedLinearOperators(1998) 214. A. A. Martynyuk,Stability by Liapunov’sMatrix FunctionMethodwith Applications (1998) 215. A. Favini andA.Yagi, Degenerate Differential Equationsin Banach Spaces(1999) and RecentAdvances 216. A. Illanes and S. Nadler, Jr., Hyperspaces:Fundamentals (1999) 217. G. KatoandD.Struppa,Fundamentals of AlgebraicMicrolocalAnalysis(1999) 218. G.X.-Z.Yuan,KKM TheoryandApplicationsin NonlinearAnalysis(1999) andN. H. Pavel, Tangency, FlowInvadance for Differential Equations, 219. D. Motreanu andOptimizationProblems(1999) 220. K. Hrbacek andT. Jech,Introductionto Set Theory,Third Edition (1999) 221. G.E. Kolosov,OptimalDesignof Control Systems (1999) 222. A. I. Prilepko et aL, Methods for SolvingInverse Problems in Mathematical Physics (1999) 223. B. Fine andG. Rosenberger, AlgebraicGeneralizations of DiscreteGroups (1999) AdditionalVolumes in Preparation

ALGEBRAIC GENERALIZATIONS OF DISCRETE GROUPS A Path to Combinatorial Group Theory Through One-Relator Products BenjaminFine Fairfield University Fairfield, Connecticut

GerhardRosenberger University of Dortmund Dortmund, Germany

MARCEL DEKKER, INC.

NEW YORK ¯ BASEL

Library of CongressCataloging-in-Publication Fine, Benjamin. Algebraicgeneralizations of dis.crete groups: a path to combinatorialgrouptheory through one-relator products / BenjaminFine, Gerhard Rosenberger. p. cm. -- (Monographsand textbooks in pure and applied mathematics; 223) Includes bibliographical references and index. ISBN0-8247-0319-7(alk. paper) 1. Discrete groups. 2. Combinatorialgroup theory. I. Rosenberger,Gerhard.II. Title. III. Series. QA178.F55 1999 512’.2--dc21 98-32814 CIP

This bookis printed on acid-free paper. Headquarters MarcelDekker,Inc. 270 Madison Avenue, NewYork, NY10016 tel: 212-696-9000;fax: 212-685-4540 Eastern HemisphereDistribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001Basel, Switzerland tel: 41-61-261-8482;fax: 41-61-261-8896 World Wide Web http://www.dekker.com Thepublisher offers discounts on this bookwhenordered in bulk quantities. For moreinformation, write to Special Sales/Professional Marketingat the headquartersaddress above. Copyright © 1999 by Marcel Dekker,Inc. All Rights Reserved. Neither this book nor any part maybe reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying,microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Currentprinting (last digit) 1098765432 1 PRINTED IN THE UNITED STATES OF AMERICA

To our families: Linda, Carolyn, and David Katariina,

Anja, and Aila

PREFACE

A one-relator group or a group with a single defining relation is a group that has a presentation of the form < Zl, ..., :~n; iz~ >. Moregenerally if {G~} is a class of groups, then a one-relator product of the G~ is a group G of the form G = (,G,~)/N(R) where *G~ is the free product of the Ga and N(R) is the normal closure in this free product of the single element R. The G~ are called the factors and R is the relator. In this context a one-relator group can be viewed as a one-relator product of free groups. One-relator groups have played a central role in the combinatorial study of groups. The reasons for this are both historical and algebraic. From a purely algebraic viewpoint one-relator groups axe a natural generalization of free groups. However, the centrality and interest in one-relator groups goes deeper than the straightforward algebra and is rooted in the history of combinatorial group theory. Group presentations arose in paxt as a way to deal with the i~ffinite discrete groups introduced in combinatorial topology by Poincaxe. Amongthe first topological objects studied were surfaces and it was discovered quite early that the fundamental group of a compact surface, orientable or non-orientable, had only one defining relation. From covering space theory it became clear that subgroups of fundamental groups of surfaces were again fundamental groups of surfaces or free groups. From a group theoretical point of view this gave the remarkable theorem that a subgroupof a group in a large class of one-relator groups - the surface groups - was either a free group or a one-relator group of the same type. This result was arrived at in a different manner by Fricke and Klein IF-K] in their work on Fuchsian groups (see Chapter 4). Muchof the early combinatorial work in one-relator group theory was devoted to giving purely algebraic proofs of the above results. This tie with combinatorial and low-dimensional topology has been a two way street. It is standaxd that results in. topology could be derived from the corresponding groups and group presentations. Also, however, algebraic properties of the groups could be derived from the topological context. The purpose of this book is to give a detailed, relatively self-contained account of certain natural algebraic generalizations of discrete groups - that

PREFACE is, discrete subgroups of PSL2(C). The material grew out of a research project, initiated by the authors in 1985, to extend the algebraic study of Fuchsian gxoups to the more general context of one-relator products and related group theoretical constructions. To present these generalizations a great deal of preliminary material on combinatorial group theory, onerelator group theory and discrete groups has to be introduced. For this reason this book becomes an excellent text or supplementary text for a seminar or course in combinatorial group theory geared toward discrete groups. We would like to thank the many people who have at various times worked on this research project with us and who have worked on the same general project. In particular, we would like to thank Reg Allenby, Gilbert Baumslag, Kati Bcncsath, Tony Gaglione, Andrea Hempel, Jim Howie, Frank Levin, Colin Maclachlan, Alexei Myasnikov, Frank Roehl, Dennis Spellman, Michael Stille, Francis Tang, Rick Thomas and E.B. Vinberg. We would also like to thank the referee for manyhelpful suggestions, including the addition of the material on geometric group theory. Benja~nin Fine Gerhard Rosenberger

CONTENTS

v

Preface Chapter I. Introduction: One-Relator Groups and Discrete Groups Preliminaries from Combinatorial Group Chapter II. Theory 2.1 Preliminaries from Combinatorial Group Theory 2.2 Free Groups and Free Products 2.3 Group Amalgams 2.4 Subgroup Theorems for Amalgams 2.5 Bass-Serre Theory 2.6 Nielsen Transformations 2.7 Geometric Group Theory: Hyperbolic Groups 2.8 Geometric Group Theory: Arboreal Group Theory 2.9 Geometric .Group Theory: Automatic Groups Chapter III. One-Relator Groups 3.1 One-Relator Groups and the Freiheitssatz 3.2 The Freiheitssatz and Magnus’ Method 3.3 Geometric Versions of the Freiheitssatz 3.4 Subgroup Theory of One-Relator Groups 3.5 Cyclically Pinched One-Relator Groups 3.6 Conjugacy Pinched One-Relator Groups 3.7 Fully Residually Free Groups and the Tarski Problem 3.8 Small Cancellation Theory 3.9 SomeFt~rther Results Chapter IV. Discrete Groups 4.1 Linear Fractional Transformations 4.2 Discrete Groups 4.3 Algebraic Analysis of Fuchsian Groups 4.4 NEC Groups 4.5 Classification of Two-Generator Discrete Groups in PSL2 (IR) vii

1 5 5 5 11 15 16 18 25 31 34 39 39 40 46 49 53 59 64 70 74 77 77 79 89 103 106

ooo

vln

CONTENTS

Chapter V. One-Relator Products 5.1 One-Relator Products 5.2 The Freiheitssatz and Locally Indicable Groups 5.3 The Freihcitssatz for High Powered Relators 5.4 The Kervaire Conjecture and Klyachko’s Solution

111 111 113 121 130

Chapter VI. One-Relator Products of Cycllcs 6.1 One-Relator Products of Cyclics 6.2 Essential and Essentially Faithful Representations 6.3 Essential Representations of One-Relator Products 6.4 Freiheitssat.z for One-Relator Amalgamated Products 6.5 Faithful Representations of One-Relator Products

137 137 139 144 149 152

Chapter VII. Linear Properties of One-Relator Products of Cyclics 7.1 Linear Properties of One-Relator Products of Cyclics 7.2 The Tits Alternative and SQ-universality 7.3 The Generalized Triangle Groups 7.3.1 The Ordinary Triangle Groups 7.3.2 The Tits Alternative - Generators of Finite Order 7.3.3 The Tits Alternative - Generators of Infinite Order 7.3.4 The Finite Generalized Triangle Groups 7.4 The Virtual Torsion-Free Property 7.5 Free Product with Amalgamation Decompositions 7.6 Euler Characteristics Chapter VIII. Groups of F-Type 8.1 Groups of F-type 8.2 Freiheitssatz and Subgroup Theorems 8.3 Linear Properties of Groups of F-Type 8.4 Additional Results 8.5 Decision Problems in One-Relator Products of Cyclics Chapter IX. Related Generalizations of Discrete Groups 9.1 Related Generalizations of Discrete Groups 9.2 Groups of Special NECType The FHS and Essential Representations for Groups of 9.2.1 SN-Type 9.2.2 Linearity Results for Special NECGroups 9.2.3 Rank Conditions for Certain NECGroups 9.3 The Generalized Tetrahedron Groups 9.3.1 Essential Representations 9.3.2 Lineaxity Properties for Generalized Tetrahedron Groups 9.3.3 A Freiheitssatz and the Virtual Torsion-Free Property 9.3.4 Euler Characteristic for Generalized Tetrahedron Groups

159 159 160 166 168 171 194 201 208 211 216 221 221 223 231 238 248 253 253 254 255 258 264 266 269 274 284 288

CONTENTS 9.3.5 Finite Generalized Tetrahedron Groups Bibliography and References Subject Index Index

of Names

ix

289 291 309 313

CttAPTER I INTRODUCTION: GROUPS

AND

ONE-RELATOR

DISCRETE

GROUPS

A one-relator group or a group with a single defining relation is a group which has a presentation of the form < xl, ...,x,~;R >. More generally if {Go} is a class of groups, then a one-relator product of the Go is a group G of the form G : (*G,~)/N(R) where *Go is the free product of the Go and N(R) is the normal closure in this free product of the single element R. The Go are called the factors and R is the relator. In this context a one-relator group can be viewed as a one-relator product of free groups. One-relator groups have played a central role in the combinatorial study of groups. The reasons for this are both historical and algebraic. From a purely algebraic viewpoint one-relator groups are a natural generalization of free groups. Through tim Magnus method of dealing with one-relator groups {described in section 3.1} it can be shown that one-relator groups share similar properties with free groups. From the perspective of onerelator products this says that a one-relator group shares manyproperties with its underlying factors. Although presentations are a concise way to deal with infinite discrete groups, in general very little information can be obtained from a presentation. However one-relator groups have been amenable to a general treatment. The centrality and interest in one-relator groups goes deeper than the straightforward algebra described above and is rooted in the history of Combinatorial Group Theory. Group presentations arose in part as a way to deal with the infinite discrete groups introduced in Combinatorial Topology by Poincare. Amongthe first topological objects studied were surfaces and it was discovered quite early that the fundamental group of a compact surface, orientable or non-orientable, had only one defining relation. From covering space theory it became clear that subgroups of fundamental groups of surfaces were again fundamental groups of surfaces or free groups. From a group theoretical point of view this gave the remarkable theorem that a subgroup of a large class of one-relator groups - the surface groups - was either a free group or a one-relator group of the same type. This result was

2

ALGEBRAIC GENERALIZATIONSOF DISCRETE GROUPS

arrived at in a different manner by Fricke and Klein [F-K] in their work on Fuchsian groups {see Chapter 4}. Muchof the early combinatorial work in one-relator group theory was devoted to giving purely algebraic proofs of the above results. The purpose of these notes is to give a detailed, relatively self-contained account of certain natural algebraic generalizations of discrete groups that is discrete subgroups of PSL2(C). These algebraic generalizations arise primarily as one-relator products of cylics - that is one relator products where each factor is a cyclic (possibly finite) group. As might expected general one-relator products of cyclics share manyproperties with one-relator groups. Howeverthe introduction of torsion in the generators can both produce manysurprising results and can force the use of different techniques. In particular a powerful method developed by Magnusto deal with one-relator groups does not carry over to general one-relator products of cyclics. The genesis of these notes was the tie-in between Fuchsian groups and one-relator products. Recall that a Fuchsian group F is a non-elementary discrete subgroup of PSL2(R) or a conjugate of such a group in PSL2(C). If F is finitely generated then F has a presentation, called the Poincare presentation, of the form

(2.1)

F=<el,..,ep,

hl,..,h,,al,bl,...,ag,

where R = e~..ephl..ht[a~,bl]...[ag, and m, ~_ 2 for i = 1, ...p. The Euler Characteristic

bg, e ~ =l,i=l,..,p,R=l>

bg] and p >_ O,t >_ O,g >_ O,p+t +g > O,

of F is given by x(F) = -#(F) where p

#(F)

= 2g-

2+ t÷ ~(1i~1

An F-group G is a group with a presentation of the form (2.1). An group G with/~(G) > 0 has a faithful representation as a Fuchsian group and 2n#(G) represents the hyperbolic area of a fundamental polygon for G. From the above presentation, Fuchsian groups thus fall into the class of one-relator products of cyclics. The basic question then is which properties of Fuchsian groups are shared by one-relator products of cyclics. Fuchsian groups are of course linear and therefore satisfy many "linearity" properties - for example they are virtually torsion-free and satisfy the Tits alternative {see section 4.3}. One-relator groups are also known to

I.

INTRODUCTION: ONE RELATORGROUPSAND DISCRETE GROUPS 3

satisfy many of these same linearity properties even though they may or may not be linear. The specific questions explored are: (1) Which properties of Fuchsian groups are shared by all one-relator products of cyclics? (2) If a property of a Fuchsian group does not hold in all one-relator products of cyclics, then is there a subclass - specifically a special form of the relator - in which it does hold? The first five chapters are .preliminary and the basic work does not begin until Chapter 6. In Chapter 2 we present some basic preliminaries from combinatorial group theory. Our techniques depend in large part on group amalgamsand Nielsen reduction, so in this chapter we give the basic definitions and properties of these constructions which are the essential building blocks for infinite discrete groups. Wealso give a brief introduction to the basic concepts and definitions in geometric group theory. In Chapter 3 we review the material from the theory of one-relator groups which is most relevant to our subsequent work on algebraic generlizations of Fuchsian and other discrete groups. In particular we go through the proof of the Freiheitssatz and its applications and present the linear properties of the subgroup theory of one-relator groups. Wealso mention the tie between certain one-relator group constructions and the recent solution of the Tarski conjectures. In Chapter 4 we review the material necessary from the theory of discrete groups. Wefirst define and classify the linear fractional transformations. After this we define discrete groups and discuss their general properties, in particular the existence of fundamental domain and how it can be used to obtain a presentation. Wethen look at the Poincare presentation for a Fuchsian group and see how it determines the algebraic properties of such groups. Wenext introduce NECgroups which are a natural generalization of Fuchsian groups and which have presentations which can be handled by our methods. Finally we give a complete classification of the two-generator discrete subgroups of PSL2 (R) and discuss some examples in PSL2 (C). In Chapter 5 we discuss general one-relator products. Here we follow closely the work of J. Howie. Wediscuss the meaning of a Freiheitssatz for such constructions and the existence of such a theorem if the factors are locally indicable. Wenext introduce a major result of Howie’s on the Freiheitssatz for one-relator products when the relator is a proper power of order four or greater. This uses the concept of pictures which we discuss. Finally we mention the tie-in to the Kervaire conjecture on equations over groups. In Chapter 6 we introduce representation methods for one-relator products of cyclics. These methods were motivated by results of Ree and

ALGEBRAICGENERALIZATIONSOF DISCRETE GROUPS Mendelsohn, Helling, Rosenberger, Culler and Shalen, and Baumslag, Morgan and Shalen. Weintroduce the concepts of essentially faithful and essential representations and show that groups which admit such represenations satisfy manylinearity properties. Finally in that chapter we show that onerelator products of cyclics with a proper power relator admits an essential representation into PSL2 (C). From this we deduce a Freiheitssatz for such groups. In Chapter 7 we use the existence of a complex essential representation to examine the linearity properties of general one-relator products of cyclics with proper power relators. Wefirst discuss the Tits Alternative - that is that such groups contain non-abelian free subgroups or are virtually solvable. The Tits alternative for one-relator products of cyclics reduces to the case of generalized triangle groups which are two-generator one-relator products of cyclics. Wethen discuss the theory of such groups and their connection to the ordinary triangle groups. Wealso show the existence in many cases of so-called Ree-Mendelsohn pairs. These are generating pairs such that sufficiently high powers generate free groups. In Chapter 8 we introduce the concept of a group of F-type. This can be described in group theoretical terms most succintly as a cyclically pinched one-relator product of cyclics. This class of groups is very close in properties to the class of Fuchsian groups and we explore these similarities. In Chapter 9 we introduce two additional classes of groups which also generalize classes of discrete groups and whose properties can be studied by the same representation methods as the one-relator products of cyclics and groups of F-type. The first is called a group of special NECtype or SN-type and is a multi-relator version of a one-relator product of cyclics. The form of the presentation is an extension of the presentation for NEC-groups. The second class is called the class of generalized tetrahedron groups and were named by E.B. Vinberg as an extension of the tetrahedron groups of Coxeter.

CHAPTERII PRELIMINARIES COMBINATORIAL 2.1 Preliminaries

FROM

GROUP

from Combinatorial

THEORY Group Theory

Combinatorial group theory is roughly that branch of group theory which studies groups via their presentations, that is by generators and relations. The two important areas of combinatorial group theory which are most crucial to our work on algebraic generalizations of discrete groups are group amalgams and Nielsen reduction. The remainder of this chapter is devoted to recalling and reviewing the basic ideas in these subjects. Group products or group amalgams are the key constructions in infinite group theory. These play a pivotal role in the theory of one-relator groups and one-relator products. The general idea in the theory of group amalgams is to decompose(if possible) an infinite group G into an amalgam (in a way which we will describe shortly) of some of its subgroups. These subgroups are then called the factors of G. Information about G can then be deduced from the corresponding infor~nation on the factors. Thus amalgamdecompositions play a role in infinite group theory similar to a prime factorization theorem - although the amalgam decomposition of a group G need not be unique. There are essentially two different types of group amalgams- free products with amalgamation and HNNgroups. An infinite group, however, may decompose as both a free product with amalgamation or in a different manner as an HNNgroup. Free products are a special case of free products with amalgamation, so we discuss these first. Before beginning we note that there are two main approaches to the theory of group amalgams. The first is a classical combinatorial approach which deals primarily with presentations for the group and its factors. The second approach is a geometric-topological technique which depends on how the group acts (as a group of isometries ) on a graph. The second method is due to Bass and Serre. ~ 2.2 Free Groups and Free Products There are several standard references

on group amalgams. For most of

ALGEBRAICGENERALIZATIONSOF DISCRETE GROUPS the proofs in this chapter we refer to these. A complete treatment of the combinatorial approach is given in the books by Magnus-Karrass-Solitar [M-K-S] and Lyndon-Schupp [L-S]. The Bass-Serre Theory can be found in Serre’s book Trees [Se]. Other approaches to Bass-Serre theory can be found in the articles by G.P.Scott and C.T.C. Wall[Sc-W], M.Tretkoff [T] and in the Springer lecture notes by Dicks[D]. Much of the theory of group amalgams arises from the theory of free groups. In addition free groups play a dual role in the theory of onerelator groups. On one hand the study of one-relator groups depends on a knowledge of free groups. On the other hand many properties of free groups are shared by one-relator groups. DEFINITION 2.2.1. A group F is free on a subset X /f every map f : X -~ G can be extended to a unique homomorphismf : F --* G. X is a free basis for F and in general a free group is a group with a free basis. The cardinality of a free basis IX[ /:or a free group F is unique and is called the rank of F. If [X[ < oc, F is o/:finite rank. If F has rank n and X = {xl, ...,xn} we say that F is free on (Xx, ...,x~}. Wedenote this Combinatorially F is free on X if X is a set of generators for F and there are no non-trivial relations. In particular: THEOREM 2.2.1. F is a free group if and only if F has a presentation of the form F =< X; >. This theorem depends on the concept of a freely reduced word. If F is free on X, a freely reduced word in F on.X is a word of the form x -- x,lxv~l e~..... xv~n where x.~EX,x,~~x.~+ = -iT1 and e~= d=l. 1if i The free length of this word x is then Ix[ -- n. The word 1, that is the word with n = 0, is also considered a reduced word called the empty word. Theorem1.2.1.1 is equivalent to: THEOREM 2.2.1’. F is a free group if and only if there is a generating set X such that every element of F has a unique representation as a freely reduced word on X. This reduced word gives a normal form for elements in F. An important concept is the following: a .freely reduced word W is cyclically reduced if vl ~ v,~ or if Vl = v,~ then e~ ~ -e,~. Clearly then every element of a free group is conjugate to an element given by a cyclically reduced word. The significance of free groups stems from the following result which is easily deduced from the definition. Let G be any group and F the free group on the elements of G considered as a set. The identity map f : G -~ G can be extended to a homomorphismof F onto G, therefore:

2.2 FREE GROUPS AND FREE PRODUCTS THEOREM 2.2.2. group.

7

Every group G is a homomorphic image of a free

The theory of free groups has a large and extensive literature. several simple properties.

Westate

THEOREM 2.2.3. A free group is torsion-free. Next it can be shown that in a free group two elements gl,ge are conjugate if and only if a cychcally reduced word for gl is a cyclic permutation of a cyclically reduced word for 99- From this we deduce: THEOREM 2.2.4.

An abelian subgroup of a free group must be cyc//c.

Finally a celebrated theorem of Nielsen and Schreier states that a subgroup of a free group must be free. As we will see there is a corresponding type of subgroup theorem for each of the amalgam structures. THEOREM 2.2.5. (Nielsen-Schreier) a free group.

A subgroup of a free group is itself

There are several different proofs of this result (see [M-K-S]) with the most straightforward being topological in nature. A proof using Bass-Serre Theory is found in [Se]. A complete version of the original combinatorial proof appears in [M-K-S]and in the notes by Johnson [J]. The combinatorial proof also allows for a description of the free basis for the subgroup. In particular, let F be free on X, and H C F a subgroup. Let T = {t~} be a complete set of right coset representatives for F rood H with the property that if t~ = z~l~= z~" E T, with e~ = +1 then all the initial segments 1, x~11, x~l~ x~, etc. are also in T. Such a system of coset representatives can always be found and is called a Schreier system or Schreier transversal for H. If g E F let ~ represent its coset representative in T and further define for g ~ F and t e T, St~ = tg(t--~ -~. Notice that St~ ~ H for all t; g. We then have: THEOREM 2.2.5’. (Explicit Form of Nielsen-Schreier) Let F be free X and H a subgroup of F. If T is a Schreier transversal for F rood H then H is free on the se~ {Sty; t e T, x 6 X, St~ 7~ 1}. EXAMPLE. Let F be free on {a, b} and H = F(X ~) the normal subgroup of F generated by all squares in F. Then F/F(X~) =< a, b; a2 = b2 = (ab) ~ = 1 >= Z2 x Z2. It follows that a Schreier system for F rood H is {1, a, b, ab} with ~ = a, ~ = b and b--~ = ab. From this it can be shown that H is free on the generating set xl = a2,x2 -1 = .bab-ia-l,x3

-- b2,x4 = abab-l,x5 = abg~a

8

ALGEBRAICGENERALIZATIONSOF DISCRETE GROUPS

The theorem also allows for a computation of the rank of H given the rank of F and the index. Specifically: COROLLARY 2.2.1. Suppose F is free of rank n and IF : HI = k. Then H is free of rank nk - k + 1. From the example we see that F is free of rank 2, H has index 4 so H is free of rank (2)(4)-4+1= The second version of the Nielsen-Schreier theorem is the basis for the Reidemeister-Schreier process. This is a method to find presentations for subgroups of a group G when given a presentation of G. At various points in our study of one-relator products of cyclics and related types of groups we will utilize this process. Here we give a brief description. A complete description and a verification of its correctness is found in [M-K-S]. Let G be a group with the presentation < a~, ...a,~; R~, ...Rk >. Let H be a subgroup of G and T a Schreier system for G modH defined analagously as above. Reidemeister-Schreier H is generated by the set {Sta.;t

Process

Let G, H and T be as above. Then

@ T~av E (al,...an};Sta.

~ 1)

with a complete set of defining relations given by conjugates of the original relators rewritten in terms of the subgroup generating set. . In order to actual rewrite the relators in terms of the new generators we use a mapping T on words on the generators of G called the Reidemeister rewriting process. This map is defined as follows: If ~2 e~ with ei = +1 defines an element of H W.~- avl~1 av2 ..... av~ then

T(W)

= ~1 l~1~av S~,oo 2......Sit

where ti is the coset representative of the initial segment of Wpreceding a.~ if ei = 1 and ti is the representative of the initial segment of Wup to and including a~~ if ei = -1. The complete set of relators rewritten in terms of the subgroup generators is then given by {T(tR~t-~)} with t E T and R~ runs over all relators

in G.

2.2 FREE GROUPSAND FREE PRODUCTS

9

EXAMPLE. Let G = A4 the alternating group on 4 symbols. Then a presentation for G is G -- A4 =< a, b; a2 = b3 = (ab) 3 = 1 >. Let H = A~ the commutator subgroup. Weuse the above method to find a presentation for H. Now G/H = A4/A~ --< a, b; a2 = b3 = (ab) 3 = [a, b] = 1 >=< b; b3 = 1 >. Therefore tA4 : A~I = 3. A Schreier system is then (1, b, b2}. The generators for A~ are then X1 =

~,~la

=

a, X2 = S~,~ = bab-1, X3 = S~2~ = b2ab

while the relations are

(1) (aa)= SlaS

(2) T(baab-~) -- X~ (3) ~’(b2aab-2) = X2a (4) r(bbb) = (5) T(bbbbb-~) = 1 -2) = 1 (6) T(bUbbbb (7) T(ababab) = S~,~Sb,~Sb:a = X~X2X3 (8) T(babababb -~) = S~,~S~S~,~ = X2X3X1 (9) T(b2abababb -2) = ~b2aSlaSba = X3X1X 2 Therefore after eliminating redundant relations and using that X.~ -- XIX2 we get as a presentation for A~, < X,,X2;X~ = X~ = (X, X2) 2 = 1 >. Closely related to free groups in both form and properties are free products of groups. Let A =< a~,. .... ; RI~.... > and B =< b~,....; S~,. ..... > be two groups. Then: DEFINITION2.2.2. The free product of A and B denoted A * B is the groupG with the presentation < a~, ....b~,. .... ; R~, ..., S~, ... > - that is the generators of G consist of the disjoint union of the generators of A and B with relators taken as the disjoint union of the relators R~ of A and Sj of B. A and B are called the factors of G. In an analagous manner the concept of a free product can be extended to an arbitrary collection of groups. DEFINITION2.2.2’. If A~ =< gens Aa; rels A~ >, a ~ I , is a collection of groups then their free product G = *A~ is the group whose generators consist of the disjoint union of the generators of the A~ and whose relators are the disjoint union of the relators of the A~. Free products exist and are non-trivial.

Wehave:

10


THEOREM 2.2.6. Let G = A * B. Then the maps A -~ G and B -~ G axe injections. The subgroup of G generated by the generators of A has the presentation < generators of A; relators of A >, that is, is isomorphic to A. Similaxly for B. Thus A and B can be considered as subgroups of G. In paxticulax A ¯ B is non-trivial if A and B axe. Free products share many properties with free groups. First of all there is a categorical formulation of free products. Specifically we have: - THEOREM 2.2.7. A group G is the freeproduct of its subgroups A and B if A and B generate G and g~ven homomorphismsfl : A -~ H, f2 : B -~ H into a group H there ex/sts a unique homomorphismf : G -* H extending fl and Secondly each element of a free product has a normal form related to the reduced words of free groups. If G -- A * B then a reduced sequence or reduced word in G is a sequence glg2....gn with gi ~ 1 , each gi in either A or B and Hi, gi+l not both in the same factor. Then: THEOREM 2.2.8. Each element g E G = A * B has a unique representation as a reduced sequence. The length n is unique and is called the syllable length. The case n = 0 is reserved for the identity. A reduced word Hi--. gn E G -- A * B is called cyclically reduced if either n _< 1 or n _> 1 and gx and g,~ axe from different factors. Certainly every element of G is conjugate to a cyclically reduced word. From this we obtain several important properties of free products which carry over to more general amalgams. THEOREM 2.2.9. An element of finite order in a free product is conjugate to an element of finite order in a factor. In particular a finite subgroup of a free product is entirely contained in a conjugate of a factor. THEOREM 2.2.10. If two elements of a free product commute then they are both powers of a single element or are contained in a conjugate of an abelian subgroup of a factor. Finally a theorem of Kurosh extends the Nielsen-Schreier theorem to free products. THEOREM 2.2.11. (Kurosh) A subgroup of a free product is also a free product. Explicitly if G = A * B and H C G then H = F ¯ (*As) * (*B~) where F is a free group and (*As) is a free product of conjugates of subgroups o~e A and (*B~) is a free product of conjugates o[ subgroups of B. Wenote that the rank of F as well as the number of the other factors can be computed. A complete discussion of these is in [M-K-S] and [L-S].

2.3

GROUPAMALGAMS

2.3

Group

11

Amalgams

By a group amalgam we mean either a free product with amalgamation or an HNNgroup which we define below¯ As mentioned earlier these constructions are the essential building tools for infinite groups. They generalize the concept of a free product while at the same time preserving manyof the basic properties. In this section we introduce these amalgamconstructions. Wefirst discuss free products with amalgamation. Let A =< al,. .... ;R1, ..... > and B --< bl, ...... ;$1,. .... > be two groups with H C A,K C B proper subgroups and f : H ~ K an isomorphism. Then: DEFINITION 2.3.1. The free product to K is the group G with the presentation

of A and B amalgamating

H

G=. Similarly for B. Thus A and B can be considered as subgroups of G and A n B = H. The proof of this theorem depends upon a normal form for elements of free products with amalgamation. Let G -- A *H B and let L1 be a set of left coset representatives for A modH and let L2 be a set of left coset representatives for B rood H, normalized in both cases by taking 1 to represent H. Then a reduced sequence or reduced word or normal form in G = A *H B is a sequence of the form

12


where h E H, 1 # gi E L1 U L~ and g~...g, is a reduced word in the free product A * B, that is gj+l ~ ~5i if .q~ ~ L~. THEOREM 2.3.2. (Normal Form Theorem for Free Products with Amalgamation). HG= A *~ B then every g ~ G h&~ a unique representation as a reduced sequence. Extending the concept from free products, a reduced word .ql...g~h in G = A *H B is called cyclically reduced if either n _< 1 or n >_ 2 and g~ and g,~ are from different factors. Certainly every element of G is conjugate to a cyclically reduced word. ~omthis we obtain properties analagous to those in free groups and free products. Specifically: THEOREM 2.3.3. (1) An element ofG = A *H B of finite order must be conjugate to an element of finite order in one of the factors. Thus a finite subgroup or more generally a bounded subgroup must be entirely contained in a conjugate of a factor. (2) An abelian subgroup of G = A ,~ B (a) conjugate of an abelian sub group of A or B or (b) countable as cending un ion of conjugates of subgroups of H or (c) di rect pr oduct of an inf inite cyc lic gro up anda conj ugate of a subgroup of H. The concept of a free product with amalgamation can be extended in a straightforward manner to more than two factors. DEFINITION2.3.2. Let {Gi},i e I, be a family of groups. Let A be a group and for each i, f~ : A --~ G~ a monomorphism. Then the free product of the Gi amalgamating A is, the quotient group of the free product G = *iGi modulo the normal subgroup of *~Gi generated by all relations fi(a) = fj(a) with a ~ A and i,j As in the case of two factors, each G~ injects into G, and each element can be expressed as a normal form. Before moving on to HNNgroups we mention that there is a categorical formulation of free products with amalgamation extending theorem 1.3.7. This is the following result. THEOREM 2.3.4. Suppose G is a group, G1, G2 subgroups and A a group together with injections O1 : A --~ (]1,92 : A -~ G2. ]"hen G = G1 *A G~ if for every group H and every pair of homomorphismsfl : G~ -~ H, f~ : G2 --~ H making the following diagram commute

2.3 GROUP AMALGAMS

13

there exists a unique homomorphismF : G --* H extending fl, Our second basic amalgam construction is that of an HNNgroup. This construction has properties very nearly parallel to those of free products with amalgamation. As pointed out in [Se] they are really two different aspects of the same idea. DEFINITION 2.3.3. Let G be a group, {A~}, i E I, a family of subgroups of G, and for each i ~ I, fi : A~ -~ G a monomorphism. Then an HNN extension of G is a group G* of the form G* =< ti, i ~ I, gens G; rels G,t~, ~A~t~ = f~(A~),i ~ I G is called the base, {ti}iEi the free part or stable letters and { Ai, fi (Ai) } the associated subgroups. G* is an HNNgroup if it can be expressed as an HNNextension of some base. The base group G embeds in the HNNextension in the obvious manner. THEOREM 2.3.5. Let G* be an HNN extension of base G. Then G embedded in G* by g --~ g, that is the subgroup of G* generated by the generators of G, has the presentation < gens G; rels G >. Further the/rec part {t~} is a basis for a/tee subgroup of G*. As in all previous cases this depends on a normal form for elements of HNNgroups. However this is somewhat more intricate than that for frec products with amalgamation. Suppose G* is an HNNextension of G with associated subgroups {Ai,fi(Ai)} and suppose we choose a fixed set of left coset representatives for A~ and .f~(Ai) in G where all A~ and fi(Ai) are represented by 1. Then a normal form in G* is a sequence gl t~: g~t~2....... t~ gk + ~, e~ = +l,

14


where gl, ...,ga+l are elements of G such that for j _< k if ej = 1 then gj is a left coset representative for A~ in G while if e~ = -1 then g~ is a left coset representative for f~(A~) in G and e~ = ei+l whenever gj+l = 1 and i~ = i~+1. THEOREM 2.3.6. Every element w in G* has a unique representation as a normal form. ~omthis as before we obtain a classification as a classification of abelian subgroups.

of torsion elements as well

THEOREM 2.3.7. Let G* be an HNN extension oTG. (1)Elements of finite order in G* are conjugate to elements of finite order in the base G. Further finite subgroups must be contained in conjugates of the base. (2)An abelian subgroup H of G* is one of the following: (a) A subgroup of a conjugate oT the base. (b) A countable ascending union of subgroups of conjugates of the associated subgroups. < t’,H’; rels H’,t’-XH’t ~ = (c) An HNNgroup with presentation ~ ~ H" > with H" C H and H is the intersection of the abelian subgroup H with ~nitely many conjugates of the associated subgroups. Wenote that it is possible for a group to be both an HNNgroup and a free product with amalgamation. Consider the group

c =< a, t, as = = u] = > Let G1 --< a, t; a 2 --- (at) 3 >. This is a ~ee product of a cyclic group of order 2 generated by a and a cyclic group of order 3 generated by at. Therefore t has infinite order in G. Further let G2 --< t, u; It, u] = 1 > a free abelian group of rank 2. The identification t --~ t is then an isomorphism and G is a free product of G1 and G2 with the infinite cyclic subgroup generated by t amalgamated. Now write G as < u,a,t;a ~ = (at) 3 = 1,u-ltu = t >. Again let G] =< a, t; a s = (at) 3 = 1 >= Z2 * ~"3. Then G is an HNNextension of G1 with the single pair of associated subgroups < t > and < f(t) > where f(t) ---Wenote that the above presentation is a presentation for the groups PE~(Od), the two dimensional projective elementary matrix group with entries in Od, where Od is the ring of integers in the quadratic imaginary numberfield Q(x/-z~) and d ¢ 1, 2, 3, 7, 11 (see [F]). HNNgroups were originally developed by G. Higman, H. Neumann and B. Neumann(whence the name) in order to prove several important embedding theorems. In particular:

2.4

SUBGROUPTHEOREMSFOR AMALGAMS

THEOREM 2.3.8. (H.N.N.) Every countable a two generator group.

15

group can be embedded

Out of this theorem developed the concept of SQ-universality. DEFINITION 2.3.4. A group G is SQ-universal if every countable group can be embedded isomorphically as a subgroup of a quotient of G. Thus the Higman, Neumann, Neumann theorem above says that a free group of rank 2 is SQ-universal. Many linear groups are SQ-universal and therefore this is a property that we will frequently consider relative to our algebraic generalizations of discrete groups. SQ-universality might be thought of as a measure of "largeness" of an infinite group. 2.4

Subgroup

Theorems

for

Amalgams

The Nielsen-Schreier theorem says that subgroups of free groups are themselves free while the Kurosh Theoremsays that subgroups of free products are free products. There are similar results for the other amalgam constructions. In particular we have that subgroups of free products with amalgamation and subgroups of HNNgroups both have a special type of HNNconstruction which we will describe shortly. There are several versions of these results. In terms of combinatorial group theory the most straightforward versions are due to Karrass and Solitar [K-S 1], [K-S 2], [K-P-S 1]. The subgroup theorems (as well as the Kurosh Theorem) are actually special cases of a structure theorem for groups acting on trees. Werefer the reader to Serre’s book [Se] for a complete discussion of this. Wepoint out that it requires a detailed analysis in each case to actually recover the Karrass and Solitar results from the general structure theorem. In order to present the Karrass-Solitar subgroup theorems we first need to extend the concept of a free product with amalgamation. DEFINITION 2.4.1. Let T be a tree, that is, a connected oriented graph without dosed paths. Suppose that for each vertex P of T there exists a group Gp and for each edge y of T a group Gy together with an injection Gy -~ G~(y) where t(y) is the ~,erminal vertex of y. Suppose further G~ = Gy where ~ is the oppositely oriented edge to y. Then the group G formed by taking the free product of the groups Gp for all vertices P of T modulo the identifications induced by the edge groups is called the tree product of the {Gp} amalgamated along the G~. A group G is a tree product if it can be decomposed as above. The groups {Gp} are called the vertex groups. Notice that an ordinary free product with amalgamation G -- G1 *A G2 can be considered as a tree product with two vertex groups G1 and G2 ¯ If

16


the numberof vertices and edges of the tree is finite we call G a finite tree product. Tree products possess many of the same properties as ordinary free products with amalgamation. For our purposes what is important is that each vertex group Gp injects into the tree product. Wenow state the Karrass-Solitar result for subgroups of free products with amalgamation. THEOREM 2.4.1. Let G = G1 *A G2 be a non-trivial free product with amalgamation. I£ H is a subgroup of G then H is an HNNgroup H -= .

Then any subgroup H of G is a treed HNNgroup. Further the vertex groups of the base of H are conjugates of the base K intersected with H while the amalgamated subgroups in the base of H are conjugates of the Li intersected with H and the non-trivial associated subgroups in H are conjugates of K or of the Li intersected with H. 2.5 Bass-Serre

Theory

The approach we have taken to amalgams in the previous sections has been purely combinatorial. A second approach which is a very powerful

2.5 BASS-SERRETHEORY

17

geometric technique for handling group amalgams has been developed by Bass and Serre. It involves the action of a group G on a tree X. By analyzing this action the amalgam structure of G can be deduced. The theory also recovers, in a relatively easy manner, the main theorems of group amalgams- Nielsen-Schreier, Kurosh, Karrass-Solitar. This recovery involves a case by case analysis by amalgamtype. In this section we give a very brief description of Bass-Serre Theory. A complete treatment is found in the book by Serre [Se]. Weconsider a group G acting on a tree X, that is a graph without circuits. Weassume further that G acts without inversions that is gy ~ ~ for all g E G and y and edge in X and ~ the inversely oriented edge to y. Wefirst consider the connection between this action and free groups. Wesay that G acts freely on the graph X if it acts without inversion and fixes no vertex. Wethen get. THEOREM 2.5.1.. on a tree X.

A group G is a free group if and only if G ac~s freely

From the above result we obtain perhaps the simplest proof of the Nielsen-Schreier theorem. Let H be a subgroup of a free group G. From the above theorem there exists a tree X on which G acts freely. H must clearly also act freely on this tree and therefore H is free. The explicit form of the Nielsen-Schreier theorem giving the form for the generators of H can also be obtained. Weagain refer the reader to [Se]. Theorem 2.5.1 establishes an equivalence between free groups and free actions on trees. Wenow establish such an equivalence between amalgams and other actions on trees. If G acts on a tree X then a fundamental domain for X rood G is a subgraph T of X such that T -~ G/X is an isomorphism (G/X is the quotient graph). For groups acting on trees a fundamental domain exists if and only if G/X is also a tree. A segment is a graph of the form

Wethen get. THEOREM 2.5.2. (1) Let G act on a tree X and suppose P ~ Q is segment of X. Suppose that this segmen~ is a ~undamental domain for X rood G. Then where Gp, GQand G~ are the stabilizers edge y respectively.

in G of the vertices P, Q and the

18


(2) Conversely suppose that G = G1 *A G2. Then there exists a tree such that G acts on X with fundamental domain a segment P ~ Q, and such that ~p = G1, GQ, -~- ~2 and Gy =- A for the respective stabilizers. This theorem establishes an equivalence between free products with amalgamation and groups acting on trees with a segment as a fundamental domain. The next result establishes a similar equivalence between HNN groups and group actions on trees with loops as the quotient graph. A loop is a graph of the form

THEOREM 2.5.3. (1) Let G act on tree X with a loop {as above} as the quotient graph for X rood G. Then G = Gp is an I-INN group with base (2) Conversely an I-INN group can be made to act on a tree so that it has a loop as a quotient graph. The ideas of Bass-Serre Theory have been extended to a theory of groups acting on more general tree-like structures. This has evolved, along with the notion of a hyperbolic group and an automatic group into geometric group theory. Weintroduce these ideas formally in the final three sections of this chapter. 2.6 Nielsen

Transformations

One of the main tools in the study of free groups and related constructions involving infinite groups is the linear cancellation method using Nielsen transformations. Introduced originally by Nielsen [Nie] to prove the subgroup theorem for free groups they can be considered as the basic transformations in moving from one generating system of any finitely generated group to another generating system. Along these lines Nielsen also proved that the Nielsen transformations generate the automorphism group for any free group of finite rank. Nielsen transformations can be considered as the non-commutative analogs of row reduction of matrices and have proved to be indispensible in the theory of free groups.

2.6 NIELSEN TRANSFORMATIONS

19

The theory of Nielsen transformations has been extended to free products with amalgamation by H.Zieschang [Z 2], and this theory has been further refined by aosenberger[a 1],JR 2],JR 6],[R 7], Kalia and aosenberger [K-R] and Collins and Zieschang [C-Z 1]. It has also been extended to HNNgroups by Peczynski and Reiwer[P-R]. Using the theory many subgroup results on these constructions can be deduced. In addition several important results, such as the Kurosh theorem, can be reproved using the Nielsen technique. In this section we first describe the basic notation of Nielsen transformations for free groups and some applications. Wethen show how the theory can be extended to group amalgams. Details can be found in either the original papers cited above or in the survey paper by Fine, Rosenberger and Stille [F-R-S 2]. Other results on the Nielsen method which we will use throughout these notes will be introduced when needed. For our purposes, we consider all groups G to be countable and only consider finite subsets {xl,...,x,~},n >_ 1, in G. For a finite subset {xl,...,x~,},n _> 1, in G we define an elementary Nielsen transformation as a transformation of one of the following five types: (N1)replace {xl, x~, x3, ..., x,~} by {x~, x~, x3..., x,~} (N2) replace {~l,...,~n--l,tgn} by {~,,~1,-.-,:~n--1} (N3) replace {Xl,X2, ...,Xn} by {x~l,x2...,x,~} (N4) replace {Xl,X2, ...,xn} by {XlX,2,x~...,x~,} (N5) delete some xi where x~ = 1, 1 < i < A Nielsen transformation is a finite product of elementary Nielsen transformations. It is a regular Nielsen transformation if there is no factor of type (N5) otherwise it is singular. Each elementary Nielsen transformation of type (N1),(N2),(N3) or (N4) has an inverse which is a Nielsen transformation. It follows then that the regular Nielsen transformations form a group which contains every permutation of the set {x~, ..., x,~}. For a subset {x~,...,x,,} we let < xl,...,x,, > denote the subgroup of G that they generate. If {x~,...,x,~} is carried by a Nielsen transformation into {yl,...,y,~}, 1 _< m _< n, then clearly < x~,...,x,, >=, that is they generate the same subgroup of G. If this Nielsen transformation is regular then we must have that m = n in which case we say that {x~, ..., x,~} is Nielsen equivalent to {Yl, ..., y,~} In connection with Nielsen transformations in finite generating sets of a subgroup of a group, we always assume that this subset never contains both an element and its inverse, uuless they are equal. Now let F be a free group with fixed basis A. The length L(w) of an element from F is the length of the reduced word for w. This clearly depends on the basis A. Wewrite ul...uq ~ V~...Vm for the equality together with the fact that i(vl...vm) -- L(Vl) + L(v2) + ... L(vm), all ui, vj E F.

20


Let X = {x~, ..., x,~}, n _> 1, be a finite subset of the free group F. X is called Nielsen reduced if for all triples of elements u, v, w from X±1 of the form x~~ , ¢~ = q-l, the following conditions hold: (R1) u ~ (R2) uv ~ 1 implies L(uv) >_ n(u), (R3) uv ~ 1 and vw ¢ 1 implies that L(uvw) > L(u) - L(v) Being Nielsen reduced implies that there is not too muchcancellation in multiplying elements from X. Using in F any fixed order relative to the basis A which does not distinguish between inverse elements then if X = {x~, ..., x,~} is finite then X can be carried by a Nielsen transformation into some Y = {Yl, .., Y,~}, 1 _< m _ ±1 1, is Nielsen reduced then for each u E X there are words p~, q,,, k,~ with k~ ~ 1 such that u --- p~,k~q~ is reduced and such that if w -- u~...uq,q >_ 0,u~ E X+1, all u~u~+~~ 1, then k~l, ...,k~q remain uncancelled in the reduced form of w and L(w) >_ q. From this it is straightforward that if X is Nielsen reduced then < X > is free with X as a basis. Hence every finitely generated subgroup of a free group is free. This was Nielsen’s original proof. It can be extended to remove the finitely generated restriction {see [L-S]}. Further if F has finite rank m and X = {x~, ...,x~}, 1 < n < m, is Nielsen reduced and generates F then ra = n and X is a basis for F. If X -- {x~, ...,x,~} is Nielsen reduced then x~,...,x,~ are the shortest generators of < X > which exist, that is if y~, ...,y~ are other free generators of < X > and both sets are ordered according to the length L then L(y~) >_ L(x~), 1,.. ., n. It is clear that a Nielsen transformation applied to a basis of a finitely generated free group defines an automorphism. Nielsen also proved that the Nielsen transformations generate the automorphism group of a free group of finite rank. Wenow describe an application which we used frequently. This application is related to the study of surface groups and co-compact Fuchsian groups. Recall again that a surface group has a presentation < a~,b~,. .... ,ag,b~;[a~,b~].... [a~,b~] = 1 > in the orientable case or

inthenon-orientable case

.

Any automorphism of a surface group moves the relator to a Nielsen equivalent word. What is of interest then is how free group words of the


21

form of the surface group relator behave under Nielsen transformations. This was studied by Rosenberger [R 4],JR 15] and what was obtained was the following: THEOREM 2.6.1. [R 4],[R 15] Let F be the Tree group on al, ...,a,~ P(al,...,an)=

and

l~l...ap~p[av+l, ap+2]...[an-l,an] ¯ F

with 0 ~_ p ~_ n,n-p even and ~ _> 1 Tori = 1, ...,n. Let X = {xl, ..., x,~} be any finite system in F and let H =< X >. Suppose tha~ H contains some conjugate o[ P(ax, ..., an). Then (a) (xl, ..., x,~} can be carried by a Nielsen transformation into a Tree basis for H which contains a conjugate of P(al,..., aN); (b) {x~, ..., x,~} can be carried by a Nielsen ~ransformation into a Tree basis {Yl,.... Yk} ~ -1 1, ~_~/i ~ o~i, , for H with m > k > n, y~~ za~r~z -~ ~/~la~ for i = 1,...,p, y~ = za~z for j -- pq- 1,....,n and z ¯ F. A proof of this theorem using the Nielsen method can be found in either [R 15] or [F-R-S 2]. In these references are further extensions of the above result. COROLLARY 2.6.1. Le$ F, P(a~,..,an), X and H be as in Theorem 2.6A. Suppose ~hat all (~ are primes (not necessarily different) and tha$ contains P(a~, ..., aN). Then either P(al, ..., aN) is a memberof a basis HorH=F. H. Zieschang [Z 2] developed the Nielsen cancellation method in free products with amalgamation. This was refined by G.Rosenberger [R 1], [a2],[a 6], [R 7] and R.N. Kalia and aosenberger [K-R]. A further refinement of this technique was given by D.Collins and H. Zieschang in [C-Z 1] which we do not consider here in detail. Werestrict ourselves in these notes to the free product of two groups with an amalgamated subgroup, although the method works more generally. Let G = H1 *A H~, H1 ¢ A ¢ H:, denote the non-trivial free product of the groups H~ and H~ with the amalgamated subgroup A -- H~ ~ Hu. If A = {1) then G is just the free product G = H1 * Hu of H~ and H:. Wechoose in each H~, i = 1, 2, a system Li of left coset representatives of A in Hi normalized by taking 1 to represent A. Each x ~ G has a unique representation x = h~...h,~a with a ¯ A, 1 ?~ h~ ¯ L~ tA L2 and h~+l ~ Li if hj ¯ Li. The length of x denoted L(x) is then defined to be n and G is then (partially) ordered by length. In order to obtain results analagous to those in flee groups it is found that the ordering defined by the length L is too coarse. Therefore, as in the free group case, we need a finer pre-ordering of G. For this purpose we

22


-1 define a symmetric normal form for elements x ¯ G. Take the inverses L~ of the left coset representatives as a system of right coset representatives. Then each x ¯ G has a unique representation

with m _> 0, kx ¯ H1 U H2, 1 ¢lj ¯ L1 i2 L2, 1 ¢ rj ¯ L~-1 l.J L~1 and lj+l it Li ifl~ ¯ Li, rj+~ it L~-~ if r~ ¯ L~-~. Further if k~ ¯ A then l,~ and r,~ belong to different Hi (if m _> 1), and if k~ ¯ H~ \ A then 1,~ it Hi, r,~ it Hi (if m_> 1). We then have L(x) = 2m if kx ¯ A and L(x) = 2m + 1 if kx it A. We call l~...1,~ the leading half, rm...r~ the rear half and kx the kernel of x. One advantage of this symmetric normal form is that in forming products, cancellations can usually be reduced to free cancellations. Wenow introduce an ordering _< on G. Weassume that for each Hi, the system Li of left coset representatives has a strict total order. For our applications we may assume that the groups are countable. This is no restriction if one considers a given finitely generated subgroup of G or a given finite system in G. If G is countable then just enumerate the system Li and order it correspondingly. Let the elements of L~ precede those of L2. Then we order for each m the product ll...l,~ of left coset representatives (where 1 ~ lj ¯ L~ ULuand l~+~ it Li if/j ¯ Li), first by length and second ! lexigraphically. Henceif l~...1,~

23

singular Nielsen transformation into a system {hi, ..., hm} with hi = 1 for somei E {1, ...,m} or there is no system Nielsen equivalent to {gl, ...,g,~} which is shorter. If G is countable then every finite system can be carried by a Nielsen transformation into a minimal system. In general, as already mentioned, for a given finite system, a suitable order can always be chosen such that this finite system can be carried by a Nielsen transformation into a minimal system. The Nielsen reduction method in G now refers to Nielsen transformations from given systems to shorter systems and the resulting investigation of minimal systems. An analysis of the result of H. Zieschang [Z 2] for G {see also Rosenberger [R 6]} produces the following result. THEOREM 2.6.2. Let G = HI *A H2. If{x1, ..., x,~} is a finite system of elements in G then there is a Nielsen transformation from {xl, ..., x,~} to a system {yl, ..., y,~} for which one of ~he following cases hold: (i) y~ = for so me i ~ {1, .. .,m} ]-]’q - el (ii) Each w ~ < yl,...,y,~ > can be written as w = il~=l Y~,{~ ~ = d=l, e~ -- e~+l if u~ = ui+ ~ with L(yu~) 0, e~,~ ~ {±1}, l~ ~ R~,r~ ~ R-~ and k~ = h~t~h~,h~,h2 ~ B, ~ = 4-1, if L(x) is odd or k~ ~ B if L(x) is even. In this representation llt~l...lmt ~ is called the leading half, t~rm...t~lr~ the rear half and k~ the kernel of x respectively. The above reduced representation is then called a symmetric form for x. We now introduce, as in the amalgamated free product situation, an ordering on K. We may assume that the groups are countable. This is no restriction if one considers a given finite system in K, for given a finite system a suitable order can always be chosen so that this system can be carried by a Nielsen transformation into what we will call a Nielsen reduced system. Choose a total order of the transversals R~, R_I, and order products l~t~l...l,~t ~’~ by using the lexicographic order on the sequences (/~, ...,lm). Next we extend this order to the set of pairs {g,g-~},g ~ K, where the notation is chosen such that the leading half of g precedes that of g-~ with respect to the above ordering. Let {g, g-~} < {h,h-~} if either L(g) < L(h) or L(g) = L(h) and the leading halfofg strictly precedes that of h or L(g) = L(h) and the leading halves of g and h coincide while the leading half ofg -1 precedes that of h-~. Hence if {g,g-1} < {h, h-1} and {h, h-~} < {g,g-~} then at most the kernels of g and h may be different. For g ~ K let the leading half of g~(~), ~(g) = 4-1, precede that of g-~(~). A finite system {g~,...,g,~} in K is called shorter than a system {h~, ..., h,~} if~g~(~),g~-~(~’) } < (~), h~-~(h~)} holds forall i ~ {1, ..., m} and at least for one i ~ {1,...,m}, {h~(h~),h~ ~(~)} ~-~(~) < ~ ,g~ --~(9~)~~ fails to hold.

2.7 GEOMETRICGROUPTHEORY: HYPERBOLICGROUPS

25

A system {gl, ..., gin} in K is said to be Nielsen reduced or minimal with respect to < if either {gl, ...,g,~} cannot be carried into a system {hi, ..., h,~} with hi = 1 for somei E {1, .., m}or there is no system Nielsen equivalent to {gl, ...,g,~} which is shorter. If the group K is countable, then each finite system, as in the case of a free product with amalgamation, can be carried by a Nielsen transformation into a minimal system. In general for a given finite system a suitable order can always be chosen so that this finite system can be carried by a Nielsen transformation into a minimal system. The following theorem is a slightly refined summaryof the results of Peczynski and Reiwer [P-R]. THEOREM 2.6.3. Let K =< t,B; rel.B,t-lKlt = K-1 > be an HNN group. If {Xl,--., Xm} iS a finite system of elements in K then there is a Nielsen transformation from {Xl, ...,Xm} to a system {y~, ..,y,~} for which one o£ the [ollowing cases holds: (i) y~ = 1 /or somei E {1, ..., m} (ii) Each w ~< Yl, ...,Y,~ > can be written as

w=IIy:: with ei = :kl, and ei = ei+l if ~ = b~iq_l with L(y~) _ 1, of the yi and some product of these yi is conjugate to a non-trivial element of K1 or K-1. The Nielsen transformation can be chosen in finitely many steps such that {Yl, ..,Y,~} is shorter than {xx, ...,x,~} or the lengths of the elements of {xl,..., x,~ } are preserved. We mention that the theory over HNNgroups has been used extensively in studies by Fine, Rohl and Rosenberger [F-R-R1],[F-R-R2] and Fine, Gaglione, Spellman and Rosenberger IF-G-S-R] on freeness properties of subgroups of HNNgroups. 2.7

Geometric

Group Theory:

Hyperbolic

Groups

It is a classical idea to study a geometric object by looking at its group of isometries. Gromov[Gr] somewhat reversed this notion by making a group itself a geometric object by considering properties of groups whose Cayley graph (see section 3) satisfies certain geometric properties. In particlar introduced hyperbolic groups whose Cayley graph satisfies a geometric property of hyperbolic geometry. This ushered in a whole new branch of

26


combinatorial group theory called geometric group theory. Gromov’s original stated purpose was twofold : to generalize small cancellation theory (see section 3.8) and to extend to hyperbolic three manifold groups the group theoretic techniques used with Kleinian groups (see [G]). The scope of geometric group theory has been expanded beyond solely hyperbolic groups to also include the theory of groups acting on more general trees and the theory of automatic groups. What ties these all together is the realtionship between the group structure and the geometric structure of its Cayley graph. Geometric group theory will appear occasionally so in this section and the next two sections we survey the main definitions and ideas, looking first at hyperbolic groups. MaxDehn in his pioneering work on combinatorial group theory [De 1] introduced the following three fundamental group decision problems. (1) Word Problem: Suppose G is a group given by a finite presentation. Does there exist an algorithm to determine if an arbitrary word w in the generators of G defines the identity element of G? (2) Conjugacy Problem: Suppose G is a group given by a finite presentation. Does there exist an algorithm to determine if an arbitary pair of words u, v in the generators of G define conjugate elements of G? (3) Isomorphism Problem: Does there exist an algorithm to determine given two arbitary finite presentations whether the groups they present are isomorphic or not? All three of these problems have negative answers in general (see [L-S]) but attempts for solutions and for solutions in restricted cases have been of central importance in the field. For this reason combinatorial group theory has always searched for and studied classes of groups in which these decision problems axe solvable. For finitely generated free groups there are simple and elegant solutions to all three problems. If F is a free group on xl, ..., xn and Wis a freely reduced word in Xl, ...,x,, then W~ 1 if and only if L(W) >_ Since freely reducing any word to a freely reduced word is algorithmic this provides a solution to the word problem. Further a freely reduced word W=--l~el--2~e~ ..... x.,e" is cyclically reduced if Yl ~ Vn or if vl = v, then el ~ -e,~. Clearly then every element of a free group is conjugate to an element given by a cyclically reduced word called a cyclic reduction. This leads to a solution to the conjugacy problem. Suppose V and W are two words in the generators of F and V, Waxe respective cyclic reductions. Then V is conjugate to Wif and only if V is a cyclic permutation of W. Finally two finitely generated free groups axe isomorphic if and only if they have the same rank. Dehn in 1912 [De 2] provided a solution to the word problem for a finitely generated orientable surface group. Dehn proved that in a surface group

2.7

GEOMETRICGROUPTHEORY: HYPERBOLICGROUPS

27

Tg any non-empty .word w in the generators :which represents the identity, must contain at least half of the original relator (see section 3.5), R = [al, bl] .... lag, b~] whereal, bl,..., a~, b~ are the generators, that is if w = 1 in Sz, then w = bed where for some cyclic permutation R’ of R, R’ = ct with L(t) < L(c) where L represents free group length. It follows then that w = bt-ld in Tg and this word representation of w has shorter length than the original. Given an arbitrary w in Tg we can apply this type of reduction process to obtain shorter words. After a finite number of steps we will either arrive at 1 showing that w = 1 or at a word that cannot be shortened in which case w ~ 1. This procedure solves the word problem for T~ and is known as Dehn’s Algorithm for a surface group. Dehn’s original approach was geometric and relied on an analysis of tim tessellation of the hyperbolic plane provided by a surface group. The idea of a Dehn algorithm can be generalized in the following manner. Suppose G has a finite presentation < X; R > (R here is a set of words in X). Let F be the free group on X and N the normal closure in F of R, N = NF(R) so that G = FIN. G, or more precisely the finite presentation < X; R >, has a Dehn Algorithm, if there exists a finite set of words D C N such that any non-empty word w in N can be shortened by applying a relator in D. That is, given any non-empty w in N, w has a factorization w = ubv where there is an element of the form bc in D with L(c) < L(b). Then applying bc to w we have w = uc-lv in G where L(uc-~v) < L(ubv). By the same argument as in the surface group case the existence of a Dehn Algorithm leads to a solution of the word problem. Further Dehn also presented an algorithm based on the word problem algorithm to solve the conjugacy problem in surface groups. The general idea of a Dehn algorithm is clearly that there is "not much cancellation possible in multiplying relators". Although Dehn’s approach was geometric the idea can be phrased purely algebraically. This is the basic notion of small cancellation theory. Wedescribe small cancellation theory in more detail in section 3.8. Here we just mention that Lyndon[L 3],[L 4],[L 5] placed the study of small cancellation theory in a geometric context and this is the way it is most often looked at. Essentially Lydnon’s geometric techniques (see section 3.8) lead to geometric tesselations and the solutions to the various decision problems aoccur when these tesselations are non-Euclidean. This led Gromovto define negatively curved or hyperbolic groups. In hyperbolic geometry there is a universal constant A such that triangles are A-thin. By this we mean that if XYZ is any geodesic triangle then any point on one side is at a distance less than A from some point on one of the other two sides. Nowsuppose G is a finitely generated group with fixed finite generating set X. Let F be the Cayley graph of G relative to

28

ALGEBRAIC

GENERALIZATIONS

OF DISCRETE

GROUPS

this generating set X equipped with the word metric. A geodesic in the Cayley graph is a path between two points with minimal length relative to the word metric. A geodesic triangle is a triangle with geodesic sides. A geodesic triangle in F is 5-thin if any point on one side is at a distance less than 5 from some point on one of the other two sides. F is 5-hyperbolic if every geodesic triangle is 5-thin. Finally G is word-hyperbolic or just hyperbolic if G is 5-hyperbolic with respect to some generating set X and some fixed 5 _> 0. Gromovfurther showed that being hyperbolic is independent of the generating set although the 5 may change, that is if G is hyperbolic with respect to one finite generating set it is hyperbolic with respect to all finite generating sets. For a full account of hyperbolic groups see the original paper of Gromov[G] or the notes edited by H. Short [Sho

1]. Further suppose G is a finitely generated group with finite presentation < X[R >. If W is a freely reduced word in the finitelygenerated free group F(X) on X of length L(W) and W = 1 in G then there are words P~ E F(X) and relators R~ E R such that N

W = HP~R;’P~-I

in F(X)

where e~ = +1 for each i. G then satisfies a linear isoperimetric inequality if there exists a constant K such that for all words Wwe have N < KL(W). A summaxyof results of Gromov [G] and independently and Shapiro (see [Sho 1]) ties all these ideas together

of Lysenok [Ly]

THEOREM 2.7.1. [G],[Ly],[Sho 1] The following conditions on a t~nitely presented group are equivalent: (1) G is hyperbolic (2) G satisfies a linear isoperimetric inequality (3) G has a Dehn algorithm If G is hyperbolic it must have a Dehn algorithm. Further finitely presented and Dehn algorithm gives hyperbolic and hence all the orientable surface groups of genus greater than or equal to 2 are hyperbolic. Further Gromovshowed that all fundamental groups of closed compact hyperbolic manifolds axe hyperbolic. From the the existence of the Dehn algorithm we get the following corollaries. COROLLARY 2.7.1.

A hyperbolic group G is t~ni~ely presented.

2.7 GEOMETRICGROUPTHEORY: HYPERBOLICGROUPS COROLLARY2.7.2.

A hyperbolic

29

group G has a solvable word problem.

In fact if G is 5-hyperbolic with generating set X and if we define R = {W E F(X);L(W) is a Dehn presentation for G, that is has a Dehn algorithm. Gromov [Gr] further has proved that hyperbolic groups have solvable conjugacy problem while Sela [Sel 1] has shown that the isomorphism problem is solvable for the class of torsion-free hyperbolic groups. THEOREM 2.7.2. The conjugacy problem is solvable for hyperbolic groups. The isomorphism problem is solvable for the class of torsion-free hyperbolic groups. Hence, relative to the decision problems the class of hyperbolic groups behaves like the class of finitely generated free groups. Further hyperbolic groups, especially torsion-free hyperbolic groups, share many additional properties with free groups. Wemention some of them. THEOREM 2.7.3. A hyperbolic group can contain no subgroup isomorphic to Z x Z. In particular an abelian subgroup of a hyperbolic group which contains an element of infinite order is locally cyclic by finite. COROLLARY 2.7.3. group is cyclic.

An abelian subgroup of a torsion-free

hyperbolic

Free groups are torsion-free. Hyperbolic groups can have elements of finite order. Howeverthe existence of a Dehn algorithm assures that there are only finitely manyconjugacy classes of torsion elements. THEOREM 2.7.4. In a hyperbolic group there are only finitely conjugacy classes of elements of finite order.

many

Since subgroups of finitely presented groups need not be finitely generated it follows that subgroups of hyperbolic groups need not be hyperbolic. However there are certain geometrically defined subgroups which must themselves be hyperbolic. If X is a geodesic metric space then a subset A is quasiconvex if there is a constant e such that for any geodesic ab with endpoints a, b E A then a~ is within an e neighborhood of A. A subgroup of a finitely generated group is quasiconvex if the vertices in the subgroup form a quasiconvex set in the Cayley graph. THEOREM 2.7.5. perbolic.

A quasiconvex subgroup of a hyperbolic

group is hy-

Thinking of the Cayley graph of a free group as a tree it is clear that a finitely generated subgroup of a finitely generated free group is quasiconvex.

30


Further an infinitely generated subgroup is not hyperbolic and hence not quasiconvex. Therefore we have. COROLLARY 2.7.4. A subgroup of a finitely generated free group is quasiconvex if and only iT it is finitely generated. Gromov[Gr] states that if H is a quasiconvex subgroup of a hyperbolic group G and g @H then there is a large enough power n such that the subgroup generated by H and 9’~ is their free product. This is fairly clear for finitely generated free groups. Fine and Rosenberger [F-R 5] and independently Cohen and Lustig [Co-L] studied groups with a restricted version of this property. Specifically a group G is a restricted Gromovgroup or RG group if given any non-trivial elements x, y E G the subgroup < x,y > is cyclic or there exists a positive integer t with g~ ~ i, h~ ~ 1 and < g~,h~ >_-, < h~ >. Bestvinaand Feign[Be-F]haveshownthatan amalgamof two hyperbolicgroupsovera cyclicsubgroup is stillhyperbolic. Kharlamapovich and Myasnikov [Kh-M3] havea moregeneralresultthatthe amalga~mof two hyperbolic groupsis againhyperbolic whenever one of the ama]gaznated subgroups is quasiconvex andmalnormal in its respective factor.Related resultswereprovedby JuhaszandRosenberger [J-R]. THEOREM2.7.6.If Ill,H2are hyperbolic and H1 g~ H2 -- H is a quasiconvex subgroup, malnormal in either HI or H2, then the amalgamated product H1 *H H2 is hyperbolic. In paxticular if W1, W2 axe elements of infinite order in HI and H2 respectively and neither is a proper power then HI *wl=w=H2 is hyperbolic. Since finitely generated subgroups of free groups are quasiconvex it follows that if A and B are free groups and A N B = H is a finitely generated subgroup malnormal in either A or B then the amalgamated product A *H B is hyperbolic. Both Kharlamapovich and Myasnikov [Kh-M 3] and Bestvinna and Feign [Be-F] have unrelated results concerning the hyperbolieity of HNNextensions. TItEOREM2.7.7.

([Kh-M 3] ) A separated HNNextension < G,t : t-lAt

=b >

of a hyperbolic group G is hyperbolic if the associated subgroups A and B axe quasiconvex in G and at least one is malnormal. THEOREM 2.7.8. basis ai, ...,a,~.

([Be-F]) Let bea f in itely gen erated fre e gro up wit h Let f ~ Aut(F). Then the mappingtorus

M----< a~, ...,a,~,t;t-~a~t

= f(a~),i =1 .... ,n >

2.8

GEOMETRICGROUP THEORY: ARBOREALGROUP THEORY 31

is hyperbolic if and only if the automorphismf has no non-trivial conjugacy classes.

periodic

Recall that a free group acts freely on a tree and in fact this tree can be taken as its Cayley graph. Rips constructed for hyperbolic groups a simplicial complex, now called the Rips complex, on which the group acts. In particular Rips proved the following [Ri]. TttEOREM 2.7.9. A hyperbolic group G acts simplicially on a simplicial complex P satisfying (1) P is contractible, locally finite and finige dimensional (2) on the vertices of P, G acts freely and transitively (3) the quotient complex PIG is compact Wedose the section by mentioning a conjecture due to Gersten. PROBLEM 5. (Gersten Conjecture) A torsion-free one-relator group hyperbolic if and only if it contains no subgroup isomorphic ~o a BaumslagSolitar group Bm,n =< x, y; yxmy-1 = x ~, mn 7~ 0 >

2.8

Geometric

Group Theory:

Arboreal

Group Theory

In Bass-Serre Theory (see section 2.5) the amalgamstructure of a group is determined by its action as a group of isometrics on a tree. Here a tree is a connected graph or one-dimensional simplicial complex without loops. Particularly striking is the result for free groups which completely classifies free groups as those groups which admit free actions on trees. The use of groups actions on trees has been extended to the study of groups actions on more general tree-like structures. In this wider context a standard tree as above is called a simplicial tree or a Z-tree. The name arboreal group theory has been adopted to cover all the studies of groups on acting on trees. Chiswell [Ch 1] and independently Tits [Ti] introduced the construction of an R-tree. An R-tree is a connected metric space which is "tree-like". A precise definition will be given below. The theory of R-trees becameprominent because of work of Morganand Shalen [Mo-S 1]. Their work concerned studying a finitely generated group G by considering the space of discrete faithful representations of G as a group of orientation preserving isometrics of hyperbolic space. This space factored out by equivalent representations had a compactification. The ideal points are obtained from certain actions of G on R-trees. The basic question on these R-trees became whether there was a classification of groups admitting free actions, in analogy with the

32


Bass-Serre results. The answer is yes, in the finitely generated case, (Rips Theorem below ). An lR-tree T is a non-empty metric space with metric d such that there is no subspace homeomorphic to a circle and such that if u, v E T with r = d(u, v) then there exists a unique isometry a : [0, r] -~ T with a(0) = and a(r) ----v. A segment in a R-tree is the image of an isometry a : [0, r] -~ T. a(0), a(r) are the endpoints of the segment. To see that such a structure is "tree-like" it can be proved that the above definition is equivalent to the following, which says that there is a type of branching at every point. A non-empty metric space is an R-tree if (1) Given u, v E T there is a segment with endpoints u, (2) The intersection of two segments with commonendpoint is a segment (3) If two segments intersect in a single point which is an endpoint both then their union is a segment. Now suppose a group G acts on an R-tree T. We say g ~ G is an inversion if g leaves a segment invariant but g has no fixed points. As before G acts freely on an JR-tree if there are no fixed points. An R-free group is a group which acts freely and without inversions on an R-tree. Clearly free groups are R-free. Further free abelian groups and all orientable surface groups of any genus as well as all non-orientable surface groups of genus _> 4 also act freely on R-trees. In fact, in a sense these are the only finitely generated examples. The following result, given by Rips, (see [Levi or [Ch 5]) gives the classification of finitely generated groups acting on l~-trees. THEOREM 2.8.1. (Rips Theorem) A finitely generated ~-~ee group a free produc~ of finitely many finitely generated free abelian groups and surface groups. Morganand Shalen [Mo-S1] further extended the concept of an ll~-tree to a A-tree where A is an arbitrary ordered abelian group. This concept arose in their paper from an example of valuation on a field where the ordered abelian group is the valuation group. In Serre’s original work on groups acting on trees [Se], certain trees were constructed from discrete valuations and Morgan and Shalen’s construction can be considered a generalization of this. A complete discussion of A-trees can be found in the survey articles by Morgan[Mo],Shalen [Sh 1],[Sh 2], the work of Bass [B]and Alperin and Bass [A-B] or the papers of Chiswell [Ch 3],[Ch 4],[Ch 5]. Let A be an ordered abelian group written additively. If for each a, b ~ A with a > 0 there exists a positive integer n such that b < na then A is an

2.8 GEOMETRICGROUP THEORY: ARBOREALGROUP THEORY

33

archimedean ordered abelian group. In particular all additive subgroups of the reals R are archimedean. If A1 and A2 are ordered abelian groups then the direct sum A1 @A2 can also be made into an ordered abelian group with the lexicographic ordering. However the archimedean property is not necessarily preserved under this construction.. For example under the lexicographic ordering Z ® Z is a non-archimedan ordered abelian group. In particular if a = (1, 0) and b = (0, 1) then b < na for all positive integers n.

If A is an ordered abelian group and X is a set then an A-metric on X is a A-valued function d : X × X --* A satisfying the ususal metric space properties; (i) d(x, y) >_and d(x, y)= 0if f x = y (i i) d(x, y) -- d(y, z and (iii) d(x, y) be a one-relator group and suppose that the relator R is cyclically reduced in the free group on xl,... ,x,, and that R involves all the generators. Then the subgroup of G generated by any proper subset of the generators is free on that subset. In coarser language the theorem says that if G is as above, then given xl,... , xn-~ the only relations involving them are the trivial ones. Equivalently the only relations on Xl,... ,x,~_l are the "obvious ones" from the presentation, that is in the case of a one-relator group the trivial ones. In this chapter we review both the Freiheitssatz of Magnus and other basic results about one-relator groups which are relevant to our exten39

40


sions to algebraic generalizations of discrete groups. Other relevant discussions of one-relator groups are in the excellent survey article by G. Bauinslag in Groups St Andrews 1985 [G.B. 1] and in Lyndon and Schupp’s Combinatorial Group Theory_ [L-S]. In the next section we go over the classical Freiheitssatz of Magnusand Magnus’ method, as well as some consequences of this method. In particular we show that a one-relator group is torsion-free unless the relator is a proper power, in which case the relator and its powers provide a complete collection of conjugacy class representatives for the torsion elements. Dehn’s original suggestion for the proof of the Freiheitssatz was topological so in section 3.3 we discuss several geometric and topological interpretations and versions of this theorem. Someof these topological ideas have evolved into "pictures" over groups which are important in the study of the Freiheitssatz for one-relator products. Wewill look at these ideas again in chapter four. In section 3.4 we consider the subgroup theory of onerelator groups. In particular we consider those subgroup properties which can be considered linear group properties, such as the Tits alternative and the virtual torsion-free property. In section 3.5 we discuss the properties of cyclically pinched one-relator groups which are generalized to groups of F-type. In section 3.6 we consider the HNNanalog of cyclically pinched one-relator groups while in section 3.7 show the connection between these constructions and the Tarski problem - a question in logic. In section 3.8 we briefly introduce small cancellation theory and its ties to the Freiheitssatz and finally in section 3.9 mention some additional results which are not directly related to the subsequent generalizations. 3.2 The Freiheitssatz

and Magnus~ Method

Wefirst put the Freiheitssatz in a more general context and then present Magnus’proof of the classical result. Let X, Y be disjoint sets of generators and suppose that the group A has the presentation A ----< X; Rel(X) > and that the group G has the presentation G = < X, Y; Rel (X), Rel(X, Y) >. Then we say that satisfies a Freiheitssatz which we abbreviate by FHS {relative to A} if < X >a----- A. In other words the subgroup of G generated by X is isomorphic to A. As above, in coarser language this says that the complete set of relations on X in G is the "obvious" one from the presentation of G. An alternative way to look at this is that A injects into G under the obvious map taking X to X. In this language Magnus’ original FHS can be phrased as a one-relator group satisfies a FHSrelative to the free group on any proper subset of the generators. In the setting above we say that the group A is a FHS factor of G. From this point of view, for any group amalgam, as described in the

3.2 THE FREIHEITSSATZ AND MAGNUS’METHOD

41

last chapter, an amalgamfactor is a FHSfactor. Thus any factor in a free product with amalgamation and the base in an HNNgroup embed as a FHS factors in the resulting groups. This then becomesthe basic idea in Magnus’ method for the classical FHS and certain extensions of it. The method is to embed the group into an amalgam in such a way that the proposed FHS factor embeds into an amalgamfactor which in turn contains the proposed FHSfactor as a FHS factor. The result can then be obtained by applying the FHS for amalgams. Wenow present Magnus’ method for proving the classical FHS. Actually what we give is an equivalent version which uses the amalgamconstructions explicitly. These were implicit in Magnus’ original proof but the actual constructions were not yet defined. Magnus’ first proof used the concept of a staggered presentation, about which we’ll say more later (see [MK-S] and [L-S]), and then he proved directly that certain subgroups had the properties of free products with amalgamation, a concept which had been introduced by Schreier [Schr]. WhenMagnus realized that he was using Schreier’s constructions, he added a footnote to the proof explaining that several of Magnus’ technical lemmas could be avoided by the use of Schreier’s results (see [C-M], Chapter II.5]). The proof we outline below based on McCool and Schupp [Mc-S] and Moldavanskii [Mo]. TItEOREM 3.2.1. (F~eihei~ssa~z) Le~ G = < xl,... ,x~;R -- 1 > where R is a cyclically reduced word which involves all the generators. Then the subgroup generated by xl,... ,xn-~ is free on these generators. PROOF.Wedo an induction on the length of the relator R. If R involves only one generator, the theorem holds. To avoid using multiple indices, suppose G =< a,... , c, t; R = 1 > where t = x,~ renaming the generators without indices. Suppose first that t has exponent sum zero in the relator R. For any generator g, let g~ = t~gt -~, i E Z. Then the relator R can be expressed as a new word S in the generators ai,... , c~,.... Let m be the minimumof the a-subscripts appearing in S, M the maximumof the asubscripts appearing in S, n the minimumof the c-subscripts appearing in S, N the maximumof the c-subscripts appearing in S. The word S is then a cyclically reduced word in the generators a,~,... , aM,... , c~,... , c~v and further this word S is of shorter length than R. Let H be the one-relator group < a,~,... ,aM,... ,c~,..., CN; S = 1 >. Conjugates of the original a,... , c appear in H and by induction generate a free subgroup, so we must show that H injects into G. Again from the inductive hypothesis, in the group H, the subgroups generated by {a,~,... ,aM-x,C~,... ,CN-x} and (a,~+x,. ¯ ¯ , aM, c~+~,... , CN}are free. Therefore the group G* given by G* ~ ~ ~:,H;tamt-1 ~ am+l,.-- ,taM--1 t-1 --~ aM,... ,tCnt -1 ---- Cn+l,. .. , tCN--I$-1 ~---CN~

42


is an HNNgroup with base H. But G* is isomorphic to G, so H injects in G. Up to relabeling of generators this covers all situations except when all generators have non-zero exponent sum in R. Now suppose that no generator has exponent stun zero in R. Let a be the exponent sum of t in R and let fl be the exponent sum of a in R. Let L be {a, b, ...c} and let K be the one-relator group with the presentation K -< y,x,b,...c;R(yx-~,x~,b,...,c) >. The map t -~ yx -[~ ~, , a -~ x b -~ b,..., c -~ c defines a homomorphismof G into K. Let R1 be the cyclic reduction of R(yx-~, x~, b,... , c). Then the exponent sum of x in R1 is zero and y occurs in R~. Then as in the case above K can be expressed as an HNNgroup with a one-relator group H as the base and stable letter x. Let S be as before the single relator in H. S then has length less than R since all x symbols have been eliminated. By induction the subgroup of K generated by {x, b, ..., c} is freely generated by this set, and therefore the subgroup generated by {x~, b,... , c} is also freely generated by this set. Since a -~ xa,b -~ b,... ,c -~ c maps L onto these, L freely generates a subgroup of G. This completes the proof. Notice that the one-relator group K in the second part of the proof is actually the free product of G with an infinite cyclic group < x > and then identifying a with xa. By the Freiheitssatz a has infinite order and thus this identification gives a subgroup isomorphism and hence K is the free product with amalgamation of G and < x >. It follows that G injects into K. Thus we have actually proven the following theorem of Moldavanskii [Mo] which in turn implies the Freiheitssatz. THEOREM 3.2.2. Every one-relator group whose relator is cyclicaJ1y reduced and involves at least two generators can be embedded into an HNN group with a single stable letfer. The base of this HNNgroup is a onerelator group whose defining relator is of smaller length than the original relator and whose associated subgroups are free. This technique of embedding a one-relator group into an HNNextension with a one-relator base group with shorter relator and then using inductive arguments is what is called Magnus" Method. It has been used ex~ensively to prove results about one-relator groups, for example, to show that one-relator groups have solvable word problem, a concept we will define shortly. The theory of HNNgroups was not available to Magnus ~ the original proof is somewhat different. Magnus used the concept of a staggered presentation. This arises in the following manner. Let G be a one-relator group with relator R, let f be an epimorphism from G onto an infinite cyclic group (formed as in the proof we gave by finding a generator with

3.2 THE FREIHEITSSATZ AND MAGNUS’METHOD

43

exponent sum zero in R) and let H = Kerf. Then from the ReidemeisterSchreier method H has presentation which is "staggered" in the sense that it has the form < yi(i e Z),zj(j I) ;R~(i E Z)> f orsomeindex set I and that there are constants m and M such that each Ri is a word in {Yi+m,... ,Yi+M, zj(j ~ I)} properly involving y~+,~ and Yi+M. Further identifications between staggered layers lead to free products with amalgamation so that the layers inject. In his first proof Magnusproved the necessary embeddings for free products with amalgamation directly, without referring to the work of Schreier. Wemention this concept of a staggered presentation because it has been reflected in various geometric arguments concerning one-relator groups which use the concept of a staggered complex (see [H-1]). Further some results about one-relator groups can be generalized to groups which have staggered presentations {see [L-S]}. There are several immediate consequences of the Magnus method which are relevant to our further generalizations. First we have that a one-relator group has a solvable word problem. Recall this means that if G is a onerelator group then there exists an algorithm to determine if an arbitrary word w in the generators of G defines the identity element of G (see section 2.7). THEOREM 3.2.3.

Every one-relator

group has a solvable word problem.

Wenext give the proof of the following which characterizes torsion in a one-relator group. THEOREM3.2.4. Let G = < xl,... , xn; R = 1 > be a one-relatorgroup. Suppose R : Sm with m maximal. If m = 1, that is R is not a proper power, then G is ~orsion-free. If m _> 2 then S has exact order m and every torsion element in g is conjugate to some power of S.

PROOF.Let G -- < Xl,....,x,~,...;R = 1 >. If R omits a generator, then G is the free product of the free group on the omitted generators with the one-relator group on the generators involved in R. Since torsion in a free product is determined by torison in the factors we can without loss of generality assume that R involves all the generators. As in the proof of the Freiheitssatz to avoid using double indices write G --< a,..., c, t; R -- 1 >. Wenowuse induction on the length of R. If the length is 1 then G is cyclic and the result is clearly true. Suppose then that the length is greater than 1. If one generator, say t, has exponent sum zero in R then, as in the proof of the Freiheitssatz, G can be considered as an HNNgroup with free part generated by t and base H which is a one-relator group with relator S where S is the rewritten form of R. As before S has shorter length than R since the occurrences of t axe omitted. Further, S is an mth power if

44


and only if R is an mth power. By the inductive hypothesis the base H is torsion-free unless S is an ruth power with m > 1. Since torsion in an HNNgroup is determined by torsion in the base, the same is true for G. If R = U"~, m > 1, and U is not a proper power, then S = V"* where V is the rewritten form of U. Again by the inductive hypothesis V has order m in H so U has order m in G. Further any torsion elements in H must be conjugate to powers of V. Since all torison elements in an HNNgroup are conjugate to torsion elements in the base the result follows for G. In the case where no generator has exponent sum zero in R we again consider the group K = < y, x, b, . . . c; R(yx-~, x~, b, ..., c) >,wherea is the exponent sum of t in R and f~ is the exponent sum of a in R. NowR1 the rewritten form of R is a proper power if and only if R is a proper power. Consider the map f : G --~ K given by t -~ yx-~ , a -~ x~, b -~ b,..., c --* c. As explained after the proof of the Freiheitssatz this map gives an embedding of G into K. From the previous case, K is torsion-free unless R1 and hence R is a proper power. ~ince G embeds into K in this case G is also torsion-free. If R -- U"~, m > 1, then R1 -- U~ and from the previous case U1 has order m in K. However, f(U) is conjugate to U1 so U also has order m in G. The only thing left to show is that if R -- U"~, then given a torsion element in G it must be conjugate to a power of U. In K every torsion element is conjugate to a power of U~, since K is an HNNgroup, so therefore we must show that if two elements of f(G) are conjugate in K then they must already be conjugate in G. We saw that K is a free product with amalgamation K = G*A < x >. where A ---< a >=< x~ >. Now suppose that cgc -1 -- gl with g, gl E G and c E K. Write c = cl...c~h, h e< a > in normal form as an element in a free product with amalgamation. Wedo an induction on n, the length of c. If the length of c is 0 then c = h must be in the amalgamated subgroup and hence in G. The result then follows. For length n _> 1 we may assume that h = 1 and we have Cl ....

-1 ---- glcngc~l-..C~

Without loss of generality we may assume that c~ = x j ~ G. From the normal form theorem for amalgamated free products we must have g ~< a > and hence g = x~. It follows that xJgx-j = xJx~x-~ = x~ --- g. If n _> 2 the above equation becomes 1 -~ gl. Cl.... Cn-lgC~11...C’~

3.2 THE FREIHEITSSATZAND MAGNUS’METHOD

45

Prom the normal form expression c,~-1 = 92 E G so we have Cl.... en-2(g2gg~l)c,~12...c-~l

= gl.

Since g2gg~1 ~ G the result then follows by the inductive hypothesis, completing the theorem. One-relator groups with torsion have proved to be somewhat more tractable than general one-relator groups. This is carried over to one-relator products in general and one-relator products of cyclics in particular whose properties can be muchmore easily handled if the relator is a proper power. Weclose this section by mentioning some of these results. A one-relator group has a solvable word problem. This can be significantly sharpened for one-relator groups with torsion. In particular we have the following "Spelling Theorem" of B.B. Newman. [Ne 1] THEOREM 3.2.5. (Spelling Theorem) Let G be a one-relator group with torsion so that G =< a, b,c......; Rn = 1 > with R cyclically reduced and n > 1. Suppose W = V in G where.W is a free/3, reduced word in the given generators and V omits one of the generators occurring in W. Then W contains a subword S such that S is also a subword of R±~ and the length of S is greater than (n- 1)/n times the ±~. length of R As a consequence of the spelling theorem B.B. Newmanwas able to prove THEOREM 3.2.6. gacy problem.

One-relator groups with torsion have solvable conju-

In general A.Juhasz [J 2] has described that the conjugacy problem for one-relator groups is solvable, howeverno complete proof is available. Along these lines Pride [P 4] proved that the isomorphism problem for two-generator one-relator groups with torsion is solvable (inside the class of one-relator groups). THEOREM 3.2.7. The isomorphism problem for two-generator onerelator groups with torsion is solvable, that is, given a two-generator onerelator group G with torsion there exists an algorithm to determine if an arbitrary one-relator group is isomorphic to G or not. There are other instances of classes of one-relator groups which have solvable isomorphism problem which we will discuss later. One-relator groups with torsion are also hyperbolic in the sense of Gromov[G] (see section 2.7). Recall that if G is a finitely generated group with fixed finite generating set X, and F is the Cayley graph of G relative to this generating set X equipped with the word metric, then a geodesic in the Cayley graph is a path between two points with minimal length relative to

46


the word metric. A geodesic triangle is a triangle with geodesic sides. A geodesic triangle in F is (~-thin if any point on one side is at a distance less than (~ from some point on one of the other two sides. F is b-hyperbolic if every geodesic triangle is b-thin. Finally G is word-hyperbolic if G is (~-hyperbolic with respect to some generating set X and some fixed/i >_ 0. Wethen have the following THEOREM 3.2.8. A one-relator group with torsion is (word)-hyperbolic. Finally a group is commutativetransitive if it is centerless and commutivity is transitive. A centerless group with the property that centralizers of elements are cyclic is commutative transitive so for example flee groups and torsion-free hyperbolic groups are commutative transitive. B.B. Newmanproved the following: THEOREM 3.2.9. A one-relator group with torsion is centerless and has cyclic centralizers. In particular it is commutative transitive. A torsionfree one-relator group with cyclically reduced relator which involves at least three generators also is centerless and has cyclic centralizers and hence is commutative transitive. The second part of the theorem follows from a result of Murasugi [Mu] which states that a one-relator group is centerless if it has 3 or more generators. Wemention also that there is an algorithm, developed by G. Baumslag and Taylor [B-T], to determine the center of a one-relator group. 3.3 Geometric Versions of the Freiheitssatz Dehn’s original conception of the FHS was geometric. As reported by Magnus[C-M], Dehn’s approach was as follows. Suppose as in the proof we gave for the Freiheitssatz that one generator t in the group G has exponent sum zero in the single relator R, for which we assume that it is cyclically reduced and involves all the generators, and let Go be the subgroup of G generated by the remaining generators. Dehn visualized the graph of G as a layered structure with each layer a copy of one of the subgroups tnGot -n = Gn for n E Z. The union of these layers would then form the graph of a normal subgroup N of G with the powers of t as coset representatives. The problem was to prove that each layer is a tree and thus the corresponding subgroup, each isomorphic to Go, is free. Several people have returned to this geometric-topological approach both in reproving the original Freiheitssatz and in extending it. Belowwe present a method due to Jim Howie (see [H 5]) of using the method of Papakyriapopolous towers from 3-manifold topology to give a proof of the FHS. This technique is the topological mirror version of the algebraic Magnus technique and therefore is a return to Dehn’s ideas. After describing Howie’s

3.3 GEOMETRIC VERSIONSOF THE FREIHEITSSATZ

47

tower method, we will give another topological approach due to Bogley [Bo] and Bogley and Pride[Bo-P]. Finally we mention a third (different) geometric approach due to Lyndon [L 4], which has beeen extensively used in the study of general one-relator products. A tower is a map g = ioPlil ... phih : KI --~ K between connected CWcomplexes such that each ij is an inclusion map and each pj is a covering projection. For the proof of the FHSwe restrict the p~ to be infinite cyclic coverings, that is regular, connected coverings with infinite cyclic covering transformation groups. If S is another connected CW-complex, then a tower lifting of f is a commutativetriangle, where g is a tower (see Figure 3.3.1). It is a proper tower lifting if g is not an isomorphism, and it is maximalif f~ does not have a proper tower lifting.

~ K

S

f

~-~K

Figure 3.3.1. A TowerLifting. Notice that Magnus’ method corresponds geometrically to a tower. Initially we have a 2-complexwith a single 2-cell. This is lifted to an infinite cyclic cover and then restricted to a finite subcomplexof that cover. Then by an inductive argument we reduce to the case of a subcomplex containing a single 2-cell which is "simpler" than the original single 2-cell. For the FHSor other properties of one-relator groups which are proved using Magnus’ method, the above procedure is continued until after finitely many repetitions we arrive at a situation where there are no more infinite cyclic covers and the general case can be deduced. Howie’s tower proof of the FHSdepends on the following three lemmas

In5]forproofs). LEMMA 3.3.1. Let S be a/inite CW-complex. Then any cellular f : S --* K has a maximal tower lifting unique up ~o isomorphism.

map

For the other two lemmas we need the concept of a staggered 2complex. This is closely related to the idea of a staggered presentation as discussed earlier. A 2-complexis staggered if all of its 2-cells are attached by cyclically reduced paths of positive length and if the sets of 1-cells and of 2-cells are equipped with linear orderings which are compatible in the following sense: whenever a < fl are 2 -cells, then (min a) < (min ~)

48


(max a) < (max/3). Here rain a (respectively max a) denotes the (greatest) 1-cell involved in the attaching map for LEMMA 3.3.2. Let K be a staggered 2-complex and g : K --~ K’ a tower. Then K’ is staggered. LEMMA 3.3.3. Let K~ be a finite staggered 2-complex with at least one 2-cell such that HI(K~) = O, and suppose that the greatest 2-cell ~ of K’ is not attached by a proper power On ~r~(K’(1)),where K’(1) is the one-skeleton of K~). Then ~ collapses across awith fr ee ed ge max a. HOWIE’S TOWER PROOF OF THE FHS. As before let G = < xl,... ,x,~;R = 1 > where R is a cyclically reduced word involving all the generators. Wecan without loss of generality assume that R is not a proper power; if R = S"~, then replace R by S. Let K be the 2-complex model of the given presentation (see [H 5]), and let F be the subgraph of its 1-skeleton K(1) consisting of the unique 0-cell and the 1-cells xl,... , xn_l. It must be shownthat 7rl(F) --* ~r~(K) is injective. If not, then there exists a map f : D2 --~ K with f(S ~) c F representing a non-trivial element of ~rl (F). Suppose f is cellulex, and consider the maximal tower lifting of (see Figure 3.3.2). Notice K’ is staggered and we may assume that g(max a) = x,, for each 2-cell a of K’. Further K’ is finite for otherwise f’ would factor through an inclusion, contradicting maximality, and H~ (K’) = 0 for otherwise f’ would lift over an infinite cyclic covering contradicting maximality. Finally, no 2-cell in K’ is attached by a proper power. Repeated applications of Lemma3.3.3 show that K’ collapses to a 1-complex F’ (necessarily a tree) such that g(y) = x,~ for e~h 1-cell y in K~ \F’. In particular )~’(s~) c g-’(r) c r’, so ~¢(s’) is nullhomotopic in g-~(r) and f(S 1) is nullhomotopic in F. This is a contradiction so the map must be injective.

Figure 3.3.2 A Tower Lifting. We mention that this tower approach has also been used by Howie to reprove other results about one-relator groups as well as to derive some entirely new results (see [H 5] and the referencesthere).

3.4 SUBGROUPTHEORYOF ONE-RELATORGROUPS:

49

A second topological approach was used by Bogley[Bo] and Bogley and Pride [Bo-P] to prove a multi-relator version of the FHSgeneralizing work of I. Anshel [A]. They used the following geometric version of the classical FHS, as applied to "aspherical complexes" (see [Bo-P]). Let L be a CWcomplex with a single two-cell, and suppose that L has a maximal tree that does not contain an open one-cell c ~ of L. If the attaching map for the two-cell of L strictly involves c 1, (that is, is not freely homotopic in the one-skeleton L1 of L to a map into L~ \c 1) then the inclusion of L~ \ 1 i n L x i nduces a monomorphism offundamental gro ups. A third geometric-topological approach was due to Lyndon who attempted to translate Magnus’ original proof into combinatorial geometry. Lyndon gave a proof based on Van Kampendiagrams and the so-called maximummodulus principle for such diagrams [L 3]. Wenote that Lyndon’s method was further adapted to one-relator products by H. Short [Sho]. 3.4 Subgroup Theory of One-Relator

Groups

There is a large literature on the subgroups of one-relator groups. In this section we review those results which are of direct relevance to "linear properties", that is properties which are satisfied by linear groups. Our results on one-relator products of cyclics and related groups are based upon representing such groups in PSL2(C). Although these representations are not necessarily faithful the groups inherit manylinear properties. Onerelato~ groups satisfy several of these properties although proving them does not ordinarily involve representations. As before we refer the reader to the survey article by G. Baumslag[G.B.-1] and the sections in the book of Lyndon and Schupp [L-S] for more information on the subgroup theory. The first linear property we consider is the Tits alternative. By a linear group we mean a subgroup of GL,~ (F) or PGL,~ (F) for some commutative field F. For these notes we restrict F to have chaxacteristic zero although this is not necessary in all cases. A theorem of J. Tits [Ti] states that a finitely generated linear group either contains a non-abelian free subgroup or is virtually solvable, that is contains a solvable subgroup of finite index. In general we say that a group G satisfies the Tits alternative if it either contains a non-abelian free subgroup or is virtually solvable. From a result of Karrass and Solitar [K-S 2] any one-relator group, and more generally any subgroup of a one-relator group, satisfies the Tits alternative. THEOREM 3.4.1. A subgroup of a one-relator group is either solvable or contains a free subgroup of rank two. Closely related to the Tits alternative is SQ-universality. Recall from Chapter 2 that a group G is SQ-universal if every countable group can be embeddedin a subgroup of a quotient of G. All non-abelian free groups are

50


SQ-universal and from a theorem of Levin [Le] any non-trivial free product except Z~ * Z2 is SQ-universal. If IG : HI < (x) then H being SQ-universal implies G is. Further if G maps onto a non-abelian free group or contains a subgroup of finite index which maps onto a non-abelian free group, then G is SQ-universal. The tie to the Tits alternative is that having many non-abelian free subgroups is a good indicator that the group should be SQ-universal. Recall that the deficiency of a finite presentation for a group is the difference between the number of generators and the number of relators. Crucial to our work on one-relator products of cyclics and applicable to one-relator groups is the following result of B.Baumslagand S. Pride [B.B.P 2]. THEOREM 3.4.2. Suppose a group G admits a 1:mite presentation with deficiency d >_ 2. Then G contains a normal subgroup of finite index which maps onto a free group of rank 2. In particular G is SQ-universal COROLLARY3.4.1.

Any one-relator group with at least three generators

is SQ-universal. Further related to the presence of flee subgroups is the following result of Ree and Mendelsohn[R-M], whose proof we give. This proof will serve as an introduction to the representation theoretic methods we will be utilizing from Chapter Five on. THEOREM 3.4.3. Let G =< a,b;R m = 1 > where R is a cyclically reduced word of length at least ~wo involving bo~h a and b and m > 1. Then for sufliciently large integers n, a and b" freely generate a free subgroup of rank two. PROOF. Let

be projective matrices in PSL2 (C) with x a variable in A and B are parabolic (see chapter 4) so they have infinite order. Suppose R(a, b) = atlb ~l.,.at"b~",all ti ~ 0, ui ~ 0, and consider R(A, B), the projective matrix formed by substituting A and B into the expression for R. Then R(A,B) ---- :l:(~~fl(X) f3(x) f4(x) where fl(x), f2(x), fz(x), f4(x) axe real polynomials Recall that if T E PSL2(C) then T"~ = 1 if and only if tr(T) -t-2 cos(k~r/m) where god(k, m) =

3.4

SUBGROUPTHEORYOF ONE-RELATORGROUPS

51

Nowlet f(x) -- fl(x) ÷ f4(x) = tr(R(A,B)). This is non-constant in x so by the fundamental theorem of algebra there exists a solution x0 to the polynomial equation f(x) 2cos(r/m). Fo r th is x0 in A andB we would then have R’~(A, B) = 1 and hence the map a -~ A, b ~ B gives representation of G into PSL2 (C). Now

u=+

o ’

Z

01)

generate a free group whenever [a~[ _> 4. Then if n is chosen large enough A=±(~ x°)Bn=±(11 ’

nxo ~)

will be the basis for a free group of rank two. The basic idea in Ree and Mendelsohn’s proof will be applied to general one-relator products of cyclics in chapter six and is really the kick-off point for all the algebraic generalizations. Helling [He] and Culler and Shalen [Cu-S] have studied the affine set of all possible representations of a finitely generated group G into SL2((~). By examining this set, especially its dimension as an affine algebraic set, a great deal of information about G can be deduced. Wewill say more about this later. A third linear property we consider is that of being virtually torsionfree, that is having a torsion-free normal subgroup of finite index. A classical result of Fenchel and Fox {[see L-S]} says that a Fuchsian group is virtually torsion-free. This was extended by Selberg [Se] to any finitely generated linear group. Here characteristic zero is essential; the Selberg theorem is not true over characteristic a prime. Fischer, Karrass and Solitar haven proven the corresponding result for one-relator groups. THEOREM 3.4.4. Any o~e-relator group is virtually

torsion-t’ree.

This is clearly true if the group itself is torsion-free and therefore the work goes into showing the result when/the relator is a proper power. Wegive a proof of this result using representation theoretic techniques in chapter six. Discrete subgroups of PSL2(R) have the property that abelian subgroups must be cyclic. B.B. Newman[Ne 2] has determined similar restrictions on the abelian subgroups of one-relator groups. THEOREM 3.4.5. Let H be an abelian subgroup o~e the one-relator group G. Then H is either (1) cyclic, (2) /tee abelian oT rank two or

ALGEBRAIC GENERALIZATIONS OF DISCRETE GROUPS

52

(3) m-adic for so,he positive integer m, ~ha~ is isomorphic ~o ~he additive subgroup of ~he rationals of ~he tbrm ,j/m ~ wi~h ’m ~ed and d, k arbitrary infegers. ~r~her if U has ~orsion, ~hen H mus~ be cyc~c. The final linear property we consider is ~ha~ of being produc~ with amalgamation or an HNNgroup, hi"ore the following res~ of H. Bass [Ba 1], in a seuse, "Mmost all" subgroups of CL~(C) are such amalgam. THEOREM3.4.6. Le~ G be a ~i~ely Then one of ~he tbllo~ing m~s~ occur:

~nerated

~ab~’oup

of ~2(C).

There is an ephnorphism f : ~ ~ ~ such ~ha~ f(U) unipo~ent elements U ~ ~. (2) ~ is ~ amalgama~e~ free pro~uc~ ~ = ~ *~ ~ ~i~h G2 such ~ha~ every ~ni~eley genera~e~ ~mipo~en~ subgroup of contained in a condu~’a~e of~] or G~. (3) G is conjugate ~o a ~’ro~p o~ upper ~riang~lar ma.~rices ~II of whose ~iagonal elen]en~s are roo~s of uni~y. (4) G is conjugate ~o a subgroup orGan(A) where A hraic integers. (I)

Another reason Lha~ amalgam constructions can be considered a~ linear properties is given tl~ough the variety of representation techniques of Helling and Cu~er and Shalen mentioned earlier. %Ve now make this more precise. Suppose ~ is a finitely generated group with generators ~], ...,~. Any r~pr~s~nL~tion ~ : ~ ~ ~(~) c~n then be considered as a point

(~(~),...,~(~)) ~ (s~(c))"’. These~ep~ntationpoints ~in be su~ jected to v~ious conditions reflecting the relations in the group. These relations are polynon~al relations on the matrix entries and hence the set of all possible representation points for ~ define an ~ne algebraic set in C~. Call t~s R~(~). ~2(C) acts on R~(G) by conj~ation in the w~y. Thus we can form the categorical quotient X2 (~) of R2 (~) under action. X~ (G) is called the affine algebraic set of characters or character space of G. It can be thought of as the paralnetrization of the inequival~nt 8~u~-silnpl~ r~pre~nt~tions of ~ in ~2 (¢). CuH~r ~nd Sh~len h~ve proven: THEOREM 3.4.7. Le~ G be a finitely genera~e~ group. ~ ~he ~imension of ~he ch~’ac~er space of~ is positive, ~hen ~ decomposes ~ a non-~rivial fi’ee p~c~ wi~h amalgamation or as an ~N~ group. Thus if there are "n]aiKy" representa~ions m~st decompose as an amalgam.

into ~G~(~) then the group

3.5 CYCLICALLYPINCHEDONE-RELATORGROUPS

53

Wehave already seen that a one-relator group is either an HNNgroup or embedded into an HNNgroup. In fact, from a result of Sacerdote and Schupp [S-Sch] a one-relator non-cyclic group is an HNNgroup. From the following result of G.Baumslagand Shalen [B-S 2] it can be deduced that many one-relator groups are also non-trivial free products with amalgamation. A free product with amalgamation decomposition A *c B is called proper if C is a proper subgroup of both A and B and has index greater than 2 in at least one of them. THEOREM 3.4.8. Let G be a finitely presented group of deficiency d at le&~t 2. Then G admits a proper free product with amalgamtion decomposition G = A *c B where A, B and C are all finitely generated. COROLLARY3.4.2. Every one-relator group with at least three generators admits a proper Tree product with amalgamation decomposition.

Weclose this section by stating a result of Brodskii [Br 1],[Br 2] which has great significance for general one-relator products. A group is locally indicable if every non-trivial finiteley generated subgroup can be mapped homomorphicallyonto an infinite cyclic group. Local indicability is a very strong form of torsion-freeness. Brodskii [Br 1],[Br 2] proved that torsionflee one-relator groups are locally indicable a result reproved by Howie [H 6] THEOREM 3.4.9. Every torsion-free ble. 3.5 Cyclically

one-relator group is locally indica-

Pinched One-Relator

Groups

A surface group is the fundamental group of a orientable or nonorientable surface. If the genus of the surface is g then we say that the corresponding surface group also has genus g. It can be shown (see Chapter 4) that an orientable surface group Ta of genus g _> 2 has a one-relator presentation of the form Tg= < al, bl, ..., ag~bg; [al, bl]...[a~, b~] = 1 > while a non-orientable surface group U9 of genus g also has a one-relator presentation - now of the form U~ -~ < al,a2,...,a~;

~ = 1 >. ala~2. ~..ag

In the orientable case, if we let U = [al,bl]...[aa-~,ba-~],V = [aa,b~] then T.q decomposes as the free product of the free groups F1 on a~, bl, ...., aa~, bg_~ and F2 on aa, b~ amalgamatedover the maximalcyclic

54


subgroups generated

by U in F1 and V in F2. Hence Tg = F1 *=

F2. The algebraic generalization of the above arguments leads to the concept of a cyclically pinched one-relator group. These groups have the general form of a surface group and have proved to be quite amenable to study. The generalization in turn of cyclically-pinched one-relator groups, where we allow torsion in the generators, leads us to groups of F-type which can be considered as a natural algebraic generalization of finitely generated Fuchsian groups. Wewill discuss these in detail in chapter seven. Here we discuss some of the important results concerning cyclically pinched one-relator groups. DEFINITION3.5.1. A cyclically pinched one-relator relator group of the following form G = < al, ...,

%, av+~,...,

group is a one-

an; U = V >

where 1 # U = U(al,...,av) is a cyclically reduced, non-primitive (not part of a free basis) word in the free group F1 on a~,..., v and 1# V = V(av+l, ..., a,~) is a cyclically reduced, non-primitive word in the free group F2on ap+1, ..., an. Clearly such a group is the free product of the free groups on a~, ..., av and %+1, .-., a= respectively amalgamated over the cyclic subgroups generated by U and V. An orientable surface group Tg has the property that any 2g - 1 elements generate a free subgroup. A simple topologically motivated proof of this is as follows. By abelianizing it is clear that the rank (minimumnumber of necessary generators) of 9 i s 2 g. S uppose His a s ubgroup of Ta,thenfrom covering space theory H = ~rl(S) where S is a cover of So, the orientable surface of genus g. If ]T~ : HI < oc, then S must be another orientable surface of genus gl > g and hence H = T~. If H has infinite index in T9 then homotopically S is a wedge of circles and H is a free group. Now suppose we have Xl, ...x,, ETa with n < 2g - 1, and let H =< xl, ...,x,~ >. If H had finite index in Tg, then the rank of H would be greater than 2g which is impossible since H has rank < n. Therefore H must have infinite index in Ta and hence must be a free group. In general we make the following definition. DEFINITION 3.5.2. A group G is n-free if any set of_2 and for i = 1, .., n let W~= W~(B~)be a non-trivial word the free group on B~, neither a proper power nor a primitive element. Let G = < BI, ...,

B,~; WiW2...W,~--- 1 >.

Then G is n-free. The proof of this theorem uses the Nielsen reduction techniques discussed in the last chapter. A similar result can be obtained if the words Wi are proper powers. THEOREM 3.5.4. Let BI, ..., Bn be pairwise disjoint non-empty sets of generators, and for i -~ 1, .., n let W~= Wi(Bi) be a non-trivial word in the free group on B~. Let

c =

with ti >_ 1. Then G is (n-1)-free. This result is the best possible since a non-orientable surface group of genus g is (g - 1) free but not free.

56


These results were used in conjuction with a study by Gaglione and Spellman [G-S 1,2,3] and Fine, Gaglione, Rosenberger and Spellman [F-GR-S] on the universal theory of non-abelian free groups (see the references listed above). We’ll discuss this further in the next section. Recall that a group G is residually finite if given g E G there exists a finite quotient G* of G with the image of g non-trivial in G*. G. Baumslag [G.B. 2] has shown that all cyclically pinched one-relator groups are residually finite. THEOREM 3.5.5. Let G be a cyclically G is residually finite.

pinched one-relator

group, then

The residual finiteness of one-relator groups in general is undecided. However it has been conjectured by G. Baumslag that one-relator groups with torsion are residually finite. Several special cases of this conjecture have been handled by Allenby and Tang [A1-T 1]. In addition to residual finitenss cyclically pinched one-relator groups satisfy several stronger separability properties. A group G is conjugacy separable if given elements g, h in G either g is conjugate to h or there exists a finite quotient where they are not conjugate. J. Dyer [Dy] has proved the conjugacy separability of cyclically pinched one-relator groups. Note that conjugacy separability in turn implies residual finiteness. THEOREM 3.5.6. A cycfically pinced one-relator group is conjugacy separable, that is two elements of a cycfically pinched one-relator group G are conjugate/f and only if they are conjugate in every finite factor group of G. It was conjectured that this result could be extended to general Fuchsian groups [Ko]. Building on work of Stebe[St] and Allenby and Tang [A1-T 1], Fine and Rosenberger [F-R 4] proved the conjugacy separability of general Fuchsian groups. Wewill give the proof of this in the next chapter. A group G is subgroup separable or LERFif given any finitely generated subgroup H of G and an element g E G, with g ~ H, then there exists a finite quotient G* of G such that image of g lies outside the image of H. P.Scott [Sc] proved that surface groups are subgroup separable and then Brunner, Burns and Solitar [Br-B-S] showed that in general cyclically pinched one-relator groups are subgroup separable. Free groups themselves are subgroup separable and Tretkoff [Tr], Gitik[G], Tang[T], Kim[K],Niblo [Ni 2], Aab [Aa] and others have worked on the general question of when free products with axnalgamation of subgroups separable groups is again subgroup separable. THEOREM 3.5.7. Let G be a cyclically pinched one-relator group. Then G is subgroup separable. That is ff H is any finitely generated subgroup of

3.5 CYCLICALLYPINCHEDONE-RELATORGROUPS G and g e G, g q~ H, then there exists a finite image of g lies outside the image of H.

57

quotient G* of G such that

The decision theory of cyclically pinched one-relator groups is also very well determined. The word problem is of course solvable since it is a onerelator group. Lipschutz [Li] using small cancellation theory (see next section) has proved that cyclically pinched one-relator groups have solvable conjugacy problem. Juhasz using an extension of small cancellation theory has stated that all one-relator groups have solvable conjugacy problem [J

2]. THEOREM 3.5.8. A cyclically conjugacy problem.

pinched one-relator

group has a solvable

Rosenberger [R 21], again using Nielsen reduction methods, has given a positive solution to the isomorphism problem for cyclically pinched onerelator groups, that is, he has given an algorithm to determine if an arbitrary one-relator group is isomorphic or not to a given cyclically pinched one-relator group. THEOREM 3.5.9. The isomorphism problem for any cyclically pinched one-relator group is solvable; given a cyclically pinched one-relator group G there is an algorithm to decide in finitely manysteps whether an arbitrary one-relator group is isomorphic or not to G. More specifically let G be a non-free cyclically pinched one-relator group such that at most one of U and V is a power of a primitive element in F1 respectively F2. Suppose Xl, ...,xv+ q is a generating system [or G. Then one of the following two cases occurs: (1) There is a Nielsen transformation from {xl, ...,xp+q} to a system {al,...,ap,yx,..,yq} with Yl,..-,Yq E F2 and F~. =< V, yl,..,yq >. (2) There is a Nielsen transformation from (xl, .--,xv+q} to a system {yx,...,yp, bl,..,bq} with Yl, ...,Yp E El and F1 =< U, yl,..,yp ). For xl, ...,xp+q there is a presentation of G with one-relator. Thr~her G has only finitely many Nielsen equivalence classes of minimal generating systems. As we saw in the last section, one-relator groups in general share many properties with linear groups. Wehrfritz [We 1] has shown that manycyclically pinched one-relator groups are actually linear. THEOREM 3.5.10. Let G be a cyclically pinched one-relator group with the property that U and V are not proper powers in the respective free group on the generators which they involve. Then G has a faithful representation over a commutative field.

58


Using a result of Shalen [Sh] this can be extended to show that a cyclically pinched one-relator group of the type above has a faithful representation in PSL2(C). Wewill use this result again in conjunction with groups of F-type. THEOREM 3.5.11. Let G be a cyclically pinched one-relator group with the property that U and V are not proper powers in the respective free group on the generators which they involve. Then G has a faithful representation in PSL2(C). The small cancellation theory used by Lipschutz and Juhasz is closely tied to hyperbolicty in the sense of Gromov(see [G] and section 2.7). a consequence of the results of Bestvinna and Feighn and Kharlamapovich and Myasnikov (Theorem 2.7.6) we obtain that a cyclically pinched onerelator group, where not both U and V are proper powers, is hyperbolic. THEOREM 3.5.12. Let G be a cycfically pinched one-relator group with the property that not both U and V are proper powers in the respective free group on the generators which they involve. Then G is word hyperbolic. Theorem 3.5.12 was further generalized by Juhasz and Rosenberger (see [J-R]). From a result of G.Baumslag and Shalen [B-S 2] a one-relator group G with at least three generators admits a free product with amalgamation decomposition G -- A *c B with A, B and C all finitely generated. Such a decomposition is called a Baumslag-Shalen decomposition. Clearly cyclically pinched one-relator groups are straightforward examples of such decompositions for one-relator groups. Howeverin general very little is known about the exact nature of the factors. A study of the factors was done by Fine and Peluso IF-P] who gave the following partial converse to the cyclically pinched case. THEOPREM 3.5.13. Let G be a torsion-free one-relator group with Baumslag-Shalen decomposition A *c B with both A and B Tree groups. Then G must be cycfically pinched if either C has finite index in both factors or C is in the derived group in both factors. Further, if C has finite index in both factors then the groups A, B, C are all infinite cyclic and G has a presentation of the form < a, b; an = bm > with my n > 1. A result of Bieri [Bi] is that if G = A *c B is a torsion-free one-relator group with A, B finitely presented and C of finite index in both A and B then A and B must be free groups. Combining this with Theorem 3.5.13 gives us:

3.6

CONJUGACY PINCHED ONE-RELATORGROUPS

COROLLARY 3.5.1. Let G be Baumslag-Shalen decomposition nite index in both A and B then and G has a presentation of the

a torsion-free A *c B. If C is the groups A, B, form < a, b; an

3.6 Conjugacy Pinched

59

one-relator group with a free group and of fiC axe all infinite cyclic = bm > with m, n > 1.

One-Relator

Groups

The HNNanalog of cyclically pinched one-relator groups, which we term conjugacy pinched one-relator groups, also arise in many different contexts and share manyof the general properties of the cyclically pinched case. DEFINITION 3.6.1. A conjugacy pinched one-relator one-relator group of the form

group is a

a,~, t; rut -1 = V >

G = < a~, ...,

where 1 ?~ U = U(a,, ..., an) and 1 # V = V(ap+b ..., reduced in the free group F on ai, ..., a,~.

an) axe cyclically

Structurally such a group is an HNNextension of the free group F on al, ..., an with cyclic associated subgroups generated by U and V and is hence the HNNanalog of a cyclically pinched one-relator group. If we return to the surface group Tg (3.)

Tg =< al,b~,...,%,b~;

[al, bll...[%,b~] = 1 >

and we let bg = t then Ta has the form Tg ---< al,bl,

...,a~,t;tUt -1 = V > .

where U = a~ and V = [al,bl]...[ag-l, bg-1]a9 and hence T9 is also a conjugacy pinched one-relator group. The question arises as to which of the general properties of cyclically pinched one-relator groups can be extended to the class of conjugacy pinched one-relator groups. Given the structural similarities of free products with amalgamation and HNNextensions, many similaxities in properties are expected, of course somewhatmodified. Besides the natural ties with surface groups conjugacy pinched onerelator groups arise independently in other contexts as well. Recall that a group acts freely on a tree T if it acts as a group of isometrics on T with no fixed points or inversions (see sections 2.5 and 2.8. If the tree is an ordinary simplicial tree then from Bass-Serre theory G must be a free group. If T is an R-tree (see [section 2.8) then Rips Theorem (Theorem 2.8.1) says

60


G must be a free product of abelian groups and surface groups. Bass studied the general concept of a free action on a A-tree where A is an ordered abelian group (see section 2.8). At present there is no general structure theorem for groups acting freely on arbitrary -trees, however Alperin and Bass [A-B] have provided as examples of groups which allow free actions on A-trees cyclically pinched one-relator groups and conjugacy pinched onerelator groups except for the Klein bottle group < a, b; aba-1 = b-1 >. In particular they prove the following result. THEOREM 3.6.1. [A-B] (1) Let G =< al,...,ap,ap+~,...,an;U =V be a cyclically pinched one-relator group and suppose neither U nor V is a proper power. Then G is tree-free. In paxticular G is Z @Z-free where Z ® Z has the lexicographic ordering. -1 = V > be a conjugacy pinched one(2) Let G =< al,...,an,t;tUt relator group and suppose that neither U nor V is a proper power and that U is not conjugate to V-1 within the free group on al, ...,a,~. Then G is tree-free. In particular G is Z ® Z-free where Z @Z has the lexicographic ordering. Further conjugacy pinched one-relator groups appear in the classification of fully residually free groups and the study of CSAgroups (see F-G-M-RS]). These results are important in the solution of the Tarski problem. This will be examinedmore carefully in the next section.. The 2-free and 3-free results for cyclically pinched one-relator groups carry over with modifications to conjugacy pinched one-relator groups. The results for cyclically pinched one-relator groups used Nielsen reduction in free products with amalgamation as developed by Zieschang, Rosenberger and others (see Section 2.6 and [F-R-S 1]). The corresponding theory Nielesen reduction for HNNgroups was developed by Pecyzski and Reiwer [P-R] and is used in the analysis of eonjugacy pinched one-relator groups. The best applications of Peczynski and Reiwer’s results are in the case that the associated subgroups are malnormal in the base. Recall that H C G is malnormal if xHx-1 CI H = {1} if x ~ H. For a cyclic subgroups of a free group F this requires that U is not a proper power in F. Using this, Fine,Roehl and Rosenberger [F-R-R 1], proved the following two-free result. -1 = V > be a conjugacy THEOREM 3.6.2. Let G =< a~,...,a~,t;tUt pinched one-relator group. Suppose that neither U nor V are proper powers in the free group on al, ...,a,~. IT< x,y > is a two-generator subgroup of G then one of the following holds: (1) < x, y > is free of rank two (2) < x, y is abelian (3) x, ’y > has a presentation < a,b; aba-~ = b-1 >.

3.6 CONJUGACY PINCHED ONE-RELATORGROUPS

61

As a direct consequence of the proof, the following is obtained. COROLLARY 3.6.1. Let G be as in Theorem 3.6.1 and suppose that U is not conjugate to V-1 in the free group on al,...,an. Then any twogenerator subgroup of G is either free or abelian. The extension of Theorem3.6.2 to a 3-free result proved to be quite difficult and required some further modifications. A two-generator subgroup N of a group G is maximal if rankN = 2 and if N C M for another two-generator subgroup M of G then N = M. A maximal two-generator subgroup N =< U, V > is strongly maximal if for each X E G there is a Y E G such that < U, XVX-1 >C< U, YVY-1 > and < U, YVY-~ > is maximal. Building upon, and extending the theory of Peczynski and Reiwer, the following was proved by Fine,Roehl and Rosenberger [F-R-R 2] -1 = V > be a conjugacy THEOREM 3.6.3. Let G =< al .... ,a,,,t;tUt pinched one-relator group. Suppose that < U, V > is a strongly maxima/ subgroupof the free group on al, ...,an. The G is 3-free. If < U, V > is not strongly maximal we can further obtain that a subgroup of rank 3 is either free or has a one-relator presentation. -1 = V > be a conjugacy THEOREM 3.6.4. Let G =< ai,...,an,t;tUt pinched one-relator group. Suppose that neither U nor V is a proper power in the free group on al,...,an and in this free group U is not conjugate to either V or V-~. Let H =< Xl,X2,X3 >C G. The H is free or has a one-relator presentation on < xl,x2,x 3 >. An analysis of the techniques used in the proof of Theorem3.6.3 leads to several partial solutions of the isomorphism problem for conjugacy pinched one-relator groups. First: -~ -~ V > be a conjugacy THEOREM 3.6.5. Let G =< al,...,an,t;tUt pinched one-relator group and suppose that neither U nor V is a proper power in the free group on al, ..., an. Suppose further that there is no Nielsen transformation from { al , ..., aN} to a system {bl, ..., bn} with U ~ {bx, ..., bn-1} and that there is no Nielsen transformation from {al, ..., an} to a system (cl, ...,cn} with V ~ {cl, ...,cn-1}. Then: (1) G has rank n d- 1 and for any rninimal generating system for G there is a one-relator presentation. (2) The iso~norphismproblem for G is solvable, that is it can be decided algorithmically in finitely many steps whether an arbitrary given one-relator group is isomorphic to G. (3) G is Hopfian

62


Wenote that the results of the above theorem hold when n = 2 and U, V ~ is the are elements of FVF~ where F is the free group on al, a2 and FVF subgroup generated by the pth powers (p _> 2) and the commutators. The techniques developed for the proofs of Theorems 3.6.3,3.6.4 and 3.6.5 were also used in the study of the isomorphism problem for a class of para-free groups introduced by G. Baumslag. In particular in [GB 3] Gilbert Baumslag introduced the class of groups G~,~ for natural numbers i, j, defined by the presentations G~,j --< a,b,t;a -1 = [bi,a][b~,t]

>

This class is of special interest since the groups are para-free, that is they share many properties with the free group F of rank 2. In particular, if "y~(G~5) are the terms of the lower central series of Gis, then for all Gi,j/~(G~5) ~- F/~I~(F) and further the intersection over all n of the ~/~(a,,~) is {1}. Magnus and Chandler [C-M] in their History of Combinatorial Group Theory mention the class G~,i to demonstrate the difficulty of the isomorphism problem for torsion-free one-relator groups. They remark that as of 1980 there was no proof showing that any of the groups G~,j are nonisomorphic. S. Liriano [Lir] used representations of Gi5 into PSL(2, pk), k N, to show that GI,~ and Gao.a0 are non-isomorphic. Notice that if we let U --- a[b~, a]bi and V -- b~ then Gi5 is a conjugacy pinched one-relator group G~,~ =< a,b,t;t-~Ut

--- V >.

If in addition j -- 1 then and < a[bi,a]b > are maximal cyclic in < a, b; > and hence malnormal and therefore the techniques of Theorems 3.6.3,3.6.4 and 3.6.5 can be applied. Fine,Rosenberger and Stille [F-R-S 2] then proved: THEOREM 3.6.6. Let i be a natural number. Then: (1) the isomorphismproblem for G~,I is solvable, that is it can be decided algorithmically in finitely many steps whether or not an arbitrary onerelator group is isomorphic to G~,~. (2) Gi,1 is not isomorphicto GI,I ~¢or i >_2. (3) ff i, k axe primes then Gi,1 -~ Gk,1 if and only if i = k. (4) [or all natural numbers i, Gi,~ is Hopfian, every automorrphism o[ Gi,~ is induced by an au~omorphism o[ the Tree group F* = F*(A, B, T)

3.6

CONJUGACY PINCHED ONE-RELATORGROUPS

63

of rank 3, with respect to the epimorphism A -* a, B -~ b, T --* t, and the autornorphism group AutGi,1 is finitely generated. These techniques can also be applied to analyze a second class of groups further extending surface groups. This was done in [F-R-S 4]. These have the presentations (2)

Ka, a,n ==. G has reg~ quotients at {u, v} if there e~sts a positive imeger m such that for each positive integer s there is a finite index normal subgoup N of G with N~ =< u~ > and N~ < v >=< vm~ >. Niblo [Ni] and Kim [K] proved that if F is a ~ goup and F h~ reg~ quotients at {U, V} then the corresponding conjugacy pinched on,relator ~oup < F, t; rUt -1 = V > is sub~oup se~ arable. D.Wise [Wis] subsequently showed that a ~ goup has regular quotients at a pair of elements u~ess the elements have conjugate powers. Combining these we have the following. -~ = V > be a conjugacy THEOREM 3.6.8. Let G =< al,...,a~,t;tUt pinched on,relator group wi~h U,V non-~ri~al elements of infinite order in ~he ~ee group on al,..., a, whi~ do no~ ha~ conjugate powers in ~his ~ goup. Then G is subgroup separable. This result has been ~ended in v~io~ ways. Rosenberger and S~se [R-S] proved the following.

64


THEOREM 3.6.9. Let A be a group and U,V be elements of infinite order in A and let G =< A,t; tUt -1 > be the HNNextension associating U and V. Suppose further ~hat A has regular quotients at {U, V} and ~hat A is both -separable and < V >-separable. Then G is residuMly finite. D.Wise [Wis] extended this in the following way. A Baumslag-Solitar group BS(n, m) (see [L-S]) is the group < a, t; ta=t -1 = a m >. If n ~ ±m then BS(n, m) is not subgroup separable (see [Me]). Wise in [Wis] gave the following characterization of when multiple cyclic extensions of free groups are subgroup separable. THEOREM 3.6.10. G=< al, ...,

Let a,~, tl, ...,

t,~; tiU~t~~ = V1, ...,

tUmt-~ = V,~ >

be a multiple cyclic HNNextension of the free group on al, ..., a,~ where U1, Vi, ..., Urn, V,~ are non-trivial cyclically reducedwordsin $his free group. The G is subgroup separable unless it contains a subgroup isomorphic to

BS(m, n) ~th n +m . Wise [Wis] and Aab [Aa] independently, also considered the subgroup separability of graphs of free groups with cyclic edge groups, continuing the work of Brunner,Burns and Solitar [B-B-S], Tretkoff[T] ,Gitik [Gi], Kim[K] and Niblo [Ni 2] mentioned in the last section. 3.7 Fully

Residually

Free Groups and the Tarski Problem

Conjugacy pinched one-relator groups also arise in the study of fully residually free groups. This study has become quite important because of its relationship to the recent solution of the Tarski problem. The Tarski problem lies at the intersection of group theory and logic and remained open for fifty years. While the solution is beyond the scope of these notes in this section we describe the problem and the ties to conjugacy pinched one-relator groups. Wemust first define some necessary concepts from both group theory and logic. Let L0 be the first order language with equality containing a binary operation symbol, a unary operation symbol -1 and a constant symbol 1. This is clearly a first order language appropriate for group theory. A universal sentence of L0 is one of the form V~{¢(~)} where ~ is a tuple of distinct variables, ¢(~) is a formula of L0 containing no quantifiers and containing at most the variables of 5. Similarly an existential sentence is one of the form 3~{¢(~)} where ~ and ¢(5) are as above. universal-existential sentence is one of the form W2~{¢(~, ~)}. Similarly for an existentialuniversal sentence. It is known that every sentence of Lo is logically

3.7 FULLYRESIDUALLY FREE GROUPSANDTHE TAP, SKI PROBLEM 65 equivalent to one of the form Qlxl...Q~x~¢(~) where ~ = (Xl, ...,x~) tuple of distinct variables, each Qi for i -- 1, ...,n is a quantifier, either ~’ or ~, and ¢(5) is a formula of Lo containing no quantifiers and containing free at most the variables Xl, ...,x~. Further vacuous quantifications are permitted. Finally a positive sentence is one logically equivalent to a sentence constructed using (at most) the connectives V, A, ~¢, If G is a group then the universal theory of G consists of the set of all universal sentences of Lo true in G. Since any universal sentence is equivalent to the negation of an existential sentence it follows that two groups have the same universal theory if and only if they have the same existential theory. The set all sentences of L0 true in G is called the first order theory or the elementary theory of G. We denote this by Th(G). We say that two groups G and H are elementarily equivalent if they have the same first order theory, that is Th(G) = Th(H). If f : H -~ G is a monomorphism then f is an elementary embedding provided whenever ¢(x0, ..., xn) is a formula of Lo containing free at most the distinct variables xo, ..., x,~ and (h0, ..., h~) E ~+1t hen ¢(h0,, . .., h ~) i s t rue i n Hif andonly if ¢(f(ho),, ..., f(h~)) is true in G. The existence of an elementary embedding f : H -~ G is a sufficient condition for H and G to be elementarily equivalent. Tarski made several conjectures concerning the elementary theory of the non-abelian free groups, TARSKI CONJECTURE 1. Any two non-abelian free groups are elementarily equivalent. That is any two non-abelian Tree groups satisTy exactly the same first order theory.

TARSKICONJECTURE 2. If the non-abelian free group H is a free factor in the Tree group G then the inclusion map H --~ G is an elementary embedding. Clearly the second conjecture is stronger than the first and implies the first. If true then the theory of the non-abelian free groups would be complete. That is given a sentence ¢ of L0 then either ¢ is true in every nonabelian free group or ¢ is false in every nonabelian free group. In addition to the completeness of the theory of the nonabelian free groups the question of its decidability also arises. By this we mean does there exist a recursive algorithm which, given a sentence ¢ of Lo decides whether or not ¢ is true in every nonabelian free group. Tarski further conjectured that the theory of the nonabelian free groups is decidable. TARSKICONJECTURE 3. The elementary theory of the nonabelian free groups is decidable,

66


Kharlamapovich and Myasnikov [Kh-M 4] have recently am~ounced that all of the above Tarski conjectures are indeed true. Their proof grew out of work on residually free groups and on the algebraic geometry of groups. THEOREM 3.7.1. [Kh-M4] If the nonabelian free group H is a free factor in the free group G then the inclusion map H --~ G is an elementary embedding. Hence any two non-abelian free groups are elementarily equivalent. Moreover the theory of the nonabelian free groups is decidable. Their proof is beyond the scope of this survey but we now backtrack a little and consider some of the connections to previous work in these notes.. Since any two countable non-abelian free groups F and G are embeddable in each other it follows that F and G satisfy precisely the same universal sentences. It can be deduced that the same is true for any two free groups of infinite rank. A universally free group is a group which has the same universal theory as the non-abelian free groups. A group is elementarily free if it has the same elementary theory as the nonabelian free groups. Recent work has shown remarkable ties between these logical concepts, especially universal £reeness, and certain purely group theoretical constructions. First we exhibit a tie with groups actions on trees. Gaglione and Spelhnan [G-S 5], using work of Alperin and Bass [A-B], and independently Remeslennikov [Re 3] using a different technique, have shown that a universally free group must be tree-free. THEOREM 3.7.2. Let G be universally

free. Then G is tree-free.

The converse of Theorem 3.7.2 is not true. Examples ofnon-universally free, tree-free groups were given by Fine,Gaglione,Rosenberger and Spell1nan [F-G-R-S] and independently by Remeslennikov [Re 3]. To explain these examples and get a better understanding of universally free groups certain other group theoretic properties must be introduced. A group G is residually free if for each non-trivial g E G there is a free group Fg and an epimorphism ha : G -~ Fg such that ha(g ) ~ 1 and is fully residually free provided to every finite set S C G\ {1} of non-trivial elements of G there is a free group Fs and an epimorphism hs : G -~ Fs such that hs (g) ~ 1 for all g E S. Clearly fully residually free implies residually free. It is an easy observation that being fully residually free implies commutative transitivity. B.Baumslag [B.B. 2] proved a converse - commutative transitivity together with residually free implies fully residually free. Gaglione and Spellman [G-S 1] and independently Remeslennikov [Re 3] then were able to show that these conditions in the presence of residual freeness are equivalent to universal freeness in the non-abelian case. Thus there is the following rather remarkable theorem.

3.7 FULLYRESIDUALLYFREE GROUPSAND THE TARSKI PROBLEM THEOREM 3.7.3. Let G be non-abelian and residually following axe equivalent: (1) G is fully residually free (2) G is commutative transitive (3) G is universally free

free.

67

Then the

Further Gaglione and Spellman [G-S 3] proved if G is finitely presented then universally free implies residually free, while Remeslennikov [Re 3] showedfinitely generated and universally free implies residually free. Chiswell [Ch 5] as a corollary then gives this characterization of universally free groups. THEOREM 3.7.4. A group G is universally free if and only if it is nonabelian and locally fufly residually free. Based on these results we can present an example of a tree-free group that is not universally free. Let (g, n) be a pair of integers both >_ 2. Let G(g, n) be the group with presentation g--1

G(g, n) = ¯ i~l

B.Baumslag,F.Levin and G.Rosenberger [B-L-R] showed that G(2, n) is not residually free whenever n >_ 4g. Hence from Theorem3.7.3 for n _> 4g the group G(g, n) is not universally free. Howeverit is a cyclically pinched one-relator group satisfying the non-proper power condition. Hence from Bass’s results it is tree-free. The non-residual freeness of the group G(g, n) is also interesting in another context. Suppose both G and H are residually free. Then the free product G, H may not be residually free. However if both G and H are fully residually free then G * H is residually free. The question then arises as to whena generalized free product of fully residually free groups is residually free. The simplest exampleof such a generalized free product of fully residually free groups would be a cyclically pinched one-relator group. We will return to this question at the end of this section. Using the newly developed theory of algebraic geometry over groups (see [Kh-M1,2]) Myasnikov and Kharlamapovih [Kh-M1,2] prove that any finitely generated fully residually free group embedsin the free exponential group F~z[x]. [~l They also show that the finitely generated subgroups of F~ are finitely presented and hence any finitely generated fully residually free group must be finitely presented. This had been conjectured in several different places. Sela [Sel 2] has also announceda proof of this last result.

68


THEOREM 3.7.5. (/) Any finitely generated fully residually free group embeds in the free exponential group z[x] .F~ (2) Every finitely generated subgroupof F~z[x] is finitely presented. COROLLARY 3.7.5. Any finitely must be finitely presented.

generated fully

residually

free group

Similar techniques using ultrafilters combined with Nielsen reduction techniques were used by Fine,Gaglione, Myasnikov, Rosenberger and Spellman IF-G-M-R-S]to give a complete classification of fully residually free groups of rank 3 or less. The following construction, which for free bases is a special type of conjugacy pinched one-relator group, is needed. Let G ~ 1 be a commutative transitive group. Let u e G \ {1} and let M = Zc(u) where Zc(u) is the centralizer of u in G. Then G(u,t) =< G,t; rel (G),t-lzt

= z, for all z 6 M >

is the free rank one extension of the centralizer

M of u in G.

THEOREM 3.7.6. Let G be a fully residually free group. Then (1) if rank(G) = 1 then G is infinite cyclic. (2) if rank(G) = 2 then either G is free of rank 2 or free abelian of rank 2 (3) if rank(G) = 3 then either G is free of rank 3, free abelian of 3 or a free rank one extension of centralizers of a free group of rank 2. That is G has a one-relator presentation G-~

element of the free group on xl,x2

In the course of proving the classification is also obtained.

the following startling

result

THEOREM 3.7.7. Every 2-free, residually free group is 3-free. The proof of the classification result depends upon Theorem 3.7.7 and the following construction. Wedefine the class ~" as the smallest class of groups containing the infinite cyclic groups and closed under the following four "operators": (1) Isomorphism (2) Finitely Generated Subgroups (3) Free Products of Finitely ManyFactors

3.7 FULLY RESIDUALLYFREE GROUPSAND THE TARSKI PROBLEM69 (4) Free Rank One Extensions of Centralizers The class ~- is then classified as the class of all finitely generated groups embeddable in (F~) zM. Further every 3-generator fully residually free group lies in ~’. THEOREM 3.7.8. (1)The class ~ is precisely the class of all tinitely erated groups embeddable z[" in ]. (F~) (2) Every 3-generator f~dly residually free group lies in

gen-

Remeslennikov, jointly with Kharlamapovich and MyasIfikov (see [KhM1,2]) has given the following more inductive characterization of the class of finitely generated fully residually free groups. This class is properly contained in the class of groups which start with flee abelian groups of finite rank and are constructed by repeated iteration of the following four operations: (1) free products (2) amalgamated free products with abelian amalgamated subgroups least one of which is maximal abelian (3) free extensions of centralizers (4) separated HNNextensions with abelian associated subgroups at least one of which is maximal abelian. An HNNextension H --< G, t; t-~At -B > is a separated HNNextension ifg-~Ag ~ B = {1} for all g ~ G. This construction allows for the following inductive characterization of finitely generated fully residually free groups and allows for inductive type proofs. A fully residually free, group G is of level n if it can be constructed from an infinite cyclic group by n iterations of the above operations and not n - 1 such iterations. Remeslemfikov[Re 2] proved that every finitely generated fully residually free group acts freely on someZ’~-tree with someorder for a suitable nat~zral number n. He asked if such a group acts freely on a Z~-tree with the lexicographic order. This was answered affirmatively by Kharlamapovich and Myasnikov ([Kh-M 1,2]). Wenow summarize some of the various inclusions and equivalences of the properties discussed. All the inclusions are proper. First of all we have

(1) free groups C fldly residually free groups C residually free groups We also have

(2) mfiversally free groups = non-abelian locally fully residually free groups

(a) universally free = residually free together with cmmnutative transitive

70

ALGEBRAICGENERALIZATIONSOF DISCRETE GROUPS (4) tree-free

groups C CSAgroups C commutative transitive

3.8 Small Cancellation

groups

Theory

As indicated earlier,Magnus, using the FHS and the method developed for its proof, was able to provide a solution to the word problem for onerelator groups. In fact his proof established the stronger result that if G -< X; R > is a one-relator group and w an arbitrary word in G, then there is an algorithm to determine if w belongs to the subgroup generated by any arbitrary subset of the given generators. This is called the generalized word problem. Magnus’ work on the word problem was motivated in large part by solution given by MaxDehn [De] (see [MiD to the word problem for finitely generated orientable surface groups (see section 2.7). Recall that these groups have one-relator presentations of the form ~g ~- < Xl, Yl,-..

, Xg, yg; R ----

1 > where R -- Ix1, Yl] .. ¯ [xa, y~].

Dehn proved that in a surface group S9 any non-empty word w in the generators which represents the identity, must contain at least half of the original relator R, that is if w = 1 in $9, then w -- bed where for somecyclic permutation R~ of R, R’ -=- c¢ with Itl < Icl where I I represents free group length. It follows then that w --- bt-ld in S~ and this word representation of w has shorter length than the original. Given an arbitrary w in S9 we can apply this type of reduction process to obtain shorter words. After a finite numberof steps we will either arrive at 1 showingthat w --- 1 or at a word that cannot be shortened in which case w ¢ 1. This procedure solves the word problem for Sa and is known as Dehn’s Algorithm for a surface group. Dehn’s original approach was geometric and relied on an analysis of the tessellation of the hyperbolic plane provided by a surface group. Recall that the idea of a Dehn algorithm can be generalized in the following manner. Suppose G has a finite presentation < X; R > (R here is a set of words in X). Let F be the free group on X and N the normal closure in F of R, N = NF(R) so that G = FIN. G, or more precisely the finite presentation < X; R >, has a Dehn Algorithm, if there exists a finite set of words D C N such that any non-empty word w in N can be shortened by applying a relator in D. That is, given any non-empty w in N, w has a factorization w = ubv where there is an element of the form bc in D with Icl < [b I. Then applying bc to w we have w = uc-lv in G where luc-lvl < [ubv t. By the same argument as in the surface group case the existence of a Dehn Algorithm leads to a solution of the word problem.

3.8 SMALLCANCELLATION THEORY

71

The existence of a Dehn algorithm for a presentation is closely tied to being word-hyperbolic as revealed in the remarkable Theorem 2.7.1 which said that If G is finitely presented, then G is word hyperbolic if and only if G satisfies a Dehnalgorithm (see section 2.7). As pointed out in section 2.7, the general idea of a Dehn algorithm is clearly that there is "not much cancellation possible in multiplying relators". Although Dehn’s approach was geometric, the ideas can be phrased purely algebraically. This is the basic notion of small cancellation theory. This theory was initiated in 1947 by Tartakovskii IT] who showed, using purely algebraic methods, that certain groups, besides one relator groups, also satisfy a Dehn Algorithm. His conditions were, that in these groups again there is not muchcancellation in multiplying relators. Greendlinger [Gr], Schiek [Sch] and Britton [Bri] introduced other "small cancellation conditions" and also obtained Dehn Algorithms, and thus greatly expanded the class of groups with solvable word problem. Lyndon [L 2], in the mid 1960’s placed the whole theory in a geometric context and thus returned to Dehn’s original approach. Lyndon used this geometric approach to reprove the FHS. The geometric constructions used by Lyndon, now called Lyndon - Van Kampen diagrams, have been extended and modified for use in proving the FHS for one-relator products. Wewill return to this in Chapter 5. A complete and readable account of small cancellation theory can be found in Chapter 5 of Lyndon and Schupp’s book [L-S]. The proofs, both algebraic and geometric, are quite complex. Whatwe will do in this section is define the small cancellation conditions for free groups and then introduce the geometric constructions that go along with them. In Chapter 5, in conjunction with our discussion of one-relator products we will give the extensions of small cancellation theory to various amalgamconstructions. Suppose F is free on a set of generators X. Let R be a symmetrized set of words in F. By this we mean that all elements of R are cyclically reduced and for each r in R all cyclically reduced conjugates of both r and r -1 are in R. If rl and r2 are distinct elements of R with r~ -- bc~ and r2 -- bc2, then b is called a piece. Pieces represent those subwords of elements of R which can be cancelled by multiplying two non-inverse elements of R. The small cancellation hypotheses state that pieces must be relatively small parts of elements of R. Wedefine three small cancellation conditions, the most commonbeing the first which is a metric condition. The first is a metric condition denoted CI(A) where A is a positive real number. This condition asserts that if r is an element of R with r -- bc and b a piece, then Ibl < ~lcl. If G is a group with a presentation < X; R > where R is symmetrized and satisfies Cr(A), then G is called a ~- group. So for example, if A -- 1/6, G is a sixth group, etc.

72


Ifp is a natural number, the second small cancellation condition is a nonmetric one denoted C(p). This asserts that no element of R is a product fewer than p pieces. Notice that C’(A) implies C(p) for ), 1/( p - 1). The final small cancellation condition is also a non-metric condition denoted T(q) for q a natural number. This asserts the following: Suppose rl,... ,rh with 3 with R a symmetrized set of words. As before let F be free on X and let N be the normal closure in F of R so that G = FIN. If w is a word in G, then w = 1 if and only if w as a word in F is a product of conjugates of elements of R, that is, w = ulrlu~ 1 ... u,,r,~u~, ~ where the u~ are words in F and the r{ are elements of R. The sequence ulriu-~,... , umr,,u~, ~ is called an R -sequence of length m for w. A minimal R-sequence for w is an R-sequence of minimumlength. Wewill associate to any R-sequence for w a connected, simply-connected diagram in the Euclidean plane called an R-diagram. The small cancellation hypotheses are analyzed by analyzing these diagrams. A Lyndon-Van Kampen diagram for a group F consists of a collection of Mof pairwise distinct vertices, oriented edges and regions in the Euclidean plane together with a labelling function f assigning to each oriented edge e an element f(e) of F. This labelling function must satisfy f(e -1) = f(e) -1 where e -1 is the oppositiely oriented edge of e. Further if a is a path in Mwith o~ = e~--- en, then f(a) is defined as f(el)"" f(en). If D is a region in M, a label of D is an element f(a) for any boundary cycle of D. Wehave the following result which summarizes many of the geometric properties and existence of these diagrams: THEOREM 3.8.1. (see [L-S]) Let F be a free group and el,... ,c~ sequence oze non-trivial elements of F. Then there exists a diagram M = M(cl,... , cm) over F satisfying the following properties. (i) H is an edge of M, f(e ) ~ 1 (ii) M is connected and simply connected with a disting~zished vertex 0 on the boundary of M. There is a boundary cycle el,... ,en of M ( a cycle in M o£ minimal length which contains all the

3.8 SMALLCANCELLATION THEORY

73

edges in the boundary of M) beginning at 0 such that the product f(el).." f(e~) is reduced without cancellation and f(el).., f(e~) Cl " " " Era

(iii) If D is a region of M and el,... ,ej is a boundary cycle olD, then f(e~).., f(ej) is reduced without cancellation and is a cyclically reduced conjugate of some ci. The next provides a converse to the above theorem and also allows us to relate this to R-sequences. THEOREM 3.8.2. (see [L-S]) Le~ M be a connected, simply connected diagram over a group F with regions D1,... , Dra. Let a be a boundary cycle oTMbeginning at a vertex vo on the boundary of M and let w = f(a). Then there exists boundary labels ri of Di and elements ui off, 1 < i < m, SUCh ~hat

w -~- ?txrl~t~

-1.-.

1. Umrrn~t~n

Nowsuppose that R is a symmetrized subset of words in a free group F. An R-diagram is a diagram Mover F such that if 0 is any boundary cycle of any region D of M, then f(O) is in R. If G = FIN as before, then from the two theorems we obtain the following fact. A word w in F is in N if and only if there exists a connected, simply connected R-diagram Msuch that the label on the boundary of M is w. Thus connected, simply connected diagrams provide an adequate tool for studying membership in normal subgroups of free groups. The analysis of the small cancellation conditions lies in analyzing the structure of R-diagrazns under these conditions. Historically, Van Kampendiscovered these diagrams in the 1930’s [VK], but they were apparently not used to any great extent until Lyndon ILl] and Weinbaum[W] applied them to small cancellation theory. In doing this Lyndonprovided manycombinatorial generalizations of regular tessellations of the plane (see also [L-S]), again returning to the ideas of Dehn. In particular Lyndon used Van Kampen diagrams to translate Magnus’ original proof of the Freiheitssatz into combinatorial geometry. The main tool employed in this was the maximum-minimummodulus principle for these diagrams, which can be described in the following manner. Consider a presentation G = < X; R >, and assume that X is a disjoint union of subsets X~- . The generators in each are said to be of the same type. Assumefurther that there is an integer-valued function assigning a subscript to each generator in X. If w is a word on X and T is a type, then maxT(w) will denote the generator of type 7" with maximumsubscript which occurs in w (when w involves a generator of type T). Similarly define minT"(w). The presentation < X; R > is staggered if every relator in R contains at least one generator of each type and the following condition holds for every type T: If i < j, then maxT(ri) < maxT(rj)

74


and minT(r~) < minT(r~). ( x~ if x~ andxj are generat ors of the same type with i < j.) Let < X; R > be a staggered presentation and R, the symmetrized set generated by R. Let M be a connected, simply connected reduced R*-diagram (see below). For any type 7, maxT(M) (respectively minT(M) will de note th e ge nerator of typ e T wit h maximum(resp. minimum)subscript which occurs as the label on an edge of The maximum-minimum modulus principle then says that for each type T, there are edges in the boundary OMlabelled by maxT(M) and maxT(M). Lyndon’s proof of the Freiheitssatz not only worked for one-relator groups but also for one-relator products where the factors are subgroups of the additive group of the reals. Wewill return to these ideas in Chapter 5 when we discuss general one-relator products. A diagram over F is a reduce which admit such faithful complex representations are greatly limited. Wewill see this in Chapter 5 when we begin our examination of representations of one-relator products. As a result of the Freiheitssatz a one-relator group < xl, ..., x~; R > with n > 1 and R cyclically reduced is always non-trivial. Howeverit is possible that it can itself be a free group. Essentially this is only true when the relator is either trivial or a primitive - that is part of a free basis. The following theorem, originally due to Whitehead [Wh] completely classifies this situation. THEOREM 3.9.2. Let G =< X; R > be a one - relator group. (1) G =< X; R > is a free group, then either R is trivial or R is a memberof a free basis for the free group F on X. (2) If X {xl, .. .,x,~} th en G has ra nk n - G cannot be generated by n - 1 or [ewer elements - unless R is a membero[ a free basis for the free group F on X. In section 3.4 we examined the structure of abelian subgroups of a onerelator group. The following theorem summariizes when the whole one relator group can be abelian. Part (1) is due to Murasugi [Mu], part (2) to Baumslag and Taylor [B-T] and part (3) is a direct consequence of the other two parts. THEOREM 3.9.3. Let G =< X; R > be a one-relator group and let F be the free group on X. (1) If IX[ >_ 3 then G has a trivial center. If IX[ = 2 and G is non-abelian then the center oze G is trivial or is infinite cyclic. (2) There is an algorithm for determining the center o/~ G. (3) If G is abelian then I is a one relator group with R cyclically reduced. A Magnus subgroup of G is a subgroup M generated by a proper subset L of X where L omits at least one gerator occurring in R. Bagherzadeh [Bag] proved: THEOREM 3.9.6. Suppose G ~ X; R ~ is a one relator group with R -1NM cyclically reduced. If M is a Magnus subgroup and g ~ M then gMg is trivial or cyclic. Weclose this section by noting that the corresponding property in onerelator products of cyclics and groups of F-type is still open.

CHAPTER IV DISCRETE

4.1 Linear Fractional

GROUPS

Transformations

By a discrete group we mean a subgroup of PSL2(C) which has no sequence of non-trivial elements converging elementwise to the identity. As indicated in Chapter One these notes arose out of certain algebraic generalizations of discrete groups. These algebraic generalizations are via presentations, called Poincare presentations (see section 4.3), which make large subclass of the discrete groups, the co-compact Fuchsian groups, look quite a bit like one-relator groups. In this chapter we review muchof the material on discrete groups in general, and Fuchsian groups in particular, which is needed to understand these generalizations. The elements of PSL2 (C) can be considered as projective matrices ± ( ac bd ) with ad - bc = l and a’ b’ c’ d ~ or as linear fractional transformations z~ _ az ÷ b with ad _ bc = l and a, b, c, d ~ C. cz + d Wewill use either interpretation as needed but in this chapter we concentrate on the latter. Linear fractional transformations are also knownas Moebius transformations or homographic substitutions. Multiplication of such maps is done via matrix multiplication. That is if T : z ~-~ z~_a~+bc~+dand U : z ~--~ Zt~- ~Cz+D then the product TU : z ~ z ~ = ~+~ where

These transformations are conformal maps of the extended complex plane to the extended complex plane. It is fairly easy to show that they preserve the class of circles and lines. That is if T ~ PSL2(C) and L is 77

78


a circle or a line in the complexplane then T(L) is also a circle or a line. Considering lines as extending circles we see that the Moebiustransformations preserve the class of circles. If T(L) = L we say that L is a fixed circle for T. If P E C and T(P) = P then P is a fixed point of T. If T(z) ~z+b cz+d then we can see from the equation az+b -- z ==~cz 2 + (d- a)z- b = cz + d that every non-trivial element of PSL2(C) has at least one and at most two fixed points in the extended complex plane (if c = 0 and a = 1 the fixed point is at infinity). From this we can also see that if Tz = Uz at more than two points then T - U. Homographicsubstitutions can be classified by their fixed points. If there is only one fixed point, the transformation is called parabolic. A parabolic map is conjugate within all of PSL2(C) to a translation

z~z’=z+~, ), E C\ {0}. If there axe two fixed points for a non-trivial map T : z ~-* z’ az+b’ ~ cz+d say zi, z~, then we define the multiplier of T to be K "~- a - CZ1

a - CZ2 (actually thepair(K,l/K)sincethereis no natural ordering of fixed points; however, thisdoesnotaffect theclassification). Theimportance of themultiplier stemsfromthefactthatif T hasmultiplierK, thenT is conjugate withinPSL2(C)to the mapping z’ = Kz. Manyof themapping properties of thesetransformations canbe deduced fromthisfact.To seetheabovenotethatT(z)takesthepointsZl,z2,oo toZl,z2,a/csoby thecross-ratio z’ - zl _ K z - zl wherez’ = T(z). Zt -- Z2

Z -- Z2

Letting G(z) --- z--zth is th en ta kes th e fo rm 2

GT(z) = KG(z) or GTG-~(z) = Kz. Suppose K = re ~ is the multiplier of T. If K = r > 0 then T is called hyperbolic; if K = e ~a, T is called elliptic while in the remaining cases T is loxodromic. The values of K and the classification in general is handled via the trace where tr(T) = a + d. Wesummarizethis classification by trace in the following theorem.

4.2

DISCRETE

GROUPS

79

THEOREM 4.1.1. Let T(z) = ~cz+d T ~non-~rivial, then T is parabolic ff and only ff tr(T) = (i) (2) T is hyperbolic if tr(T) is real and Itr(T) l > (3) T is elliptic ff tr(T) is real and Itr(T)l < (4) T is loxodromic if tv(T) is non-real. From the above we get that Moebius maps are completely determined (up to inverses) by their fixed points and multipliers. Further we obtain the following result. THEOREM 4.1.2. U and T commute only ff U, T have the same fixed points unless both have order 2 and their product has also order 2. Farther T has finite order n if and only if T is elliptic with multiplier K = e2~k~/" with gcd(k, n) = Finally the classification governs the relationship between the fixed points of a map and its fixed circles. Werestrict our attention here to maps with real traces. In particular. THEOREM 4.1.3. (/) The fixed points of a hyperbolic map lie on its fixed circles. (2) The fixed point of a parabolic maplies on its fixed circles. (3) The fixed points of an elliptic map are inverse with respect to its fixed circles. Weclose this section by mentioning that the upper half-space C × R can be made into a model of hyperbolic 3-space H3 in such a way that an element of PSL2(C) acts as an isometry. If the elements of the Moebius maps are restricted to be real so that we are in PSL2(R) then the upper half-plane can be made a model of the hyperbolic plane H2 so that elements in PSL2(R) are isometries. This will be important in the next section to our geometric analysis of Fuchsian groups. Werefer the reader to the books by Beaxdon[Be] or Katok[Ka] for a complete discussion of the relationship between hyperbolic geometry and Moebius maps. 4.2 Discrete A subgroup G C PSL2(C) is discrete non-trivial elements

Groups if G contains

no sequence of

a’~c~d,~b’~)’ T’~(z)=a~z+b-----~c~z+d,, T’~=+( which converges of discrete

to the identity

I=±(~ 0)1 elementwise.

Asexamples

groups we have the Modular group PSLu(Z) and PSL~(A)

80


where A is any discretely normed subring of C. Thus the Bianchi Groups Fd = PSL2(Od) where Od is the ring of integers in the quadratic imaginary numberfield Q(v~), d a square-free positive integer, are all discrete. (see Closely related to discreteness is discontinuity. Let G be a group of topological mappings of a topological space S into itself. If D is an open subset of S then G is discontinuous in D if the transforms of no point of D have an accumulation point in D. For subgroups G of PSL2 (C) this can be phrased - if S is an open subset of the complex plane C, then G is discontinuous in S if for all z ¯ S the set {T(z);T ¯ G} has accumulation point in S. G is a discontinuous subgroup of PSL2 (C) it is discontinuous in some open subset. The limit points of G are the accumulation points of the sets {T(z); T ¯ G, z ¯ C} If G is discontinuous it must be discrete for if {T,~} C G with all Tn ~ I and T,~ ~ I then Tn(z) ~ for al most al l z ¯ C and T~(z) ~ for all z ¯ C. Then every point z ¯ C would be a limit point and G could not be discontinuous. On the other hand it is possible for a subgroup of PSL2(C) to be discrete but not discontinuous in (:. The Bianchi Groups as defined above provide examples for this - that is each F~ is discrete but nowhere discontinuous in (~. To illustrate this, consider the Picard Group F1, the group of linear fractional transformations with Gaussian rational entries (see IF]). Every point in C is a limit point of this group. If z ¯ C, then z a limitof Ganssian rationals~~, } where bn,d~¯ Z[i]and(b~,dn) 1,since the Gaussian rationals are dense in C. Since b,~ and dn are relatively prime, there exist a,~, c~ such that a~d~ - bnc~ -- 1 and thus the transformations T~ : z ~-~ z~ = a~z ÷ bn c~z + d~ are in F1. But then T,(O) = b~/d~ -* z, so z is a limit point of F1. The above result cannot hold for real subgroups. For subgroups of PSL2(R) discreteness is equivalent to discontinuity. THEOREM 4.2.1. Let G be a s~bgroup o[PSLu(R). Then G is discrete ff and only ff G is discontinuous in the upper haft-plane. The proof of Theorem4.2.1 consists of showing that the limit points of a discrete subgroup G of PSL2(R) must lie on the real axis. It follows then that G is discontinuous in the upper half plane. Further, subgroups of PSL~(R) preserve the upper half-plane. It can be shown that PSL: (C) can be made to act on hyperbolic 3-space ~a in such a manner that the above theorem extends. That is subgroups of PSL2(C~)will be discrete if and only if they are discontinuous in ~3. We refer to Beardon’s book [Be] for a discussion of this.

4.2 DISCRETEGROUPS

81

A subgroup G of PSL2(C) is called elementary, if for g, h E G with infinite order their commutator[g, h] has trace two, that is tr([g, hi) = 2, equivalently, if any two elements of infinite order in G have a commonfixed point (considered as linear fractional transformations). The elementary, discrete groups can be easily completely classified (see [Fo]). A Fuchsian group is a non-elementary, discrete subgroup (and thus discontinuous) of PSL~(R) or a conjugate of such a group in PSL2(C). Since the real axis can be mappedon any given circle by a T e PSL2 (C), can equivalently define a Fuchsian group as a non-elementary, discontinuous subgroup of PSL2(C) which fixes a circle C and maps the interior of C on itself. C would be called the fixed circle of the group. From the definition it is clear that the elements of a Fuchsian group must have real trace (since conjugation preserves trace ). Therefore if is a Fuchsian group and T ~ G, then T is hyperbolic, elliptic or parabolic. If T were hyperbolic or parabolic, its fixed points would lie on the fixed circle C for G while if T were elliptic, its fixed points wouldbe inverse with respect to C. Further if T were elliptic, then T must bc of finite order or G would not be discrete. The theory of Fuchsian gro~lps is extensive. In this section and the next we touch on those ideas which we will need for further developments. First it follows easily that a subgroup of a Fuchsian group must itself be Fuchsian. Further we have the following theorem. THEOREM 4.2.2. A non-elementary subgroup of a ~-hchsian group is itself Fuchsian. Further if [G : H[ < oo and H is discontinuous, then G is also discontinuous. It follows that a Fuchsian group cannot be subgroup of finite index in any non-discontinuous group. Wewill consider Fuehsian groups as non-elementary, discrete subgroups of PSL~(R). Therefore, if G is a Fuchsian group, it is discontinuous in the upper half plane H and its limit points are on the real line R ~A oe. Let l~ = l~ U R U oc. An elementary, discrete subgroup G of PSL.2 (R) has finite G-orbit in H. The elementary, discrete subgroups of PSL2 (R) can completely classified. THEOREM 4.2.3. Let G be an elementary, discrete subgroup o[ PSL2 (~). Then G is either cyclic or conjugate in PSL2(R) to a group generated by T : z ~ T(z) = kz(k > 1) and U : z ~ U(z) = -1/z. Fuchsian groups can be further classified by their limit point set. If G is a Fuchsian group, we let A(G) denote the set of limit points in C of G. The following can be proven. THEOREM 4.2.4. (1) A(G)

There are two possibilities

for A(G):

82

ALGEBRAICGENERALIZATIONSOF DISCRETE GROUPS (2) A(G) is a nowhere dense perfect set in

A Fuchsian group is of the first kind if A(G) = R. Otherwise it is of the second kind. Our algebraic generalizations are motivated by presentations of finitely generated Fuchsian groups. This presentation is obtained from a classical geometric construction due to Poincaxe which we now outline. A complete description of this procedure is itself almost book length and for complete details we again refer to the books of Beardon [Be], Lehner [Leh] or Katok [Ka]. The older book by Ford [Fo] on Automorphic Functions also contains a highly readable description of the basic procedure. The upper half plane H can made into a model for the non-Euclidean hyperbolic plane H2 by defining lines to be circles perpendicular to I(. If metric is defined on H2 by

ds = IdzllY where z

= x ÷ iy

then PSL2 (I() and its subgroups act as groups of isometries. Further PSL2(R) maps the class of lines in H2 to lines in H2. Thus a discrete group G C PSL2 (If() can be considered as a discrete discontinuous group of isometrics on H2. The general program for determining a presentation for G can be outlined as follows: (1) Given a discrete group G we can associate a connected (convex) 2region N in H (2) This region N has a boundary consisting of non-Euclidean lines and possibly some line segments on R and the sides in ~ti u axe paired {S~, S~}. Further for each such side pair there is a g~ ~ G such that g~ S~ -- S~. (3) The side pairing transformations generate (4) A complete set of relations in terms of the side pairing transformations can be deduced from the structure of the vertices and angles of N. Wenote further that if the quotient ~/G is compact, then the region N is a non-Euclidean polygon. Conversely a method due to Poincare allows one to reverse this whole procedure. That is given a non-Euclidean polygon N satisfying certain conditions and whose sides axe paired under elements of PSLu(I¢,), then the subgroup of PSLu (IR) generated by the side pairing transformations discrete and N is the region associated with it. Wenow clarify the basic ideas in the program above. For any subgroup G C PSL2(C) we say that zl, z2 ~ C axe congruent under G or G-congruent if there exists a g ~ G with gzl = z2. If G is discrete, a fundamental region for G is a connected open region R together with parts of its boundary satisfying the following two conditions:

4.2 DISCRETEGROUPS

83

(1) No two points zl, z2 E R are congruent under (2) Every point z E C is congruent to some point in the closure of For discrete subgroups G of PSL2(R) the following classical construction guarantees the existence of a fundamental region for G. Weoutline the construction and for proofs we again refer to [Be], [Fo] or [Ka]. If T : z ~, T(z) =cz+d ~ e PSL~ (C) its isometric circle I(T) is defined as the circle Icz + dl = 1. This is the locus of points about which there are neighborhoods in which areas and lengths are unaltered in magnitude by T. Straightforward computations show that for any transformation T, T maps I(T) on I(T -1) and ext I(T) on Int I(T-I). Further for a discrete group G the radii of all the isometric circles from the group are bounded while the limit points of G coincide with the points of accumulation of the centers of the isometric circles. For a discrete group G in PSL2(R) let R be the region in C exterior to all isometric circles from G and let Ro = R N Hr. Ro is called a Ford domain for the group G. The following can be proven. (1) Ro is a fundamental region for G. Any compact subset of 2 can b e covered by finitely manytransforms of R0 by G (2) The transforms of R0 fill H2 without overlap. Wesay these regions form a tiling or tesselation of H2. The boundary of R0 consists of circular arcs and/or line segments perpendicular to ]R possibly together with line segments on R. These are called the sides of Ro. If a side lies on R it is called a free side. A point in the intersection of two sides is called a vertex. An ordinary vertex is a vertex lying in 1HI2 while a real vertex is a vertex on R. (3) For a discrete group G, ORoconsists of a countable number of sides and vertices. (4) The sides of Ro which lie in 2 are c ongruent i n pairs, t hat i s f or each side S~ of Ro in ]HI2 there is another side S~ and a g~ ~ G such that g~ : S~ -~ S~ (we remark, that S~ could be equal to S~). (5) The side pairing transformations {g~ } generate the group We give a short proof of this last statement. Suppose g e G and let Zo ~ Ro. Let C be a curve joining zo and g(zo) { which is also in ~2 }. C can be embedded in a compact set A . Let G* be the group generated by the side pairing transformations {g~}, A can then be covered by a finite number of transforms of Ro by elements of G*. Then in particular there is a h e G* carrying R0 into a region containing g(zo) Then th e tr ansforms of R0 by g and h overlap. But each transform of Ro is also a fundamental region so transforms which overlap actually coincide. It follows that g = h, So g ~ G* and thus G is generated by the {g~}. We thus have a set of generators for G. Weconcentrate on the case where G is finitely generated and R0 has no free sides.

84


(6)If G is finitely generated then Ro has only finitely many sides and ordinary vertices and if in addition R0 has no free sides, then Ro is a convex non-Euclidean polygon lying in the closed upper haft-plane. Nowwe let G be finitely generated and suppose Ro has no free sides. To obtain a presentation for G we need defining relations and these depend on the angle and vertex structure of R0. Consider an ordinary vertex Zo of Ro ¯ A transform of zo under the generating side-pairing transformations must be another vertex since these transformations carry sides to sides. The restriction of the G-congruenceclass of Zo to those elements which lie in Ro or its boundary then consists entirely of vertices and is called an ordinary cycle. If z0 is a fixed point of an elliptic mapin G, this cycle is called an elliptic cycle otherwise an accidental cycle. Wethen have: (7) The numberof vertices in any ordinary cycle is finite. This is clear since R0 has finitely manysides¯ Let zo E lt~ 2 be an ordinary vertex and {Zo, zl, ..., z~} be its ordinary cycle in somegiven order so that T1 : zo --~ z~,T2 : z~ ~ z2 etc. Then T~T,,-1 .... T1 fixes zo. If this is an elliptic cycle then TnT,~-I .... T1 must be an elliptic mapsince the fixed points of parabolics and hyperbolics do not lie in l~ 2. Further since G is discrete, then T,~...T~ must be of finite order, say m. This determines a relation (Tn .... T1)"~ = 1. If tl~is cycle is an accidental cycle, then T~ .... T~ must be the identity since the only non-trivial elements of G having fixed points in lt~ 2 are elliptic. This gives a relation T,~....T~ = 1. Relations such as those given above are called cycle relations. It may also happen that we have an elliptic fixed point z0 for T of order 2 where z0 lies on the midpoint of a side. In this case zo is considered as a vertex. T interchanges the two separated semiarcs and these are also considered sides with side pairing T, Wethus have a relation :/-2 ___ 1. Cycle relations of this form are called reflection relations. (8) Each ordinary vertex determines (T,~ .... TI)m=l,m >_l

a cycle relation

of the form

The cycle relations are actually determined by the angle structure at z0. The transforms of R0 fill up the hyperbolic angle around z0 since the transforms of R0 tesselate lI~ 2. Wepicture the regions around z0 in figure 4.2.1. The angle sum must then be a rational factor of 2~r. If the sum is 2Trim, then Zo is a fixed point of an elliptic map of order m and its cycle is an elliptic cycle. Otherwise its an accidental cycle and the angle sum is 2rr.

4.2 DISCRETEGROUPS

Figure 4.2.1.

85

Transform Regions Around a Point

The real vertices - those that lie on ]~ - are fixed points of parabolic maps and thus are called parabolic vertices. If Ro has finitely manysides then each parabolic cycle is finite. As with ordinary vertices we obtain a map Tk...T1 which fixes a parabolic vertex P but now this does not determine any non-trivial, additional relations - only that the mapis parabolic. However, the side pairing mappings, which fix P, generate Gp the stabilizer of P. The crucial result for finding presentations is that the cycle relations (including the reflection relations) are a complete set of defining relations for G. (9) For a finitely generated discrete group G such that R0 has no free sides, a complete presentation is given by: (i) Generators; side-pairing transformations for a Ford domainR0. (ii) Relations; cycle relations (including reflection relations) each ordinary vertex. Thus to find a pre~ntation for a discrete subgroup G of PSL~(R) we would first use the method of isometric circles to determine the Ford domain Ro. Unless G is given in some special way (such as the Modular Group which we will examine below), this is a non-trivial task. From R0 presentation is derived from the side pairing transformations and the vertex cycle ( or angle) structure. Generally one intermediate step is done if o~ is a fixed point of a parabolic element in G. Let G~ = stabilizer in G of {~}. Then G~----- (:k (l 0 nlA), > 0} These are translations, so a strip in H2 would give a fundamental region for G~. Thus let F --- {z -- x q-iy;~ < x < ~ q-~} be

86


a fundamental region for Go¢. Then a Ford domain for G would be F f3 { exterior of all isometric circles in G - Goo}. The following theorem is a summaryof the above discussion. After the theorem we will give an example. THEOREM 4.2.5. Let G be a finitely generated discrete subgroup o[ PSL2(R) and suppose Ro is a Ford domain for G having no free sides. Then (1) Ro is a non-Euclidean polygon whose sides are congruent in padrs under G and there are a finite number of sides. (.9)The side pairing tranMorma~ions generate G and a complete set relations [or ~ are given by the cycle relations at each ordinary vertex. As an example we find a presentation for the Modular Group M = PSL2(Z). Since the entries are integers this is clearly a discrete subgroup of PSL2(~,). The translations in Mare of the form z ~ z ~ = zq-n which are generated by z ~ z~ = z + 1. A fundamental region for Moocan be taken as any strip with width 1 in H2. We choose -1/2 < x < 1/2. Nowconsider the remaining elements of Mand we find the largest isometric circles. Elements of Mare of the form z ~ z~ = az + b/cz + d with a,b,c,d E Z, ad-bc = 1. Ifc = +1 thenad+b = 1. For any integer values a, d we can determine an integer b. It follows that the center +d of ] =t= z - d[ = 1 can be any integer. Thus there a~e isometric circles of radius one centered at each integer. These circles enclose all points within a distance of x/~/2 of the real axis. For any other transformations [c[ _> 2, so the radius of its isometric circle does not exceed 1/2. Since the center lies on R, it follows that these isometric circles lie inside the circles of radius one centered on the integers. Thus the exterior of these isometric circles intersected with the strip -1/2 < x < 1/2, y > 0, gives the Ford domain for M. Thus we have the diagram as in figure 4.2.2. Here P = i, P1 = -1/2+x/-~/2i, P2 = 1/2+v~/2i. Wesee that Ro has 4 sides $1, $2, $3, $4 where $3, $4 are chosen as sides because P is an elliptic fixed point of A : z ~ z~ = -1/z of order 2. There is also a parabolic vertex at o~. S1,S2 are paired by T : z ~-~ z ~ = z+l while $3,$4 are paired byA : z ~ z ~ = -l /z. ThusMisgenerated byA andT. The elliptic cycle at P is just {P}. This gives the relation A2 = 1. The elliptic cycle at P1 is {P1, P2}. We have T(Pi) = P2 and A(P~) = PI so that the map AT fixes P1. This gives the cycle relation (AT)3 = 1. The cycle relation at P2 is equivalent. Therefore a complete presentation for Mis < A, T; A~ = (AT)"~ = 1 >.

4.2 DISCRETEGROUPS

Figure 4.2.2.

87

Fundamental Region for the Modular Group

A theorem of Poincare allows one to reverse the whole procedure and thus construct Fuchsian groups. The first fully correct proof of this result seems to have been given by Maskit. [Mas 2]. Suppose we begin with a non-Euclidean polygon D in H~ whose sides are paired by a set of isometrics. Wecall D a polygon with identifications. If for some side S we have S = S~, then the side pairing isometry must be of order 2 and there is a relation A~ = 1. Relations of this form are called reflection relations. Let D be a polygon with identifications and D. the identified polygon obtained by identifying the sides of D. Then there exists a surjection H : D -~ D* where II(x) = II(x ~) if there is a side pairing isometry A with A(x) = x’. For z, x’ e D* set

wherep is the hyperbolic metric and the infimumis taken over all n and over all 2n-tuples of points of D where II(x) = zl, H(z~) = H(z~+l), H(z’~) D is then a complete polygon if (1) For each x E D*,II-l(x) is a finite set in which case p* metric on D* (2) D* is complete in this metric. In such a complete polygon each vertex determines a cycle as before. If {Zl, ..., Zk} is such a cycle and A1, ..., Akare the correspondingside pairing congruence maps, then z~ is a fix point of AkAk-1...A1. Then we say the polygon D satisfies the cycle condition if for each cycle {zt, ..., zk} there is an integer t so that Y] a(zi) 2rr/t where a( zi) is theinte rior hype rbolic angle measure at zi. A cycle with sum 2rr/t determines a cycle relation of the form (Ak...A~) t = 1. A Poincare polygon is a complete polygon

88


with identifications then the following.

satisfying

the cycle condition. Poincam’s theorem is

THEOREM 4.2.6. (Poincare) Let D be a Poincare polygon and G the group generated by the side pairing transformations. Then G is discrete and a complete set of relations is given by the cycle relations and reflection relations. Poincare’s theorem allows one to construct Fuchsian groups from geometry. It also has the following use as applied to the problem of finding presentations for unknown groups. Suppose G is a discrete subgroup of PSL2(R) and {gl, ..., gk} is a set of generators for G. Then to determine a presentation for G we try find a polygon {again this is a non-trivial task} whose sides are paired by gl, ...,gk. If we can showthat this is a Poincare polygon then the cycle relations on gl, ...,gk determine a presentation. In the next section we will show that Poincaxe’s Theorem not only leads to a presentation for a Fuchsian group but in the case where it is finitely generated to a very special type of presentation. Wenote at this point that there is another construction of Ford domains and Poincare polygons for discrete groups which does not use the isometric circles. Let z0 E H2 and 1, V1, V2, ...Vn, ... be the elements of discrete G - which is by necessity countable. Let L~ be the hyperbolic half-plane closer to zo than to V~(z0),that is L~ = {z p(z, Zo ) < p(V~(z0))}. Let No = N~Li. Then No, called a Dirichlet region for G, is also a Poincare polygon for G. It can be proven that the hyperbolic area of a Dirichlet region for a Fuchsian group G is finite if and only if G is finitely generated and the region has no sides on R. For a region D C H2 we let #(D) stand for its hyperbolic area. If G is a Fuchsian group, then G acts as a discontinuous group of isometries on 1t~2. Therefore the quotient space G/~~ is a Riemannm~rface which can be identified with a Dirichlet region D with the side identifications. (see [Ka]) Wedefine #(D) = #(G/H2). A Fuchsian group G is said to be cocompact if G/H2 is compact. In this case G is finitely generated and we get the following. THEOREM 4.2.7. A Fuchsian group G is cocompact ff and only if p(G/~2) < (:x) and G contains no parabolic elements. Wewill see in the next section that if G is finitely generated with parabolic elements, then group theoretically it must be a free product of cyclics. Weclose this section by noting that them are many results on determining when a finite system of elements in PSL2(R) actually generate a Fuchsian group. In section 4.5 we return to this question, relative to

4.3 ALGEBRAICANALYSISOF FUCHSIANGROUPS

89

two-generator subgroups. These results will not play a part in our generalizations. Here we state one theorem which gives the type of result that one can obtain. THEOREM 4.2.8. (see [Jo], [R 5]) Let G be a non-elementary subgroup of PSL2(R). Then the following are equivalent: (1) G is discrete - thus a Fuchsian group. (2) 2. G acts discontinuously in H (3) The 2. fixed points of elliptic elements in G do not accumulate in H (4) Each elliptic element in G has finite order. (5) Every cyclic subgroup of G is discrete. 4.3 Algebraic

Analysis

of Fuchsian Groups

Weconcentrate now on finitely generated Fuchsian groups such that R0 has no free sides. These then have finite sided Poincare polygons with finite hyperbolic area. By a careful analysis of Poincare’s theorem we can obtain the following set of results knownalso collectively as Poincare’s Theorem. These give a special presentation for cocompact Fuchsian groups which is the starting off point for our algebraic generalizations. If an elliptic element g E G is part of a generating set it is called an elliptic generator. Analogously we have hyperbolic generators and parabolic generators. THEOREM 4.3.1. (See [Be, Ka, Leh]) (1) Let G be a non-trivial finitely generated Fachsian group such that Ro has no free sides. Then there exist s,s >_ O, parabolic generators pl,...,ps, 2g, g >_ O, hyperbolic generators al, bx,..., a9, b9 and t, t >_O, elliptic generatorsel,..., et of respective finite orders ml,...,mt if t > 0 with n = t + g + s > 0 such that G has the following presentation on these generators. (a.3.1) G --( pl, ..,pt, el, ..., es, al, bl,..., a9, 59; e~x ..... es m~ = R = 1 > where R 9, =pl ...psel...etlal, bl]...[a This presentation is called a Poineare presentation for G. The sequence (g; m~, .., ms; t) is called the signature of G. The hyperbolic area of the Dirichlet region N for G is given by t

#(N) = 2r(2g - 2 + t + ~(1 - 1/m~)) Further lz(N) is the area for any fundamental domain for G and hence may define #(G) = #(N). The number 1 X(C) = -~(a)

90


is cared the Euler characteristic of G. (2) Conversely given a signature (g; ml, ..., ms; t) with rai :> 2 then there exists a non-elemen~axy Pachsian group with ~hat signature only if t

2~r(2g - 2 + t + ~ 1 - 1/m~)

0

If t > 0 the group is group theoretically a free product of cyclics. We just solve for one of the pi to get a presentation of this form. Wewill now ignore this situation. If a Fuchsian group had a parabolic element, it would clearly have a parabolic generator. Since the pi are precisely the parabolic generators, then, if t = 0, there are no parabolic elements and the corresponding group is cocompact. Therefore we now consider cocompact Fuchsian groups G which have signatures (g;ml, ...,ms;0) and Poincare presentations (4.3.2)

G----< el,...,es,

al,bl,...,ag,

b.q;e~1 .....

e~" ---- R---- 1 >

where (4.3.3)

R -- el...et

[al,

bl]...[ag, bg]

and n = g + s > 0, rni _> 2, i ---- 1,..,

s and #(G) = 2~r(2g- 2 + E 1 1/mi) > O.

The number g is called the genus of the Fuchsian group. Notice first that if s = 0 in the Poincare presentation so that there are no elliptic elements, then a Fuchsian group is isomorphic to a surface group of genus g and is thus a one-relator group. Wheng = 0 we get the following form for the Poincare presentation < el,...,es;e~

1 .....

e~~ = O...e~ = 1 >

(4.3.4) :< ¢1, ..., Of special interest

vs-1; e~nl_-- ... __ es-lm’-~ = (el...es_l)m~ = 1 is when m = 3 so that the presentation

4.3.4 becomes


91

These are known as the ordinary triangle groups or just triangle groups. Wewill discuss them at length in Chapter 6. A group H which has a presentation of form (4.3.2) with the special relator of form (4.3.3) is called an F-group and actually represents a Fuchsian group if and only if p(F) as defined above is positive. The F-groups with #(F) _< 0 can be easily classified. (see [L-S]). If F is an F-group then we define the Euler characteristic of F as x(F) = -~#(F) if F is infinite x(F) -- if F is fi nit e. Notice howsimilar an F-group is to a one-relator group. Morespecifically we call a group with a presentation of the form (4.3.4)

< al, ...,

an; a~l .....

a~" = R -- 1 >

where mi = 0 or rni :> 2 for i -- 1,..,n and R = R(al, ..,a~,) is a cyclically reduced word of length _> 2 in the free product of cyclics on al, ..., a,, a one-relator product of cyclics. Wewill handle such groups in general in the next chapter. Notice now however that a Fuchsian group is precisely a one-relator product of cyclics with the special relator 4.3.3. As pointed out above, if there are no elliptic elements so that s -- 0 in the Poincare presentation then a Fuchsian group is isomorphic to a surface group of genus g and is thus a one-relator group. As for Fuchsian groups, of particular interest in the class of one-relator products of cyclics is when n = 2 and the relator is a proper power. These then are groups with presentations of the form (4.3.6)

2, m2=0orm2_>2, m_>2andR=R(a~,a2) isa cyclically reduced word in the free product of cyclics on al, a2 which involves both al and a2. These are called the generalized triangle groups and will be studied in detail in Chapter 6. The above comments lead to two separate approaches to studying Fuchsian groups: (A) Consider them as discrete subgroups of PSL2(R) or a conjugate in PSL2 (C) and then use properties of homographic substitutions and hyperbolic isometries to determine group theoretic properties. (B) Consider them as part of the wider class of one-relator products cyclics and study them in this more general context. A third approach by studying their corresponding Cayley complex is outside the realm of these notes and is discussed in [L-S].

92


Relative to these approaches there are several algebraic questions raised. Specifically (1) Given a group theoretic property of the class of Fuchsian groups, proved either by approach (A) or approach (B), is it true for the whole class of one-relator products of cyclics? (2) If a given Fuchsian group property is not true in the whole class one-relator products of cyclics are there special forms of the relator R other than (4.3.3) for which it is true? (3) Given a group theoretic property of the class of Fuchsian groups, proved by approach (A), can a purely algebraic proof using approach (B) or a variant be given? Relative to approach (B) we can refine it somewhatfor certain Poincaxe presentations. If s > 0 so there is a generator el of order ml, then the Poincare presentation can be rewritten as <e2,...,es, al,bl, .... ,aa, ba;e~~ .. e~m: = (e2...e~[al,bi]...[aa,

ba]) m~ = 1 >

This is of the general form (4.3.4)

< al, ...,an;a~ 1 .....

a~n’~ = R"~ = 1 >

where mi = 0 or m~_> 2 for i = 1, ..,n and R = R(a~,..,a,~) is a cyclically reduced word in the free product of cyclics on al,..., a,~ and m _> 2. In this case we say that the relator is a proper power. We can now ask questions (1),(2),(3) relative to the subclass of one-relator products cyclics with proper power relators. Wedo this in detail in Chapters 6 and 7. Finally ifg _> 2or g = 1, s _> 2 org = 0, s _> 4then an F-group G decomposes as a non-trivial free product with amalgamation. For example suppose g > 0, s > 1 let G1 =< el, ..,

es;

e7~"

.....

esm~---- 1 >, G2~--

Then G is the non-trivial free product of G1 and (]2 with the infinite cyclic amalgamation generated by el...e~ in G~ and [al,bl]...[ag,bg] in G2. In general we define a group of F-type to be a group with a presentation of the form

(4.3.5)

.~1 < al, ..., a,~,¯ a~

...

a’~"--UV=I>


93

where rni -- 0 or mi _> 2 for i = 1,..,n and U = U(al,..,ap),l 1 and s > 2 so that the group decomposesas a non-trivial free product with amalgamation. Specifically G -- G1 *A G2 with G~ =< e~, ...,e~;e~ 1 = ... G2 ---( al,bl,

...,ag,bg;

= eTa = 1 >

> and A =< c~...e~ >G1--G: ¯

G~ is a free product of cyclics, G2 is a free group and the amalgamated subgroupA is infinite cyclic. Before continuing we make some comments on the cases we’ve excluded with this method. If g >_ 1 but s -- 1 the Poincare presentation reduces to a one-relator group with torsion and the results follow from the analysis of such groups as in the last chapter. If g = 0 the theorem can ultimately be reduced to the case of the triangle groups which are usually handled by the analytic methods which we describe next. This is a commonsituation concerning results on Fuchsian groups: groups with genus g :> 1 are handled relatively easily, while the triangle groups are handled separately. This dichotomy will be carried over to general one-relator products of cyclics

94


where the generalized triangle groups are usually handled as a separate class. Now let H be an abelian subgroup of G. From Theorem 2.3.3, H is either a conjugate of an abelian subgroup of G1 or G2, an ascending union of conjugates of subgroups of A or a direct product of an infinite cyclic group and a conjugate of a subgroup of A. Since G1 is a free product of cyclics, G2 a free group and A infinite cyclic, the first two cases imply directly that H is cyclic. The final case says that H is either cyclic or free abelian of rank two. Wenow apply part of the analytic approach to say that if H =< Xl,X2; [xl,x2] = 1 > then by a direct discreteness argument H is not a Fuchsian group. Hence H must be cyclic. Next suppose H is a finite subgroup of G. Then from Theorem 2.3.3, H must be conjugate to a finite subgroup of one of the factors. G: has no finite subgroups so H must be conjugate to a subgroup of G1. Since G1 is a free product of finite cyclics, its finite subgroups axe precisely the conjugates of subgroups of the cyclic groups generated by the generators. Finally G1 is centerless, G2 is centerless so G must be centerless. Nowwe outline a second proof based on homographic substitutions. Suppose H is a Fuchsian group and here we make no restrictions on the genus or on the numberof elliptic generators. If H were abelian then all elements of H must have the same fixed points (see Theorem 4.12). Therefore must consist of only parabolic elements or only hyperbolic elements or only elliptic elements. Supposefirst that H is totally parabolic, all having the same fixed point. By conjugation we can assume that this fixed point is at (x) and hence all elements of H are translations z ~ ~ -- z+ ~, ~ ER. Since H is discrete there is a minimumpositive A0, for otherwise 0 would be a limit point of the group. Suppose z ~ z ~ = z + A1 E H with A~ > 0. Then for any n ~ Z, z ~ z ~ = z + (A~ - nA0) is also in H. If)~ ¢ for some m ~ Z then there exists an n e Z with 0 < A~ - nAo < A0 which contradicts the minimality of )~o. Therefore )~ = mA0for some m e Z and z ~-~ z~ -- z + A~ is a integer multiple of z ~ z~ -- z + )~o. HenceH consists of all multiples of z ~-~ z~ = z + A0 and is therefore cyclic. This handles the case where H is totally parabolic. If H is totally hyperbolic, then we can assume the commonfixed points are at 0 and ~. Therefore the elements of H all have the form z ~ z ~ = Kz with K > 0. Then log z~ -= log z + log K and the same argument as for parabolics applies. Finally if H is totally elliptic, then we may assume, that each element has the form z ~ z ~ = e~°z and the argument works for log z~ = log z + iO. Thus each abelian Fuchsian group is cyclic and therefore any abelian subgroup of a Fuchsian group is cyclic.

4.3 ALGEBRAICANALYSISOF FUCHSIAN GROUPS

95

If H is a finite group, it must consist entirely of elliptic elements. It can be shownthat all these elements must have the same fixed point. Therefore this group must be abelian and from the above must be cyclic. Finally suppose H is a Fuchsian group and 1 ~ T E H is in its center. Then since T commuteswith all elements of H, T must have the same fixed points as all elements of H. It follows that all elements of H have the same fixed points and hence all commute, that is H is abelian. Therefore H is centerless, since by convention a Fuchsian group is non-elementary. This last remark also shows the following: if T, U, V E G with G Fuchsian then if T commutes with U and U commutes with V then T also commutes with V. Combiningthis with the centerlessness of G we get the following. COROLLARY 4.3.1.

Any non-abelian

Fuchsian group is commutative

transitive. Wenext consider the other linearity properties. Recall that a group G satisfies the Tits alternative if either G contains a non-abelian free subgroup or G is virtually solvable. From the theorem of Tits [Ti] any finitely generated linear group and hence any finitely generated Fuchsian group has this property. THEOREM 4.3.3. Any finitely generated l~uchsian group satisfies the Ti~s alternative, tha~ is either contains a non-abelian free subgroup or is virtually solvable. Furthermore a subgroup of a finitely generaeed ~-hchsian group will contain a non-abelian free subgroup. The second part of the theorem also follows easily from the following important result which we will use repeatedly in our general analyses of free subgroups of one-relator products of cyclics. THEOREM 4.3.4. (see [Be]) A non-elementary discrete PSL2 (C) will contain a non-abelian free subgroup.

subgroup

Wenote that Theorem 4.3.3 can be extended, with appropriate modifications, to the wider class of F-groups, that is, groups G with Poineare presentations but allowing ~u(G) 2. In either case such a group maps onto a non-abelian free subgroup and is thus SQ-universal. Wemention that SQ-universality can be taken as a measure of the largeness of an infinite group. COROLLARY 4.3.2. universal

Any t~nitely

generated

Fuchsian group is

Wenext turn to the general subgroup theory. From the geometric def-


97

inition as a discrete subgroup of PSL2(R) it is clear that any subgroup of a Fuchsian group must be Fuchsian, cyclic or infinite dihedral. However, although Fuchsian groups were studied first in the nineteenth century, it wasn’t until 1971 and 1972 that a purely algebraic proof of this fact was given. Hoare, Karrass and Solitar [H-K-S 1,2] using the ReidemeisterSchreier method proved the following more general results for F-groups. THEOREM 4.3.7. (4.3.1)

Let G be any F-group with a presentation

G --< el, ...,

es, al, bl, ..., ag, bg; 1 .. ...

of the form

e, "n" -- R -- 1 >

where R =e~...es[al, bl]...[a~, b~]. Then any subgroup of finite index is again an F-group of the same type. Further any subgroup of int~nite index is a ~ee produc$ of cyclics. Wenote that Hoare, Karrass and Solitar proved also that the result is still true if there are no b~ and the relator is replaced by R -- el ~ esa~ a This is equivalent to considering the torsion-free groups to be non-orientable surface groups. In their papers they also gave an algebraic proof of the following result called the l~iemann-Hurwitz formula. THEOREM 4.3.8. (Riemann-Hurwi~z Formula) Le~ G be any F-group and H a subgroup of finite index. Then #(H) -- p(G)IG : HI equivalently in

~erms of Euler characteristic

x(H) : X(G)[G : For Fuchsian groups, Theorem4.3.8 is a consequence of a classical result which says that the fundamental domain for a subgroup of finite index j consists of j copies of a fundamental domain for G. The Riemann-Hurwitz formula imposes tremendous restricitions on the structure of subgroups which can appear in a Fuchsian group. Along these lines it can be proved that the Fuchsian group with the fundamental domain of smallest hyperbolic area is the triangle group < a,b;a 2 = b3 = (ab) 7 = 1 >. This is called the (2, 3, 7)-triangle group and Higman[Hi 4] proved that almost all alternating groups are quotients of this group. The Ricmann-Hurwitz formula and Euler characteristic has been extended in many directions. A discussion of this is in Lyndon and Schupp

98


[L-S],Bass [Ba 3], Chiswell [Ch], Wall [Wa] and Brown[Brn]. Wewill extend this formula in Chapters Six and Seven to one-relator products of cyclics and groups of F-type. Weclose this section with three additional algebraic results. First we state a Freiheitssatz for Fuchsian groups. This follows from the free product with amalgamation decomposition and the Hoare,Karrass,Solitar result on subgroups of infinite index. THEOREM 4.3.9. Let G be a Fuchsian group with Poincare presentation (4.3.1) G= where R ~-- Pl ...ptex...~s

[ax, bl]...

[ag, bg].

H n = t + s + g and n >_ 3, then any subset o[ (n- 2) o[ the g/yen generators generates a free product of cyclics. Wewill give a proof of this whenwe discuss the general Freiheitssatz for one-relator products of cyclics and groups of F-type. Next recall that in a cyclically pinched one-relator group two elements are conjugate if and only if they are conjugate in every finite quotient (Theorem 3.5.6). If G is any group and g, h ~ G then we use the notation g ,-~ to denote g conjugate to h. If g ~ G we say that g is conjugacy distinguished if for every h ~ G either g -- h or there exists a finite quotient of G where they are not conjugate. The group G is conjugacy separable if every element is conjugacy distinguished. Dyer’s result - Theorem3.5.6 can be rephrased by saying that all cyclically pinched one-relator groups are conjugacy separable. Fine and Rosenberger [F-R 4] were able to complete the proof of the following theorem. THEOREM 4.3.10. separable.

Any finitely

generated Fuchsian group is conjugacy

This theorem was conjectured by Allenby and mentioned in the Kourovka notebook [Ko]. Stebe in 1970 [St 1] proved that elements of infinite order in Fuchsian groups were conjugacy distinguished. Allenby and Tang [A1-T ¯ 3] proved that certain triangle groups were conjugacy separable. The proof of Theorem 4.3.10, which we outline below, involved reducing the result to the triangle group case and proving it there by a detailed case by case analysis. In their paper Fine and Rosenberger conjectured that this result should be true for a certain class of one-relator products of cyclics including the groups of F-type. This was subsequently proved by Allenby [A1 3]. (See also IF-He-R]).


99

PROOF.(Theorem 4.3.10) From Stebe’s work [St 2] the result is true G is either a free product of cyclics or has positive genus. Therefore we consider the case where the genus g = 0 and G is not a free product of cyclics. G then has a Poincare presentation of the form (1)

G =

with s _> 3, each mi _> 2 and ~(G) > Nowsuppose g E G. We must show that g is conjugacy distinguished which we abbreviate now as c.d. If g has infinite order, Stebe [St 1] has shown that g is c.d. Thus we assume that g has finite order and suppose h E G. If h has infinite order, clearly h ~ g and then from Stebe there exists a finite quotient G* with ¢ : G -~ G* such that ¢(h) ~ ¢(g). we consider h of finite order. If g ~ h then since the conjugacy classes of elements of finite order in groups of form (1) are given by the conjugacy classes {< el >}, ..., {< c8 > } we can assume without loss of generality that g and h are powers of generators, that is g = e~’,h = e~ with 1 _< i,j of T(m, m, q) which is isomorphic to the finite cyclic group of order m . Now let h = b-~. Ifm = p_> 5 we have that the Euler phi function ~(m) _> 4. It follows from results of Langer and Rosenberger [L-R] that can find a finite field K and a representation p : T(m, rn, q) -~ PSLu(K) with a -+ A, b --* B such that tr(A) ~t ±tr(B) = :l:tr(B-1). Therefore A is not conjugate in PSLu(K) to B-~ and hence the images of g and h are not conjugate in the finite factor group PSLz(K). This handles case 2. Case 3: g=a ~ and h=a a with 1_< a,/~ < rn, a~flandm =por m = q. Assume without loss of generality that m = p. Then the images of g and h are not conjugate in the factor group G =< a, b; a "~ = b"~ = ab = 1 >= Z,~. Case 4:g=a ~ andh=a awithl_ 3. If (l/m) + (l/p) + (l/q) > 1 T(m,p, q) is finite and therefore conjugacy separable. Thus let (1/m)+(1/p)+(1/q) If (l/m) + (l/p) + (l/q) = 1 then since a ~ f~ and m ~ p ?t q we T(m, p, q) = T(2, 3, 6). In this case T(2, 3, 6) has the cyclic group Z6 epimorphic image where the images of g and h are not conjugate. Finally suppose that (l/m) + (l/p) + (l/q) < 1. T(m,p,q) is a Fuchsian triangle group. Suppose first that p -- q and we must have p _> 3. Mapthe generators b, (ab) of T(m, p, p) onto the permutations ~ = (12...p) and ~ -- (1,p,p- 1, ..., 2). This defines an epimorphism of T(m,p,p) onto the finite permutation group < ~, ~ :>. From a result of Singerman [Si] we then obtain a subgroup H of index p in T(m, p, p) with the presentation

Since each x~ in T(m,p,p) is conjugate to a power of a, we may assume that g and h are powers of xt. From this we get that both g and h are

102


conjugacy distinguished in H. Let k E H. If k has infinite order, then g is conjugacy distinguished in H using Stebe’s result on elements of infinite order in Fuchsian groups. If k has finite order then k is conjugate to a power of some xj and we can reduce to the case T(m, m, m) which is handled by Allenby and Tang [A1-T 3]. Another result of Stebe’s [St 2] says that if H is a subgroup of finite index in G and a is an element of H which is conjugacy distinguished in H, then a is conjugacy distinguished in G. Then from this result g and h are also conjugacy distinguished in the overgroup T(m, p, p). Therefore there is a finite epimorphic image where the images of g and h are not conjugate. This handles the case when p -- q. Nowsuppose p ¢ q. From Singerman’s results [Si] we get a subgroup H of finite index in T(m, p, q) which is not a triangle group and which contains a, g and h. Without loss of generality H has genus g > 1 or has at least two conjugacy classes of cyclic subgroups of order m. (If we find a subgroup of genus 0 and only one conjugacy class of cyclic subgroups of order m, then we can map onto a finite factor group with torsion-free kernel and take the pull-back of an element of order m.) If H has positive genus, then g and are conjugacy distinguished in H from Stebe[St 1]. Wemust now consider the case where H has genus g ---- 0 and at least two conjugacy classes of cyclic subgroups of order m. We show that both g and h are conjugacy distinguished in H. Let k E H with k not conjugate to g. If k has infinite order then from Stebe’s work on Fuchsian groups there is a finite image where g and h are not conjugate¯ So suppose that k has finite order. H then has a presentation g =< xl,

¯ fl _~ ....

~ Xl ~ ~ Xl...Xv

~---

1>

with xl = a, fl = f2 = m, v >_ 3. If at least one f~ >_ 3, i _> 3 we can reduce the problem to a triangle group T(m, m, r) for some r :> 3 which we handled in our previous arguments. So therefore let f3 ..... f, = 2. If m _> 4 the problem can be reduced to a triangle group T(m, m, 2) which is already handled. So now we consider the case where m -- 3. Then v _> 4 since this is a Fuchsian group. Since k has finite order, k is conjugate to a power of some xj, 1 _< j < v. Assume k--x~ with0_< ~ < fj, l_<j_< v. Ifj ~ 2 we may reduce the problem to a triangle group T(3, r, 2) with r -- 2 or r =- 3. This was handled in our previous cases. Now let j-- 2sothat k=x2~,0_< ~ < 3. If ~ --- 0 then since H is residually finite, we can find a finite image in which g and k are not conjugate. If ~ ~ 0 then H has the factor group --

H =<x2, ....,

x,; x~ = x] .....

x,2 = x~...x~ ---- 1 > .

In H the image of g is the identity while the image of k is non-trivial. Then from the residual finiteness of H there is a finite image of H and thus of H

4.4 NEC GROUPS

103

in which g and k are not conjugate. Thus g and by the same arguments h are conjugacy distinguished in H and from Stebe’s results also in T(m, p, q). It follows that there is a finite epimorphic image of T(m, p, q) in which the images of g and h are not conjugate. This completes case 4 and the proof for the triangle groups (n = 3). Nowwe consider the situation where n _> 4. If i ~ j, that is, g and h are not conjugates of the same generator, consider the factor group G of G

Remember,we assumed here that (i,j) = (1, 2). Since g is a conjugate a power of el the image of g is trivial in F while the image of h (since is conjugate to a powerof e2) is non-trivial. Fromthe residual finiteness of G it follows that there is a finite image of G and thus of G where g and h axe not conjugate. Therefore suppose that i = j. Therefore g and h are both conjugates of powers of el. If n _> 4 we combine the epimorphism G ---* G =<e2, ...,

e,~; e’~~ .....

e~" = e2...e~

=1 >

together with the Fenchel-Fox theorem for G to obtain a subgroup H of finite index in G with el and therefore g and h in H. This subgroup H has either genus g >_ 1 or at least two conjugacy classes of cyclic groups of order ml. Therefore g and h axe conjugacy distinguished in H and again from Stebe also in G. Therefore we can find a finite image of G where g and h are not conjugate completing the theorem. Finally a group G is subgroup separable or LERF(=Locally extended residually finite), if for any finitely generated subgroup H and any g ~ there exists a subgroup H* of finite index in G with H C H* and g ~ H*. Theorem3.5.10 said that cyclically pinched one-relator groups are subgroup separable. G. Scott [Sco] proved that finitely generated Fuchsian groups are subgroup separable. THEOREM 4.3.11. separaNe.

Any finitely

4.4

generated Fuchsian group is subgroup

NEC Groups

We now introduce an additional class of discrete groups, the nonEuclidean crystallographic or NEC groups, related to the class of Fuchsian groups. This class also leads to a class of presentations that generalize Poincare presentations. Our techniques for handling one-relator

104


products of cyclics also handles generalizations of certain presentations in this class. This will be discussed in Chapter Nine. A non-Euclidean crystallographic group or NEC group G is any discontinuous group of isometries of the hyperbolic plane. Recall that if the upper half-plane is taken as a model for the hyperbolic plane then each Fuchsian group is also a discontinuous group of isometries and hence an NECgroup. However, elements of PSL~(R) considered as hyperbolic isometries are all orientation preserving, so a general NECgroup maydiffer from a Fuchsian group by allowing orientation reversing hyperbolic maps. In particular an NECgroup may contain hyperbolic reflections. Again using the upper half plane model, the simplest hyperbolic reflection is given by the complex transformation z ~ z ~ -- -~. With these ideas, an NECgroup consists of elements of PSL~(R) together with transformations of the form z~,

z~= ahq-b c~ + d’ a, b, c, d E IR, ad - bc = - 1.

NECgroups were introduced by Wilkie [Wi] in 1966 who further gave a classification of the finitely generated ones. From this point of view they were further studied by Macbeath [Mac], Macbeath and Hoare [Mac-HI, Hoare,Karrass and Solitar [H-K-S 3], Martinez [Mar], Zieschang [Z 1] and others. A fairly complete discussion is in the book by Zieschang, Vogt and Coldeway [Z-V-C]. The geometric analysis of an NECgroup, which in turn leads to a group presentation, is handled in an manner analagous to a Fuchsian group. If G is an NECgroup, then it acts discontinuously on the upper half plane considered as a model of the hyperbolic plane H2. As such, it admits, as in the Fuchsian group case, a fundamental domain No with, possibly infinitely many, non-Euclidean lines and segments of ~ U ~ as the sides. The segments of IR t3 o~ on the boundary of No are the free sides. G is of finite type if No has finitely many sides. In particular this is true if No is compact in H2, in which case No is a non-Euclidean polygon. As in the finitely generated Fuchsian group case the side pairing transformations, including transformations of order two which fix sides, generate a finitely generated G, and the angle structure at the vertices together with certain reflection relations, which we describe below, determine relations. A signature, which leads to the classification of NECgroups of finite type, can be obtained in the following way (see [Z-V-C] for more details). Let be an NECgroup and assume that its fundamental polygon is compact. Let B = [’I[2/G the orbit space of H2 modulo the action of G with the quotient topology. This is called an orbifold, ll~ is then a surface of finite type obtained from a closed surface of genus g by cutting out a finite number of open disks. Weobtain then that 1~ consists of either an orientable compact

4.4 NECGROUPS

105

surface of genus g or a non-orientable compact surface of genus g together with a finite number C1, ..., Ca of boundary components. The boundary points of ~ are the images of points of H2 lying on axes of reflections of G. The quotient map r : H2 -~ ~ behaves as a covering map at all points except at those which are not fixed points of rotations or reflections of G, while at a rotation center, which does not lie on a reflection axis r, behaves as a branched covering at a branch point. On ~ there are finitely many m of points of this type with branching numbers hi, ..., hm. If two reflection axes Cl,C2 intersect at x E H2 at an angle ~, then the product ClC2 is a rotation about x of order 2(~. On the boundary component Ci, 1 _ 0, images of inequivalent rotation centers, each of which defines a cycle (h~,l, h~,2, ..., h~,m~)of rotation orders. If the surface of ~ is orientable we fix an orientation of ~ and this induces an orientation on every boundary component. If ~ is orientable let e = 1 while if ~ is not orientable let e = -1. The signature of G is then (4.4.1)

(g : hl, ...,hm;q; {(h1,1, ... , h~, ,,1), ... (hq,~,...,hq,,~q)}).

Without changing the topological type of the NECgroup G we can do the following things: (i) arbitrarily change the enumerations of the hi and the C~; (ii) if the surface ~ is non-orientable the cycles (hi,l, ..., hj,,,j) be independently reversed; (iii) if the surface 1~ is orientable all the cycles (hi, l, ..., h~.,,~j) can be simultaneously reversed. It was shownby Wilkie that two NECgroups with compact fundamental polygons are topologically equivalent if and only if their signatures are related via the above transformations. Further in [Z-V-C] it is shownthat this is also a necessary condition for two NECgroups with compact fundamental domain to be isomorphic. The fundamental domain for such an NECgroup G is then obtained from ~ in the following manner. Cut ~ along suitable arcs to obtain a disk which does not contain the images of rotation centers in its interior. Lifting the cut arcs to H2 gives the fundamental polygon No. Each of these arcs is covered by two sides of No except when the arc ends in the image of a rotation of order two. The other sides of the boundary of No are in oneto-one correspondence with the different segments on the C~. Putting the information contained in the signature together we get a presentation for an NECgroup which reduces to the Poincare presentation when the group only has orientation preserving maps. Specifically we have the following result. (see [L-S] where it is described in a different memnerin terms of planar complexes. ) THEOREM 4.4.1. Let G be an NECgroup. Then G has a presentation of the following form:

106


(a) Generators: X U J where X and ar e di sjoint se ts an d J i~self is the disjoint union of ordered sets Jj = (x1,~1, ..., xj,~j), n~ >_ (b) Relators: Ro U RI U R2, the union of three sets of relators

(i)ao

e

(ii)R1 is the union o[seSs R~5 = x j,~ ,...,tj,~¢ ) wherem~,h are positi~ integers ~d tj,h = Xj,haj,hXj,h+~a~ (induing h cyclically modulo nj) with aj,h words containing only generators ~o~ X. m~ where ~t k are positive (iii) Ru is a set of wordsof the form sk integers and sa are words containing only generators &omX. Finally in the set of aH aj,~ together wi~h aH sk each generator &ore X occurs exac$ly $wice. In the presentation, if the s~face is orientable we can thi~ of the X generators as the ~chsian ~oup part and the J generators as reflection. If the s~face is orientable and there are no reflection, this reduces to the struct~e we can get via the Poincare presentation. Wemention one other study relative to both Fuchsian groups and NEC groups. If N0 is a fundamental polygon for an NECgroup G with compact fundamental polygon, then the side pairing tra~formations generate G. CM1 these g~metric generators. The minimum number of necessary geometric generators is called the geometric rank of G denoted gr(G). For any group G the minimumnumber of necessary generators is the rank of G denoted r(G). Cl~ly r(G) 5 gr(G). Nie~en posed the question as to whether the two concepts coincide for Fuchsian ~oups. It has b~n shown that for a Fuchsian ~oup geometric ra~ can be larger than ra~ however results of Peczy~ki, Rosenberger and Zieschang [P-R-Z] show that there is only one t~e of such examples. For NECgroups in general, it is more complicated. Ka~ma~ and Zieschang [K-Z] have shown that there e~st NEC~oups G with r(G) ~ ~gr(G). R.Weidmann [Wei] has extended these r~ts to examples where gr(G) = 2n + 1 and r(G) = n + 4.5 Cl~sification

of Two-Generator

Discrete

Groups in PSL2(R)

In this final section of Chapter 4 we present without proof several resets w~chgive a complete classification of the possibilities for generating pairs of two-generator non-elementary Fuchsian groups. These results have appeared in many d~erent locations but a summa~of the results, together with aH the proof, can be found in the pa~r of Fine and Rosenberger [F-R 10] (s~ a~o the papers of P~zitsky [Pu], P~zitsky and Rosenberger [PuRl and Rosenberger [R 12]; in the latter the complete results are a~eady summ~ized). The techniques employed in the proofs are straightforw~d and depend only on the properties of linear ~actionM tra~formations and their tr~es. There have b~n several other approaches to the classification

4.5 TWO-GENERATOR DISCRETEGROUPSIN PSL2(R)

107

of two generator Fuchsian groups, however the results as stated in this section seem to be the ones most in spirit with the rest of these notes. We mention that Maskit and J.Gilman [Gi-M] have taken in certain cases a more geometric approach to the classification problem but we note that the geometric movesappearing in tiffs paper are in essence geometric realizations of Nielsen transformations. Before stating the theorems we need some notation. Suppose G =< a, b > is a two-generator group. Wewrite (a, b) ~N (U, if there exists a Nielsen transformation from (a, b) to (u, v) (see section 2.6 for the necessary terminology on Nielsen transformations). If the element a has finite order n then a transformation (a, b) --* "~, b)wit 1 _< m < n and gcd(m, n) -- 1 is called an E-transformation. An extended Nielsen transformation is a finite sequence of Nielsen transformations and E-transformations. Wewrite (a, b) ~N (u, v) if there exists an extended Nielsen transformation from (a, b) to (u, v). If (a, b) %N then < a,b >=< u,v >, that is, the pairs (a,b) and (u,v) generate the same group. Wenow state our main theorems. First of all by a direct computation with the Poincare presentation we obtain Theorem 4.5.1 which gives the total possibilities, up to isomorphism, for presentations of two generator Fuchsian groups. THEOREM 4.5.1. A two-generator (z~on-elementary) Fuchsian group has one and only one of the following descriptions in terms of generators and relations: (1.1) G =< A,B; >, G is a free gro~ip on A,B (1.2) G=< A,B;A p= I > for2 for2 is non-elementary. Then G is discrete if a.nd only if there is an extended Nielsen transformation from (U, V) to a pair (R, which satisfies {a~ter a suitable choice of signs} (1) 0 _< tr(R) 5 with gcd(r, 2) = 1 and (U, V) is Nielsen equivalent in a trace minimizing manner to a pair (R, S) which satisfies tr (R) = tr (S) and tr (RS) 1 (f) tr([U,V]) = -2cos(3~/r),r N,r > a with gc d(r, 3) -= 1 a (U, V) is Nielsen equivalent in a trace minimizing manner to a pair (R, S) which satisfies tr (R) -~ tr (S) = tr (g) tr([U, V]) = -2 cos(4~r/7) and (U, V) is Nielsen equivalent in a trace minimizing manner to a pair (R, S) which satisfies tr (S) = tr

= tr + 1. Moreover, if G is discrete then G is of type (1.1) in case (a), of type(1.5) in case (b), of type (1.6) in case (c), a (2, 3, r) triangle groupin case (d),a (2, 4, r) triangle groupin case (e),a (3, 3, r) triangle groupin case (f), and a (2, 3, 7) triangle group in case (g). COROLLARY 4.5.1. Suppose G ~ U, V > C PSL2(R) is a Fuchsian group. Then [tr([U, V]) - 21 >_ 2- 2cos(~r/7). Weclose this section with a result based on a theorem of Majeed [Mail, (See Rosenbcrger [R 3]), concerning free subgroups in discrete groups. Notice the similarity of this result with the Ree-MendelsohnTheorem (Theorem 3.4.3). Wewill be using this result in our analysis of free. subgroups in generalized triangle groups. Recall that a subgroup of PSL2 (C) is elementary if any two elements of i~ffinite order (regarded as linear fractional transformations) have at least one commonfixed point. A subgroup of PSL2(C) is called elliptic if it consists of solely elliptic maps. THEOREM4.5.5. Suppose A,B ~ PSL,2(C) with = an d assume that G is non-elementary and non-elliptic. Then there is a generating pair (U, V) of G which is Nielsen equivalent to {A, B) such that < U~, V*~ > is a discrete free group of rank 2 for some large integer n. Wemention that a corresponding classification for two generator NEC groups has been done by E.Klymenko and M.Sakuma [K1- S].

CHAPTER V ONE-RELATOR

5.1 One-Relator

PRODUCTS

Products

As we saw in Chapter 3 there is a vast theory of one-relator groups generalizing the theory of free groups. Further, one-relator groups were tied to linear groups via their linearity properties. In this chapter we generalize one-relator groups to an extended construction called one-relator products. This construction also arises from the theory of discrete groups via the Poincare presentation. As we will see, many of the linearity properties shared by discrete groups and one-relator groups, will carry over, under appropriate conditions, to this more general class. Let {Ai}, i in some index set I, be a family of groups. Then a onerelator product is the quotient, G = A/N(R), of the free product A = ¯ iAi by the normal closure N(R) of a single non-trivial word R in the free product. Weassume that R is cyclically reduced and of syllable length at least two. The groups Ai are called the factors while R is the relator. In analogy with the one-relator group case we say R involves Ai if R has a non-trivial syllable from A~. If R = S"~ with S a non-trivial cyclically reduced word in the free product and m _> 2, then R is a proper power. Wethen also call S a relator. As in the one-relator group case the proper power case is somewhateasier to work with. In this context a one-relator group is just a one-relator product of free groups. From the Freiheitssatz a one-relator group with at least two generators in the given presentation is never trivial. On the other hand a one-relator product of non-trivial groups maycompletely collapse. For example, consider A =< a > and B = to be finite cyclic groups of relatively prime order. Then the one-relator product G = A * B/N(ab) is a trivial group. Because of examples such as this, a natural question to ask is under what conditions the factors actually inject into a one-relator product. Wesay that a Freiheitssatz holds for a one-relator product G if each factor injects into G via the identity map. More generally let X, Y be disjoint sets of generators and suppose that the group A has the presentation A -- < X; Rel(X) > and that the group G has the presentation G =< X, Y; Rel (X), Rel(X, Y) >. G satisfies a Freiheitssatz, abbrevi111

112


ated FHS, {relative to A} if < X >G= A, that is the subgroup of G generated by X is isomorphic to A. Alternatively this means that A injects into G under the identity map taking X to X. In coarser language this says that the complete set of relations on X in G is the "obvious"one from the presentation of G. If G satisfies a FHSrelative to A we say that A is a FHSfactor of G. In this language a Freiheitssatz holds for a one-relator product if each factor is a FHSfactor. As with one-relator groups, the starting off point for a study of onerelator products is to determine a Freiheitssatz. The example above shows that there is no such result in general and therefore some restrictions must be imposed. There are two approaches. The first is to impose conditions on the factors while the second is to impose conditions on the relator. Wewill discuss conditions on the factors first. Recall that a group H is locally indicable if every finitely generated subgroup has an infinite cyclic quotient. It turns out that local indicability of the factors is a strong enough condition to allow most of the results on one-relator groups to be carried over to one-relator products. In the next section we will give a proof of the FHSfor one-relator products of locally indicable factors (a result proven independently by Brodskii, Howie and Short). Wewill also mention some further results on local indicability and one-relator products with locally indicable factors. Local indicability is a strong form of torsion-freeness and them have been some succesful attempts to prove a FHS for one-relator products of torsion-free factors. Wediscuss these also. The second approach is to impose restrictions on the relator. The most commonrelator condition is that R is a proper power of suitably high order, that is R = S"~ with m _> 2. Wewill discuss this approach in section 5.3. If m > 7 then the relator satisfies the small cancellation condition C1(1/6) {see section 3.8} and a FHScan be deduced from small cancellation theory. A FHSdoes hold in the cases m --- 4, 5, 6 (m -- 6 due to GonzalezAcuna and Short [Go-S], m = 4, 5 due to Howie [H-3,4]) but the proofs are tremendously difficult. The cases m = 2, 3 are still open in general although specific cases where a FHSdoes hold have been proved. In particular if the factors admit representations into a suitable linear group, a FHScan be given. This will be done in Chapter 6. The technique for handling these proper power situations is combinatorial geometric and closely tied to small cancellation diagrams. Wewill only summarize these geometric results and a more complete treatement of the geometric techniques can be found in the excellent survey articles by Howie[H5] and Duncan and Howie [D-H 1] as well as the original papers. One-relator products and the Freiheitssatz are closely tied to the solution of equations over groups, also called the adjunction problem. Basically the adjunction problem is the following. Let G be a group and

5.2 THE FREIHEITSSATZAND LOCALLYINDICABLEGROUPS

113

W(Xl,.. ¯ , x,~) be an equation (or system of equations) with coefficients G. This question is whether or not this can be solved in some overgroup of G. In the case of a single equation W(x) = 1 in a single unknownx a result of F.Levin [Le] shows that there exists a solution over G if x appears in Wonly with positive exponents. To see the connection with the FHS consider the one-relator product G = A, < t > IN(R) where R is a cyclically reduced word in the free product A* < t > of syllable length at least 2. Consider R then as a word R(t) in the variable t with coefficients in A. The equation R(t) = has a solution over A in some overgroup containing A if and only if the natural map from A into G is injective. More generally Baumslag and Pride [B.B.- P 1] have shownthat if X is a class of groups which contains the infinite cyclic group and is closed under free products with finitely many factors, then the existence of a FHSfor one-relator products of X-groups is equivalent to the fact that any equation is solvable over an X-group. By this we mean that if G = (.~H~)/N(R) where the Hi are arbitrary X-groups and R is a cyclically reduced word in the free product on the Hi of syllable length at least 2, then each Hi injects into G if and only if any equation is solvable over an X-group. In connection with the adjunction problem we mention the Kervaire Conjecture (also called the Kervaire-Laudenbach Conjecture). This says that if G = A* < t > IN(R) is trivial, then A is trivial. From the FHS this is clearly true if A is a free group. Recently A.Klyachko [K1] proved that the Kervaire conjecture is true wheneverA is a torsion-free group. In this paper he also proved a Freiheitssatz. Specifically if A is torsion-free and the exponent sum of t in R E A, < t > is :i:1, then A injects into A* < t > IN(R). Klyachko’s result is a strenghtening of the FHS for locally indicable factors. Wewill discuss this and give Klyachko’s solution in section 5.4. 5.2 The Freiheitssatz

and Locally

Indicable

Groups

As with one-relator groups the starting off point for the study of onerelator products is to provide a FHS. As indicated in the last section there are two ways to proceed: to place restrictions on the factors or to place restrictions on the relator. The factor condition which is not only strong enough to give a FHSbut also to allow the extension of most of the theory of one-relator groups is local indicability. Recall that a group G is locally indicable if every non-trivial finitely generated subgroup has an epimorphism onto Z. In particular locally indicable groups are torsion-free and hence local indicability can be considered as a very strong form of torsionfreeness. Locally indicable groups were introduced by Higman in work on group rings and the zero divisor question. Specifically, in his thesis Higman

114


[Hi 2] proved that if K is an integral domain and G is locally indicable, then the group ring KGhas no zero divisors, no idempotents other than 0 or 1 and no units other than those of the form ug with u a unit in K and g E G. The following theorem was discovered independently by Brodskii[Br 1,2], J.Howie [H 7] and H.Short[Sho]. It is interesting that all three proofs are entirely different. THEOREM 5.2.1. A one-relator isfies a Freiheitssa~z. That is, if G = *~A~/N(R), is a cyclically reduced word in least two, ~hen each A~ injects factor.

product of locally indicable factors satwhere each A~ is locally indicable, and R the free product *~A{ of syllable length at into G under the identiky map, i.e. is a FHS

Before giving a proof of this major result we briefly describe the evolution of the theorem. Magnusproved the classical FHSfor one-relator groups in 1929 using the algebraic techniques as described in Chapter Three. There were simplifications of this proof by Moldavanski[Mo], McCooland Schupp [Mc-S] and others once the theory of group amalgams was better understood. In 1972 Lyndon [L 4] gave a proof of the FHS using combinatorial geometry. This proof arose out of ideas and diagrams in small cancellation theory. As a by-product of this proof, Lyndon showed that the FHS held for one-relator products whose factors were additive subgroups of the real numbers R. S.Pride [P 1] then showed that the FHS holds for one-relator products of locally fully residually free factors while this was extended to locally residually free factors by B. Baumslagand S. Pride [B.B. - P 1]. The above results axe all examples of locally indicable factors and pointed to the main result. In 1980 Brodskii [Br 1] announced Theorem 5.2.1, but it was not published until 1984. Brodskii’s method was algebraic and mimicked Magnus’ original treatment. B.Baumslag[B.B. 1] rediscovered this algebraic proof. J. Howie independently proved Theorem 5.2.1 by giving a straightforward modification of his tower proof of the classical FHSwhich was presented in section 3.3. Finally H.Short [Sho], adapting Lyndon’s combinatorial geometric arguments to the context of one-relator products of locally indicable groups, gave a third independent proof of Theorem5.2.1. Short’s technique was important because it introduced to the study of one-relator products the concept of pictures over a group. These pictures are the duals of Lyndon - Van Kampendiagrams for one-relator products and proved instrumental in handling the proper power case. We will return to these ideas in the next section. Wenow give a proof of Theorem5.2.1 along the algebraic lines of Brodskii and B. Baumslag.


115

PROOF.(Theorem 5.2.1) The proof essentially mimics Magnus’ original treatment. Weprove the case of two factors, the general result then follows easily. Therefore assume that G = (A * B)/N(R) with A and B locally indicable and R a cyclically reduced element in the free product A * B of syllable length at least two. By symmetryit suffices to showthat A injects into G. Let Ao be the subgroup of A generated by those elements which appear in R and similarly let B0 be the subgroup of B generated by those elements which appear in R. Let Go -= (Ao * Bo)/N(R). If Ao and B0 inject into Go, it would follow that G is the amalgamated product G = A *Ao Go *Bo *B and hence A and B inject into G. Hence we can assume without loss of generality that A = A0 and B = B0. Recall that A and B now are finitely generated, and hence each has an infinite cyclic image. The proof now proceeds by an induction on the syllable length L of the relator R. If L = 2 then both A and B are infinite cyclic groups and the result holds since G is just a two-generator one-relator group. Thus we can assume that R has syllable length L greater than 2. Weconsider first the case where B = is infinite cyclic. Then since A is locally indicable there is an epimorphismof A onto an infinite cyclic group C =< c > with kernel N. Suppose further that A/N =< aN >. This induces a homomorphismof A ¯ B onto C ¯ B and suppose that under this homomorphismthe relator R is mapped to R1. Wenow have the group G1 = (C * B)/N(R1) and suppose here first that c has exponent sum zero in R1. Thus R lies in the normal closure of N tO B in A, B. Let b~ = a-~ba~ and write R in terms of the b~ and the elements of N. At least two of the b~ must be involved in writing R since R is of length at least 4 (cyclically reduced of length greater than 2) and some am, with r E Z, r ~ 0, n E N, must occur in R since we are assuming that A -- A0. Let s be the minimumindex of a b~ appearing in R and let t be the maximum. Let K = N, < bs > * < bs+l > * ¯ ¯ ¯ * < bt >, Ko = N* < b~ > * < b~+~ > ,...*

< bt-~ > and

K1 = N* < bs+l > *.." * < bt > ¯ R is of length smaller, than L as a word in the free product K0* < bt >. Hence by the inductive hypothesis K0 is embedded in K = KIN(R). Similarly, R as a word in the free product < b~ > ,K1 is of length less than L and so K1 is embedded in K. Thus the original one-relator product G

116


is the HNNextension of K with stable letter a and associated subgroups Ko, K1 with a-lbia = b~+l,S . From the homomorphismdefinition of free products it follows that A * B = A * B. Further in the image of R under the homonmrphism induced by a from A * B into C * B the exponent sum of E is zero. Therefore the result follows from the argument above when there is zero exponent stun since, as in the case of a one-relator group, G embeds into G = (A ¯ B)/N(R). Finally, for the case B = B0 ---, suppose that b has zero exponent sum in R1. Then b has also zero exponent sum in R. R lies in the normal closure of A which has the form M= .iA~, i E Z, where A~ = b-*Ab~. Since B = B0 and R is cyclically reduced of length at least four, at least two of the Ai are involved in R. Suppose s is the minimumindex such that Ai is involved in R and t is the maximumindex. Define Ko -- A~+I *-.- * At and K1 = A~ *... * At-~. Then R belongs to Mo ---- A~ * Ko = At * K1, and the A~, K0, K~ and M0are all locally indicable. Let L1 denote the length of some cyclically reduced conjugate of R, regarded as a word in A~ * K0. We have 2 _< L1 < L since s ~ t and 2 which further induces a homomorphsim from A ¯ B onto A * D. Let R2 be the image of R under this map. If d appears with non-zero exponent sum in R2 then if the length of R~ is one then (A * D)/N(R2) is the free product of A and a finite cyclic group and hence A is embeddedin it which implies the required result. Otherwise if R2 has length at least 2 then we can use the first case, where B is infinite cyclic to establish that A is embeddedin (A * D)/N(R2) and hence in (A * B)/N(R) which gives the result. As alluded to, local indicability of the factors is strong enoughto not only allow the FHSas above but to extend muchof one-relator group theory. We


117

close this section by mentioning someof these extensions. Further material in this area can be found in the articles by Howie[H 1,5,6]. Theorem 5.2.1 can be iterated and extended in several ways to obtain the FHSfor groups formed by more general constructions than one-relator products. These extensions are generally of the form (*Ai)/N(Rj) where the A~ are locally indicable and the relators Rj are "staggered" [H 5]. We present one result of this type. This involves a straightforward iteration using a result of Howie (see Theorem5.2.3 below). Wemention that there are many other results along these lines especially by Howie [H 5], Howie and Pride [H-P] and M. Edjvet [E 1,2,3]. THEOREM 5.2.2. Let A1,... , An be locally indicable groups, and/’or each i = 1,... , n - 1, let R~ be a non-trivial word in the [ree product G~ = AI*A2*... ,A~+I o[ syllable length at least ~vo and involving A~+~. Assume fureher that [or i = 1,... , n - 2, R~ not a proper power. Then each A~ is a FHSfactor for G ~- (*iAi)/N(R1,... , P~-l). PROOF. Suppose n = 3 so that G -- (A,B,C)/N(R~, R2) with A, B, C locally indicable and R~ involving A and B not a proper power and R2 involving C. From a result of Howie [HI(see Theorem 5.2.3 below) one-relator product of locally indicable factors is locally indicable if the relator is not a proper power. Therefore GI = (A*B)/N(R~) is locally indicable. Now G = (GI*C)/N(R2) is then a one-relator product with locally indicable factors so both G~ and C inject. Further A and B inject into G1 and thus into G. The general result then follows easily by induction. This type of iteration will be used in Chapter 9 in the study of groups of special NECtype. The next result, due to Howie[H 6], generalizes a result of Brodskii, and answers a question posed by Pride. This is that torsion-free one-relator groups are locally indicable. THEOREM 5.2.3. Let G = (A * B)/N(R) be a one-re/ator product o[ locally indicable groups with the relator R being cyclically reduced of syllable length at least 2 in the free product A * B. Then the [ollowing are equiwlent: (i) is locally ind icable. (ii) ~ is torsion-free. (iii) is not~ pr oper power in A * B. COROLLARY 5.2.1. Aft torsion-free cable.

one-relator groups are locally indi-

Theorem 3.2.4 was the Spelling Theorem of Newman. This said in essence that in a one-relator group ~ with relator R = S~, m ~_ 2, any

118


non-empty word which represents the identity in G must contain a cyclic subword of R or R-1 longer than S"~-1. There have been improvements and more precise versions of this result for one-relator groups given by Gurevich [Gu], Schupp [Sc] and Pride [P 1]. The following, due to Howie and Pride [H-P] extends this to one-relator products of locally indicable groups. Further this result in turn allows for the solution of the word problem in one-relator products of locally indicable groups when the relator is a proper power. THEOREM 5.2.4. Let G = (A * B)/N(R) where A and B are locally indicable and R = S"~, m >_ 2. Let W be a non-empty word in A. B which lies in N(R). Then either W is conjugate to R or -1 or Wcontains tw o almost disjoint cyclic subwords, each o/: which is a cyclic subword oze R or R-~ "-~. longer than S COROLLARY 5.2.2. I/: m > 1 and A and B have solvable word pro.blems in the above theorem ~hen G also has a solvable word problem. Brodskii and Mazurovskii have also solved the word problem for G in the case where rn -- 1 but under the stronger hypothesis that A and B are effeetlvely locally indicable. By this they mean that there exists an algorithm which, given a finite set of generators for a subgroup H, will decide whether or not H is trivial and if not will exhibit an epimorphism of H onto Z. Corollary 5.2.2 and the extension cited above lead to a consideration of the general decision problems in one-relator products with locally indicable factors. Recall that for one-relator groups, the solvability of both the word problem and the generalized word problem was obtained by Magnus. The conjugacy problem is also solvable and was handled in the torsion case by Newmanand in general by Juhasz. These standard decision problems are subsumed in a class of decision problems called the genus problems. (see Duncan and Howie [D-H 1]). The genus problem GP(g, n) is the following algorithmic problem: given a group G, a set of generators for G and n words W1, ..., W,, in these generators decide whether or not the equation XlW1XF1...XnWnX~1[y1,

Z1]...[Y9,

Zg]

= 1

can be solved for Xi, Y~, Zj in G and if so to find an explicit solution in terms of the given generators. GP(0, 1) is then the word problem while GP(O, 2) is the conjugacy problem. Pride [Pr] termed the class GP(0, n) the dependence problems. In general we say the genus problem is solvable for a given group G if GP(g, n) is solvable for all g, n. For one-relator products with locally indicable factors the main result is the following.

5.2 THE FREIHEITSSATZANDLOCALLYINDICABLEGROUPS

119

THEOREM 5.2.5. Let G ~- (A * B)/N(S "~) where A and B -axe locally indicable, m >_ 2, and S is cyclically reduced in A * B of length at least 2. Suppose that A and B are g/yen by recursive presentations for which the genus problem GP(g’, n’) is solvable for all pairs of integers (g’, n’) that 0 where R = S "~ and S is not a proper power. Suppose T is a transversal for the subgroup < S > N(R) F(X). Then {tRt -1, t ¯ T} is a free basis for N(R). COROLLARY 5.2.2. Let G =< X; R >. then the cohomological dimension of G is 2 (1) H G is torsion-free, unless G is free. (2) H R = Sr~ with m >_ 2 and S not a proper power, then for n >_ 3 and any ZG-module M, H~(G, M) ~ H~(W, where W i s the cyclic subgroup of G generated by the element s represented by S. These ideas can be extended somewhat to one-relator indicable factors. In particulaz we have.

products of locally

THEOREM 5.2.9. Let G = (A * B)/N(R) where A and B are locaJly indicable, and R -- S"~ with m >_ 2, S cyclically reduced in A * B of length a~ leas$ 2 and S is not a proper power. Then N(R)/[N(R), N(R)] is isomorphic as a ZG-module to the cyclic module ZG/(1 - s)ZG generated by R[N(R), N(R)] and where s is the element of G represented by S. COROLLARY 5.2.3. Let G, A, B be as in Theorem 5.2.9 and suppose C, the subgroup generated by s is a cyclic group of order m. Then for each Ha(G;-)

= Ha(A;-)

+ Ha(B;-)

+ Hq(C;-)

Ha(G; -) = Ha(A; -) + Ha(B; -) + Hq(C;-). As a consequence of these cohomological results, an extension of the theorem of Bagherzadeh [Bag] on Magnus subgroups in one-relator groups (see Theorem3.7.6) is obtained [H 2]. THEOREM 5.2.10. indicable, and R is any subgroup H of or B ~ gBg-~ with g any subgroup of this

Let G = A * B/N(R) where A and ar e lo cally cyclically reduced in A * B of length at least 2. Then the form AAgBg-l, g ¯ G or AAgAg-~ with g ¯ G\ ¯ G \ B is cyclic. Further if R is a proper power then form is trivial.

There is also an analog of the strong Cohen-Lyndon Theorem (Theorem 5.2.8) for one-relator products of locally indicable groups due to Edjvet and Howie [E-HI. THEOREM 5.2.11. Let G = (A * B)/N(R "n) where A and B are locally indicable, and R is cyclically reduced in A * B of length at leas~ 2. Let C be the cyclic group generated by R. Then there is a set U of double coset

5.3 THE FREIHEITSSATZ FOR HIGH POWERED RELATORS represeneatives

121

of N(R’~)A * B/C such Chat N(R"~) is freely generated by

{uR’~u-1;u ~ U}. Finally, locally indicable groups are torsion-free. in the theory of one-relator products is: CONJECTURE. The Freiheiessaez eorsion-free faceors.

A standard conjecture

holds for one-relator

products

of

Brodskii and Howie [Br-H] and Klyachko [K1] have proved some results on special cases supporting this conjecture. Let x = (xl,... ,x~) be sequence of elements in a group G. x~ is isolated in x if no xj belongs to the cyclic subgroup generated by xi for i ~ j. THEOREM 5.2.12. ([Br-H]) Lee G = (A.B)/N(W) where A and eorsion-free groups and W -- a~b~.., a~bk is a cyclically reduced word in A.B such that some ai is isolated in (ax,... , ak) and some bj is isolated in (b~,... ,bk). Then A and B naturally inject into G. If W= al Zml ¯ ¯ ¯ akx rn~ is a word in A* < x >, then the sign-index a is the number of sign changes in the cyclic sequence (ml,... , mk). THEOREM 5.2.13. ([Br-H]) Let G = A* < x > IN(W) where eorsion-free and W = a~xTM ... a~x"*~ is a cyclically reduced word in A* < x > of length at least 2 and sign-index a 0 (ii) a~...a~ is in A\ {1} (iii) a = rnk q- mla~ + m2(a~a2)+... + mk_~(al . . . ak_~) is not a in QG

(iv) ~ _< Then < z > naturally in~ects into G, that is z has infinite

order in G.

COROLLARY 5.2.4. Let G = (A*B)/N(W) where A and B are torsionfree groups and W= a~bl . . . akbk is a cyclically reduced word in A* B such that ai 7~ 1 ~ b~ and k 1/~ for all r in R. There are similar analogues for C(p) and T(q) but these are not necessary for our FHSresult. Wedefine a small cancellation product to be a group G = FIN where F is either a free product, free product with amalgamation or HNNgroup and N =- N(R) is the normal closure of a symmetrized set of words in F satisfying a small cancellation condition C~(A) for some Our major result is the following which is actually a summaryof several results (see [L-S]). T~EOaEM 5.3.3. (The FHS for Small Cancellation Products) ( see[L-S] and the references there) Let G be a small cancellation product satisfying C’(A) for some A < 1/6. Then G satisfies a FHS relative to any of the amalgamfactors, that is, any factor of the underlying area/gain injects into G under the identity homomorphism. The proof of these results follows the same general outlines as for small

126


cancellation theory over free groups, that is, the analysis of appropriate diagrams. The approach given in Lyndon and Schupp[L-S] is to treat each amalgam construction separately. J. Corson [Cor] has considered Small Cancellation Theory over graphs of groups. His approach both unifies and extends this classical treatment. He defines non-spherical sets of words which are the analogues of symmetrized sets of words satisfying small cancellation conditions. He then proves that if F is a graph of groups and N is the normal closure in F of a non-spherical set of words, then the group natural map F --* FIN embeds the vertex and edge groups of F. Similar results have been done independently by R.M.S. Mahmud[Ma]. Wemention also that Collins and Perraud have proven a FHS for small cancellation products (over a free product) satisfying different small cancellation conditions [C-P]. The relevance of the above discussion to the case of one-relator products is the following. Suppose G = (A * B)/N(S "~) with A, B arbitrm’y groups and m _> 2. If m>_ 7 then the resulting group is a 1/6-th small cancellation product and the FHS holds. Further the geometric pictures and analyses used in small cancellation theory can be utilized in this case to extend most of the results on one-relator products of locally indicable groups. The case m = 6 was handled by Gonzalez-Acuna and Short [Go-S] in conjunction with a topological problem. Their technique provided an adaptation of Lyndon’s combinatorial geometric proof of the FHSto the situation of one-relator products. In doing this they adapted the concept of pictures which axe duals of Lyndon Van-Kampendiagrams. We now briefly describe these objects. Pictures were introduced as a technique in group theory by Rourke[Ro]. We mention also the FHS for m = 6 was done independently by Collins and Perraud [C-P] who showed that in this case the relator m S satisfies the non-metric small cancellation condition C(6). A picture F of a one-relator product G = (A * B)/N(I~), m _> 2, on a compact surface E consists of: (i) A disjoint union of (small) discs Vl,... ,V n in int(~), called the vertices of F. (ii) A properly embedded 1-submanifold ~ of To = ~,, \ int(Uv~). components of this submanifold are called the arcs of F. (iii) An orientation of the boundary 0~0 and a labelling function that associates to each componentof 0~0 \ ~ a label which is an element of AUB. A picture over 82 is called a spherical picture. The data above is also required to satisfy a numberof properties reflecting the fact that it is to represent the one-relator product. (a) In any region A of F--componentof ~ \ (t3v~ (~ ~), either all labels

5.3 THE FREIHEITSSATZFOR HIGH POWEREDRELATORS

127

belong to an A-region or a B-region accordingly. (b) Each arc separates an A-region from a B-region. (c) The vertex label of any vertex v~-the word consisting of the labels of Ovi read in the direction of orientation from somestarting pointis identically equal to Rm in the free monoid on A U B up to cyclic permutation. (d) Suppose A is an orientable region of F of genus g with k boundary components. Then each boundary component has a boundary 1label a~ ¯ ..a~~ , ~ = =t=1, where al,... ,a~ are the labels of that boundary component in the cyclic order induced from some fixed orientation of A and e~ -- ÷ 1 if the orientation of the segmentof 0~0 labelled a~ agrees with that of the boundary component of A, and -1 otherwise. If al,..., ak are the boundary labels of A, then the equation XlalX~"1... XkakX[~[Y~, Z1]’" [Y~, Zg] --- 1 is solvable for (X~, Y~, Zj) in A if A is an A-region or in B if A is a B-region. Under the conditions (a) through (d) above, pictures can be constructed over G. Further there are certain allowable operations on pictures which, when applied, yield an essentially equivalent picture. These arc called bridge moves and insertion or deletion of floating dipoles or exceptional spherical pictures. We refer to [D-H 1] or [Re] for a precise definition of bridge moves and floating dipoles but describe exceptional presentations which are crucial to some further discussions. As will be seen these axe related to triangle groups. Wesay a relator R is exceptional and the corresponding picture is exceptional if R = xUyU-1 for some word U and letters x,y up to cyclic permutation. If p,q are the orders of x, y respectively, then we say that G is of type E(p, q, m) and we then call R"~ of the form E(p, q, m). In the case where U is trivial, A =< x >, B =< y >, and then G is the triangle group T(p, q, m) (see Chapter 4). If liP-t- 1/q ÷ 1/m - 1 = s > 0 then T(p, q, m) is finite order 2Is. In this case, that is, G -~ T(p,q, m), s > 0, there is a canonical spherical picture F(p, q, m) arising form the action of G on 2. In general, if G is exceptional of type E(p, q, m) with s > 0, then there is a natural homomorphismfrom T(p, q, m) to G and F(p, q, m) induces a spherical picture over G. Wenote that a one-relator product can be exceptional in more than one way and so great care must be taken in handling such exceptional presentations. (see [D-H1]). Two pictures over G on the same surface are equivalent if each can be obtained from the other by a sequence of these allowable operations. Pictures can be associated to maps from the surface E to a certain space with fundamental group G (see [D-H 1]) and equivalent pictures have maps which differ only by a certain homotopy.A picture on ~. over G is efficient if it has the least numberof vertices in its equivalence class.

128


For the proof of the FHS,the crucial condition on pictures is the following: CONJECTURE F. Suppose G = (A * B)/N(R "~) is a one-relator product where S is a cyclically reduced word in the free product A ¯ B of syllable length at least 2 and suppose m _> 2. Let F be an e//icient picture on the ~ by disc D~ over G such that at most 3 vertices of F are connected to OD 2 by arcs. arcs. Then every vertex of F is connected to OD TI~EOREM5.3.4. ([H 3,4])Suppose that G = (A* B)/N(R) and that Conjecture F holds for pictures over G on the disk. Then the FHS holds for G. The techniques in handling the FHSfor high powered relators then reduce to showing that Conjecture F holds. The geometric arguments however become tremendously complex. The main results obtained by Howie are given in the next theorem. An analysis of pictures yielded the FHSfor m = 5 [H 3] while a much deeper analysis did the same for m = 4 [H 4] and certain cases when m = 3 [D-H 2] (The last result joint with A. Duncan). TItEOREM 5.3.5. Suppose G ---- (A * B)/N(Rm), m >_ 2, is a one-relator product with R cyclically reduced of length at least two. If at least one of the following conditions hold, then G satisfies conjecture F:

(i) > 4; (ii) m ---- 3 and no letter oeeuring in R has order 2. THEOREM 5.3.6. Suppose G = (A, B)/N(R m) is a one-relator product where R/s a cyclically reduced word in the free product (,Ai) of syllable length at least 2 and suppose m >_ 4. Then the FHSholds, that is, each factor Ai natura/ly injects into G. Further if m -- 3 and the relator R contains no letters o[ order 2, then the FHSholds. There are no known examples for which the FHS fails therefore we have the following conjecture.

for m _> 2 and

CONJECTURE. The FHS holds for any one-relator product G = (A * B)/N(R’~), where m >_ 2 and where R is a cyclically reduced word in the free product A * B of syllable lengeh at least 2.

COROLLARY 5.3.1. Let G = (A* B)/N(R~), m >_, where R is a cyclically reduced word of leng’~h at least two in A. B and is not a proper power. If conjecture F holds for G, the no proper cyclic subword of R"~ represents the identity in G. In particular, r, the element of G represented by R, has order m in G. The technique of pictures and geometric analysis has also been used to further extend many of the results on one-relator products of locally

5.3 THE FREIHEITSSATZ FOR HIGH POWERED RELATORS

129

indicable factors to the situation of high powered relators. However in most of these results, special care must be taken in the case of exceptional relators. There is first a generalization of the LyndonIdentity Theorem and the Cohen-Lyndon Theorem. THEOREM 5.3.7. Let G = (A*B)/N(Rm), m >_ 2, where R is a cychcally reduced word in A * B of length at least 2 which is non-exceptional. conjecture F holds for G, then N(R)/[N(R), N(R)] is isomorphic as a module to ZG/(1 - r)ZG where is the element of G represented by R. COROLLARY 5.3.2. Let G, A, B, R, r and m be as in Theorem 5.3.7. Let conjecture F hold for G. Then for q >_ 3 the restriction induced maps Hq(G;-)

-,

Ha(A;-)

+ Hq(B;-)

+ Hq(C;-)

are naturaJ isomorphisms of functors on ZG-modules, and for q = 2 a natural epimorphism, where C =< r > is the cyclic group of order m generated by r. Dually the maps Hq(A; -) + Hq(B; -) + Hq(C; -) --~ Hq(G; are natural isomorphisms for q _> 3, and it is a natural monomorphismfor q=2 THEOREM 5.3.8. Let G = ( A* B) /N ( ~) where R is cyclicedly red uced in A * B of length at least 2. Suppose that m >_ 6 or m >_ 4 and letter of R has order 2. Let C be the cyclic group generated by r, r the element represented by R, except in the case where R is exceptional of type E(2,2,m), that is R = xWyW-l,x 2 = y2 = 1, in which case C is -1. Then there is a set U of the diheclral group generated by x and WyW double coset representatives of N ( R’* ) A * B / C such ~ha$N ( m) i s f reely generated by {uRmu-x; u E U}. Using the cohomological results together with a result of Serre [Se] a classification of the finite subgroups of one-relator products can be obtained. TI~EOREM 5.3.9. Let (7, A, B, R, m, C be as in Theorem 5.3.8. Let conjecture F hold. Suppose K ~ {1} is a finite subgroup of G. Then K C gag -1, K C gBg-1 or K C gCg-1 for some g E G. Moreover precisely one of these occurs and the left coset gA (respectively gB, gC) uniquely determined by K. In particular if A, B are locally indicable and m >_ 2, then any element of ~nite order is conjugate to a power of R. Note that the second part of the theorem is the direct analog of the torsion theorem for one-relator groups. The theorem of Magnus on normal closures of elements can also be extended.

130


THEOREM 5.3.10. Suppose R1,R2 are cyclically reduced non-trivial words of length at least 2 in the free product A * B where ml, rn~ are positive integers. Suppose that for each i -- 1, 2 at leas~ one of the foflowing conditions holds: (i) m~_> and R~is notof t he formE(2, 3, 4) or E(2, 3 , 5); (ii) ml _> and nolet ter of R~ hasorder 2. If R~1 and R’~2 have the same normal closure in A. B, then m.~ = m2 and 1. R2 is a cyclic permutation of R2 or R~ There are also various versions of the Spelling Theorem. Theorem5.3.8 gives the result when m _> 4. There is a corresponding technical result for m >_ 3 (see [D-H1]). THEOREM 5.3.11. Let R be a cyclically reduced word of length I >_ 2 in the free product A * B. Assume that m >_ 4 and that R is not of the form E(2, 3, 4) or E(2, 3, 5). Let W be a non-empty, cyclically reduced word in the normal closure of R~. Then either (1) W is a cyclic permutation of R~"~; or (2) W has two disjoint cyclic subwords U1, U2 such that each Ui is identical to a cyclic subword Vi of R±’~, L(UI) = L(U2) _> (m 1)/- 1 and W has a cyclic permutation xtU~x2U2 for some elements Xl,X2 of the pregroup A U B; or (3) W has k disjoint cyclic subwords U1,...,Uk for some k, 3 _< k _< 6 such that each Ui is identical to a cyclic subword Vi of R±’~, and Vi has length at least (m - 2)/- 1 for i < 6- k and at least (m - 3)/fori>6-k. Weclose this section with the next result which mirrors Theorem 5.2.5 on the genus problem for one-relator products of locally indicable groups. TtiEOREM5.3.12. Let G = (A * B)/N(R"~) where R is cyclially reduced in A ¯ B of length at least 2. Suppose that A and B are g/yen by recursive presentations for which the genus problem GP(g~, n’) is solvable for all pairs of integers (g’, n’) such that < g’< gand 1 < n’< n +2(g - g’) and R is given explicitly in terms of the generators for A and B. Suppose further that one of the following conditions holds (i) m > 5 and G is not of type E(2, 3, 5) or E(2, 3, (ii) > 4 and nolet ter in R hasorder 2. Then GP(g, n) is solvable for 5.4 The Kervaire

Conjecture

and Klyachko~s

Solution

As described in section 5.1, one-relator products and the Freiheitssatz are closely tied to the solution of equations over groups, also called the

5.4 THE KERVAIRECONJECTUREAND KLYACHKO’SSOLUTION131 adjunction problem. The adjunction problem is the following. Let G be a group and W(xl,... ,x,~) = 1 be an equation (or system of equations) with coefficients in G. This question is whether or not this can be solved in some overgroup of G. To see the connection with the FHS consider the one-relator product G = (A. < t >)/N(R) where R is a cyclically reduced word in the free product A* < t > of syllable length at least 2. Consider R then as a word R(t) in the variable t with coefficients in A. The equation R(t) = 1 has solution over A in some overgroup contaaining A if and only if the natural map from A into G is injective. More generally B.Baumslag and Pride [B.B.-P 1] have shown that if X is a class of groups which contains the infinite cyclic group and is closed under free products with finitely many factors, then the existence of a FHSfor one-relator products of X-groups is equivalent to the fact that any equation is solvable over an X-group. By this we mean that ifG = (*iH~)/N(R) where the Hi are arbitrary X-groups and R is a cyclically reduced word in the free product on the Hi of syllable length at least 2, then each Hi injects into G if and only if any equation is solvable over an X-group. There are three standard conjectures concerning equations over groups. CONJECTURE 5.4.1. (The Kervaire Conjecture) - Given any non-trivial group A, then the one-relator product (A* < t >)IN(R) is non-~rivial. CONJECTURE5.4.2.

Any single equation over a torsion-free

group A is

solvable. CONJECTURE 5.4.3. Any single power equation, that is an equation of the form (W(t)) k = g, is solvable over an arbitrary group A. Fromthe Freiheitssatz for one-relator products of locally indicable factors it follows that all three conjectures are true whenA is a locally indicable group. The obvious question then is can this be extended to torison-free groups. The main result of this section is to give the solution, due to Klyachko[K1], of the Kervaire conjecture when A is a torsion-free group. THEOREM 5.4.1. Let G = (A, < t >)IN(R) where each letter in R from A has infinite order. Then G is non-trivial. In particular G is non-trivial if A is torsion-free. Before giving the proof of Theorem 5.4.1 we make some brief comments on the history of the adjunction problem and equations in general. For more information we refer to the articles by Howie [H 1,5], Duncan and Howie [D-H 1], the older survey article by Lyndon [L 5] and the relevant sections in [L-S]. The study of the adjunction problem was initiated by B.H. Neumann [Ne] who proved that given a group G and g E G the equation x"~ = g

132


for m E Z can always be solved over G. Gerstenhaber and Rothaus [G-R] showed that any equation, or more generally any system of equations, over a compact connected Lie group A is solvable in some overgroup of A. As a corollary they obtain that any equation over a finite group is solvable in some finite overgroup. F. Levin proved that any equation W(x) = is solvable over an arbitrary group if x appears in Wo~fly with positive exponents. We can also consider systems of equations. To any finite system of equations Z : r~(tl,... ,tn) ..... rm(tl,... ,t,~) = 1 over a group, in variables {t~}, we can associate the integer matrix M(E) (#~j) where #~j is the exponent sum of t~ in the word (equation) rj. If the rank of M(E) is equal to the number of equations in E, then E is called independent. Howie [H 7] as a consequence of a result of Gersten [Ge] has proved that every independent system of equations over a locally indicable group has a solution in some overgroup. Welist and summarize some of these results and then give the proof of Klyachko’s Theorem. THEOREM 5.4.2. (Gerstenhaber and Rothaus) Every independent system of equations over a i~ni~e group has a solution in some [inite overgroup. Tn~,OREM 5.4.3. (Levin) Any single equation W(x) = i is solvable an arbitrary group G if x appears in W with only positive exponents. TH~,OR~,M5.4.4. (Howie) Every independent system of equations over a locally indicable group has a solution in some overgroup, In connection with this last result we mention another result, due to Howieand Short [H-S] along the same lines but yielding a Freiheitssatz. THEOREM 5.4.5. Let G = (A, < tl,... ,tn >)/N(rx,... ,r~). Then injects naturally into G under either of the following two conditions: (i) A is locally indicable and the system (rx,... , rn)/s independent (ii) A is locally indicable, n = 1 and rl is not conjugate to an element of A. Wenow turn to the proof of Klyachko’s Theorem. The proof depends on the following non-evident but relatively elementary topological observations. Suppose we have a simply connected domain D on an oriented surface and a moving point - Klyachko refers to this as a car - on the boundary. The car is said to move properly around the domain if it moves along the boundary continuously in the positive direction with no stops, no reverses and passing through every point on the boundary infinitely often. If two or more such moving points meet, it is called a collision. The observations are then:

5.4 THE KERVAIRECONJECTUREAND KLYACHKO’SSOLUTION 133 LEMMA 5.4.1. Suppose there is an oriented 2-sphere and a finite connected graph on it which divides the sphere into a finite number of simplyconnected domains. Suppose, for each domain, there is a car moving properly on the boundary of each domain. Then there exist at least two points of the sphere where there is a complete collision. A collision is complete if at a point of multiplicity k, k cars collide simultaneously. LEMMA 5.4.2. Suppose a11 the conditions of Lemma5.4.1 are satisfied but stops are allowed. Then there will also exist at least two points of complete collisions provided the following condition holds: Suppose in a vertex V, n cars Cl, ..., c~, listed in counter-clockwise order, stop. Then cars ci, ci+l rood (n) are never simultaneously situated in V. The proofs of these lemmas are somewhattricky but otherwise relatively straightforward. Weleave them as exercises. Wealso need the following result of Howie [H 8]. LEMMA 5.4.3. (/H 8]) Suppose W~(t), i E I, is a system of equations over a group A and suppose that the natural map A-~< A,t : W~(t) = 1,i ~ I is not that (i) (ii) (iii) (iv)

a monomorphism.Then there exists a tesselation

of a 2-sphere such

The edges are oriented. The corners axe labelled with elements of A. The exterior vertex has non-trivial label. Each interior vertex has trivial label. "(v)" The/abe/of each face is equal to some Wi up to permutation and inverse.

From these lemmasthe following two additional results can be obtained: LEMMA 5.4.4. Let H be a group with P, P1 isomorphic subgroups of H with P1 = ¢(P) where ¢ is a given isomorphism. Suppose that ai, bi are elements of infinite order in H for i = O, 1, ..., k with the property that < ai,P >= *P,< hi,P1 7=< bi > *PI fori -- 0,...,k andsuppose c ~ H. Then the sy~em of equations k (H bit-Xa~t)ct -i~o {t-lpt

= ¢(p),p ~

is solvable over H. PROOF.If this system was not solvable over H then from Lemma5.4.3 we could construct a tesselation of the 2-sphere satisfying those conditions.

134


Weshow that on an arbitray tesselation of this form we can determine a motion of cars satsifying the conditions of Lemma5.4.2 with fewer than two complete collision points. This contradiction will prove the result. A tesselation of the form of Lemma5.4:3 has three types of faces. We picture these in figure 5.4.1. Suppose each car moving around a face of type 1 (2) is situated at some initial time period at corner b0 (b~1) and move with velocity one edge per unit of time away from bo (b~1) but stay at corner c(c-1), the corner immediately prior to b0 for 2k - 1 time periods. For cars moving around faces of type 3 let them move without stops and with the same constant velocity and pass the corner ¢(p) at the inital time period.

Figure 5.4.1 Tesselation Face Types

It is easy to see that cars of different kinds never stop simultaneously and it is evident also that during the time intervals (0, 1) + 2/, 1 E Z, no car movesin the positive direction with respect to orientation of the edge and

5.4 THE KERVAIRE CONJECTUREAND KLYACHKO’SSOLUTION135

Figure 5.4.2 Source and Sink Vertices

Figure 5.4.3 Corners c and c-1 alternate - no stops in other corners during the intervals (1, 2)+2/, l E Z, no car movesin the negative direction. In particular there can be no collision outside vertices. If there is a collision in an interior source (or sink) vertex (see figure 5.4.2), it means a relation in < a~, P > (or < b~, ¢(P) >). This relation must be trivial and the tesselation is reducible. For vertices of other kinds it is clear that the schedule of cars stopping in such a vertex satisfies the conditions of Lemma 5.4.2. (figure 5.4.3) Therefore the only point which can be a point of complete collision is the exterior vertex. This contradiction then proves the lemma. For any group A consider B -- A* < t >. We use the notation A~ -t-~At ~ C B, P,~ the subgroup of B generated by (A~, 0 B, R,~ =< (A~}, 1 B and ¢ : P,~-I -~ R,~ the natural isomorphism p -~ t-lpt. LEMMA 5.4.5.

Let W ~ A * t and suppose the exponent sum oft in W

136


is one. Then W is qonjugate to a product k i~-0

where ~or some m each a~, b~, c e P,~ (1) each ai¢ Pro-1 (2) each b~ ~ R,~, PROOF.Clearly Wis conjugate to a product vt, v ¯ P~, v ~ Rs for some s. Choose the conjugate of Win this form such that s is minimal and the length of v as an element of Ps-~ *riB-1 Rs = P~ is minimal for this s. If we change each R~ factor h in the normal form of v by t-l¢-~(h)t, we will obtain a product satisfying the required conditions for m-- s - 1. PROOF.(Theorem 5.4.1) We now give the proof of the main result that is that the Kervaire conjecture is true for A a torsion-free group. In particular we assume G = A. < t > IN(R) and assume that each letter of R has infinite order in A. Consider R then as an equation W-- 1 with each coefficient having infinite order in A. Wecan assume that t has exponent sum 1 in Wfor otherwise G has a non-trivial cyclic quotient. Further from Lemma5.4.5 we can assume that W has form 5.4.1. The equation W-- 1 given in form 5.4.1 can be considered as an equation over H = P,~ = Ao * ... * Am the free product of m ÷ 1 copies of A. Let P = P,~_~ and ¢(P) -- R,~. Then from Lemma5.4.4 the system equations over H (5.4.2) W = 1, (t-~pt = ¢(p),p has a solution t in an overgroup H of H. The conditions of Lemma5.4.4 hold since if u ¯ (B ¯ C) \ B and u has infinite order, then < u, u>*B. Clearly W(t) = 1. Hence H is an overgroup of A containing a solution of the sytem 5.4.2 and therefore the map A -~ G is a monomorphism. It follows that G is non-trivial. Notice that the proof contained the following Freiheitssatz. COROLLARY 5.4.2. Suppose A is torsion-free and the exponent sum of t in R ¯ A, < t > is 4-1, then A is a FHS factor ofG = (A* < t >)/N(R) Weclose by noting that Klyachko’s paper contained several other results on equations over torsion-free groups. In particular., THEOREM 5.4.6. The sy~em of equations Iv(t), G] -- over a group G is solvable over G if every coefficient of v(t) has infinite order - in particular if G is torsion-free.

CHAPTER VI ONE-RELATOR

PRODUCTS

6.1 One-Relator

OF CYCLICS

Products of Cyclics

Wenow turn to the main focus of these notes - one-relator products of cyclics, their linearity properties and their generalizations. A one-relator product of cyclics is a one-relator product where each factor is a cyclic (possibly finite) group. Thus these are groups with presentations of the form (6.1.1)

V ~-~

Xl,...~xn,

.x ~1 -~- .... 1

x~,~=S m~ 1>

where n >_ 2, e~ -~ 0 or ei >_ 2 for i = 1, ..., n, S is a non-trivial cyclically reduced word, not a proper powerin the free product of cyclics on Xl, ..., xn and m _> 1. If m _> 2 then the relator R = S"~ is a proper power. As was the situation with one-relator groups and general one-relator products, the proper power case is somewhat easier to handle. If each e~ = 0, then G is just a one-relator group and hence the class of one-relator products of cyclics clearly generalizes the class of one-relator groups. Further each finitely generated co-compact Fuchsian group falls into this class via the Poincare presentation. Therefore the class of onerelator products of cyclics provides a natural algebraic generalization of Fuchsian groups. The next few chapters are primarily concerned with the following general question: General Program on One-Relator Product of Cyclics: Given an algebraic property true in all Fuchsian groups, must it be true in all onerelator products of cyclics. Further, if not, what specific conditions on the relator (if any ) are sufficient so that it will hold. In regard to this general program we concentrate on the linearity properties exhibited also by one-relator groups. In particular, the Tits Alternative, the virtual torsion free property, SQ-universality, amalgamdecompositions and separability properties. These will be taken up in subsequent chapters. In this chapter we consider the representation theoretic methods used in handling the class of one-relator products of cyclics. 137

138


Since we allow torsion, the results on local indicability of factors do not apply. Further we do not restrict the relator to have a power higher than 4. Thus we go beyond what we have previously discussed. The basic technique to handle these groups is to consider certain special representations (defined in the next section) into some suitably chosen matrix group. For most applications this object group is PSL2(C_.). In the next section we examine these special representations, which are in a sense "close" to faithful. It is from the properties of the image group coupled with the special properties of these representations that our results will be derived. Of special interest is the case when n -- 2. The groups then have the form (6.1.2)

G =< a, b; an = bq .= Srn = 1 >

where as before S is cyclically reduced, not a proper power in the free product of cyclics on a, b involving both a and b. These groups are called the generalized triangle groups and clearly generalize the ordinary triangle groups (see Chapter 4). Wewill discuss them in detail in the next chapter. Historically the interest in one-relator products of cyclics, as separate from general one-relator products, arose out of a result of G.Baumslag, Morgan and Shalen [B-M-S] showing that if m _> 2 the resulting generalized triangle groups are always non-trivial. Their result grew out of a topological question which reduced to the group theoretical question of whether it is possible to kill a free product of two cyclic groups with a single high-powered relation. If the relator is a proper power, m _> 2, then almost all the linearity questions have a positive solution. Howeverif m = 1 and there is a generator of finite order { not all ei = 0} then almost nothing is true in general. (see [FH-R 1],[F-L-R 1,2]). Therefore specific restrictions must be imposed on the relator. The strongest algebraic analog (via presentations) of a Fuchsian group seems to be a group of F-type. This is a group with a presentation of the form (6.1.3)

e--< Xl, ...,xn;x~ ~" .....

Xenn = 1, UV = 1 >

where n _> 2,e~ = 0 or e~ _> 2, i = 1,...,n,1 _< p _< n-1,U = V(Xl, ...,Xp), V = V(Xp+l,..., Xn) with U a cyclically reduced word in the free product on xl, ...,xp which is of infinite order and V a cyclically reduced word in the free product on Xp+l,...,x,~ which is of infinite order. These groups are the analog for the class of one-relator products of cyclics of the cyclically pinched one-relator groups and satisfy manyof the algebraic properties of Fuchsian groups. Welook at these in detail in Chapter 8.

6.2 ESSENTIAL ANDESSENTIALLYFAITHFULREPRESENTATIONS 139 6.2 Essential

and Essentially

Faithful

Representations

The special representations needed in the study of one-relator of cyclics are defined in the following way. Suppose G is a group with the presentation (6.2.1)

G =< al,

..-,

an; a~ 1 .....

a~’~ ---- R~1 .....

products

R~~ = 1 >

where ei = 0 or e~ _> 2 for i = 1, ..,n, mj >_ 1 for j = 1,...,k and each R~ is a cyclically reduced word in the free product of the cyclic groups < al > ,..., < a,~ > of syllable length at least two. A representation p : G -~ Linear Group over a field of characteristic zero is an essential representation if for each i = 1, ..., n, p(a~) has infinite order if e~ = 0 or exact order e~ if ei >_ 2 and for each j = 1, ..., k, p(Rj) has order mj. For any group G a linear representation p over a field of characteristic zero is an essentially faithful representation if p is finite dimensional with torsion-free kernel. In much of the work on one-relator products of cyclics and related groups certain essential representations will be shownto be essentially faithful. For a finite presentation < X; R > the deficiency d is the difference between the number of generators and number of relators. For presentations of the form 6.2.1 we further define the extended deficiency d* to be n-k. Note that if the orders of the generators (al, ...,aN} and the additional relators {R~,..., R~}are exactly as they appear in the presentation, then an essentially faithful representation must be essential. On the other hand if the conjugacy classes of torsion elements are precisely given by the powers of the generators (if of finite order) or the powersof the other relators, then an essential representation is essentially faithful. In this section we consider some basic results on groups which admit essential and essentially faithful representations. Our techniques then will evolve into showing that a particular class of groups of interest do admit such representations. The first results are straightforward and handle the situation where the group G is virtually torsion-free. THEOREM 6.2.1. Let G be a finitely generated group. Then G admits an essentially faithful representation if and only if G is virtually torsion-free. PROOF.Suppose G is finitely generated and p : G --~ Linear Group is an essentially faithful representation. Since G is finitely generated, p(G) is a finitely generated linear group. From a result of Selberg [Se] p(G) is then virtually torsion-free. Let H be a torsion-free normal subgroup of p(G) of finite index and let H* be the pull-back in G. H* has finite index in G. If g ~ 1 has finite order, then p(g) has exactly the same order since p is an essentially faithful representation. Therefore g cannot be in H* since its

140

ALGEBRA|CGENERALIZATIONSOF DISCRETE GROUPS

image would then be an element of finite order in the torsion-free group H. Therefore H* must be torsion-free and G is virtually torsion-free. Conversely suppose G is virtually torsion-free. As explained above G must then contain a torsion-free normal subgroup H* of finite index. Choose a faithful finite dimensional representation p* of the finite group G/H*. The composition of this with the natural homomorphism from G to G/H*will give the desired representation. A modification of the proof of the Ree-Mendelsohn Theorem (Theorem 3.4.3) shows that any one-relator group with torsion must admit an essential representation. Coupling this with the torsion theorem for one-relator groups yields a proof of the following corollary due to Fischer, Karrass and Solitar (Theorem 3.4.4.) COROLLARY 6.2.1.

Any one-relator

group with torsion

is virtually

torsion-free. PROOF.Let G =< X; R~ > be a one-relator group with torsion so that R is not a proper power in the free group on X and n _> 2. From the ReeMendelsohn proof G admits an essential representation p : G ---* PSL2(C). Hence p(R) has exact order n. By the torsion theorem in one-relator groups each element of finite order in G is conjugate to a power of R. Since the image of R has exact order n the image of any power of R has exactly the same order in p(G) as it had in G and hence any element of finite order has its order preserved under p. Therefore no non-trivial element of finite order can be in the kernel of p. Therefore p is essentially faithful and G is virtually torsion-free.. For groups with presentation 6.2.1 and where each mj _> 2 and where the extended deficiency _> 3, the next, result shows that admitting an essential representation implies the existence of a subgroup of finite index mapping onto a non-abelian free group. This in turn implies SQ-universality. This result extends certain deficiency results of B.Baumslagand Pride [B.B.-P 2]. (G.Baumslag,Morgan and Shalen [B-M-S] and R.Thomas IT 1] proved similar results using different methods. Wepoint out that although this conclusion extends these other results, the hypotheses are somewhatstronger. Recall that a group G is SQ-universal if every countable group can be embedded in a quotient of G. If G has a SQ-universal quotient, then G is itself SQ-universal and similarly if G has a SQ-universal subgroup of finite index then G is itself SQ-universal (see[L-S]). From the work Higman,Neumannand Neumanna free group of rank 2 and hence any nonabelian free subgroup is SQ-universal. Weneed the following result due to B.Baumslag and S. Pride [B.B.-P 2]

6.2 ESSENTIALANDESSENTIALLYFAITHFULREPRESENTATIONS141 THEOREM 6.2.2. Let G be a finitely presented group of deficiency greater than or equal to 2. Then there exists a subgroup of finite index mapping onto a Tree group of rank 2. In particular G is SQ-universal. THEOREM 6.2.3. Suppose G is a finitely tation of the form (6.2.1)

G =

with each Ri cyclically reduced in the Tree product of the cyclic groups < al >, ..., < a,~ >, of syllable length at least 2 and for i = 1, ...,n,e~ = 0 or e~ > 2. Suppose each m~ > 2 and that the extended deficiency d* = n - k > 3. Then if G admits an essential representation, G must contain a subgroup of finite index which maps onto a non-abelian free subgroup. In particular G is SQ-universal and contains a non-abelian Tree subgroup. The result is still true if d* ---- 2 but mj > 2 for somej or ei = 0 for somei or ei > 2 ]:or somei. PROOF.Suppose G has a presentation of form 6.2.1 as in the statement of the theorem. If a quotient of G contains a subgroup of finite index which maps onto a non-abelian free subgroup, then the group G does also. Hence we can, without loss of generality, assumethat ei _> 2 for i = 1, ..., n passing to a quotient if necessary. Suppose that mj _> 2 for j -- 1, ..., k and that G admits an essential representation p into a linear group V (over a field of characteristic zero) Therefore from Selberg’s theorem on finitely generated subgroups of linear groups [Se] there exists a normal torsion-free subgroup H of finite index in p(G). Thus the composition of maps ( where r is the canonical map) G --~P p(a) -~ p(a)/H gives a ¢ of G onto a finite group. 1 ..... a,~ * -- 1 > be the free product of the Let X =< al,...,a~;a~ cyclic groups < al >, ..., < an >. There is a canonical epimorphism/~ from X onto G. Wetherefore have the sequence X --~ G -~¢ p(G)/H Let Y -- ker(¢ o fl). Then Y is a normal subgroup of finite index X and Y is torsion-free. Since X is a free product of cyclics and Y is torsion-free, it follows that Y is a free group of finite rank r. Suppose [X : Y[ = j. Regard X as a Fuchsian group with finite hyperbolic area #(X). Every finitely generated free product of two or more cyclics can

142


faithfully represented as such a Fuchsian group if it is not isomorphic to the infinite dihedral group. From the Riemann- Hurwitz formula we have that j#(X) = #(r) where #(Y) -- (2~r)(r and #(X) -- (2~r)((n - 1) - (~

~- ~)).

Equating these expressions we obtain r=l-j(---+

.....

+-- - (n- 1)). en

G is obtained from X by adjoining the relations R~I, ..., R~ X/K, where K is the normal closure of R~ ..., Rk . Since K is contained in Y, the quotient Y/K can be considered as a subgroup of finite index in G. Applying the Reidemeister- Schreier process or repeated applications of Corollary 3 in [B-M-S] Y/K can be defined on r generators subject to (j/m1) + .... ÷ (j/mk) relations. The deficiency d of this presentation for Y/K is then J d--r J Since each ei >_ 2 and each mj >_ 2 we have the inequality d_>

l÷j(

n-k

2 1)

By assumption the extended deficiency n-k _> 3 and hence the deficiency d of the above presentation is at least 2. From the result of Baumslagand Pride, Theorem 6.2.2, Y/K, and hence G, has a subgroup of finite index mapping onto a free group of rank 2. Therefore Y/K is SQ-universal and since this has finite index in G, G is also SQ-universal. Further an SQuniversal group must contain a non-abelian free subgroup. This completes the main parts of the theorem. If the extended deficiency is 2 but not all mj -- 2 or not all e~ -- 2, then the inequality above becomes proper - that is: n-k d>l÷( ~ 1)=1. Hence d > 1 and since d is an integer d _> 2 and the proof goes through as before.

6.2 ESSENTIALANDESSENTIALLYFAITHFULREPRESENTATIONS143 This type of argument using deficiency and the Baumslag-Pride result was used by G.Baumslag, Morgan and Shalen in their study of generalized triangle groups. Wereturn to this in the next chapter. Theorem6.2.3 is closely related to the following result of R.Thomas[T1] proven by looking at the Cayley graph. THEOREM 6.2.4. (6.2.3)

Let G be a group with a presentation

G --(

Xl,...,xn,

W~ .....

Wrmr-~ l >

where W~axe non-trivial words in the generators xl,...,x~ and ml,..,mr are integers _> 2. Suppose that W~, ..., Wr have exact orders m~, ..., mr respectively in G. Let a = (l/m1) + ... q- (1/mr). H G is finite then a - n÷ 1 > 0 and IGI >_ 1/(a - nq- 1). In particular this is true fig admits an essential representation. PROOF.If F is a graph, a walk in F consists of an alternating sequence Po, 11,p1,12, ...,p,~-~,In,p,~ of points pi and edges li such that li is incident with p~_l and pi. The walk is closed if P0 = P~ and open otherwise. If there is a closed walk with n > 2 distinct points Po, ...,Pn, then the walk is called a cycle. The cycles of F under the operation of symmetric difference span a vector space over the field of two elements knownas the cycle space of F and if F is a connected graph with p points and q edges then this vector space has dimension q - p + 1. Werefer to the book by Harary. [Har] for these results and terminology. Nowlet G be a group with finite presentation < X; R > where X and R are finite sets of respective orders n and r. Let Y denote the elements of X together with their inverses, elements of order 2 occurring twice so that [Y[ = 2n. The Cayley graph of G is defined as follows. The vertices of F are the elements of G and there is an edge {a, b} in F if and only if the elements a, b E G satisfy ay = b for some y E Y. The edges of F are undirected and each pair of points a, b ~ F is connected by as many edges as there are elements y ~ Y with ay = b; thus there may well be multiple edges in F. A trivial generator would give rise to a pair of loops at each point of F. Nowsuppose that G is a finite group of order m. The Cayley graph as defined above then has m points and mn edges and is regular of degree 2n. If g ~ G and w is a word in the elements of Y which equals the identity in G, then the edges of F starting at the point g and corresponding to the elements of w form a closed walk in F. Thus each relator gives rise to m closed walks in F although there may well be repetitions amongthese. From [Har] we have that since F is connected with m points and mn edges, the cycle rank of F equals mn - m + 1 {[Har], corollary 4.5}. Since

144


each relator in G can be defined from R, we have that the r closed walks at each point of F corresponding to the elements of R span the cycle space of Fandhencethatmn-m+l 0 and m > 1/(a - n + 1) as required. If G admits an essential representation, then the orders of the elements Wi are exactly mi. Wenote that it is not always possible to find an essential representation. Consider the group G defined by < a, b; a3 = b12 = (ab) 2 = (a-lb) 13 = 1 >. If G had an essential representation~ it would be infinite from the last theorem. However a-~b = aab = ab-la -1 so that (a-lb) 12 = ab-12a-1 = 1. It follows that a-Xb = 1 so that a = b and hence a3 = aa = 1 and so a = 1. Thus G is the trivial group. 6.3 Essential

Representations

of One-Relator Products

Wenow show that every one-relator product of cyclics with proper power relator admits an essential representation into PSLg.(C). From this we can deduce easily that these groups are never trivial and in most cases infinite. Our main result is the following concerning one-relator products whose factors admit faithful representations into PSL~(C). THEOREM 6.3.1. Let G = (A * B)/N(R "~) where A and B are groups admitting faithful representations into PSL2 (C) and where R is a cyclically reduced word in the free product A * B of leng~h at least 2 and m >_ 2. Then G admits a representation p : G --* PSL2(C) such tha~ PlA and PlB are faithful and p(R) has order m. In particular A -~ G and B -~ G are injective, that is, the b-~eiheitssatz holds. Before proving this result we will prove a special case of it (Theorem 6.3.2) where each factor is a free product of cyclics. The proof of this theorem is based on the technique of Ree and Mendelsohn (Theorem 3.4.3)

6.3 ESSENTIAL REPRESENTATIONS OF ONE-RELATORPRODUCTS 145 and is a refinement of a technique used by Baumslag, Morgan and Shalen [B-M-S] who used it to prove that all generalized triangle groups are nontrivial. Wewill return to the Baumslag, Morgan and Shalen results in the next chapter when we discuss the generalized triangle groups. Wefirst note that Theorem 6.3.1 has the following corollary which greatly extends the classes of factors for which a Freiheitssatz holds in a one-relator product. COROLLARY 6.3.1¯ Let G = (A * B)/N(R "~) where R is a cyclically reduced word in the flee product A * B of length at least 2 and m >_ 2. Then the t~eiheitssatz holds whenever both A and B are in any of the following classes of groups: (1) Fuchsian groups (2) Kleinian groups (3) Cyclic groups (4) Surface groups (5) Free abelian groups of rank 2 (6) Free metabelian groups of rank 2 (7) Cyclically pinched one-relator groups with malnormal cyclic amalgamated subgroups in both factors (8) Mixed combinations of any of the above The corollary is a consequence of the fact that any group in the classes cited admit faithful representations in PSL2 (C). THEOREM 6.3.2. Suppose G is a one-relator product of cyclics proper power relator. That is, G has a presentation V -~< al,...,an;a~’

= a~2 .....

with

a~"~ = Rm(al,...,an) -~- 1

with n >_2, m >_ 2,ei = 0 or e~ >_2 for i -- 1,...,n and R(al,...,a,~) is cyclically reduced word in the ~ree product on al,..., a,, which involves all the generators. Then G admits an essenital representation into PSL2(C) which is faithful on the free product on a~, ..., a,~-l. In particular ~ 1 al~...~an-1 that is < al, ...,a,~-i

~--~ al,a ¯ el -- 1 > *..-* ( an-l; ae"-I -- 1 > > is the free product ofcyclics of the obvious orders.

PROOF.(Theorem 6.3.2) From our results on Fuchsian groups (Chapter 4 ) we can choose projective matrices A1, A2, ..., A,_I E PSL2(C)such that the subgroup generated by these is the appropriate free product of cyclics. That is < A1,...,A,~_~

>-~< A1, Ai = 1 > *...*

< A,~_I;An_1 -~ 1 >

146


so that < A1, ...,An-1 > faithfully represents the free product of cyclics e~-~ ~ 1 >. ~ al, ...~ an-l; a1 ~ a2 e~~- ... = an-1 We will determine a projective matrix A,~ E PSL2(C) of order e~ so that the subgroup generated by A1, ...,A,~ provides a representation p : G -~ PSL~(C) with p(ai) = Ai for i = 1, ...,n and p(R) having order m. The representation will thus be essential. Further, since the image of < al, ..., a,~-i > will then be a free product of cyclics with each of maximal possible order, it follows that p restricted to the subgroup generated by al, ..., a,_l is faithful and this subgroup is the appropriate free product of cyclics. Choose

where t = 2cos(~/e,~) and w is to be determined. Since tr(A~) = t 2cos(~/en) we have A~- = 1 in PSL2(C). Consider the relator word R(al, ..., as). Substituting A~, A2, ..., A~ into R we obtain the projective matrix

wheref~, f2, f3, f4 are polynomialsin the coefficients of A1, As, ..., A,~. Considering w as the only unknown,f~, f~, fa, f4 are then polynomials in w. If tr(R(A~, ..., A~)) = f~ + is not a co nstant poly nomial in w , t hen the polynomial equation

+

= cos( /,n)

can be solved for w by the Fundamental Theorem of Algebra. For a particular solution wo in A~ we would then have R’~(A~, ..., A~) 1 because of the trace. Therefore the subgroup generated by A1,..., A,~ provides an essential representation p: G -~ PSLu(C~) under p(a~) = A~,i 1, ..., n. F~her by the choice of A~, ..., A~_~we have that al, .., a~_~ is the appropriate ~ product of cyclics and p restricted to the subgroup < a~..., a~-i > is faithful. What is le~ in order to complete the proof is to show that there is a choice of A~, ..., A~_~such that tr(R(A~, ..., ~)) is nonco~tant in Since R(a~, a:, ..., a~) is cyc~cally reduced, without loss of generafity we can ~ite the relator matrix R(A~, ..., A~) R(A~, A2,...,

A~) = B~A~~ B:...BkA~ ~

with B~ non-trivial words in A~, A2, ..., andl~t~<e~.

A~_~and hence non-trivial ~trices

6.3 ESSENTIAL REPRESENTATIONS OF ONE-RELATORPRODUCTS147 1

has lower left entry c + ya - yd - y2b which is non-zero for all but finitely many choices of y. Similarly for B2, ...,Bk. Then by conjugating all the B1,...,Bkbyasuitable(ly ~) if necessary, we can assume that the lower left entries of B1,..., Bk are non-zero. By considering the diagonalization of An we see that A~ has the form

where gl,g2,ga, g4 are polynomials in w of respective degrees 1, 2, 0, 1. It follows that each factor B~A~in the expression for the relator R must have the form

where hi, h2, ha, h4 are also polynomials in w. Since each B~ has non-zero lower left entry, we must have that the degree of h4 is exactly 2 while the degrees of h~, h2, ha are be

Since m >_ 2 in the definition of a generahzed triangle group, the corollary follows directly since the non-trivial cyclic group on a injects into G. Baumslag, Morgan and Shalen proved more, depending on the values of p, q, m. Wewill return to this in the next chapter. Wenote that Corollary 6.3.2 and a version of Theorem 6.3.2 were proved independently by S.Boyer /By], who used essential representations into SU(2) rather than into PSL~(C). By extending the proof of Theorem 6.3.2 we obtain a proof of Theorem 6.3.1.

148


PROOF.(of Theorem 6.3.1) Let G = (A * B)/N(R "~) where A and B are groups admitting faithful representations into PSL~(C,) and where R is a cyclically reduced word in the free product A * B of length at least 2 and m _> 2. Write the relator as R = albl...akbk

with ai E A, bi ~ B.

Since R is cyclically reduced of length _> 2, we may assume that ai ~ 1 and bi ~ 1 for all i. Choose faithful representations PA : A --} PSL~(C) and PB : B PSLu(C,) such that (after a suitable conjugation if necessary), pA(ai)

= ( : x~ wit hxi~0

pB(bi) ---- ( ** Yi wit hyi~0 for all i -- 1,..., k. w E C let p] denote the representation

/ PA conjugated by (~

That is

Considering w as a variable define T(w) = tr(p~4 (al)pB (bl)...p~4 (ak)pB (bk)). Then T(w) is a polynomial in w. As in the proof of Theorem 6.3.2 the coefficient of wek is equal to ±xlyl...xkyk and is therefore non-zero because of the choices of #A and pB. Then as in the proof of Theorem 6.3.2 there exists a w0 with T(Wo) = 2cos(r/m). Now define the representation p : G -~ PSL:(C) by PlA = #A and PiB = pB. Then p(R) has trace T(wo) - 2 cos(r/m) and hence p(R) has order m. Furt~h~er PlA = p]O is faithful on A and PIB = PB is faithful on B and therefore p is the desired representation. Since a one-relator product of cyclics with proper power relator has now been shown to admit an essential representation, the results, appropriately phrased, of section 6.2 can be applied. This will be the focus of much of the remainder of these notes. For the rest of this chapter we look at several other results on essential representations.

6.4 FREIHEITSSATZ FOR ONF~-RELATORAMALGAMATED PRODUCTS149 6.4 Freiheitssatz

for

One-Relator

Amalgamated Products

In analogy with the one-relator products we define a one-relator gamated product as a group of the form

amal-

G = (A *c B)/N(R) where R is a non-trivial, cyclically reduced element of length at least two in the amalgamated product A *c B. In this section we generalize the results and techniques of the previous two sections to obtain a Freiheitssatz for a certain class of one-relator amalgamated products. In particular suppose that G = (A *c B)/N(S "~) with C cyclic and m >_ 2. If (A *c B) admits a complex two-dimensional representation which is faithful on A and B, then under certain conditions, this representation can be extended to a representation p : G -~ PSL2(C)which is also faithful on A and B. It will then follow that A a.nd B both inject into G. Our proof is related to ~ result of Vinberg [V]. Once we establish this Freiheitssatz, we can prove a series of results about these one-relator amalgamated products mirroring some of the previous material on one-relator products. If A, B E PSL2(C), then we say that the pair {A, B} is irreducible if A, B, regarded as linear fractional transforlnations; have no commonfixed point, that is, tr [A, B] ¢ 2. The main result of this section is the following which gives a Freiheitssatz for certain one-relator amalgamatedproducts. THEOREM 6.4.1. { The Freiheitssatz } Suppose H = H1 *A H2 with A =< a > cyctic. Let R ~ H\A bc given in a reduced form R = alb~...a~b~ with k >_ 1 and ai ~ Hi \ A, bi ~ 2 \ A for i = 1, ...,k. As sume that th ere exists a representation ~ : H -~ PSL2(C) such ~hat ¢IH~ and ¢lt~ are faithful and the pairs {¢(a~), ¢(a)} and (¢(b~), ¢(a)} irre ducible for i =1, ..., k. Then the group G = H/N(R~), m >_ 2, admits a representation p : G -~ PSL2(C) such that Ht -~ G ~-~ PSL2(C) and H2 -~ G ~-~ PSL2(C) are faithE~1 and p(R) has order m. In particular Ht -~ G nnd H2 -~ G are injective. PROOF.The proof generalizes somewhat the proofs of Theorem 6.3.1 and Theorem 6.3.2. Let ¢ : H -~ PSL2(C) be the given representation of H such that ¢1H1and ¢IH~ are faithful and the pairs {¢(a~), ¢(a)} {¢(b~), ¢(a)} are irreducible for i = 1, ..., We may assume that ¢(a) has the form ¢(a)=

0)

s_1 or

150

ALGEBRAICGENERALIZATIONSOF DISCRETE GROUPS ¢(a) ---- (~

Suppose first that ¢(a)

1

(: 0)

s_ 1 and let

-1 T---

0 t

with t a variable whose value in C is to be determined. Define (1) p(hl) ¢(hl) fo r hi e H1 and (2) p(h2) --- T¢(h2)T-~ for h2 E H2. Since T commutes with ¢(a) for any t, the map p : G -~ PSL~(C) will define a representation with the desired properties if there exists a value of t such that p(R) has order m. Recall, as in the last section, that a complex projective matrix A in PSLu(C) will have finite order m >_ 2 if trA = :]=2cos(~r / m). As in the statement of the theorem assume R = albl...akbk with k _> 1 and a~ E H1 \A,b~ E H2 \A for i = 1,...,k, and assume that the pairs {¢(a,), ¢(a)} and {¢(b~), ¢(a)} are irreducible for i = 1, ..., k. f(t) = tr(¢(a~)T¢(bi)T -1 .... ¢(ak)T¢(bk)T-~). f(t) is a Laurent polynomial in t of degree 2k in both t and t -1. The coefficients of t 2k and t -2k are non-zero because the pairs {¢(a~), ¢(a)} {¢(b~), ¢(a)} are irreducible for i -- 1, ..., k. Therefore by the fundamental theorem of algebra there exists a to with f(to) = 2cos(n/m). With this choice of to we have tr(p(R)) -- 2cos(~r/m) and thus p(R) has order m. Therefore p is a representation with the desired properties. Now assume ¢(a)

. In this

case define

T=0

again t a variable. Again t commuteswith ¢(a) and the proof goes through as above with tr(p(R)) polynomial in t o f degree 2k (se e the proof of Theorem6.3.1) giving the desired representation. Exactly the same proof as that used for Theorem 6.4.1 can be employed to prove the following two generalizations. THEOREM 6.4.2. Le$ H = H1 *A H2 with A = H~ AH~. Let R E H\ be given in reduced form R = albl ...akbk with k ~_ 1 and a~ ~ H1 \ A, b~ H2 \ A for i = 1, .., k. Assume that ~here exists a representation ¢ : PSL2(C) with the properties that ¢IH~and ¢lu~ are faithful, for each a ~ A ~here exists

% ~ C, such that

¢(a) 0= (% ~ ~ or for

eachaeAthere

6.4 FREIHEITSSATZ FOR ONE-RELATORAMALGAMATED PRODUCTS151

each a E A, a ¢ 1, the pairs (¢(ai), ¢(a)} and (¢(bi), ¢(a)} irre ducible. Then the group G = H/N(R’*), ra >_ 2, admits a representation p : PSL2(C) such that H1 --* G ~-L PSL2(C) and H2 -~ G P-~ PSL2(C) are faith[ul and p(R) has order m. In particular H1 --~ G and H~ --* G are injective. THEOREM 6.4.3. Let H -- HI *A H2 with A = H~ NH~. Let R ~ H\ be g~ven in reduced [orm R = alb~...akbk with k >_ 1 and H2 \ A [or i -- 1, ..., k. Assumethat there exists a representation PSL2 (C) with the properties that ¢]H1 and ¢1H2are [aith[ul, [or each a ~ A

there

existsx~O,y~O,

such that

¢(a~) ----

xi

’

Yi * "

Then the group G -= H/N(R’~), m >_ 2, admits a representation p : PSL2(C) such that H~ -~ G P-~ PSL2(C) and H2 --+ G P-~ PSL2(C) are [aithful and p(R) has order m. In particular H1 -~ G and H2 -+ G are injective. The proof of Theorem6.4.2 is exactly the same as the proof of Theorem 6.4.1. The conditions on the images of the elements of the amalgamated subgroup again force the trace polynomial to be non-constant. In Theorem 6.4.3 the form of the images of the elements which appear in the relator force these images to be irreducible in pairs with the images of the nontrivial elements of the amalgamated subgroup. Again this forces the trace polynomial to be non-constant and the proof goes through as in theorem 6.4.1. Weclose this section with a general question related to the above. QUESTION.Suppose H = H~ * a H2 with A = H~ ~ H2. Suppose there exist [aithf~l representations ¢i : Hi -~ PSL2(C) [or i 1, 2. Under what conditions on H~, H2, A may we construct a representation ¢ : H --~ PSL2(C) such that ¢IH~ and ¢1~ are Wenote that it is possible to construct such a representation in the following cases: (1) H~,H~are free and A is cyclic. (2) H1, H2 are free products of cyclics and A is cyclic. (3) Each Hi, i ---- 1,2 is either free, a free product of cyclics or a nonelementary Fuchsian group with positive dimensional Teichmuller space and A is cyclic. (4) A is finite cyclic

152


A general criterion to answer the question is given by P. Shalen in [Sh] and generalized by S.Litz to GL2((2) [Lit]. Finally we note that if H1, H2 embedinto PSL2 (C) then the free product H1 * H~ also embeds into PSL~.(C). In later sections we will return to the representation of one-relator amalgamated products to consider certain linearity properties in such groups. 6.5 Faithful

Representations

of One-Relator

Products

Suppose that G = (A * B)/N(R) is a one-relator product where the relator R is a non-trivial word of syllable length at least two in the free product A * B. Wehave seen that if A and B admit faithful representations into PSL2 (C) and R is a proper power, then G admits an essential representation into PSL2 (C). This result and the techniques of the previous sections in this chapter then suggest the following two questions. (1) Under what conditions on the relator R will the one-relator product G admit a faithful two-dimensional complex representation? (2) Under what conditions on the relator R will the one-relator product G admit a two-dimensional complex representation which is faithful with a discrete image? In this section we describe some partial answers to these questions. Wefirst survey what is knownabout faithful discrete representations into PSL~ ((2). Wethen prove a necessary condition on the relator, based on a classical result of Magnus, for a two generator one-relator group to admit a faithful discrete representation into PSL~(C). Using cancellation arguments in free groups we then describe conditions on words R(x, y) in a free group on two generators x, y which allow them to meet the above condition. Recall from Chapter 4 that a finitely generated non-elementary Fuchsian group F has a Poincare presentation of the form (2.1) F -----< el,..,ep,

h~,..,h~,a~,b~,...,ag,bg;e~’ = 1, i ----- 1,..,p,R = 1 >

where R -- e~..%hl..h~[al,bl]...[ag,b~] andp> O,t > O,g > O, pq-t q-g > O, and m~_> 2 for i ~ 1, ...p. Hence as mentioned Fuchsian groups fall into the class of one-relator products of cyclics. The Euler Characteristic of F is given by x(F) = -I~(F) where p(F) -- 2g - 2 -b t q- ~ (1 1/ mi). If p(F) > 0 then 2~r#(F) represents the hyperbolic area of a fundamental polygon for F and a presentation of the above form can be represented by an actual Fuchsian group and hence admits a faithful~ discrete representation. Recall that a group of F-Type is a one-relator product of cyclics,with a presentation of the form (2.2)

G =< al,. .... ,an, ¯ ael1 .........

a~n -~ 1, UV= 1 >

6.5 FAITHFUL REPRESENTATIONSOF ONE-RELATORPRODUCTS 153 wheren _> 2,ei --- 0 or e~ _> 2, 1 _< p _< n- 1, U = U(al, .., ap) is a cyclically reduced word in the free product on al, ..., ap which is of infinite order and V -~ V(ap+l, ...,an) is a cyclically reduced word in the free product on av+l, ..., a,~ whichis of infinite order. In Chapter 8 we will prove that if neither U nor V is a proper power in the free product on the generators which they involve, then the group G admits a faithful two-dimensional complex representation. This uses a result of P.Shalen [Sh] from which we can deduce that cyclically pinched one-relator groups with malnormal cyclic amalgamated subgroups in both factors admit faithful two-dimensional complex representations. Whether the image group is discrete or not depends on the further exact structure of R. If both U and V axe proper powers then we will see that there is no faithful representation. In general a faithful, discrete representation p : G --~ PSL2(C)of a group G is said to be of finite volttme if Ha/p(G) has finite volume where Ha is hyperbolic 3-space. Helling, Kim and Mermicke [H-K-M]have shown that if m _> 4 the group G =< a, b; am = b2 = ((a-lb)2(ab)a) 2 = 1 > has a faithful, discrete image in PSL2(C). Further Helling, Mennicke and Vinberg [H-K-V] show that the groups G :- with at least one of k, l, m-- 0 or k, l, m>_ 2 and with k _ 1. In connection with these results it shown that the groups G --< a,b;a a = b3 = (aba-lbab-~) ~ = 1 > and G =< a, b; a3 = b4 = (aba-~bab-1) 2 = 1 > axe arithmetic. In a similar manner Hagelberg [Ha] and Hagelberg, Maclachlan and Rosenberger [HaMc-R] showed that the groups G =< a, b; aa = b~ = [a, b] "~ = 1 > with at least one of k, t, m= 0 or k, t, m_> 2, and k with k,t,m >_ 2 and k 1 and (l/t) + (l/t) + except for (k, t, ra) = (2, 2, m) and (2,3,2). In addition Hagelberg, Maclachlan and Rosenberger proved that for (l/k) + (l/k) + (l/m) _> (1/t)+(1/t)+(1/m) _> 1 the groups G = with (k,t,m) (3, 3, 3)~(3, 4, 2 ) o r ( 4,4,2) are axit hmetic. If (k,t ,m) = (3,3 ,3) the group G is a subgroup of index four in the Bianchi group PSLu(O3) while if (k, t, m) -- (3, 4, 2) or (4, 2), G is comme nsurable with the Picar d group PSL2(O~). On the other hand if G --< a, b; ak = b~ -- [a, b]"~ 1 > with at least one o~ k,t,m ---- 0 or k,t,m >_ 2, and k _< t, G has a faithful, discrete representation into PSL~(C) if and only if (k, t, m) (~, 3, ~), (~, ~, ~), (3, 3, ~), (~, The above examples seem to lead to a general necessary condition for

154


a generalized triangle group G to have a faithful, discrete representation into PSL2(C.) of finite volume. In [Ha-Mc-Ro]there is the following partial result. THEOREM 6.5.1. Let G =< a,b;a p = bq -~ Rm(a,b) = 1 > with p : or p > 2, q = 0 or q > 2 and m > 2 and R(a, b) a cyclically reduced word, not a proper power, in the free product on a, b which involves both a and b. Suppose one of the following holds: (1) m_>4 (2) m = 3 and the word R(a, b) does not involve a letter (with respect to the free product on a and b) oT order 2. Suppose/ur~her that G has a Taithful, discrete representation into PSL2 (C) of finite volume. Then p > 2,q > 2 and (l/p) + (l/q) + (l/m) > 1. A generalization of this theorem would be of great interest. In the special case of finitely generated one-relator groups we can directly obtain such a generalization using results of ChiswelI[Ch] and Ratcliffe IRa] on Euler characteristic. THEOREM 6.5.2. Let G =< al ..., n,R (al, ..., an) 1 > with n 2, m _> 1 and R(al, ..., an) a cyclically reduced word, not a proper power in the free group on al, ...,an involving all the generators. Suppose that G has a TaithIul, discrete representation into PSL2(C)oT finite volume, then n = 2 and m = 1 - that is, G is a torsion-free, two-generator, one-relator group. PROOF.From [Ch] G has an Euler characteristic given by x(G) = l-n+ (l/m). From IRa] if G has a faithful,discrete representation into PSL2(C) of finite volume then the Euler characteristic must be non-nesativ% hence n= 2 and m= 1. The result of theorem 6.5.2 leads us to consider the problem of classifying all the torsion-free two-generator one-relator groups which admit a faithful, discrete representation into PSL2(C) of finite vohtme. There are many known examples of such groups with this property, for instance G a, b; aba- l bab-1 = 1 >. Wenow consider the case of a two-generator one-relator group G with presentation (6.5.1)

G =< a, b; R’~(a, b) = 1

where R(a, b) is a non-trivial cyclically reduced word in the free group on a, b involving both a and b and m _> 1. in general there are no faithful representations into PSL2(C). For example it can be shown that the group

6.5 FAITHFUL REPRESENTATIONSOF ONE-RELATORPRODUCTS 155 H =< a, b; atbs = 1 > with s, t _> 2 has no faithful representation. However from certain special properties of complex projective matrices coupled with a property of conjugates in free groups due to Magnuswe get the following necessary condition for such a group to admit a faithful two-dimensional complex representation. This result was proved in a slightly different manner by Magnus [M 4]. THEOREM 6.5.3. Let G be a two-generator one-relator group with form (6.5.1) and suppose G is non-metabelian. If G admits a faithful representation into PSL2(C), then the relator R(a,b) must satisfy the property that the word R(a, b) is conjugate in the £ree group on a,b to the word R+l(a-l,b-1). PROOF.Assumethat there is a faithful representation p : G -~ PSL2(C) with a --* A, b -~ B. From the properties of complex projective matrices it is known that there is a projective matrix C with CAC-1 = A-~ and -~ = B-~. Therefore we must have that R’~(A-~, B-~) = 1. Because CBC p is faithful, we then must also have Rm(a-~,b -~) = 1 in G. Since G is a finitely generated linear group it is residually finite and Hopfian and therefore the mapa --~ a-~, b --~ b-1 defines an automorphismof G which is induced by a Nielsen transformation. From a result of Magnus{see Chapter 3} R(a, b) must be conjugate in the free group to R+l(a-1, b-l). Recall that in a free group F if X~n is conjugate to Y±’~, then X is conjugate to Y±I. Wenote that Theorem 6.5.3 does not hold ifG is metabelian. The group G(n) = where n >_ 2 has a faithful representation into PSL2(C) given by a-~A--

(0 0 ) (01 (v~)_ ~ ,b~

but bab-la -n is not conjugate to (b-la-lban) ±1 in the free group on a, b. From standard cancellation arguments in free groups we can further describe conditions on words R(x,y) in a free group on two generators x,y which allows them to be conjugate to R±~(x-l,y -~) and thus can be permissible relators in a two generator one-relator group which admits a faithful two dimensional complex representation. First, in what follows, we give some notation. Suppose F is a free group on a,b. If s,t -- 4-1 then U(aS,b t) means U(aS,b ~) = 1 or U(a~, bt) -~ a~’btl’...aSe’btl~ for some natural number r and with non-zero integers. If U(aS,bt) ?~ 1 then U-l(a~,b t) clearly means that -~. Analogously wc use the notation u-l(a s, bt) = b-~1~a-~...b-tl~a ~, U(b at), s, t = 4-1.

156


THEOREM 6.5.4. Suppose F is a free group on a, b and let 1 # U = U(a,b) be an elemen~ off or 1 ~ U = U(b,a) be an element If U =- U(a,b) assume that there is a V in F with V-1U(a,b)V U-t(a-l,b-1). If U = U(b,a) assume that there is a V in F V-1U(b,a)V = U-t(b-~,a-~). Then, possibly after replacing U by u-l(a-l,b -1) = U*(b,a) ifU = U(a,b) or U by U-l(b-l,a-1) = U*(a,b) iT U = U (b,a) - one of the following cases holds: (1) U = U(a, b) and a-*U(a, b)a* = U-l(a-l,b -1) for some non-zero integer t (2) U = U(a,b) = a~U~(b,a)bSa~U2(b,a)b q for some non-zero integers t,s,q and b-sUi(b,a)b ~ -- u~l(b-i,a -~) and b-qUu(b,a)b q = -1) u~l(b-l,a (3) U =- U(b,a) and b-tU(b,a)b ~ = U-i(b-i,a -~) for some non-zero integer t (4) U = U(b, a) = b~U~(a, b)aSb~U2(a, q for somenon-z ero integers t, s, q and a-sUl(a, b)a~ -- u~-l(a -1, 5-1) and a-qU2(a, b)aq --u~l(a-l,b-1). PROOF. Suppose, without loss of generality, that 1 ~ U = U(a,b) = a~’bA...a~rbl~,r >_ 1, with ei, fi non-zero integers. Let V = adtbkl ...ad"bk~a d’~+~with all ki non-zero integers, di a non-zero integer for i = 2, .., n and dl and d,,+l integers. Let L denote free product length in the free product decomposition F =< a > ¯ of F. Cancellation arguments with respect to L give immediately that d~ ~ 0 if and only if d~+l ~ 0. If d~ = d~+t = 0 then V = b~ ...ad"b k" and since U(a, b)V = VU-~(a-~, b-~) we have U-~(a -~, b-~)V -~ = V-~U(a, b) and so bl~ a~...b l~ a~ b-k~...b -k~ -~ b-k~...b -k~ ae~ b~. ..a~b ~. Therefore here we may replace V(a,b) by U-l(a-~,b -~) = W(b,a) (and Y by -1 ) . H ence we may assume that d~ ~ 0 ~ d,~+l ¯ Then ~ = ad~bkl...ad-bk~ad~+~b$~ae,...b$~ae~ aelbl~...ae,bl~ ad~bk~...ad~bk-ad-+ with d~ ~ 0 ~ dn+l ¯ If n > r then from the above equation V = V1V2 with V1 = U(a,b) and U(a,b)V2 V2u-i(a -1, 5- 1). He re we canrepl ace V by V2. Therefore we may assume that r _> n. -~, b-~) the first of the two possibiliIfn = 0 then U(a, b)ad~ = ad~U-~(a ties. Nowassume that r _> n _> 1. If r -- n then again from the above equation V = U(a, b)a d~+~ and therefore U(a, b)a d~+~ = ad-+iU-~(a-1, b-i). Now assume that r > n _> 1. Then U(a,b) = Wl(a,b)W2(a,b) with V = W~(a,b)ad~+t,d~+~ -- e,+l and we get W2(a,b)W~(a,b)a d~+~ = ad"+~W~(a-i,b-~)W~(a-i,b-~). Let W~(a,b) = a~U~(b,a)b ~ and $. W2(a, b) ---- adU2(b, Wethen get Ul(b, a)bkd~+~-~ bk U~l(b -1, a-l)a e.

6.5 FAITHFUL REPRESENTATIONSOF ONE-RELATORPRODUCTS 157 Hence e -- dn+l and Ul(b, a)b k = baU~t(b-t, a -1) . Analagously d = d~+l and U2(b, a)bI ~- blUf l(b -~, a-t). This completes the second possibility in the theorem. After possibly exchanginga and b if U(a, b) is conjugate to U-1 (a-1, b-1) we are left with the situation 1 ~ U = U(a, b) = ~1 bIt . ..aerb Ir , r >_1, with ei, fi non-zero integers and U ( a, b) d =adU- 1 ( a-t, b- t) forsome non-zero integer d. If r = 1 then we have the equation a~lblta ~1 = a*lblta ~1 so U(a,b)a d = adU-t(a-l,b -1) with d = el. Nowlet r _> 2. Comparing the exponents in the above equation leads to the next result. THEOREM 6.5.5. Let 1 # U = U(a,b) = aelblt...a~rbI~,r _> 2, with ei, f~ non-zero integers. Suppose U(a, b)a d = adU-t(a -~, b-t) /=or some non-zero h~teger d. (1) Hr = 2s _> 2 is even then U(a, b) = ~t bit.., a ~ bI~ a~ +t bI. a~¯ bl. - ~ .. ... (2)

a~ bit

Hr=2s-l>_3isoddthen U(a, b) =~1bit.., a ~ bf. ae. bI~-1. .... a~:bI~.

The final U(a-t,b-1).

theorem handles the case where U(a,b) is conjugate

to

THEOREM6.5.6. Let 1 ~ U = U(a,b) be in F or 1 ~ U U(b,a) be in F. HU = U(a,b) assume that there is a V in F with V-1U(a,b)V = U(a-l,b-t). H U = U(b,a) assume tha~ there in F with V-tU(b, a)V = U(b-t, a-t). Then possibly after replacing U by U(a-t,b -~) = U*(b,a) i/=U = U(a,b) or U by U(b-~,a -t) = U*(a,b) U = U(b, a) one of the following cases hold: (1) U -= U(a, b) and U(a, b) = (S(a, b)S(a -1, b-l)) d [or some na~urM number d and some elemen~ S(a, b) in F. d for some natural (2) U = U(b,a) and U(b,a) = (T(b,a)T(b-t,a-~)) number d and some element T(b, a) in F. PROOF. Suppose, without loss of generality, that 1 ¢ U = U(a,b) = a~tblt...a~rbI~,r >_ 1, with ei, f~ non-zero integers. Let V = d’+t with all ki non-zero integers, di a non-zero integer adtbk~...ad’~bk’~a for i = 2, .., n and dl and d~+t integers. Again let L denote the free product length in the free product decomposition F =< a > * of F. In this case cancellation arguments with respect to L give that dt ~ 0 if and only if d~+t = 0. If dl = 0 and d,~+t ¢ 0 then

158


V --- bkl...ad’~bk"ad’~+l and a-elb-ll...a-e"b--5"a-d’~+Ib-k’~...a-d~b -k~ -a-d,~+~ b-k,~...b k~a~Ibll ...a¢r b~r, and we may replace U(a,b) by U(a-l,b -1) = W(b,a) (and V by v-l). Therefore we may assume that dl ~ 0 and d=+l = O. Then ae~ bfl...a ~ bfr adl bkl...a d" bk" =a’~1 bk~ ...a d’~ bk’~ a-~ b-f~...a -~ b- fr with dl # 0. Ifn > r then V = V1V~with V1 = U(a,b) and U(a,b)V2 V2U(a-1, b-~) and here we may replace V by V2. Therefore we may assume that r >_ n. If r = n then V = U(a,b) = U(a-l,b -1) which is impossible. Thus we have r > n. Then U(a,b) = Ul(a,b)U2(a,b) with Ul(a,b) = and we get the equation U2(a,b)Ul(a,b) = Ul(a-~,b-~)U2(a-l,b-i). If -l) L(Ul(a,b)) = L(U2(a,b)) then U2(a,b) = Ul(a-l,b and therefore U(a, b) = Ul(a, b)Ul(a-1, 5-1). If L(UI(a, b)) < L(U2(a, thenU2(a,b) = Ul(a-~,b-~)W~(a,b) and W~(a,b)U~(a,b) = Ul(a,b)Wl(a-l,b-1). If -1) and L(U2(a,b)) < L(U~(a,b)) then Ul(a,b) = W2(a,b)U2(a-l,b Uu(a,b)W2(a,b) = W2(a-l,b-1)U2(a,b) that is W~(a,b)U2(a,b) U2(a,b)W~(a-l,b -1) with W~(a,b) = W2(a-l,b-1). In both these last two cases the desired result follows by induction.

CHAPTERVII LINEAR ONE-RELATOR

7.1 Linear Properties

PROPERTIES PRODUCTS

OF OF

CYCLICS

of One-Relator Products of Cyclics

As we saw in Chapter Six, one-relator products of cyclics with proper power relators admit essential representations into PSL2(C). Although many of these representations are far from being faithflfi their existence leads to the existence of manylinear properties in this class of groups, The existence of these linearity properties mirrors the situation for one-relator groups (see Chapter Ttn:ee). Throughout this chapter we concentrate on the case where the relator is a proper power. Therefore our groups all have the form (7.1.1)

< al, ...,a,,;a~ 1 .....

a~," = R"(al, ..,aN) ---- 1

wheren _> 2, ei = 0 or ci _> 2 for i = 1, ..., n, R is a cyclically reduced word in the free product on al, ..., a~,, involving all al, ..., a,~, and m>_ 2. For n = 2 the resulting class of groups is the class of generalized triangle groups. These all have the form

(7.1.e)

< a, b; a"p = bq = R’" (a, b) = 1

where as above R is a cyclically reduced word in the free product on a, b involving both a and b and m _> 2. For many of the linearity properties examined the generalized triangle groups must be considered separately. In section 7.2 we first consider the Tits Alternative. Weprove that if, n _> 3 then any group with a presentation of the form 7.1.1 satisfies the Tits Alternative. Related to this we consider the SQ-universality of these groups. The complete solution of the Tits Alternative and SQ-universality the generalized triangle groups is still open. In section 7.3 we discuss the general theory of this class. In particular the Tits Alternative is knownto hold for a generalized triangle group except possibly in one situation. The techniques used to study the generalized triangle groups leads to showing 159

160


the existence, in many cases, of Ree-Mendelsohn pairs. These are generating pairs (g, h} for which there is a sufficiently large power t such that < gt, ht > is free of rank two. Filmlly in section 7.3.3 we give the complete classification of the finite generalized triangle groups recently completed by Howie, Metafsis and Thonlas [H-M-T 1], and Levai, Rosenberger and Souvignier [L-R-S]. In section 7.4 we consider mffticient conditions for a one-relator product of cyclics with proper power relator to be virtually torsion-free. From work of Culler and Shalen, if a finitely generated group admits "enough" representations into SL2(C), it must decompose as a non-trivial amalgam. A result of Bass in a different direction shows that manyfinitely generated subgroups of GL2(C) are in fact non-trivial free products with amalgamation. These results are used in section 7.5 to examine the existence of amalgam decompositions tbr groups in the class of one-relator products of cyclics. Finally in section 7.6 we discuss the existence of an Euler characteristic for groups in this class. 7.2 The Tits

Alternative

and SQ-Universality

The first linear property we consider is the Tits alternative. By a linear group we mean a subgroup of GLn(F) for some commutative field F. For these notes we restrict F to have characteristic zero although this is not necessary in all cases. A theorem of J. Tits [Ti] states that a finitely generated linear group either contains a non-abclian free subgroup or is virtually solvable, that is contains a solvable subgroup of finite index. In general we say that a group G satisfies the Tits alternative if it either contains a non-abelian free subgroup or is virtually solvable. Theorem3.4.1 due to Karrass and Solitar shows that any one-relator group, and more generally any subgroup of a one-relator group, satisfies the Tits alternative. Our first result here is that if n _> 3 then a one-relator product of cyclics with a proper power relator must satisfy the Tits Alternative. THEOREM 7.2.1. Let G be a one-relator product of cyclics with proper power relator so that G has a presentation (7.1.1)

< a~, ...,

a~; a[~= ....

a~~

= R’~’at~,..,a~)"=1

with m >_ 2, ei = 0 or ei >_ 2 for i = 1, ...,n and R(a~, ...,a~) a cyclically reduced word in the free product on al, ..., a~ involving all a~, ..., a~. Then G satisfies the Tits alternative ifn > 3. I~ particular (1) /f n _> 4 or n _> 3 and (e~, e2, Ca) ~ (2, 2, 2), then G contains a non-abelian free subgroup.

7.2 THE TITS ALTEI~NATIVEANDSQ-UNIVERSALITY

161

(2) Ii’n = 3 and (el,e2, e3) ---- (2,2,2), then either G contains a nonabelian free sllbgrollp or G contains a free abelian s~ibgrollp of rank 2 and of index 2. PrtooF. If n _> 4 this is a direct consequence of the Freiheitssatz. Since n >_ 4 the subgroupgenerated by al, a2, a3 will be a non-trivial free product of rank 3. This clearly contains a free subgroup of rank 2. Nowsuppose n = 3 and (el, e2, e3) # (2, 2, 2). ~omthe l:~eiheitssatz, contains a non-trivial free. product of cyclics of rank 2 which is not infinite dihedral. Hence G has a free subgroup of rank 2. Nowsuppose that n = 3 and (~, e2, e3) = (2, 2, 2) so that G has form (7.2.1.)

G -~

If m _> 3 then from Theorem6.2.3 it follows that G contains a subgroup of finite index mapping epimorphically onto a non-abelian free group. Hence it follows that G must itself contain a non-abelian free subgroup. As a consequence G must be SQ-mfiversal, a fact that we will return to after the completion of the proof. Therefore we are reduced to the case where n -- 3, m -- 2 and (ebe2,e3) = (2,2,2) and G has the form (7.2.1). In this case a = a~a2, b = ala3 and let H =< a, b >. The index ]G : HI _< 2 and we show that H either contains a free subgroup of rank 2 or that H is free abelian of rank 2 with ]G: HI = 2. First write R(a~, a2, a3) = a~S(a, b) where e -- 0 or e -- 1 and S(a, b) is freely reduced in the free group on a, b. S(a, b) involves both a and b since R(a~a2, a3) involves all al~ a2, a3. Case 1. m--2andc--0. Choose three projective matrices of order 2 in PSL2 (C) A1-- q-

1 A2 -: =t= -1 1 0 ’ 1

2 and Aa -- + -w

with w e C. A1,A2 generate an infinite dihedral group. Let A -- A1A2 and B -AIAa. Then < A, B > cannot be an infinite dihedral group because if it were then tr(B) = tr(AB) and this imples that w2 + 2 = 0 and 3+ 2w2 - 2w = (w- 1) 2 = 0 giving a contradiction.

162


Nowas in the proof of the ~reiheitssatz for one-relator products of cyclics choose w0 so that tr(R(A1, A2, A3)) = 2 cos(~/m). Then < A1, A2, provides a representation for G in PSL2 (C). Wcclaim that A and B = B(wo) havc no commonfixcd point for if thcy did then < A, B > would be metabelian and tr([A, B]) = 2. From this would follow that Wo= 0 or w0 = -1. We then obtain that A3 --- +A1 or As = +A2. In both cases it follows that S(A, B) = ~ f or s ome k~ Z and R(AI,A2,A~,) = A~Ak. Since m = 2 and e = 0, this shows that R(AI,A2, A~) cannot have order m in < AI,A,2,Aa > contradicting the way the trace was chosen. Thereibre A and B have no commonfixed point. Further tr(A) = tr(AiA2) = 3 so tr(A) ~ If tr(B) ~ and tr (AB) ~ wemay a ssume ( aft er a suit able conjugation) that < A,B >C PSL2(R). It then follows from a result of Rosenberger [Ro] that < A, B > contains a generating pair {u, v} such that < ut, v~ > is free of ra~k 2 for a sufficiently large integer t. If both B and AB are loxodromic {tr(B) ¢ ~, and tr(AB) 6 It~} then there exists a positive integer t such that ]tr(Bt)l > 2 and Itr(AB)tl 2. From results of Majeed [Maj], < Bt, (AB)t > is free of rank 2 for a sufficently large integer t. Clearly < B, AB >=< A, B >. If tr(B) ~ but tr (AB) ~ th entr(BA B) q~ ~ sin ce tr(BAB ) = tr(B)tr(AB)-tr(A) and < BA, BAB >=< A, B >. Then from the results of Majccd thc subgroup < (BA)t, (BAB)t > will bc free of rank 2 for a suffciently large integer t. Finally iftrB ~ ~ but tr(AB) ~ th en tr (AB-1) ~ ~ si nce tr (AB-~) = tr(A)tr(B) -tr(AB) and < B, AB-~ >=< A,B >. Then a.s above < Bt, (AB-1)t > is frcc of rank 2 for a sufficicntly largc intcgcr t. So in all cases H =< a, b > contains a generating pair {u, v} such that < ut~ vt > is free. of rank 2 for a mffficiently large integer t. This completes case 1. Case 2. m=2ande=l. The relator now has the form R(al,a2,a3) = alS(a,b). H =< a, then has index 2 in G and applying the Reidemeister-Schreier process we find that H has the ~presentation H --< a, b; S(a-~, b-~)S(a, b) -~ 1 Wecan assume without loss of generality that S(a, b) is cyclically reduced in the free group on a, b and that there is no free. cancellation between S(a -~, b-~) and S(a, b). Thc cxponcnt sum on b is zcro, so wc can cxprcss H as an HNNgroup with stable letter {b}. For each i ~ Z let xi = ~. b-~ab Then S(a-~,b-~)S(a, canbe e xpressed as a fre el y reduc ed word T on the {xi}.

7.2 THETITS AUI’~RNATIVE ANDSQ-UNIVEILSALI’I’Y

163

Let M be the greatest integer such that XM appears in T and m the least integer such that Xmappears in T. Clearly IM - mI _> 1. H can then be expressed as H =< x~,~, .... x~,b;T(x,,,,. .... xaz) = 1, b-tx~b = x~+~,i = m,....M - 1 > Let K-~-

>

XM;> ¯

Then H is an HNNgroup with base K, free part generated by b and associated subgroups K1 and K2. If ]M - m] _> 2 then K1 is free of rank at least 2 and is contained in H, so in this case H contains a free subgroup of ra~k 2. If IM - m] = 1 then d~¢ = m + 1. It follows that either x,~ = x,~+l or x,, ~ x,,+l in H. If Xm = Xr~+l then b-lab = a or ab = ba. Then H =< a,b > is free abelian of rank 2 and is of index 2 in G. Finally if x,~ ¢ x,~+l then there exists no p, q ~ Z \ {0} such that p (.T.,~X~+I)P----..T.q.,, or (x,~:r.~l+~) =x,,,+~q sincetheabelianization of His frcc abclian of rank 2. Thcn from Britton’s lcmma on HNNgroups (scc Chapter Two) < x,~x~+~, b > is a free subgroup of rank 2. This completes case 2 and the proof of the theorem. As mentioned in Chapter Three, closely related to the Tits alternative is SQ-universality. Recall that a group G is SQ-universal if every countable group can be embedded in a subgroup of a quotient of G. All non-abelian free groups are SQ-universal and from a theorem of Levin [ Le] any nontrivial free product except Z~ ¯ Z2 is SQ-universal. If IG : HI < oc then H being SQ-universal implies G is. Further if G maps onto a non-abelian free group or contains a subgroup of finite index which maps onto a non-abelian free group, then G is SQ-universal. The tie to the Tits alternative is that having manynon-abelian free subgroups is a good indicator that the group should be SQ-universal. As a direct consequence of Theorem6.2.3 and the existence of essential representations ibr one-relator products of cyclics we obtain the ibllowing result. THEOREM 7.2.2. Let G be a one-relator product of cyclics with proper power relator so that G has a presentation of the tbrm 7.1.1 (7.1.1)

< al,...,a,~;a~

x .....

a~~ = R"~(al,..,an)

=1

as in Theorem7.2.1. Suppose thag n >_4 or n >_ 3 and (e~, e2, e3) ¢ (2, 2, or n ~ 3,(e~,e2,e~) = (2,2,2) and m >_ 3, t, hen G contains a subgroup

164


of finite index mapping epimorphically onto a non-abelian free group. In paxticulax under these conditions G is SQ-universal. Thus ibrn _> 3, relative to SQ-universality we are reduced to the two special cases considered in the proof of Theorem7.2.1, that is when m = 2 and (el, e2, e3) = (2, 2, 2). Thus we now consider groups of the G =< al, a2, a3; a~ = a’~ = a~ = R’2(al, a2, a3) = 1 where R(at, a2, a3) is cyclically reduced in the free product on at, a2, a3 and involves al:a2,a3. A.~ before let a = ala2, b = ala3 and let, H =< a,b > in G. Thus H has index less than or equal to 2 in G and again as in the proof of Theorem7.2.1 write R(al, a2, a3) = a~S(a, b) where ~ = 0 or e = 1 and S(a, b) is freely reduced in the free group on a, b. S(a, b) involves both a and b since R(a~, a2, a3) involves all a~: a2.. a3. Notice that the element R(at, a2, a3) is in H only if e = 0. Wethen obtain the following theorem. THEOREM 7.2.3.

Suppose the group G has a presentation

of" the tbrm

G = 1 and ai,~i axe non-zero integers for i = 1, ..., k. Suppose further that one of the following holds: (i) ->2 or [fill/-~ 2 if k = 1 (ii) gcd(a~,...,crk) > 2 or gcd(3~,...,/dk) > 2 ilk >_2 (iii) S(a, b) a p roper power in thefree groupon a, b Then G contains a subgroup of finite index mapping onto a nonabelian free subgroup and hence is SQ-universal. PROOF.Suppose first H has the presentation

that e = 0 so that R(al, a2, a3) = S(a, b). Then

H =< a, b; S’2(a, b) = S2(a-~, b-1) = 1 > . ~Yoma theorem of M.Edjvet [E 3] H has a subgroup of finite index mapping onto a non-abelian free. group and hence G has also.

7.2 TtiE TITS ALTERNATIVE ANDSQ-UNIVERSALITY

165

Nowsuppose that e = 1 so that R(al, a2, az) al S(a, b) ThenH has the presentation H =< a,b;S(a,b)S(a-~,b

-1) = 1 > .

If (i) or (ii) holds, then the group H has a free product of cyclics Zr with r _> 2 as an epimorphic image, so the result follows. If (iii) holds thcn S(a, b) = T’~(a, with 0/>2. Hthcn has a s a factor group the group H with the presentation ~ =< x,y;T’~(x,y)

= T’~(x-l,y -~) >.

The theorem of Edjvet used in the first part of the proof can be applied to thissituation to get the result. Finally suppose that k ----- 2. Wemayassume, without loss of generality (passing to ~nother generating pair if necessary and after a suitable conjugation) that 0/1,a2,fl~,f12 are all greater than or equal to 1. H has a free., product of cyclics Zr * Z~ with r >_ 2, s > 3 as an cpimorphic imagc, from which the result would tbllow, except in the tbllowing cases: (a) a~ =2, a2=fl~=l,f12_>2;

(d) 0/1 = fll = f12 = 1, 0/2 = Consider first case (a). Let a~ = 2, c~2 = fl~ = 1, fl~ > 2 and let fl = Let K be the subgroup of H generated by x = a~,y = b and z = aba-1. K then has index 2 in H and has a presentation K =< x,y,z;xyz~x-~z-ly

-~ = xzxyZx-~y-lx-lz-~

=1 >

which has the free product Zfl+l * ~2 as an epimorphic image. In case (b) where a~. = 3, a~. =/~ = 1, ~2 = 2 let K be the subgroup H generated by x = b~, y -- a and z = bab-1. K then has index 2 in H and has a presentation K =< x, y, z; y3zxz-3y-lx-1

= z3xyxy-3x-~z-lx

-~ = 1 >

which has the free product Z2 * Z4 as an epimorphic image. In case (c) where 0/~ = a2 = ~ = 1, ~ = 2 let K be the subgroup of generated by x = b2,y = a and z = bah-~. K then has index 2 in H and has a presentation K =< x, y, z; yxyz-~x-~z -1 = zxzxy-~x-ly-~x-~

=1 >

166

ALGEBRAIC

GENERALIZATIONS

OF DISCRETE

GROUPS

which has the free product Z2 * Z as an epimorphic image. Finally in case (d) where al = fll --- f12 = 1, a2 = 2 then H has the group H with presentation ~ =< x,y;x’~ = y4 = (xy’2)’~ = 1 as an epimorphic image. Let K be the subgroup of H generated by u = y and v = xyx. K then has index 2 in H and has a presentation K =< u,v;u 4 = v 4 = (u2v’2) "2 = 1 > which has the free product Z2 ¯ Z4 as an epimorphic image. This completes the proof. Weconjecture that if k _> 3 then the subgroup H in the above theorem contains a subgroup of finite index mapping epimorphically onto a free group of rank 2. From a result of Sacerdote and Schupp IS-S] we know that a group H as in Theorem 7.2.3 is SQ-universal if k _> 3. Combining Sacerdote and Schupp’s result with Theorems 7.2.1 and 7.2.2 ~ then have the following theorem. THEOREM 7.2.4. Let G be a one-relator product or" cyclics with proper power relator so that G has a presentation of form 7.1.1 (7.1.1)

< a~,...,a,~;

a~ =

.

with m >_ 2, e~ = 0 or e~ >_ 2 ~or i = 1, ...,n and R(al,...,a,,) a cyclically redfaced wordin the £r~ product on a~, ..., a,~ involving all al, ..., a,~. Then i[ n _> 3 either G is SQ-univcrsal or v/rtually ~cc abclian. In the next section we consider the generalized triangle groups. We conjecture that ibr this class each group is either SQ-universal or virtually solvable. 7.3 The Generalized

Triangle

Groups

Whenn = 2 we have the case of the generalized triangle groups. Relative to the Tits alternative, the complete solution ibr these groups is still open. Essentially what is presently knownis that the Tits alternative holds except possibly when both generators have finite order p, q, the relator has order m = 2, (l/p) + (l/q) _> (1/2) and the relator has syllable length greater than eight in the free. product on the generators. This is the content of Theorem 7.3.1 given below. First ~ fix some notation. A generalized triangle group is a group G with a presentation (7.3.1)

G =< a, b; a r = bq = R’~(a, b) = 1

7.3 THE GENERALIZEDTRIANGLEGROUPS

167

where p < q, p k 2 or p = 0, q > 2 or q = 0, R(a, b) is a cyclically reduced word in the free. product on a and b involving both a and b and m _> 2. If p >_ 2 we let s(G) = (l/p) + (l/q) + (l/m). A generalized triangle clearly generalizes an ordinary triangle group T(p, q, m) which has the presentation T(p, q, m) =< a, b; p

=b q=

(ab"~ = 1 > .

Wenote that many authors restrict the class of generalized triangle groups to the case where both generators have finite order. Wedo not make this restriction. For the generalized triangle groups the major result relative to the Tits alternative is the following. THEOREM 7.3.1. tion

Let G be a generalized

triangle

group with presenta-

G =< a, b; a p = bq = R"~(a, b) = 1

where p _< q, p _> 2 or p = O, q > 2 or q = O, R(a, b) is a cyclically reduced word in the free product on a and b involving both a and b and rn > 2. Then G satisfies the Tits alternative except possibly whenp >_ 2, q > 2, rn -2, (l/p) + (l/q) > 1/2 and the relator R(a, b) has syllable length greater than eight in the ~ree product on a, b. This result is actually a summaryof several results whose precise statements and proofs will be given in sections 7.3.2 and 7.3.3. These results generally show more than just the existence of free subgroups or solvable subgroups of finite index. If G is a 2-generator group then a generating pair (u, v} is a Ree-Mendelsohn pair or RM-pair if there exists an integer t such that < ut, vt > is a free, subgroup of G of rank 2. In particular included amongthe results summarized in Theorem7.3.1 are the following: (1) If > 3, p = 0 and q _>2 t hen G c ontains an RM-pair. (2) If m = 2,p = 0 and q _> 3 then G contains an RM-pair. (3) If ra = 2,p = 0 and q = 2 and suppose R a’ ~ba’~b...a’*~b with k _> 1, then: (a) If k >_ 2 or ]’n~] > 3 for some i = 1,...,k, then G contains a non-abelian free subgroup. ~hrther if k is even or k is odd with k > 3 and ni +... + nk # O, then G contains an RM-pair. (b) If k = 1 and In~l >_ 3 then G has an RM-pair. (c) If k = 1 and ]n~] < 2 then G is infinite and solvable. The reason that there are two types of results - ones giving the existence of RM-pairs and others only guaranteeing that G satisfies the Tits alternative, comes from the method of proof. This was seen previously in the proofs in the last section. ~lb obtain an RM-pair we consider the essential

168


representations of G in PSLu(C). Let G =< A, B > be the image group. If we can find a representaion such that < A, B > is non-elementary and non-elliptic, we can use the results of Rosenberger [R 3,13] and Majeed [M~j] as in the proof of Theorem 7.2.1 to obtain an RM-pair. If we cammt show that the image group has this property we can in the remaining cases use combinatorial arguments to get that G either has a non-abclian free subgroup or a solvable subgroup of finite index. Before going to these results we first review and highlight some of the main features of the ordinary triangle groups T(p, q, m). 7.3.1

The Ordinary Triangle

Groups

Let p, q, m be integers greater than one. Then the ordinary triangle group or just triangle group T(p, q, m) is tim group defined by the presentation (7.3.1.1)

T(p,q,m) :< a,b;a p = bq = (ab) *’~ = 1 > .

Notice first, that as abstract groups any permutation of the triple (p, q, rn) leads to an isomorphic ordinary triangle group. This is clearly not necessarily true for the generalized triangle groups. Note also that in the definition of the generalized triangle groups wc did not require the exponents p, q to be greater than one, so that we allow generators to have infinite order, l~br the ordinary triangle groups it is generally standard to require only generators of finite order. Group theoretically, if one or more of the exponents is zero, we get a non-trivial free product of two cyclics. It is often convenient to think of these free products as triangle groups. For example the Modular group PSL~.(Z) has the structure Z2 * Z3. This can be considered as the triangle group T(2, 3, oo). The terminology triangle group comes from the following construction. Let p,q, m > 1 be integers with (l/p) + (l/q) + (l/m) < 1. Then exists a non-Euclidean hyperbolic triangle A with angles ~/p, r/q, rim. Let L, .¢¢, N respectively be the reflections of the hyperbolic plane 7~ in the sides of A and let T* be the subgroup of the isometry group of 7-/ generated by L, M, N. It can be shown (see Magnus[M 6]) that T* has the presentation T*

~=Mu=Nu=(LM) ~=(MN)q=(NL)~’=1>. =. Therefore T is isomorphic to T(p, q, m). Further since now T consists of orientation preserving isometries of 7-/, the elements of T can be considered as being in PSL2(R) (see Chapter 4) and hence T(p, q, m) has a faithful representation in PSLg.(R). The union of the original triangle A with the image L(A) under the reflection L serves as a positive area fimdamental domain for T. From tlfis it follows that T is discrete and thus a Fuchsian group. Wesummarize this by saying that ibr any p, q, m > 1 with (l/p) (l/q) + (l/m) < I the triangle group T(p, q, m) has a faithful representation as a Fuchsian subgroup of PSL2(R). (This also follows from Poincarc’s Theorem (see Chapter 4)). This construction can be mirrored in both the Euclidean plane and ill the spherical (double elliptic) plane. In both these cases the resulting triples axe restricted. For a triangle in the Euclidean plane, we must have (l/p) (l/q) + (l/m) = 1. If < p < q < m,the n the onlypossi ble tripl es are (2, 3, 6), (2, 4, 4) and (3, 3, 3). Usingreflections in the sides of tile triangle generate a subgroup of the Euclidean isometry group and then taking the subgroup of orientation preserving isometries we get the Euclidean triangle groups T(2, 3, 6), T(2, 4, 4) and T(3, 3, 3). In each case these groups to tesselatior~ or filings of the Euclidean plane. (See. Magnns[M 6] for somevery pretty pictures of these tesselations as well as tesselations of from hyperbolic triangle groups.) It can be shown that these groups have faithful (but not Vhchsian) representations in PSL2(C). Wesay that a representation of a group in PSL2 (C) is a Fuchsian representation if the image group is a Fuchsian subgroup of PSL2(C). Finally for a spherical triangle we mttst have 2 < p, q, m and (l/p) (l/q) + (l/m) > 1 and if 2 _< < q _<m t heposs ible trip les are (2, 2, m), (2, 3, 3), (2, 3, 4) and (2, 3, 5). As before we get the triangle T(2, 2, m), T(2, 3, 3), T(2, 3, 4) T(2,3, 5) which are al l fi nite groups. These groups car~ be faithfully represented ir~ PSL2(C) and describe the finite groups of symmetries in R3 of the regular solids. T(2, 2, m) axe the dihedral groups D,~ and are the symmetry groups of the regular m-gon or dihedron. T(2, 3, 3) has order 12 and is the symmetry group of a tetrahedron. It is isomorphic to A4, the alternating group on ibur symbols. T(2, 3, 4) has order 24, is the symmetrygroup of a octahedron, and it i~morphic to $4, the symmetric group on four symbols. Finally T(2, 3, 5) has ordcr 60 and corrcsponds to thc symmctry group of thc rcgular icosahedron. It is isomorphic to As. ~om this discussion we see that T(p, q~ m) is finite if and only if 2 < p,q,m and (l/p) + (l/q) + (l/m) > 1. The ~mhsian triangle

170


groups are clearly infinite while one can show that the Euclidean triangle groups contain free abelian subgroups of finite index. If 2 _< p, q, mand (l/p) q- (l/q) q- (l/m) < 1 then T(p, q, m) can be considered as a group. It follows from the Fenchel-Fox Theoremthat there is a torsion-free Fuchsian group of finite indcx. This must bc an oricntablc surface group of genus g _> 2. Such groups clearly map epimorphically onto non-abelian free groups and hence in this case T(p, q, m) is SQ-universal. If 2 _< p, q, and (l/p) + (l/q) + (l/m) ---- 1 the resulting three, groups are all virtually free abelian of rank two. It follows that they are all infinite and contain subgroups of finite index mapping onto Z. Finally from the concrete construction as groups of isometries it is clear that in T(p, q, m) the element a has exact order p, the element b has exact order q and the element ab has exact order m. Wesummarize all these facts below. THEOREM 7.3.1.1. Let G = T(p,q,m) =< a,b;a v = bq = (ab) "~ = 1 > with p, q, m > 1 be a ~ria~gle group. Let s(G) (l /p) + (l /q) + (l Thdn (1) G admits a faithful representation into PSL2(C). The image group can be ~bchsian if and only if s(G) < (2) G is flnife if and only if s(G) (3) If s(C) < 1 ~hen C contains an orientable surface group of genus g >_ 2 as a subgroup of finite index. In particular G contains a subgroup of finite index mapping onfo a non-abelian free group and hence (I is universal (4) H s(G) ---- 1 ~hen~ is virtually free abelian of rank (5) The element a ha~ exact order p, the element b h~ exact, order q and $he element ab has exact order m. (6) G is virtually torsion-free. (7) G satisfie~ ~he Tits al~,ernative. The hyperbohc triangle groups are of further interest amongthe Fuchsian groups as the subclass of ~hchsian groups with fundamental domains having small hyperbolic area. Of particular interest here is the group T(2, 3, 7) which has a fimdamental domain of the absolutely minimum hyperbolic area. This is a straightforward consequence of the Poincare presentation and its relation to the hyperbolic area of a fundamental domain (see Chapter 4). The group T(2, 3, 7) is of further interest in Riemannsurface theory because of the following set of results of Hurwitz (see Magnus[M 7]). The group H of conformal self-mappings of a Ricmannsurface $ of genus g _> 2 is a finite group of maximumorder 84(g - 1). If Sa is the fundamental group of S, then H is a quotient group of a l:hchsian group F by a normal subgroup isomorphic to S.q. The order of H is the maximalorder 84(g- 1) and only if F is isomorphic to T(2, 3, 7). Thus the finite groups of maximal

7.3.2

THE TITS AEI’ERNATIVE- GENERATORS OF FINITE ORDER171

order which can be isomorphic to the group of conformal self-mappings of a Riemannsurface are precisely the finite quotients of T(2, 3, 7). Such groups are called Hurwitz groups and have been studied fairly extensively (see the survey by Conder [Co 2]). We mention that Higman [Hi 3] proved that all alternating groups An for n sufficie~flty large are Hm’witzgroups. Macbeath [Mac 1,2] proved that PSL2(q) is a Hurwitz group if and only if either (i) q = 7 ; (ii) q = ±1rood 7; or (ii i) q = p3wher e p -- ±2 or -- :t:3 rood 7. 7.3.2

The Tits Alternative

- Generators of Finite

Order

Our purpose in the next two sections is to prove Theorem 7.3.1. suppose that G is a generalized triangle group with presentation (7.3.1)

We

G --< a,b;a ~ = bq -- R"~(a,b) = 1

where p _< q, p >_ 2 or p = 0, q _> 2 or q = O, R(a, b) is a cyclically reduced word in the free., product on a and b involving both a and b and m >_ 2. We will give a series of results which combinedwill showthat G satisfies the Tits alternative except possibly whenp _> 2, q >_ 2, m -- 2, (l/p) + (l/q) _> and the relator R(a, b) has syllable length greater than eight in the free product on a, b. In this section we concentrate on the case where both generators have finite order, that is we assumethat p _> 2, q _> 2 in presentation 7.3.1. The main result, due to Baumslag, Morgan and Shalen, mirrors the situation fbr the ordinary triangle groups and also reduces the problem to a finite set of triples (although this involves i~ffinitely manypossible relators). THEOREM 7.3.2.1. with presentation

[B-M-S] Suppose G is a generMized triangle

group

G =< a,b;a ~ = bq = R"~(a,b) = 1 where 2 _ 2 and let s(G) = (l/p) + (l/q) + (l/m). (1) ITs(G) < 1 then G contains a subgroup o[ ~nite index mapping a non-abelian free subgroup. In particular G is SQ-universal and contains a free subgroupof’ rank 2. (2) If s(G) = 1 then G contains a subgroup of finite index mapping onto Z. (3) a has exact order p, b has exact order q and R(a, b) has exact order PROOF.~rom Theorem 6.3.2, G admits an essential PSL2 (C). Part (3) follows directly from this fact.

representation

into

172

ALGEBRAICGENERALIZATIONSOE DISCRETE GROUPS

The proofs of parts (1) and (2) mirror the proof of Theorem 6.2.3. s(G) _~ 1 and let p be an essential representation of G into PSL2(C). Therefore from Selberg’s theorem (Theorem 4.3.5) there exists a normal torsion-free subgroup H of finite index in p(G), the image of G. Thus the composition of maps ( where rr is the canonical map) C A p(G) -~ p(C)/H gives a ¢ of G onto a Let X =< a, b; ap < a; ap = 1 >, < b; onto G. Wetherefore

finite group. = bq -- 1 > be the free product of the cyclic groups bq = 1 >. There is a canonical epimorphism/~ from X have the sequence X ~-~ G --~¢ p(G)/H

Let Y = ker(¢ofl). Then Y is a normal subgroup of finite index in X and Y is torsion-free. Since X is a free’, product of cyclics and Y is torsion-free, it follows that Y is a free group of finite rank r. Suppose IX : YI = J- Since every finitely gmmrated free product of two or more cyclics can be faithfully represented as a Fuchsian group if it is not isomorphic to the infinite dihedral group, we may regard X as a Fuchsian group with finite hyperbolic area #(X). From the RiemannHurwitz formula we have that

jit(x) where It(Y) = (2rr)(r and #(X) = (2u)(1

- (~

Equating these expressions we obtain r = 1 -j(:

1 +- - 1). P q

G is obtained from X by adjoining the relations R’~(a, b) and so G = X/K where K is the normal clom~re of R"~. Since K is contained in Y~ the quotient Y/K can be considered as a subgroup of finite index in G. Applying the Reidemeister- Schreier process or repeated applications of

7.3.2

THE TITS AUFERNATIVE - GENERATORS OF FINITE ORDER173

Corollary 3 in [B-M-S], Y/K can be defined on r generators subject (j/m) relations. The deficiency d of this presentation for Y/K is then d=r-

jm--:l-j(~+-ql+-ml

to

_l)=l_j(s(a)_l).

If s(G) < 1 then d > 1 and since d is an integer, the deficiency of the above presentation is at least 2. From the result of Baumslag and Pride, Thcorcm 6.2.2, Y/K, and hcncc G, has a subgroup of finitc indcx mapping onto a free group of ra~k 2. Therefore Y/K is SQ-universal and since this has finite index in G, G is also SQ-universah l~lrther an SQ-universal group must contain a non-abelian free subgroup. If s(G) = 1 then the deficiency is d = 1 and the group Y/K maps onto an infinite cyclic group and is thus infinite. Therefore G is infinite. If G is a generalized triangle group with presentation G =< a, b; ap = bq -~ R’~(a, b) = 1 where 2 2, and R(a, b) is a cyclically reduced word in the free product on a and b involving both a and b, then R(a, b) can be written as a word R(a, b) = ~ bq~ . .. a ~ bq~: with k _> 1 and 1 _< p.¢ < p, 1 _< qi < q fbr j = 1,..., k. The syllable length as a word in the free product on a, b is then 2k. In the next result we show that the Tits alternative holds wheneverk = 1, that is the syllable length is two. THEOREM 7.3.2.2. sentation

Suppose G is a generalized triangle

group with pre-

G =< a, b; a v = bq = (a~bt) m = 1 > where2 _ 1. Its _> 2 andt_> 2 then the free product < a,b : a ~ = b~ ~ 1 > is a epimomorphicimage of G. If s + t _> 5 this free product is not Z2 * Z~ so it

174


has a free subgroup of rank 2 and therefore so does G. Therefore the only other cases to consider are s _> 2, t = 1 or s = t = 2 or s = 1,t_> 2. Consider the case where either s > 2, t = 1 or s = t = 2. Then p = st, r _> 2 and q _> 4 since p < q. Wemay then write G as a free product with amalgamation G = HI * H H2 where H1----~ a; ap = 1 > H2 =< a~, b; (aS) " = b~ = (a~bt) "~ = 1 > and

H =< a~; (a~y = 1 >. Let x = -1 abaSb-la and y = b. Standard cancellation methods in free products with amalgamation show that the subgroup generated by x, y has the presentation ~ x, y; x’*" ~- yq --~ 1 >. Since r _> 2 and q _> 4 this has a free subgroup of rank 2. Next consider the case where s -- 1, t _> 2. Then as in the previous situation, q = tr with r _> 2. As before G is a free product with amalgamation G--H1 ~:H H2 where H1 =< b;b q = 1 > H2 =< a, bt; ap -= (bt) " = (abt) "~ = 1 > and H =< bt;(bt) r = 1 >. Ifp _> 3 let x = babta-lb -~ and y = a. Then the subgroup generated by x, y has the presentation < x, y; xr = yP ---- 1 > which has a free subgroup of rank 2 since r _> 2 and p _> 3. If p ---- 2 and m _> 3 let x = babta-lb -1 and y = abt . Then < x,y >=< x,y;x""

= y"~ = 1 >

which has a free subgroup of rank 2. Finally, if p = m = 2 then the cyclic subgroup < bt > is normal in G sinceabta=abta -~ = b-t. ThenG/ --~, =< a,b;a 2-- b t = 1 >. If t ___ 3 then G has a free subgroup of rank 2 and therefore so does G. If

7".3.2

THE TITS ALTERNATIVE - GENERATORS OF FINITE ORDER175

t = 2 then G is infinite dihedral and so is infinite and solvable. Therefore G is infinite and solvable since both < bt > and G/ < bt > are solvable. These cases exhaust the possibilities for (s, t) and therefore complete the proof. It ibllows from the previous results that G contains a non-abelian free subgroup if s(G) < 1 and in general satisfies the Tits alternative if the relator has syllable length 2 in the free product on a and b. This reduces the Tits alternative to the case of syllable length greater than or equal to four and the followingset of triplex: (2, 2, ,~), (2, q, 2), (2, 3, 3), (2, 3, 4), (2, 4, 4), (2, 3, 6), (2, 6, 3), (2, 3, 5), (2, 5, 3), (3, 3, 2), (3, 4, 2),(3, 5, and (3, 3, 3). Most of these are covered in the next theorem. We~nention here that J.Howie [H 9] has completely settled the case when s(G) = and shown that these have non-abelian free subgroups unless they are equivalent to an ordinary triangle group. His theorem handles the triples (3, 3, 3),(2, 4, 4) and (3, 6, 2) which are done in a different manner next results. Someof the results of this section are u~d by Howie for the proof of his theorem. Wedefer the precise statement of Howie’s result until the end of the section. THEOREM 7.3.2.3. senta~ion


group with pre-

G =< a,b;a" = bq = R"~(a,b) = 1 where2 _ 2 and 1 to be finite since it has an element of order 6. Thereibre if < A, B > is elementary, it must be metabelian. It follows that the possibilities for tr(AB) are v~ or -x/~. Further tr(AB) is a zero of either gl(x) or g2(x). It follows as before that without loss of generality we can only have k odd and g~(x) = (x v/ 3) k and

as(x) = (x k which again is a contradiction. Thereibre in the case p = 2, q -- 3, m = 6, G must have a non-elementary image and hence must contain a non-abelian free subgroup. Case(5): p---2, q--4, m=3: To get an essential representation of G in PSL~.(C) with image group < A, B > wc must choosc tr(R(A, B)) -- e, e = =t=1 and wc must considcr the polynomials gl(x) f( X) + e, e = :t :1 an d

This holds wheneverp ---- 2 and m = 3. If q = 4 then an elementary < A, B > must be isomorphic to Sa and so the possibilities ibr tr(AB) are +1. Again this forces k to be odd and without loss of generality the polynomials to be g~(x)--ak(x-1)

~ and

g’~(x) = ak(x 1) a. As before this is a contradiction and we can conclude that in the case p - 2, q -- 4, m = 3, G must have a non-elementary image and hence must contain a non-abelian free subgroup. Case (6): p = 2, q-- 5,m ~To get a.n essential representation of G in PSL2(C) with ima.ge group < A, B > which is elementary we must have < A, B >~- A5. Let ,\ 2 cos(~r/5). Then we have s =~ + 1 and th e po ssibilities fo r tr (AB) ar 1,-1, A-land 1-A. Suppose first that k is even, that is k --- 21 _> 2. Then without loss of generality we can only have g~ (x) = a~(x"~ - 1)~ and

182

ALGEBRAICGENERALIZATIONSOF DISCRETE GROUPS g~(x) = ak(x"~ + ~ -- ~ 2)

which implies that l = 1. Howeverif I = 1 we obtain from gl(x) - g2(x) ±2 that a2(1 - ~) = ±2 or equivalently that 2 - ---- ±2~. This i s a . c ontra,diction sincc a2 = Sql (A)Sa2(A), that is, without loss of gcncrality, a2 is 1 or A or A ÷ 1. Thus k cannot be even. Nowsuppose k is odd. Then we must have gl(X) -- ak(x -- e)"(x -- ~(.~ 1)) s and g~(x) = ak(x e) "(X + ~(A -- ~, where r + s ---- k and ~, y -- ±1. This also gives a contradiction since -er-

~s(A-1)¢0.

As beibre we can conclude that in the case p -- 2, q -- 5, m = 3, G must have a non-elementary image and hence must contain a non-abelian free subgroup. Case (7): p = 2, q = 6, m-To get an essential representation of G in PSL~_(C)with image group < A, B > which is elementary we may have < A, B > dihedral or metabelian but not isomorphic to A4, $4 or A5. Therefore the possibilities for tr(AB) axeO, 1, --1. If k is odd then 0 cannot be a zero of either g~ (x) or of g2 (x) and hence without loss of generality we can only have g~(x)=a~(x-

1) a and

g (x) = a (z 1) which gives a contradiction. Nowsuppose k is even. Then without loss of generality

we must have

g~ (x) = a~x~ and gu(x) = a~(x"~ - ~ 1) where k ~-- 21 _> 2. This implies that k ---- 2 which gives the polynomials g~ (x) = a2x"2 and g2(x)=a2(x ~ -- 1).

7.3.2

THE TITS AUI’ERNATIVE- GENERATORS OF FINITE ORDER183

a2 can only be 1, ~/3, 2, 3 or 2v/3. Using gl (x) - g2(x) -- ±2 we get that a2 = 2 and hence we must have gl (x) = 2 and

g2(x)= 2(xUp to isomorphism these polynomials can only be realized ibr the group G =< a, b; a"~ = b~ = (abab’~)"~ = 1 > . Let H be the subgroup of G generated by u = b and v -- aba. H has a presentation H =< u,v;u ~ = v ~ -~ (vu~) "~ = (uv’~) ~ = 1 > . This then has the free product Z3 * Z3 as an epimorphic image and so has a non-abelian free subgroup. Thereibre in the case p -- 2, q = 6, m = 3 either the group has a nonelementary image or is up to isomorphism the group G above which has a non-abelian free subgroup. Hence in all cases the group has a non-abelian free subgroup. The final case covered by the theoremis p -- 2, q -- 4, m-- 4. Case (8): p = 2, q = 4, m= To get an essential representation of G in PSL2(C) with elementary image group < A, B > we must choose tr(R(A, B)) = ev~, e = ±1 and must consider the polynomials gl(x) = f(x) ~v/2, e -- -- ±1 and

Here g~(x)-g2(x) = 2x/2e and tr(AB) is again a zero ofgl(x) or ofg~(x). Since G contains an element of order 4, < A, B > cannot be isomorphic to either Aa or A5. Therefore the possibilities for tr(AB) axe 0, 1,-1, x/~ and - ~/~. Suppose first that k is odd. Then 0 cannot be a zero of either gl (x) of g2(x) and hence without loss of generality we can only have g~(x) = a~. (x - e)r(x ~, ~/2)~ and

=

+

+

184


where r ÷ s = k, 0 _< r, s and e, u = ±1. Howeverthis produces a contradiction since -er - usv~ ~ O. Therefore k must be even. Let k -- 21 _> 2 and suppose first that tr(AB) = occurs an d as sume, without lo ss of generality tha t gl( 0) --= Nowwe have the possibilities:

(a):

gl(X) ---- akxk and g2(x) = a~,(x"~ - 1)r(x u - 2) ~ with r+ s = l, r, s >_O.

(b):

gl(x) = akxr(xu - 1)~ with r + 2s -- k, r, s _> 0 and g~ (x) = a~, 2 - 2)t

(c):

g~(x) = akxr(x 2 -- 2)~ with r + 2s = k, r, s >_ 0 and g2(x) a~(z 2 - 1) t.

If case (a) holds and I _> 2 then r + 2s # 0. Hence l = 1, that is, r = or s = 1, Wecan then have only a 2 = 1, V~ or 2, Hence r = 0~ s = 1 and a.~ = "v/~2 since gl(x) - gz(z) -= d=2vf~. This gives the pol:~aaomials u and gl(X) ---- X/2X g2(x) ---- X/~(x2 - 2). Up to isomorphism these polynomials can be realized only ibr the group G =< a, b; a~ = b4 = (abab’~)~ = 1 > . Let H be the subgroup of G generated by u = b and v = aba. Then H has a presentation H--.

H then has the group K =< c, d; cu = d4 -- (cd2) 4 -- 1 > as an epimorphic image. This group K decomposes as a free product with amalgamation K = K~ *A K~

7.3.2

THE TITS ALTERNATIVE - GENERATORS OF FINITE ORDER 185

where K1 --< d;d 4 = 1 >, K2 =< c,e;c 2 = e ~ = (ce) 4 -- 1 > and A --< e;e 2 = 1 > with the identification e --= ar2. Let g ~- dcd[2cd-1 and h -- cd2. Using the cancellation method in free products with amalgamtion we get that the subgroup of K generated by g and h has the presentation

This is a non-trivial free product Z2 * Z4 and hence has a free subgroup of rank 2. Now suppose that case (b) holds. Again we must have 1 -- 1 since g2(x) -gl(X) : :t:2v/-~. This implies then that k = 2 and r + 2s = 2 which contradicts r > 0, s > 0. Therefore case (b) cannot hold. Finally suppose that case (c) occurs. Then we must have > 4 and a~. -- 2V~ since gl(x) - g2(x) -- ±2v/~. This implies that we have gl(x)

= 2V~x"(x~ - 2) s = 2V~xk + d~-2x k-e ~ + ...

+ d2x

with r + 2s = k _> 4,0 < r,s, and g2(x) ---- 2v~(x: - 1) ~ =- 2v/~x~ + c~-2xk-~ + ... + c2x~ + co. Fromdk-2 ~- Ok-2 we get that 2s ----- l and hence r -- l. Therefore we must have l = 2 since if l > 3 it would follow that d2 = 0 but c2 ~ 0. This gives us the polynomials gl (x) = 2v~x2 (x ~ - 2) and

g2(x)= 245(x- 2. Up to isomorphism these polynomials can be realized only for the group 4 -- 1 > . G -= . H then has the free product Z2 * Z4 as an epimorphic image and therefore H contains a non-abelian free subgroup and hence so does G. Now assume that tr(AB) = 0 does not occur. Then without loss of generality we must have the polynomials g~(x) = atc(x ~ - 1) ~ and

186

ALGEBRAICGENERALIZATIONSOF DISCRETE GROUPS g2(x) = aa(x~ - 2) ~ where k = 21.

If I _> 2 then l = 21, that is l = 0 contradicting that 1 > 0. Therefore we must have l = 1 and k = 2. This gives the polynomials gl(x) = a2(x~ - 1) and g2 (x) = a2 2 -- 2). As befbre since gl(x) - g2(x) = +2v~, we must have a2 = ±2v~. However this gives a contradiction because we can only have a2 = 1, v~ or 2. Hence wc can get no group hcrc with elementary image. This completes the case p = 2, q --- 4, m= 4 and the proof of the theorem. Whenthe generators have finite order, we have now reduced to the triples (2, 2, m),(2, m, 2),(2, 3, 3),(3, 3, 2),(3, 4, 2),(3, 5, 2), and (3, 6, 2). (2, 2, m) both generators have order 2 and the relator can be rewritten (ab) t for some t. This is easily seen to be a finite dihedral group. Thus the opencases are the triples (2, m, 2),(2, 3, 3), (3, 3, 2),(3, 4, 2),(3, 5, (3, 6, 2). The next result handles the case (2, 3, THEOREM 7.3.2.4. sentation


group with pre-

G =< a,b;a 2 = b3 = R3(a,b) = 1 where R(a, b) = abqiabq2 ... abq~ with k > 1 and 1 _< qj _< 2 forj = 1,..., k. (1) If k >_ 2 then one of the following holds. (a) G has a free subgroup of rank 2. (b) is fin ite of order 1440 andup t o i somorphism has the presentation G =< a, b; a2 -- b3 = (ababab~)a = 1 >. (c) G is an infinite presentation

solvable group and up to isomorphism has the

G =< a, b; a2

= b3 =

(ababababU)3 = 1 > .

(d) k = 2 and G has a free abelian subgroup of rank 2 and of index 6.

(2) If k = 1 then R(a, b) ~ with1 2. If or R(a, b) = abab2 (the two symmetric isomorphic groups). In the first case 1 >=< a, b; a~ = b3 = (ab) 6 = 1 >. This

k = 2 then either R(a, b) = abab words ab2ab2 and ab2ab lead to G =< a, b; 2 =b3= ( abab) 3 = is a Euclidean ordinary triangle

7.3.2

THE TITS ALTERNATIVE - GENERATORS OF FINITE ORDER 187

group which contains a free abelian subgroup of rank 2 of index six. In the secondcase G --< a, b; a2 = ba = (abab~)a = 1 > . This has the symmetric group Sa as an epimorphic image mapping a to a 2-cycle and b to a 3-cycle. Using Reidemeister-Scheier we find that the kernel is free abelian of rank 2 and clearly of index 6 since ]Sa[ = 6. Nowsuppose that k _> 3. The general strategy is the same as in the previous theorem. That is we attempt to find, up to isomorphism, all G such that for an essential representation p : G -* PSL2(C) given by a -~ A, b --* B with tr(A) = 0, tr(B) = 1 then p(G) is elementary, recalling that if G has a non-elementary image it has a free subgroup of rank two. Hence we assume that for each essential representation p : G -* PSL2(C) given by a -~ A, b -~ B with tr(A) = O, tr(B) --- 1 then p(G) is elementary and hence finite or metabelian. To get an essential representation of G in PSL2(C) with image group < A, B > we must choose tr(R(A, B)) -- e, e -- ±1 and we must consider the polynomials gl(x) = f(x) + e, e = ±1 and

which are both in Z[x]. Further ak = 1. tr(AB) is a zero of either g~(x) or of g2(x) and the possible values for tr(AB) are 0,1,-1,~/~,-x/~,v~,-x/~,A,-A,1A and A- 1, where 2 cos(r/5), s =A + 1. Supposefirst that k is odd. Then as in the proof of the previous theorem we must have without loss of generality gl(X) = ((X + 1)(X -- A)(X-- 1 r = (Xa -- 2X -- 1 ) r and g2(x) = ((x - 1)(x + A)(x + r = (x3 - 2x+ 1) r, r e N. This is possible only for r = 1 and then up to isomorphism the resulting polynomials can be realized only for the group G =< a, b; a2 = b3 = (ababab=)a = 1 > . This group was shown to be finite of order 1440. Wenote that M.Conder showed that G is an extension of the cyclic group Z2 by a direct product of the alternating groups At and A5. Nowsuppose k = 21 _> 4. Let y = x2 and we get g~(y) = y~ + dt_~y~-1 + ...

+ d~y + do and

188

ALGEBRAICGENERALIZATIONSOF DISCRETE GROUPS gu(y) = y~ + ct-ly ~-1 + ... +cly ~- co

with gl(Y) -g~(Y) = ±2. The possible zeros ofgl(y) and gu(y) are 0, 1, 2, 3, A2 2. or (A - 1) If As is a zero of gl(Y) then (A - ~)2 is also a zero of gl (Y) since gl Z[y]. Hence if A2 is a zero of gl(Y) then gl(y)=(y 2 - 3yT1)rh~(y),r where hi(y) E Z[y] and the possible zeros of hi(y) are 0, 1, 2, 3; and 2 and (A - 1) 2 are not zeros of g2(y)- An analogous remark holds for By using the equalities gl(Y) -g2(Y) = ±2, d~_l -- c~-i and d~_: ct_2,(the latter if l > 2) we get, in an analogous manner to that in the previous theorem, that As cannot be a zero of g~ (y) or of g2 (Y). It follows that only 0, 1, 2, 3 are the possible zeros of gl(y) and gu(y). Since g~ (y) - g2 (Y) -- ±2, we have that 0 must be a zero of one of two. Assume, without loss of generality that g~ (0) -- 0. Then 2 must a simple zero of g~(y) since gl(y) g~(y) = ±2. No w Clmust be an odd integer and hence 0 is a simple zero of gl (Y)- 3 cannot be a zero for g~ (y), again because gl(Y) g~(Y) = ±2. Si nce I _>2 we therefore obt ain g~(y) = y(y 3) t-~ and g2(Y) -=- (Y 2)(y - ~-1. This is possible only for l = 2 and hence k -- 4. polynomials gl (X) = 4 -- 3 X~ and

This leads to the

g~(x) = ~ -3x2 + 2 2. replacing y by x Up to isomorphism these polynomials can be realized only for the group G = . Weclaim that this group is infinite solvable. Let x --- ababab2ab2, y = abab2 ab2 ab, u -- ab2 ab2 abab and v = ab2 ababab2. Let H be the subgroup of G generated by x, y, u, v. H is normal in G with G/H ~- Z3 × $3 of order 18. H is center-by-abelian with commutator subgroup H~ of order less than or equal to two lying in the center of G. Hence G is solvable. G is infinite because the Euclidean triangle group T(2,3,6)----
2. If k = 1 then certainly we may asume that R(a, b) = ab so is isomorphic to the alternating group A4. Themostdifficult cases are (2, n, 2),(3, 3, 2),(3, 4, 2),(3, 5, 2) and (3, The result of J.Howie H 9 (see Theorem7.3.2.9) settles the case (3, 6, For the remainder we have only partial results depending on the length of the relator. Wefirst summarizethe situation for (3, 3, 2). THEOREM 7.3.2.5. sentation


group with pre-

G =< a,b;a 3 = ba = R2(a,b) -- 1 whereR(a,b) -- aPlbqlaP:bq2 ...aPUbqk with k _~ 1 and 1 _< Pi -< 2, 1 _< qj _< 2 for j = 1,..., k. Then if k . In this case G has order 180. (3) G -=. In this case G has order 288. PROOF.The proof is based on arguing on the syllable length 2k and then using the techniques and strategies of the previous proofs in this section. That is we attempt to find essential representations with elementary images. If the image group can only be non-elementary, then the group G must have a free subgroup of rank 2. To determine, for given syllable length k, the groups which can only have elementary images we make a detailed analysis of the trace polynomials. The case of syllable length 2, that is k -- 1, has already been handled, so we start with k = 2. Since (p, q, m) = (3, 3, 2) up to isomorphism, possible relators R(a, b) are abab, abab2 and aba2b2. The first case is the hyperbolic ordinary triangle group T(3, 3, 4) which has a free subgroup rank 2. In the other two cases we get the groups G1--

and

G2 = . Both of these groups can be shownto be finite of the given orders, either using a subgroup or a computer calculation. For example consider G1. Let N1 be the cyclic subgroup generated by (ab) 5 and let N~ be the

190


closure of a in G. An elementray computation shows that (ab) 5 commutes with both a and b and hence N1 is normal in G. Further (ab) ~5 = 1. Then G/N~ =< u, v; u3 = v3 = (uv) ~ = (uvuvg-) 2 = 1 >~- As. Now G1/N2 2< x;x 3 _- 1 > so G1 is not isomorphic to A~. Therefore G~ has order IGll = (3)(60) = 180. G2 can then be handled by Reidemeist er- Schreier method. Now consider syllable length 6, that is k -- 3 so that R(a,b) a~lbtlaS2bt2aSab t3 with 1 _< s~ _< 2, 1 . a,b;a A subgroup analysis now shows that this group contains a non-abelian free subgroup. Let H1 be the normal closure in G of ab. This has index 3 and a presentation H1 =< x, y; (xyxy2) 2 = (xay) 2 = (x-ly2) 2 = 1 > where x = aUba-~,y -- ab. Nowlet H~ be the normal closure in H~ of x. This has index 2 and a presentation H2 =< u,

V;

(u4v3)

2

= (UZV4)

2

= U2V2U3V2U2V

3

=

1 >.

The last relation implies that the words in the first two brackets are conjugates, so the second relation can be deleted. Let K be the normal closure in H2of the elements ur, uv-1. K has a transversal {1, u, u2, uz, ua, u~, u6} in H2 and is generated by zl = u~, z2 = vu-~, z3 = uvu-z, -~, za = u2vu -4, -5], -6, z5 = u3vu z6 = u4vu z7 = u5vu and z8 = u6v. If the relations z~ = z~ = z~ = Zs = 1 are added to K, the resulting factor group, after simplification, has the presentation ~ =< x,,x~,xz;x21

= x~ = x~ = (xlx~) 2 = 1 >~- 92 * Z2

where D2 is the Klein 4-group. Since this is a free product not Z2 * Z2 it follows that K is SQ-universal and therefore has a non-abelian free subgroup. Hence K and also G, has a non-abelian free subgroup. Further since K has finite index in G this argument also shows that G is SQ-universal. The final situation of where k = 4 so syllable length 8. This handled in much that same manner. Up to isomorphism all groups have essential non-elementary representations with two exceptions. The first is the group G1 =< a, b; a3 = b3 = (ab2aZbZaba2b)2 = 1 > .

7.3.2 THE TITS ALTERNATIVE- GENERATORSOF FINITEORDER

191

The normal closure H1 of b in G1 has index 3 and has a presentation H1=< x, y, z; x3 -- y3 _- z3 = Ix, y]2 = Ix, z]2 = [y, z]~ --- 1 > The normal closure K of z, yx in H~ then has index 3 and is generated by al ~ yx, a2 = xy, a3 -~ x-lyx-l~ a4 ~ z~ a5 -~ xzx -1 and a6 = -I. x--lzx Adding the relations a~ = 1 and a~ = a3 to K we get a factor group with a presentation

= (a4a~l) 2 = (alasala-~l) 2 = (alahala~l) 2 = 1 >. In K let H be the normal closure of a4,ah,a6. This has index 2 in K and is generated by xl = a4, x2 ~ ah, x3 -- a6, x4 -- ala4al, x5 -= alahal, and x6 = alasal. Adding to its presentation the relations xl = x2 = x4 = x5 gives the factor group ~ =< Xl,

Z3,

X6;

Xl 3 -~-

X33 = X6 3 = (XlZ3) 2 = (TlX6) 2 = 1 >.

This is the free product of the groups < x~,x3;x3~ = x~ = (xlx3) ~ = 1 > and < xl,x6;x~ -- x~3 : (XlXs) ~ : 1 > with the subgroup < Xl;X~ : 1 > amalgamted. From a result of Lossov [Los] H is SQ-universal and hence it follows that G~ is also SQ-universal. From this it follows that G1 contains a free subgroup of rank 2. The second exception is the group G~ =. In this case the normal closure presentation

H1 of in G~ has index 3 and a

H~ =< x,y,z;x 3 = y3 = z3 = (xyxz2)~ = (yzyx2)~ = (zyzy2)2 = To show that H~ has an essential non-elemenmtary representation in PSL2(C) it suffices to find a map p : H~ -* PSL2(C) with p(x) = p(y) = Y, p(z) su chthat the t race s of X, Y and Z are one, a nd < X, Y > is non-elementary, and the values t~ = tr(XY), t2 = tr(XZ), t3 = tr(YZ) are non-zero solutions of the system of equations

t2-t2t3-$1~-O,

192

ALGEBRAICGENERALIZATIONSOF DISCRETE GROUPS t3 -- tatl -- t2 =O.

The solutions of this system are the three (different) zeros of the polynomial f(t) = 3 - 3t2 + 3, irreducible over Q. All are real and one of them, which may assume to be tl, is greater than 2. In particulax, it follows that if ( is chosen so that ( + ~1 = tl, then with X’=( 0-1

II)’Y----(,ll

~)’

the group < X, Y ) is non-elementary. Finally, in a straightforward manner, we may construct a matrix

directly to satisfy the conditions tr(Z)

= a+d =

det(Z) = ad - bc = tr(XZ) = c - b + d = t2 and tr(YZ) = a - ~c + -1 = t3 . Hence, H~ and G2 have a f~ee subgroup of rank two. The eases (3, 4, 2) and (3, 5, 2), (also (3, 6, 2) although this is covered [H 9]) can be handled in an analogous manner. For syllable length less than or equal to eight the Tits alternative holds. A finite group can only occur when the syllable length is two, and the group is isomorphic to the symmetric group $4 or the alternating group As. The details are lengthy and can be found in [R 16,20] and [L-R 1,2]. THEOREM 7.3.2.6. sentation


group with pre-

G =< a,b;a 3 = bq-- R~(a,b) = 1

with q = 4, 5 or 6 and where R(a, b) = aPlbqlaP2bq2.., aP~bq~ with k >_ 1 and 1 2 and 1 < qj < q for j = 1,...,k. Then, if q > 7, q ~ 8, 10, 16 then G has a free subgroup of rank 2. We now present Howie’s Theorem [H 9] which settles the case where s(G) = 1. The proof uses some similar techniques to the preceding proofs, some of the preceding results and a general result (also proved in [H 9]) concerning free subgroups of the fundamental group of a finite 2-complex. For the details we refer the reader to the paper. Equivalence of R in the theorem is equivalence under the equivalence relation generated by automorphisms of the cyclic groups generated by a and b, inversions, cyclic permutations of R(a, b), and the interchange of a and b in the case where p=q. THEOREM 7.3.2.9. presentation

[H 9] Suppose G is a generalized triangle group with G =< a, b; ap = bq = Rm(a, b) = 1

where 2 2 and let s(G) = (l/p) + (l/q) + (l/m). If s(G) = 1 then G contains a free

194


subgroup of rank 2, except in the case where the relator R is equivalent to ab in which case G is isomorphic to the solvable Euclidean triangle group T(p, q, r). In closing this section we give the following conjecture. CONJECTURE. The Tits alternative holds for any generalized group where the generators have finite order. 7.3.3

The Tits Alternative

o Generators of Infinite

triangle

Order

Wenow consider where one or both of the generators has infinite order, that is where p _> 2, q = 0 or p = q = 0. Where both generators have infinite order, that is where p = q = 0, the theorem of Ree and Mendelsohn guarantees a non-abelian free subgroup. Recall that for a 2-generator group G a generating pMr {u, v} is a Ree-Mendelsohn pair or RMopair if there exists an integer t such that < u*, v* > is a free subgroup of G of rank 2. Clearly the existence of an RM-pair shows the existence of a non-abelian free subgroup. In this section we interchange the role ofp and q and consider a to have finite order and b to be of infinite order. THEOREM 7.3.3.1. sentation


group with pre-

G =< a,b;a v = R’~(a,b) = 1 where 2 < p, 2 _3 or m >_3 then G contains an P, AYl-pair. (2) Suppose p = m = 2 so that = an suppose R(a,b) = ab’~labn2...ab "k with k >_ 1, hi # O,i = 1, ...,k. Then (a) H k >_ 2 or Ini[ >_ 3 [or some i, then G has a free subgroup of rank 2. In particular ilk is even or k is odd, k >_3 and n l + n2 +...+ nk ~ O, or k = 1 and [nl[ _> 3, then G has an RM-palr. (b)/_fk = 1 and [nl[ _< 2, then G is infinite and solvable. PROOF.We consider first case (2) of the theorem where q -- m -Suppose that G =< a,b;a 2 = R2(a,b) = 1 with R(a, b) = ab’~labTM ..... ab’*~, k >_1, and ni # 0, i = 1, ....k. Wewant to show that (1) Ilk >_ 2 or [ni[ _> 3 for somei, then G has a free subgroup of rank (2) If k = 1 and In1[ _< 2, then G is infinite and solvable. These results will be a consequence of the following technical lemma.

7.3.3 THE TITS ALTERNATIVE - GENERATORS OF INFINITE ORDER195 LEMMA 7.3.3.1. Let G be as above. Suppose one of the following holds: (1) k is even (2) k is odd, k >_3 and nl + n2 + .... + nk ~ 0 (3) k: 1 and]nil >_3 Then G contains a Ree-Mendelsohn pair. PROOF.(of Lemma) Suppose first that k is even. Then R(a, b) gT(a, b) with t E Z and T(a, b) in the commutator subgroup of the free product on a, b. Now choose

and as in the proof of Freiheitssatz choose a w0 so that tr(R(A, B)) 2 cos(r/2) -- 0 so that < A, B > represents Further we can choose w0 so that A and B have no commonfixed points. To see this notice that A has the fixed point zl = i and z2 = -i while B has only the fixed point z(wo) = w0 - 1. If A and B have a commonfixed point, then the group < A, B > is metabelian. Under a suitable conjugation by C ~ PSL2 (C) we obtain

CBC-~ = B~ -- :k

1 "

Because R(a, b) = b~T(a, where T(a, b) is in thecomm utator subg roup in the free product on a and b we must have R(AI,B1)= :k (1

~) for some c ~ C.

But R(A1, B~) has order 2 in < A~, B1 >, so this is a contradiction. Therefore A and B have no commonfixed points. Further ~r(AB) ~ O. Otherwise tr([A,B]) = tr(ABA-1B -1) -~ (tr(AB)) q- 2 = 2 which would imply that A and B must have a commonfixed point. Since tr(AB) ~ and A, B have no common fix ed poi nt, exa ctly the same argument as was used in the proof of Theorem 7.2.1 can be used to show that G has a Ree-Mendelsohn pair. This handles the case when k is even.

196


Nowconsider the case whenk is odd, k _> 3 and nl Choose two projective matrices A:±(_01

~)andB---±(~

+ n2 ~- ......

~- nk ~

O.

~)withweC

with no commonfixed point. By induction we obtain tr(R(a, b)) = ±(aaw~ ÷ .......

÷ alw) with

ak =--nln2 ...... n~ ~ 0 and al -~ (-1)t+l(nl -~ ......

nk) ~ 0, k --- 2t+ 1.

Since tr(R(a, b)) is a non-constant polynomial in w of degree _> 3 with al ~ 0 we may choose w0, not equal to 0, for B such that tr(R(A, B)) Then R(A, B) has order 2 in < A, B > which then represents G. The argument used in the first case now goes through. Finally suppose that k = 1 and Inll _> 3. Then G has the form < a,b;a 2 = R2(a,b) = 1 > with R(a, b) = ’~1. This ha s as a f actor gro up ~--

with Inll >_ 3. This is a free product of cyclics and not infinite dihedral and therefore ~ has a Ree-Mendelsohn pair and hence so does G. This completes the proof of the lemma. Wecan now complete the proof of case (2) in the theorem. If k _> then the result follows from the lemmaif either k is even or k is odd with n~ + .... + nk ~ O. Supposethen that k is odd, k _> 3 and n~ + ... + nk = O. Let a~ = b-~ab~, for i ~ Z. Since the exponent sum of b in R(a,b) is zero, it follows that R(a, b) can be written as a word T on (ai}. Let n, M be the minimum and maximumi such that ai appears in T. Then G can be written as an HNNgroup G =

This has free part generated by b and associated subgroups K1---< am,....... aM-1;a2m ..... K2~--~ am+l~...... 2aM; a

a2M_l = 1 > a2M---- 1 ~

7.3.3 THE TITS ALTERNATIVE - GENERATORS OF INFINITE ORDER197 Since k _> 3 it is clear that G has a free subgroup of rank 2. Nowconsider the case when k = 1. If Inll _> 3 the result follows from the lemma. If Inll = 1 then G =< a,b;a 2 = (ab) 2 = 1 > . This is infinite dihedral (Z1 * Z2) and solvable. Finally if In~l = 2 then G has the form < a,b;a 2 = (ab2) 2 = 1 >. It follows that ab2a = b-2 and so < b2 > is a normal subgroup of G. Further the factor group G/< b2 >-~< a, b; a 2 = b2 = 1 > which is infinite dihedral and therefore solvable. Then G is solvable since both < bu > and G/< b2 > The above completes the situation when p = 2 and m = 2. We note however that the technique we used of representing the group G in PSLu (C) with tr(B) = does no t gi ve th e desired re sult in thecasewherek is o dd, k >_ 3 and n~ + ... + nk = O. To see this consider the group < a, b; a2 = (ababab-2) ~ = 1 >. This group has a free subgroup of rank 2 but the image group < A, B > with tr(B) = is sol vable und er any repr esentation a -~ A, b -* B of G in PSL:(C). Wenow turn to case (1) of the theorem where either p >_ 3 or m _> Wehandle m _> 3 first and then consider the situation where p _> 3 and m=2. LEMMA 7.3.3.2. Suppose G =< a, b; ap = R’*(a, b) = 1 > with p >_ m >_ 3 and R(a,b) a cyclically reduced word in the ~ree product on and b involving both a and b. Then G contains a Ree-Mendelsohn pair. Pr~oor. (of Lemma)The proof is similar to the proof of the previous lemma. We find a homomorphic image of G in PSL2(C), G =< A, B such that A and B have no commonfixed points. The arguments used in the preceding results then carry through. As in the proof of the previous lemmachoose A--+(

0_1

~)withy:2cos(r/p)and

where wo E C so that tr(R(A, B)) = :t:2 cos(r/m) so that < .4, B > represents G. As in the proof of the previous lemmaw0 can be chosen so that A and B have no colnmon fixed points. To see this notice that A has the fixed points zi = cos(r/p) + i sin(r/p) and z2 = ~- = cos(r/p) - i sin(r/p), has the lone fixed point z(w) = w- 1. Choose w0 such that tr(R(A, B)) 2cos(r/m) as in the previous proofs, w0 is then a zero of a real polynomial, f(x) - 2 cos(r/m) with f(x) non-constant in x. Since this is a real polynomial ~-5 is also a zero.

198

ALGEBRAIC GENERALIZATIONS OF DISCRETEGROUPS

If z(wo) ---- Wo-- 1 # Zl and z(wo) ~ z2, then A and B have no common fixed point. Suppose then that Zl = wo- 1. Wewill then find a Wl such that R(A, B) still has the appropriate order and such that Z(Wl)¢ zl z2. An analogous argument would work if z2 = w0 - 1. To do this choose wl so that tr(R(A, B)) = -2 cos(r/m). Then R(A, B) still has order m in PSL2((~) and so < A, B > still providesa representation of G. Further Wlis a zero of the real polynomial f(x) + 2 cos(r/m) with the same f(x) as before. Then Zl # w~- 1 since zl = Wo- 1 and Wo~ wl. To see that w0 ~ wl suppose they were equal. Then wo would be zero of the polynomial f(x) + 2 cos(r/m) - (f(x) - 2 cos(r/m)) = 4cos(r/m). Howeversince m > 3, 4 cos(r/m) # 0 and so Wl # W0. Suppose then that zz = wl - 1. Then wl - 1 = z2 = Zl = w0 - 1. Thus Wl= w--6 and so w~is a zero of the polynomial

f(z) + 2cos(r/m) - (f(x) - 2 cos(=/m)) = acos(r/m) as before, giving a contradiction. ThereforeZ1 # Wl -- 1 and z2 # I/)1 -- 1 and therefore we can find A and B having no commonfixed points and providing a representation. The proof nowproceeds as in Theorem7.2.1. Weconsider the product AB. Suppose AB is loxodromic (see Chapter 4). Then BABis also loxodromic since tr(BAB) = 2tr(AB) - Now {AB,BAB} is a gener ating pair for < A, B >. Wewill show that these form a Ree-Mendelsohnpair for < A, B > - that is there exists a t e Z such that {(AB)t, (BAB)t} is a basis for a free subgroupof rank 2 in < A, B >. Since these are the images of a, b ~ G, it wouldfollow that a, b constitute a Ree-Mendelsohn pair in G. There exists a positive integer s such that [tr(BAB)8[ > 2 and [tr(AB)S[ > 2. It follows, as in Theorem7.2.1, from the work of Majeed [Maj] that there is a positive integer t such that {(AB)t and t} (BAB) is a basis for a free subgroupof rank 2. Now suppose AB is non-loxodromic. Since p > 2, m > 3 and A and B have no commonfixed point it follows that AB cannot be elliptic of order 2. Then after a suitable conjugation we may assume that < A, B > c PSL2(R). The result then follows from the work of Rosenberger [R 3,13]. This completes the proof of Lemma 7.3.3.2 and the case where m > 3. The final lemmahandles the case wherep > 3 and m = 2. Its proof will complete the proof of Theorem7.3.3.1.

7.3.3 THE TITS ALTERNATIVE - GENERATORS OF INFINITE ORDER199 LEMMA 7.3.3.3. Suppose G =< a,b;a p = R2(a,b) = 1 > wi~h > 3 and R(a, b) a cyclically reduced word in ~he free produc~on a and b involving both a and b. Then G contains a Ree-Mendelsohn pair. P~tOOF. (of Lemma)The proof is similar to the arguments used in the proofs of the previous two lemmas. Weconsider several cases and in each case we find a homomorphic image G =< A, B > in PSL2(C) such that and B have no commonfixed points. The arguments used in the preceding results then carry through. Wefirst handle the case when p is not a power of 2. Case (1) p -- 2~(2k ÷ l) with k _> 1. It is enough to prove the statement for p = 2k ~ 1. Consider a homomorphicimage if necessary. Assume then that p = 2k q- 1 _> 3. Choose A = :k ( 0-1 ~1with r = 2c°s(~r/p)

B=B(w)--5:(~

2w-w~-l) 2- w

and

with w E C.

A has fixed points zl = cos(w/p) q- i sin(r/p) and z2 = ~ = cos(r/p) i sin(~r/p). B has the lone fixed point z(w) = w- 1. Choose w0 such that tr(R(A, B)) -- 2 cos(~r/2) = 0 as in the previous proofs. If A and B a commonfixed point, then < A, B > is metabelian. Conjugating by a suitable C ~ PSL2 (C) we obtain -~ = A~ --- qCAC

0)

q-1 with q -- z~ or q = z2 and

CBC_l:Ul:q_( Then R(A1, B1) + q-

¯

~ 1) 1 "

X fo r so me c e C an d

x e Z with Because R(A1,B~) has order 2 in < AI,B1 > we have either x > 0 or x = 0, c ~ 0. This leads though to a contradiction since q2X ~ +1 if x > 0 because p = 2k + 1 _> 3 and 2c ~ 0 if x = 0. Therefore A and B have no commonfixed points and the previous arguments concerning free subgroups carries through.

200


If p is a powerof 2 we must consider several different forms of the relator R(a, b). Further if p = 2v it is enoughto prove the statement for v -- 2 that is p = 4. Then: Case (2). p = 4 and R(a, b) = aTbST(a, with r, s EZ,0 < r < 3,s ¢ 0 and T(a, b) in the commutator subgroup of the free product on a and b. Choose

1

6-w

,wEC.

A has the fixed points Zl = cos(r/4) + i sin(~r/4) and z2 = z-~ cos(Tr/4) is in(r/4). B has th e fi xed po ints za (w) = w - 3 + ~ an z4(w) = w-3- vr~. Nowchoose w0 so that tr(R(a, b)) = th at is so tha t < A, B > represents G in PSL2(C). Suppose A and B = B(wo) have at least one commonfixed point. Then the group < A, B > is metabelian and tr([A, B]) = tr(ABA-1B-1) = 2. Conjugating by a suitable C e PSL2(C~) we obtain CAC_I=A,=+(q

0 with q = Zl or q = z2 and 0 q-1

CBC-1 ~- BI = +

t_ ~

with cl e (~ and t -- 3 + x/~ or t = 3 - v~. Then under this conjugation the relator R(A~, B1) must have the form R(A~,B~)

+ 0 q- rt-s "

Because s ¢ 0, R(AI, B~) cannot have order 2 in < A~, B~ > which is a contradiction. Therefore A and B = B(wo) have no commonfixed points and the method of the previous cases carries through. Case (3). p -- 4 and R(a,b) = aTT(a,b) with r ~ Z, r ¢ 2, 0 < r < 3 and T(a, b) in the commutator subgroup of the free product on a and b. As before choose A=+

-1 x (0 1) with B = B(wo) +

x= 2cos(r/4)

--

V~ and

Wo- 1) (~o 2wo- 2-Wo~

7.3.4

THE FINITE GENERALIZED TRIANGLEGROUPS

201

with Wo~ C chosen so that < A, B > represents G - that is R(A, B) has order 2. NowB has only Z(Wo) = Wo- 1 as a fixed point. If A and B had this as a commonfixed point then < A, B > is metabelian and a suitable conjugation would lead to a contradiction on the relator R(a, b). Therefore A and B would have no commonfixed points and we argue as in the previous cases. Nowthe final case. Case (4) p = 4 and R(a, b) = a2T(a, wit h T(a, b) in the comm utator subgroup of the free product on a and b. Since p = 4 the group ~ =< ~,~;~2 = R2(~,~) = 1 >=< ~,~;~2 T2(~, ~) = 1 > is a homomorphicimage of G with T(~, ~) in the commutator subgroup in the free product on ~ and ~. If T(~, ~) = 1 then ~ = 7,2 * and the statement of the lemma holds for G and therefore also for G. If T(~, ~) # 1 then T(~, ~) is conjugate to a cyclically reduced word TI(~,~) involving both ~ and ~. The result then follows from Lemm.a7.3.3.2. This completes the proof of Lemma7.3.3.3 7.3.3.1.

and the proof of Theorem

Summarziugthe statements in Theorems 7.3.2.1 to 7.3.2.6 7.3.3.1 we get our main result. THEOREM 7.3.2. tion


triangle

and Theorem

group with presenta-

G =< a, b; ap = bq = Rm(a, b) = 1 where p _2 or p = O, q >_2 or q = O, R(a, b) is a cyclically reduced word in the free product on a and b involving both a and b and m >_ 2. Then G satisfies the Tits alternative except possibly whenp >_2, q >_ 2, m = 2, (l/p) + (l/q) _~ and the relator R(a, b) has s ylla ble lengt greater than 8 in the free product on a, b. 7.3.4

The Finite

Generalized

Triangle

Groups

From the preceding section we see that a generalized triangle with presentation

group G

G =< a,b;a p = bq = R’~(a,b) = 1 with 2 _< p _< q is infinite if s(G) -- (l/p) ÷ (l/q) + (l/m) _< 1. For ordinary triangle groups (see section 7.3.1) this is enoughto characterize the finite groups - that is T(p, q, m) is finite if and only if(1/p)~-(1/q)+(1/m) 1. Howeveras we can see from the results in the previous section there are examplesof infinite generalized triangle groups for every possible triple with

202


q _> 3. This raises the question of whether the finite generalized triangle groups can be completely classified. The answer is yes and the classification was carried out by Howie, Metafsis and Thomas(see [H-M-T1]) and Levai, Rosenberger and Souvegnier [L-R-S]. The first three people determined an almost complete list which classified all the finite groups except with two presentations undecidable. The second three people determined that one of these possibilities was infinite while the other was finite of large order. This final group has been called the LRS monster. The classsification proceeds in the following manner. Let G =< a,b;a p = bq = Rm(a,b) = 1 with 2 1. As in the previous section, if G were to be finite, leaves only the followingpossible triples: (2, 2, n), (2, n, 2), (2, 3, 3), (2, (2, 4, 3), (2, 3, 5), (2, 5, 3), (3, 3, 2),(3, 4, 2) and (3, 5, 2). Suppose that R(a,b) = ~’lbql . ..a p~bq~ with k > 1 and 1 _ would be a finite subgroup of PSL~(C) generated by an element of order p and element of order q. As we used in the last section, it is knownthat the only finite subgroups of PSL~ (C) are up to isomorphism the cyclic groups for all n E l~l, the dihedral groups D,~ for all n E N, the alternating groups A4 and A5 and the symmetric group $4. Thus if G were finite, we know the possible structures of the image under the essential representation p. Recall that we can choose A,B ~ PSL~(C) such that tr(A) = 2cos(~r/p) and tr(B) -- 2 cos(w/q). It follows as before that tr(R(A, B)) is a polynomial of degree exactly k in x -- tr(AB) where k is as above, tr(R(A, B)) --- f(x) is

7.3.4


203

the trace polynomial for G. Since the possible finite subgroups of PSL2(C) are knownthis gives a finite list of possible values for tr(AB) and hence a finite list of possible values for the roots of the trace polynomial. Define a polynomial

= II(f(x) where the product is taken over all primitive m-th roots a of -1 with nonnegative imaginary part. Thus, if m -- 2, 3, 4, 5 then a(x) is respectively f(x), (/(x))2 _ 1, (f(x)) 2 - 2 or (f(x)) 4 -- 3(f(x)) 2 ÷ 1. Then a(tr(AB)) -- 0 if and only if tr(R(A, B)) --- a ÷ -1 fo r so me primitive root a of -1, which is the case if and only if R(A, B) has order m in PSL2(C). Howie, Metafsis and Thomas [H-M-T 1] then proved the following. LEMMA 7.3.4.1. Let G be a generalized triangle group, f(x) its trace polynomial and a(x) the polynomial as defmed above. If ~r(x) has a multiple root, then G is infinite. This lemma, whose proof we will refer to the paper [H-M-T 1], provides a bound on the degree of the trace polynomial and hence a bound on the length of the relator R(a, b). These reductions leave only a finite numberof possibilities in the (2, q, 2), (3, 3, 2), (3, 4, 2) and (3, 5, 2) tions to check. Building on the previous results of Baumslag,Morganand Shalen [B-M-S], Conder[Co 1] and Fine, Levin and Rosenberger [ F-L-R 1], [R 16,20],[L-R 1,2], Howie, Metafsis and Thomas [H-M-T 1] using a mixture of representation-theoretic, geometric group theoretic and computational group theoretic techniques identifed eleven finite generalized triangle groups (with k _> 2), numbers(1) through (11) in the classification theorem and showed that there were only two other possibilities,(12) and a group G1. Levai, Rosenberger and Souvignier [L-R-S] were able to show that G l was infinite and that the group (12) was finite. The final group (12) determined to be of order 424673280 = 220345 using the computational group theory system GAP. Howie, Metafsis and Thomas have called it the LRS-Monster and have studied some of its subgroups. Wenow list the classification. THEOREM 7.3.4.1. Let G be a finite generalized triangle group and not an ordinary triangle group. Then G is, ~p to isomorphism, one off (I) < a, b; 2 =ba= (abab2)2 = 1>~- A4 xZ2 oforder 24; (2) < a,b; ~ -- b 3 :(ababab2)2 = 1 > oforder 48;

204


(3) < a,b;a 2 = b 5 = (abab2) 2 = 1 >~A~ xZ2 o/:order 120. This is ~ -- 1 >; isomorphic to the group < a, b; a~ = b3 --- (abababab~) ~ 3 2 (4) < a, b; =b = ( abab2) = 1>~A5x Z3 of order 180; (5) < a,b; ~ =b4= ( ababab3)~ = 1> of order192; (6) < a, b; ~ =b3= (aba~b~)2 = 1> oforder288; ~ --- 1 > oforder 576; (7) < a, b; 2 -- b 3 --- ( ababab2ab2) 2 3 ~ = 1> of order720; (8) < a, b; =b = (ababab~abab~) 2 5 2 (9) < a, b; =b = (ababab4) = 1:>-- -< a, b ; a 2 =b5 = (abab2ab4)~ = 1 > of order 1200; (10) < a,b; ~ =b~= (ababab~)a = 1> of order1440; (11) < a,b; ~ =ba= ( ababababab2ab2)2 = 1> oforder 2880; (12) < a,b;a 2 = b3 = (abababab2ab2abab~ab~) ~ = 1 > of order 424673280-- 220345. PROOF.Weshow that the list in the statement of the theorem is complete up to equivalence under the equivalence relation generated by tomorphisms of the cyclic groups generated by a and b, inversions, cyclic permutations of R(a, b), and interchange of a and b in the case where p -- q. That is we show that any generalized triangle group presentation leading to a finite group is equivalent to one of the 14 presentations in the list. Nowsuppose that G is a generalized triangle group with presentation G--< a,b;a p -- bq -- Rm(a,b) = 1 where 2 1 and 1 2. From Theorem7.3.2.3, if G were to be finite we must have m _< 3 so we can restrict to the triples (2, q, 2), (3, 3, 2), (3, 4, 2), (3, 5, 2) and (2, 3, q_>3. The case (2, 3, 3) was covered by Theorem7.3.2.4. Here the only finite group, up to isomorphism, is G =< a, b; a~ = ba = (ababab2)~ = 1 > which has order 1440 and is number (10) on the list. In the case (3, 3, 2) a check of the possible finite groups in PSL2(C) together with the fact that the trace polynomial must have only simple

7.3.4


205

roots by Lemma7.3.4.1, if G were to be finite, shows that k _< 4. This was then handled in Theorem7.3.2.5 where it was shown that the only two finite groups are G =< a,b;a 3 = b3 = (abab~) 2 = 1 > which has order 180 and is number (4) on the list and G =< a, b; a 3 = b3 = (aba2b2)~ = 1 > which has order 288 and is number (6) on the list. In the cases of the triples (3, 4, 2) and (3, 5, 2) from Lemma7.3.4.1 also that 2 < k < 4. There situations with 2 _< k _< 4 were covered by [R 16,20] and [L-R 1,2,3] where it was shownthat there were no finite groups. Weare nowreduced to the cases (2, q, 2) with q _> 3. Supposeq _> 6, then G and hence an essential image of G in PSL2(C) will contain an element of order q _> 6. The only finite non-cyclics subgroups of PSL2 (C) having elements of order greater than 5 are the dihedral groups. This implies that if < A, B > provides an essential image of G, then the only possible value for tr(AB) = 0. Hence the trace polynomial has degree 1 and R(a, b) has syllable length 2, that is up to equivalence R(a, b) -~ abk. If gcd(k, q) 1 then this reduces to an ordilaary triangle group while if gcd(k, q) ?~ Theorem 7.3.2.2 shows that G must be infinite. It follows that if G were to be finite we must only nowconsider the triples (2, 3, 2), (2, 4, 2), (2, 5, These are completed by the following three lemmas. LEMMA 7.3.4.2. Suppose G is a generalized triangle group with triple (2, 5, 2) and relator R(a, b). Then G is finite only if, up to the equivalence as defined at the beginning of the proof,, (i) R(a, b) = ab in which case G is an ordinary triangle group o[ orderlO. (2) R(a, b) = ~ in w hich case G hasorder 120. T his i s number(3) on the list. (3) R(a, b) ~ ababab4 in which case G has order 1200. This is number (9) on the list. (4) R(a, b) abab~ab4 in which cas e G a ls o has orde r 1200. This is number (9) on the list. R(a, b) aSabab4 and R( a, b) = aba52ab4 def ine iso morphic groups. PI~OOF. (of Lemma7.3.4.2) Since G has an element of order 5, if were finite, an essential image must be either the dihedral group D5 or the alternating group As. Suppose G --< A, B > is a finite essential image of G. If G - D5 then tr(AB) ~- while if G --- A5 the poss ible valu es for tr(AB) are :1:1 or q_ 1-~_~. (Not q- ~+~-~since < A,B > is finite). It follows that the trace polynomial has degree at most 5 and so R(a, b) has syllable

206


length at most 10 - that is k _< 5. If k _< 4 this has already been handled in the the previous section giving the finite groups on the list. If k -- 5 and G is finite, then all possible roots occur in the trace polynomial. A computer search (see [H-M-T1]) then shows that the only possible relators are then R(a, b) = ababaab~ab4abk,1 _< k

where e~ = 0 or e~ _> 2 for i = 1,..,n, mj >_ 1 for j = 1,...,k and each R~ is a cyclically reduced word in the ~ product of the cyclic groups < a~ >, ..., < a~ >, of syllable len~h at least two ~nd suppose p is an essemial representation ~om G into a fi~te dimensional linear ~oup so that for each i = 1, ..., n, p(a~) has i~nite order if ei = 0 or exact order e~ if e{ ~ 2 and for each j = 1, ..., k, p(R~) has order m~. If the conjugscy cla~es of torsion elements in G ~e precisely given by the powers of the generators (if of fi~te order) or the ~wers of the other relators, then ~sential representation is essemially faithS. ~om Theorem 6.3.2 - the ~eiheitssatz for on,relator products of cyclics - a on,relator product of cyclics with proof power relator always ad~ts an essential representation. O~ first result is the following. THEOREM 7.4.1. Let G =< a~,...an, ¯ a~ ..... where n,m ~ 2, e~ = 0 or e{ ~ 2 [or i = 1,..n,and U = U(a~,..ae),V = V(a~+~, ...an), are non-tri~l words i~ $hc ~ee prod~c~s o~ a~, ...,a~ and ae+~,...,a~ respectivel~ for some 1 ~ p ~ n - 1. Then G is virtually torsion-[r~. PROOF.The proof depends on the follow~g lemma w~ch character~es the conjugacy closes in G. LEMMA 7.4.1. Le$ G be as i~ Theorem Z 4.1. Then any element order is conj~e to ~ power of UV or of some a~. PROOF.(of lena) Without loss of generality we can a~e that UV involves all the generators and that UVis cyc~cally reduced. Assume first n = 2. Then G has the form G =< a~,a~;a~ ~ = a~ (a~a2) = 1 >. We may assume that s,t ~ 1 and by sy~etry, since (UV) m = 1 an~VU)~ = 1 ~e equiwlent that s ~ t. We m~y also ~e t~t s is a divisor of e~ if e~ ~ 2 and t is a divi~r of e2 if e2 ~ 2. If s = t = 1

210


then G is an ordinary triangle group and the fact that G is virtually torsionfree follows from the Fenchel-Fox Theoremin the hyperbolic case, from the fact that its virtually free abelian in the Euclidean case and from being finite in the spherical case (see section 7.3.2). If s _> 2 then s is a nontrivial divisor of el if el _> 2. NowG is the free product of H1 =< al; a~’ = 1 > and H2 --< a2,a;a~ 2 = ae’/8 = (aat2) ’n --= 1 > with < a > and < a~ > amalgamated. Since elements of finite order in a free product with amalgamation are conjugate to elements in the factors the result follows. Nowsuppose n >_ 3. By hypothesis one of U, V has infinite order. Let a = UV and suppose without any loss of generality that p _< n - 2. Then G can be expressed as a free product with amalgamation G =/’/1 *n/’/2 where H~=< a~, ..... %, a; a"~ = a~~ =. .... = a~,~ = 1 > > ~- _e,~=l a H2-----< ap+l, .... an. ~ %+1 ap+ 1 :- ...... n and H =< U-la;

>=< V; >.

The result follows since both factors are now non-trivial free products of cyclic groups. This completes the proof of the lemma. The proof of Theorem 7.4.1 now follows directly. Let G be as in the theorem. Then G admits an essential representation into PSL2(C). Since the conjugacy classes of torsion elements in G are precisely given by the powers of the generators and the powers of the single other relator then this essential representation is essentially faithful. G is then virtually torsionfree from Theorem 6.2.1. Results of Collins and Perraud [C-P] and J.Howie [H 3,4] allow us to extend this to further one-relator products of cyclics with a relator of order at least 4. THEOREM 7.4.2. Let G =< al, ...an; a~1 ...... ae~ " = Rm(al,...an) 1 > with m >_ 4,n > 2 and e~ = 0 or e~ _> 2 for i = 1, ...n and R(a~, ..,an) a cyclically reduced word in the free product on al, .., an which involves all the a~. SupposeR(al, .., an) is not conjugate in the free product on a~, .., a~ -1 for some word U and 2 elements X and Y to a word of the form XUYU ofordersp >_ 2,q > 2 respectively where (l/m) + (l/p) + (l/q) Then G is virtually torsion-free. PROOF. First suppose m > 6. Then the symmetrized closure of R(al, .., an) satisfies a small cancellation condition. If R is not conjugate -1 for some in the free product on ax, ..,a,~ to a word of the form XUYU word U and 2 elements X and Y of orders p > 2, q > 2 respectively where

7.5 FREE PRODUCTWITH AMALGAMATION DECOMPOSITIONS 211 (l/m) -t- (l/p) -b (l/q) > 1, "~ is not conjugate in the fre e pr oduct on al, .., an to its own inverse. From results of Collins and Peraud [C-P] this insures that every element of finite order in G must be conjugate to a power of R or a power of some ai. The proof then proceeds exactly as in theorem 7.4.1. If m -- 4 or m -- 5 the result follows from analagous work on small cancellation diagrams done by J. Howie [H 3,4] which also allows the classification of elements of finite order. As a direct corollary we get. COROLLARY 7.4.1. Let G be a one-relator product of cyclics with relator Rm(al, ...a,,). If m >_ 4 and each a~ has odd order then G is virtually torsion -free. Using identical proofs we can give the following extensions of theorems 7.4.1 and 7.4.2 to one-relator products of groups which admit faithful representations in PSL2 (C). THEOREM 7.4.3. Let G = (A * B)/N(R "~) where A and B are finitely generated groups which admit faithful representations in PSL2 (C) and is a cyclically reduced word in the free product A, B of syllable length >_2. (i) If m >_ 2 and A and B are locally indicable, then G is virtually torsion-free. (2) If m >_ 2 and R = UV where U is a non-trivial element of A and is a non-trivial element orB, then G is virtually torsion-free. (3) If m >_ 4 and R is not conjugate in A ¯ B to a word of the form -~ for some word U and 2 elements X, Y of respective orders p >_ XUYU 2, q >_2 with (l/m) -b (l/p) q- (l/q) then G is virtu ally torsi on-free. From work of Duncan and Howie [D-H 1,2] Theorem 7.4.2 and 7.4.3 can be extended to the ease where m -- 3 and the relator R(ax, ..., an) contains no letter of order 2. Weclose section 7.4 with the following conjecture. The Fischer-KarrassSolitar Theorem answers it in the affirmative for free groups while the results in this section give somefurther supporting evidence. CONJECTURE. Let A and B be finitely generated groups which admit faithful representations in PSL2 (C). Let R be cyclically reduced in A ¯ B of syllable lengfh >_ 2. Then if m >_ 2 the group G = (A * B)/N(R’*) virtually torsion-free. 7.5

Free Product with Amalgamation Decompositions

Wenow consider the property of being a non-trivial free product with amalgamation. Via the Poincare presentation any Fuchsian group with at

212


least (algebraic) rank 3 admits such a decomposition and as we saw Corollary 3.4.2 any one-relator group with at least three generators also admits such a decomposition. The reason that amalgamconstructions can be considered as linear properties is through the variety of representation techniques of Helling and Culler and Shalen Werepeat the brief description given in Chapter 3. Suppose G is a finitely generated group with generators gl, .-., g,~- Any representation p : G -~ SL2(C) can then be considered as a point (p(gl), ...,p(gn)) (SL2(C))’~. These re presentation po ints wi ll be subjected to various conditions reflecting the relations in the group. These relations are polynomial relations on the matrix entries and hence the set of all possible representation points for G define an affine algebraic set in C4’~. Call this R2(G). SL2(C) acts on R2(G) by conjugation in the natural way. Thus we can form the categorical quotient X2(G) of Ru(G) under this action. X2 (G) is called the at~ine algebraic set of characters or character space of G. It can be thought of as the parametrization of the inequivalent semi-simple representations of G in SL2(C). Culler and Shalen then proved that (Theorem 3.4.7) if G be a finitely generated group and if the dimension of the character variety of G is positive then G decomposes as either a non-trivial free product with amalgamation or as an HNNgroup. Thus if there are "many" representations into (SLu(C)) then the group must decompose as an amalgam. Using this Culler and Shalen result we can prove that any one-relator product of cyclics with proper power relator and at least 3 generators must be a non-trivial free product with amalgamation. 1 ...... THEOREM7.5.1. Suppose G =< al,....an;a~ a~ ~ = Rm(al, ...an) = 1 >. Hm>_2,n >_ 3 and e~ = 0 or e~ >_2 for all i = 1, ...,n, then G is a non-trivial free product with amalgamation. PROOF.Wemay assume that each ei >_ 2. If not we introduce relations ai A = 1 with f~ >_ 2 for each ai with ei = 0 and write fj = ej if ed > _ 2. The group G* =< al, .... an;all 1 ..... aln ~ = Rm(ax,...an) = 1 > is an epimorphic image of G. It is knownthat if ¢ : G1 -~ G2 is an epimorphism and G2 is a non-trivial free product with amalgamation, then G~ is also via

¢. Hence let e~ _> 2 for all i = 1, ...n. This condition indicates that G is generated by elements of finite order. Wemay assume that R is cyclically reduced in the free product on a~, ..., an and involves all the generators for if a generator is omitted in R(a~, ...an), then G is a non-trivial free product since we are assuming that n _> 3. Since rn _> 2 and n _> 3 there exists an irreducible essential representation of G in PSL2(C), and therefore especially the space of representations of

7.5

FREE PRODUCTWITH AMALGAMATION DECOMPOSITIONS213

G in (PSL2(C)) "~ is non-empty. Here an irreducible representation means that in the image group of G there are at least two elements which have no commonfixed points considered as linear fractional trasnsformations. From the result of Culler and Shalen mentioned above if the dimension of the character space of G in PSL2(C) is positive {as an affine algebraic set}, then G admits a decomposition as either a non-trivial free product with amalgamtion or as an HNNgroup. From the proof of the existence of irreducible essential representations for such one-relator products of cyclics we have that each of n - 1 matrices have two degrees of freedom with the trace and determinant being specified while the final matrix has one degree of freedom - the trace, determinant and relator condition specified. Therefore from the work of Culler and Shalen [Cu-S] (or also from Helling [He] and Rosenberger [R 9] in a different setting) the dimension of the character space is 2(n- 1) + 1- 3 = 2nThis is positive if n _> 3 . Thus if n _> 3, G splits as a non-trivial free product with amalgamation or an HNNgroup. However G is generated by elements of finite order so its abelianization Gab is finite. Therefore G cannot be an HNNgroup and must therefore be a non-trivial free product with amalgamation. The case of 2 generators - the generalized triangle groups is quite different. These group someti~nes admit splittings as non-trivial free products with amalgamation and sometimes don’t. The ordinary triangle groups T(p, q, m) = 2,m >_ 4, ei = 0 or ei >_ 2 for i = 1,...,n and S(al,...,an) is a cyclically reduced word not a proper power in the free product on al,..., an which involves a//the generators. Suppose fi~r~her that S(al, ..., a,~) is not conjugate Jn the free product on al ,..., an to a word of the form XY for some elements X, Y ofrespective orders p _> 2, q _> 2 where (1/m)+(1/p)+(1/q) > 1. Then G is of homological type WFL, with vcd(G) 2. ~rtheriflG ://I < oc then x(H) is defined and X(H) = IG : HIx(G) {Riemann-Hurwitz formula}. PROOF.From results in section 7.4 it follows that under the conditions on the relator, and if m _> 4 , G is virtually torsion-free. This was a consequence of work of J. Howie (see section 7.4) on one-relator products with high-powered relators. This work implies that under the conditions above any finite subgroup of G is cyclic and any element of finite order in G is conjugate to a power of S(al, ..,an) or a power of some ai. Further the orders of generators ai are precisely ei if ei _> 2 and infinite if ei = 0 and the order of S is m. Construct X, the Cayley complex for the given presentation {see [Brn] }. From the work of Howie{[H 3],[H 4]} the relation module splits as a direct sum of cyclic submodules.{see IBm]} Therefore the complex X is acyclic and hence X is a two-dimensional contractible G-complex such that every isotropy group is trivial or finite cyclic and X has only finitely manycells rood G. The construction is straightforward and similar to the case of onerelator groups. {see K. Brown[ 2,Sec.II.5] and also [Hu] }.This implies then that X has an equivariant Euler characteristic xa(X). It follows that the equivariant Euler characteristic Xa (G) = Xa (X) is defined. Since the group G is virtually torsion-free it follows from K. Brownthat G is is of homological type WFLwith vcd(G) _ 2, ei = 0 or ei _> 2, 1 _< p _< n - 1, U(al, .., ap) is a cyclically reduced word in the free product on al, ..., ap which is of infinite order and V(ap+l, ..., am) is a cyclically reduced word in the free product on ap+~, ..., aN which is of infinite order. With p understood we write U for U(al, .., ap) and V for V(ap+l,..., Nowif U = a~~ then el must equal zero since we assume that U has infinite order. In this case G reduces to G =< a~, ....,

aN; a~~ ........

a~~ = 1 >

which is just a free product of cyclics and thus an F-group. Therefore if p = 1 (or p = n - 1) we restrict in this chapter groups of F-type those where U = a~ (or V = a~) with Irnl _> 2. Thus a presentation for a group of F-type clearly generalizes the Poincare presentation and the terminology group of F-type was chosen to accent this tie to Fuchsian and F-groups. On the other hand a group of F-type also generalizes, by allowing possible torsion in the generators, a cyclically pinched one-relator group

8.2 FREIHEITSSATZ AND SUBGROUPTHEOREMS

223

and therefore an alternative designation could be a cyclically pinched one-relator product of cyclics. In all cases, a group of F-type, G, decomposesas a non-trivial free product with amalgamation: G = G1 *A G2 where the factors are free products of cyclics ~1 ~---< al, ....

,ap:a

¯ e, ~ I

...

= a~~ = 1 >

~+~1 = ..... G2~-< ap+l, .... ,an:¯ ap+ A =
--
.

224


If H2 admits a representation < a2, ....,

¢ : He -~ PSL2 (C) such that

ap; a~2 .....

a~~" = 1 >-~ Ha ¢-~ PSL2(C)

and G2 ~ H~ ¢-~ PSL2 (C) are faithful then G, has a representation p :G --* PSL2(C) such that Plol and PIG2 are faithful. Therefore we can assume that UV involves all the generators. Choose faithful representations ffl : G1 ~ PSL2 (C) a2 : G2 ~ PSL: (C) such that 0 t~ -1) and~r2(V) where tl and t: are transcendental over Q. This may be done since U and V both have infinite order and if Gi, i = 1 or 2, is non-cyclic, then the dimension of the character space as an affine algebraic set is positive (see the proof of Theorem7.5.1) From work of Shalen (1emma3.2 of [Sh]) there exists an automorphism a of C such that a(tl) = t2. Wedefine a faithful representation pl of G1 by Pl (~5) (g) = (~(crl (i,j) (g)), i = 1, 2 and j = 1, 2, wherep~ (~,j) (g) a~(i5)(g)) is the (i,j) entry of p~(g) (respectively of al(g)). Further p~ = a2. Then P2 is faithful on G~. G~ and G~ generate G and let p be the representation of G induced by p~ and p2. This gives the desired representation of the theorem. We show that if neither U nor V is a proper power then the above construction leads to the existence of a faithful representation of G . In [Sh]P. Shalen proved the following result LEMMA 8.2.1. (Proposition 1.3 of [Sh]) Let G1 *H G2 be a free product with amalgamation and let n > 1 be an integer. Suppose that there exist faithful representations p~ : G~ --* SL,~(C), (i = 1, such that (a) PllH = P21H (b) p~(h) is a diagonal matrix for every h ~ H and (c) Pi(n,1)(g) £ 0fo r ev ery g ~ G~\ Uand p~(~,~)(g) ~ 0 g~G2\H.

fo r ever y

8.2 FRE|HEITSSATZ AND SUBGROUPTHEOREMS Then G1 *H G2 ha~ a faithful

representation

225

in SLy(C).

Wenote first that if the original group is centerless then this faithful representation can be extended to a faithful representation into PSL2(C). Nowconsider the representation of the group of F-type G, constructed as above. Hypotheses (a) and (b) of Lemma8.2.1 are obvious from construction. We now check hypothesis (c). Suppose Pl(2,1)(g) = some g E G1 \ H. Then Pl(g) is an upper triangular projective matrix in PSL2(C). Hence, the subgroup K =< g, U-1 > of G1 is solvable. Since G1is a free product of cyclics this implies that K is either cyclic or infinite dihedral. K cannot be cyclic since we assumed that U is not a proper power in G1 and g ~< U-1 >. Therefore K must be infinite dihedral. Then necessarily we have that g and gU-1 have order 2. From this we obtain that

pl(U_l)

~_

0 0 t~ ~)

2

andpl(gU-1)=(~

2 xt~l 1) --Wt~

with w(t2-t~ ~) = 0. This gives a contradiction because t2 ¢ t~ 1. Therefore Pl(~,l)(g) ~ 0 for all g E G1 \H. In a similar manner, since V is assumed not be a proper powerin G2 it follows that P20,2) (g) ~ 0 for all g e G~ It follows than from Lemma8.2.1 and the remark after it that G has a faithful representation in PSLg (C). Wenote that if both U and V are proper powers, then there is no faithful representation in PSL2(C). If U = Uff,a >_ 2, and V = V~,/~ _> 2, and p : G -~ PSL2(C) is a representation where p(U) and p(V) have infinite order then p(U~) and p(V1) must commute. However U1 and V1 do not commutein G. {Non-elliptic elements of PSL2 (C) commuteif and only they have the same fixed points (see Chapter 4). Therefore p(U~) commutes with p(V1~) implies that p(U1) commuteswith p(V~).} There are several immediate consequences of Theorem 8.2.1 which we describe in the next section. Wenote here that using exactly the same argument the following generalization of Theorem8.2.1 can be obtained. TI-IEOREM8.2.2. (1) Let H1 and H2 be groups and U1, U2 elements of infinite order in H~, H2, respectively. Supposethat each H~, i -- 1, 2, admits a faith&l representation p~ in PSL2(C)such that tr(p~(U~)) is transcendental. Let H = (Hi * H2) / N ( Ui U2 ) be the one-relator product of Hi, H2 relator UiU2. Then H admits a representation p : H -~ PSL2(C) such that PlH~and PlH~are faithful.

226


(2) Suppose H is as in (1) but assume further that each solvable subgroup of H~,i = 1, 2, is either cyclic, finite or infinite dihedral andUi is not a proper power in Hi. Then H ad~nits a faithful representation in PSL2 (C). Manyof the consequences which follow for groups of F-type from Theorem 8.2.1 will also hold for groups H of the form of Theorem 8.2.2. We will point these out also in the next section. The theory of one-relator products of cyclics has been built so far on the foundations of essential representations and the Freiheitssatz. The next two results give the Freiheitssatz for groups of F-type. THEOREM 8.2.3. (Freiheitssatz for Groups ofF-type) Let bea gro up of F-type. If the relator UV involves all the generators, then any subset of (n - 2) of the given generators generates a free product of cyclics of the obvious orders. PROOF.Let G be a group of F-type with decomposition (4) and let G1, G2 and p be as in the decomposition. Suppose{x l, ...., x~-2 } is a subset of {al, ...,a,~}. Suppose first that p - 1 of the x~ are elements of GI and n - p - 1 of the x~ are elements of G2. Assumewithout loss of generality that x~ -- a~, i -- 2,. .... , n- 1. Since UVinvolves all the generators, it follows that < a2, ..., ap > contains no element which is conjugate to a nontrivial power of U. Similarly < av+l, ..., a,~-i > contains no element which is conjugate to a non-trivial power of V. Recall from Theorem 2.6.2 that if G = Hi *A H2 and {Zl,...,Zm} is a finite system of elements in G, then there is a Nielsen transformation from {Zx, ...,z,~} to a system {yl, ...,y,~} for which one of the following cases hold: (i) y~ --- 1 for somei ¯ {1, ..., m}. > can be written as w = ~l~=~v~, ~ = (ii) Each w E< Yl,.-.,Ym 4-1, ei ---- ei+l if ~i ---- ~+~withL(yg~) 1 contained in a subgroup of G conjugate to H~ or H2 and a certain product of them is conjugate to a non-trivial element of A. Further the Nielsen transformation can be chosen so that {Yl, ...,Y,~} is shorter (with respect to the length and a suitable order) than (z~..., z,~} the lengths of the elements of {Zl, ..., zm}are preserved. From the above result it follows that < a~, ...,a~-i > is a free product of cyclics of the obvious orders. Next consider the case where p of the


227

xi axe elements of G1 and n - p - 2 of the xi axe elements of G2 ¯ The remailfing case where p - 2 of the xi are in G1 and n - p of the xi are in G2 is handled identically. Assumethen that xi = ai, i = 1, ..., n - 2. Then G1 =< al, ...,ap > and we have V = gWh where g,h ¯< ap+l, ...,an-2 > and the norlnal form of W(with respect to the free product of cyclics G2 ) begins with a non-trivial power of an-i or aN and also ends with a nontrivial powerof a,~_l or a,~ since UVinvolves all the generators. The system {V, ap+h.... , a,~_~} is Nielsen equivalent to {W,ap+~, .... , a,~_2}. A product u~vi .... urv~ with r _~ 1, 1 ¢ u~ ¯ G1 and 1 ~ vi ¯< ap+l,...,an-2 > for i = 1,...,r can be trivial only if a product VSlw~ .... VS~w~with k >_ 1, 1 ~ w~. ¯< %+1, ...,a~_~ > and s~ ¯ Z\ {0} for j = 1,...,k is trivial. But this latter product is trivial only if a product W~Iz~...W~’~Zmwith m_> 1, 1 ~ z~ ¯< a~+l,...,a,~-2 > and t~ ¯ Z\ (0} forj = 1,..,m is trivial. But this last product can’t be trivial. Therefore u~v~ .... u~vr is non-trivial if r > 1 and 1 ¢ u~ ¯ G~ and 1 ¢ vi ¯< ap+i,...,a,~_: > for i = 1,...,r. This implies that < a~, ..., an-2 > is a free product of cyclics of the obvious orders. Wenote that nevertheless example consider

the algebraic

rank of G may be n - 2. For

G -=. G can be generated by x = ala2 and y = a3al since [x, y] = (a~a2a3) 2 = a4 ~ 1. a3a2a

Wealso note that in general a subset of (n - 1) of the given generators need not be a free product of cyclics. For example in < al~a2~a3~a4~a

¯ ~ a~ ~ = a~~ = a~4~ = ala2aaaa = 1 > 1 ~

with e~ _> 2 the subgroup generated by a~, a2, a3 has the presentation < al, a2, a3; a~1 = a~~ = a~3 = (aia2a3) ~ = 1 > which is not a free product of cyclics. Howeverif we allow that the relator UVinvolves all the generators and that U and V are proper powers in the respective free products of generators which they involve, then any n - 1 of the given generators generates a free product of cyclics of the obvious orders. Specifically: THEOREM 8.2.4. Let G be a group of F-type with relator UV such that UV involves all ~he generators. If both U and V are proper powers (U = Uff , V = V~k with m > 2, k > 2 in the respective free products on

228


the generators which they involve), then any subset of (n 1)of the given generators generates a free product of cyclics of the obvious orders. In a orientable surface group of genus g _> 2 any subgroup with three or fewer generators is a free group. An algebraic proof of this for two generators was given by G. Baumslag [G.B. 2] and for the case of three generators by G.Rosenberger [R 8] (see Chapter 3). Our next result says that under certain restrictions any two generator subgroup of a group of F-type must be a flee product of cyclics. THEOREM 8.2.5. Let G be a group of F-type. Suppose further that neither U nor V is a proper power or is conjugate to a word of the form XY for ele~nents X, Y of order 2. Then any two generator subgroup of G is a free product of cyclics. PROOF.Without loss of generality we may assume that UV involves all the generators. Recall as in the proof of Theorem8.2.3 that from Theorem 2.6.2 it follows ifG -- HI*AH2and {xl, ..., x,~} is a finite system of elements in G, then there is a Nielsen transformation from {xl, ..., x~} to a system {yl, ..., y~) for which one of the following cases hold: (i) Yi = 1 for somei e {1, ..., m}. q (ii) Each w E< yl,...,y,~ > can be written as w = Hi=I Y~,ei = =t=l, e~ = e~+l if ui = u~+l with L(y~,) 1 contained in a subgroup of G conjugate H~ or H2 and a certain product of them is conjugate to a non-trivial element of A. Since the group of F-type G decomposes as a free product with amalgamation with amalgamated subgroup A ==< V >, the result will follow from Theorem 2.6.2, described above, if we can show that there are no non-trivial powers U~, Vt~ and elements X1 E GI\ , X~ ~ G2\ < V > such that X~U*~X~~ = U~: for some s2 ~ 0 or X2Vt~X~1 = Vt~ for some t~ ¢ 0. Suppose there exists a non-trivial power U~ of U and an element X ~ G~\ such that XU*X-~ t= U for some t 2~ 0. The argument for V works identically.. We consider G~ a discrete subgroup of PSLu (R) which has no parabolic elements. Then is hyperbolic. Let z~, z2(z~ ~ z2) be the fixed points of U. Then either X fixes both Zl and z2 or X interchanges zl and z2. If X fixes both z~ and z2 then X and U commute and s = t. Since G1 is discrete this implies that X = Uq~, U = U[ for some q, r ~ Z \ {0} and


229

U1 E G1. In this case Irl _> 2 since X ~. But this implies that U is a proper power, contrary to our assumptions. If X interchanges zl and z2 then X fixes the midpoint of the axis of U. This implies that s = -t and X has order 2. In particular XUX-1 -1 =U which implies that (XU)~ = 1 since X2 = 1. Then < X, U > is infinite dihedral. Let Y = XU. Y has order 2 and U = XY contrary to our assumptions. Therefore there is no X ~ GI\ and non-trivial power -~ = Ut for some t 7~ 0. The identical argument works U~ such that XU*X for V and therefore by Theorem 2.6.2 the theorem holds. Wenote that without the additional hold. For example suppose (7=< al,a2,a3,

hypotheses the theorem does not

aa;a~ 1 -=a~ 2 =a~3 =a~4 =(ala2)S(aaaa)S=l

Then < (alan) ~, (aaaa) ~ > is not a free product of cyclics. A less trivial example is given by G =< al,a2,a3, a4;a21 ---- a~ = ag --= a34 = ala2a3a4= 1 >. Since [a~a~,a3al] = a4 it follows that G =< ala2,ala3 > and thus < a~a2,a~a3> is not a free product of cyclics. Finally we close this section with the statement and proof of a result of A. Hempel [Hem] which gives a strengthened version of the two-generator subgroup theorem for groups of F-type (Theorem 8.2.5). THEOREM 8.2.6. Let G be a group ofF-type and let H be a non-cyclic two generator subgroup of G. Then H is conjugate in G to a subgroup < x, y > satisfying one of the following conditions. (i) < x,y > is a free product of cyclic groups (2) t i s i n =< V > fo r so me natural nu mber t and y- ~x~y is in < a~,...,ap > ory-~xty is in < ap+~,...,a,~ >. PROOF.G is a non-trivial free product with amalgamation G = GI*HG~ with G~ =< a~,...,ap >, G~ =< ap+l,...,o~ > and H ==< V >. Any subgroup entirely contained in a conjugate of either G~ or of G~ is clearly a free product of cyclics. Now suppose that U = U(a~,..,a~) = U], with q >_ 1 and U1 not a proper power in G1 and that V = V(ap+~, ..,a~) = V[, with r _> 1 and not a proper power in G2. Choose in each Gi (i = 1,2) a system Li of left coset representatives of H in Gi normalized by taking 1 to represent H and taking the inverses L~-1 of the left coset representatives as a system of right coset representatives.

230


Introduce a length L and an order in G as described in section 2.6. If g is given in the symmetric normal form g = sl...s,,kr,,...r~ then L(g) = if k E H and L(g) = 2m+ 1 if k it H. Suppose that K =< a, b > is a non-cyclic two-generator subgroup of G. Wemay assume that {a, b} is minimal with respect to L and the order. From Theorem 2.6.2 we may assume that < a, b > is either a free product of cyclics or after a suitable conjugation a is in either G1 or G2 with at E H for some natural number t. Without loss of generality we may then assume that a -- U1, and let b = sl...s,~kr,~...r~ be given in symmetric normal form. If L(b) _< 1 there is nothing to prove so suppose that L(b) > 1 and then m >_ 1. From Nielsen reduction in free products with amalgamation or from the proof of Theorem8.2.5 it follows that ifg is in G~ and gU~g-1 is in H then gU~g-~ = U~t and if g is in G2 with gU~g-~ ~ H, then t = dq for some integer d and gU~g-~ = U~t . Further if g is in G~ and gU~g-~ t= U~ for some non-zero integer t, then we must have already gUlg-~ = U~1 and ifg is in G2 and gud~qg-~ = U~dq for some nqn-zero integer d then we must have already gU~g-1 = U1~q. Again from Nielsen reduction in free products with amalgamation it follows that if L(bUd~qb)< 2L(b) - 1 for some non-zero integer d, then r~ = si -~ and r~Ud~qr{1 = U~dq. -~, More generally if ri = s~ ¯ ~dq --1 ~dq rit~ 1 ri = U1 for i = 1, ...,~, 0 _< j < m and L(bud~qb) < 2L(b) - 2j for some non-zero integer d, then r~+l = sj-~l and +dq. rj+lUdqr-~_~ = U -~. Analogous statements hold for b-~Udlqb Suppose b = Sl...s,~kr .... rl as above and suppose further that there are at least two different letters g~,g2 of the s~,.,s,,,r,~,..,rl if k ~ H and s~, .., s,~, k, r,~, .., r~ if k it H such thatgiU~gi t -1 is not inH, i=l,2, ~ d: bt: withd~,d2 for all non-zero integers t. Then in any product bt~ ud~bU 1 non-zero integers and bt~ 7£ 1 7£ bt~ the letters gl, g~ remain uncancelled in b, (consolidations to elements of length _> 1 are allowed). The analagous statement holds for products of the type bt~ud~ibt~Ud~bta with dl, d2 nonzero integers and bt* ?~ I for i = 1, 2, 3. Therefore in this case using standard cancellation arguments K ----< a, b >= is a free product of cyclics. Nowsuppose that there is at most one letter g of the Sl,., s,~,r,~, .., rl if k ~ H and Sl,..,s,~,k,r~,..,rl if k it H such that gU~lg-~ is not in H for all non-zero integers t. Then after a suitable conjugation applied to the subgroup < U1q, b > we get a conjugate < U~, b~ > of < U1q, b > with ~ in G1 or in G2 with e = :t:1. This gives the desired result. b~-~U~b Workof Rosenberger [R 8] can be used to give a complete classification in certain groups of F-type of subgroups of rank less than or equal to 4 satisfying a quadratic condition. Specifically we have. THEOREM 8.2.7.

Lef G be a group ofF-type

such ~hat neither

U nor

8.3 LINEAR PROPERTIESGF GROUPSOF F-TYPE

231

V is a proper power or is conjugate to a word of the form XY with X, Y elements of order 2. Let ul, ..,un, 1 ~_ n ~_ 4, be elements of G. Suppose that the system {ul, .., un} is not Nielsen equivalent to a system containing an element which is conjugate in G to an element of the amalgamated subgroup --< V >. Suppose W(ul, ...,u,) = 1 where W(xx, ...,x,~) is a quadratic word in the tkee group on Xx, .., xn and let H be the subgroup generated by ul, .., un. Then H is either trivial, a bee product of cyclics or a surface group (oriented or non-oriented). 8.3 Linear Properties

of Groups of F-Type

The existence of an essential representation for a finitely presented group G indicates that the group is "almost" a finitely generated linear group. This was the idea behind the investigations in the last chapter on torsion one-relator products of cyclics. Here we consider the basic linearity properties for groups of F-type. As we will see, most of the algebraic properties of F-groups carry through to this extended class. In this section we consider the linearity properties studied in the last chapter. These are (1) The virtual torsion-free property; (2) The Tits alternative; (3) SQ-universality; (4) The existence and computation of rational Euler characteristics. As already described, the fact that a group o£ F-type decomposesas a free product with amalgamation is immediate from the definition. As a direct consequence of this decomposition we get our first results which mirror exactly the situation for F-groups. Throughout this section we suppose that G is a group of F-type with presentation (8.1.3) G =< al,. .... , an; a~1 ......... a~~ = 1, U(a~,..., ap)V(av+~,.., an) = where n _> 2,ei = 0 or ei > 2, 1 _< p < n - 1, U(al,..,ap) is a cyclically reduced word in the free product on al,...,ap which is of infinite order and V(ap+~, ..., a,~) is a cyclically reduced word in the free product on %+~,...,an which is of infinite order. Ifp = 1 (or p = n - 1) we restrict groups of F-type to those where U = a T (or V = a~) with lm[ >_ 2. Welet el G1 ~ %+1-G2~ and 1 - .... -A =< U-~ >=< V >, so that G = G~ *A G2. G1, G~ axe the factors of G and UV is the relator.

232


THEOREM 8.3.1. Let G be a group ofF-type with presentation (8.1.3). Then (1) If e~ >_2 then a~ has order exactly e~. (2) Any element of finite order in G is conjugate to a power of some a~. (3) Any finite subgroup is cyclic and conjugate to a subgroup o[ some ~ai~. (4) Any abelian subgroup is either cyclic or ~ee abelian of rank PROOF.This th~rem follows directly from the ff~ product with amalgamation decomposition and the fact that the factors are ff~ products of cyclics. ~ompa~ (2) it follows that any essential representation m~t be essentially faithf~. The e~stence of an essential representation then guarantees, as we saw in Chapter 6, that any group of F-type is virtually torsion-fr~. Fu~her we saw in Theorem 8.2.1 that if neither U nor V is a proper power, then there is actually a fai~hf~ representation into PSL2(C) and hence G can be co~idered as a linear group. Therefore under these conditions G is residually finite and also Hopfian from a th~rem of Malcev [Mall. Wewill see in the next section, that relative to the prope~ies of residual fi~teness and Hopfian, we can remove the restrictions on U and V. THEOREM 8.3.2. Le~ G be a group oT F-type. Then (1) G is vir$ually torsion-[ree. (2) H nei$her U nor V is a proper power, $hen G is residua~y finite and thus Hopfian. The bu~ of the last chapter was devoted to proving that almost all torsion one-relator products of cyclics satis~ the Tits alternative. The situation for groups of F-type is even simpler. A group of F-type will either contain a ff~ subgroup of ra~ 2 or is itself solvable. This latter situation can o~y occur for t~ ~oups. THEOREM 8.3.3. Le$ G be a group ofF-type. Then either G has a ~ee sub~oup of rank 2 or G is solvable and isomorphic to groups wi~h one o[ the following presentations: O) H1 =< a,b;a2b ~ = 1 >; (ii) H2=< a, b, c; a2 = b2 = abc~ = 1 >; Oil) Ha =~ a, b, c, d; 2 =b~~ c 2 = d2 = ab~= 1 > . PROOF. Suppose n > 4. Then a flee product of cyclics injects into G and therefore G has a flee subgroup of ra~ 2. n = 4 and not all ei = 2. Then a flee product of two cyclic Z2 * Z2 injects into G and therefore G has a ff~ subgroup of

of ra~ ~ 3 Next suppose groups not ra~ 2.

8.3 LINEAR PROPERTIESOF GROUPSOF F-TYPE

233

If n = 4 and all ei = 2, then necessarily, since U and V have infinite order, we must have p = 2. Then G has a presentation < a,b,c,d;a 2 = b2 = c 2 = d2 = (ab)S(cd) t = 1 > with s > 1, t > 1. G then has as a factor group the free product ~ =< a, b, c, d; a2 = b2 = c2 = d2 = (ab) s = (cd) t = 1 > . This is a non-trivial free product ~ = * < c, d; c 2 = d2 = (cd) t = 1 >. If (s, t) # (1, 1) then G has a free subgroup of rank 2 and therefore also contains a free subgroup of rank 2. If (s, t) = (1, 1) then G has the presentation < a,b,c,d;a 2 = b2 = c2 = d~ = abcd = 1 > which is solvable. If n = 3, then at least one ei = 0, since U and V are assumed to have infinite order. Supposewithout loss of generality that et = 0 so that al has infinite order. If p = 2 then a free product of two cyclic groups not Z2 * Z2 injects into G and therefore G has a free subgroup of rank 2. Nowlet p = 1. Then U = a +s with s >_ 2. Supposefirst that (c2, ca) # (2, 2). Then a t product of two cyclic groups not Z2 * Z2 injects into G and therefore G has a free subgroup of rank 2. Next suppose that (e~, e3) = (2, 2). Then G a presentation < a,b,c;b 2=c2=aS(bc)t=

l

where s > 2 and t >_ 1. G then has as a factor group the free product ~ =< a, b, c; as = b2 = c2 = (bc) t = 1 >, ~=< a;a ~ = 1 > * < b,c;b ~ = c 2 = (bc) t = 1 > . If s > 2 or t > 2 then a free product of cyclics of rank > 2 and not Z2 * Z2 injects into G and thus G has a free subgroup of rank 2. It follows that G does also. This leaves the case where s = 2 and t = 1. G then has the presentation < a, b, c; b2 = c 2 = a2bc = 1 > which is solvable.

234


Finally suppose n = 2. Then since U and V have infinite order both generators must have infinite order and G must have a presentation < a,b;aSb ~ = 1 > with s _> 2, t _> 2. G then has as a factor group the free product of cyclics ~ =< a,b;a s = bt -- 1 >. If (s, t) # (2, 2) then G has a free subgroup of rank 2 and therefore does also. This leaves the situation where (s, t) = (2, 2). G then has presentation < a, b; a2b2 = 1 > which is solvable. Recall from our previous discussions the close ties between the existence of non-abelian free subgroups and SQ-universality. The next result says that for group a of F-type G either it is SQ-universal or isomorphic to one of the three exceptions given in Theorem8.3.3. THEOREM 8.3.4. Let G be a group ofF-type. IT G is not solvable then G has a subgroup of finite index which maps onto a [-ree subgroup of rank 2. In particular a group of F-type G is either SQ-universal or solvable. PROOF.Suppose G is not solvable¯ First assume that n _> 5 or n -- 4 and at least one ei ~ 2. Wemay assume without loss of generality that each ei _> 2 passing to an epimorphic image if necessary. The first part of the proof mirrors the proofs of Theorem6.2.3 and of Theorem7.3.2.1. Let p : G -~ PSL2(C) be an essential representation. It follows from the work of Selberg that there exists a normal torsion-free subgroup N of finite index in p(G). Let ~r be the canonical epimorphism from p(G) onto p(G)/N. The composition ¢ of the maps in the sequence G ~ p(G)

--~ p(G)/N

gives a representation of G onto a finite group. Further ¢(a~) has order in p(G)/N since p is essential and N is torsion-free. Nowconsider the free product of finite cyclic groups X :(al,...,an,

1 ~- . ...

~

.

There is a canonical epimorphism e : X --~ G. Consider the sequence X _A_+ (7 ~ p(G)IN.


235

Let Y = ker(¢e). Then Y is a normal subgroup of finite index in and YVI < a~ >= {1} for i = 1,...,n. Then by the Kurosh theorem Y is a free group of finite rank r. The finitely generated free product of cyclic groups X may be considered as a Fuchsian group with finite hyperbolic area. Therefore from the Riemann-Hurwitz formula we have

j,(x) = ,(Y) where #(X) = n- 1 - ((1/el)

.. ......

+

#(Y) = - 1. Therefore r : 1 - j((1/el) + .... + (1/en) - n + 1). The group G is obtained from X by adjoining the additional relation UV = 1 and thus G = X/K where K is the normal closure of UV in X. Since K C Y the factor group Y/K may be regarded as a subgroup of finite index in G. Using work of Baumslag, Morgan and Shalen (corollary 3 of [B-M-S]) Y/K can be defined on r generators with j relations. Thus the deficiency of this presentation is given by d = r-j = 1 - j((1/el)

+ ....

+ (1/e,~) - n+

= 1 + j(n - 2 - (1/e~) - ....

- (1/e,)).

If n_> 5or n=4andat least oneei ~ 2thend_> 2. It follows, then from the theorem of B.Baumslag and Pride, that G contains a subgroup of finite index which maps onto a free group of rank 2. Next suppose n = 4 and all ei = 2. Then necessarily p = 2 and U = (ala2) 8, Y = (aaa4)t with Is[ _> 1, It[ _> 1. Further since G is non-solvable we must have Is] _> 2 or It[ _> 2. Without loss of generality we assume that s _> 2 and t _> 1. Then G has as a factor group the free product. ~=< al,a2,aa;a

~ 8 T = a2 = (a~a2) = 1, a~ = 1 >.

G has as a normal subgroup of index 2 a group isomorphic to the free product of three cyclic groups 8=y2=z2=1>. <x,y,z;x Therefore G and hence G also has a subgroup of finite index mapping onto a free group of rank 2. Now suppose n = 3 and at least two ei ¢ 2. Then without loss of generality we may assume that p --= 1, al has infinite order and U = a~

236


with s >_ 2. Suppose first that a2 also has infinite order and V = a~t with t _> 2. Then G has as a factor group the free product of cyclics G=.

G and therefore G also has a subgroup of finite index mapping onto a free group of ramk 2. Nowsuppose that V ---- a~, e = ~:1. Then G is isomorphic to the free product Z * Z¢3 which has a subgroup of finite index mapping onto a free group of rank 2. Therefore we may now suppose that V involves both a2 and a3. Choose a number m _> 7 such that m is relatively prime to s. Then G has as a factor group G " "~ a = a ~:=a~ 2 =< a, a2, a3~

s=aV _- 1> .

Eliminating a = V-~ we get that G =< a2, a3;a~~ = a~3 -~ V(a2, aa) "~ = 1 >=< x,y;x p -- yq = V(x,y) m = 1 > with mk 7 and_ (p, q) -- (eu, e3) ¢ (2, 2). Thus ~ is a generalized triangle group with s(G) < 1. It then follows from Theorem7.3.2.1 that ~ and thus G has a subgroup of finite index mapping onto a free group of rank 2. Next suppose n -- 3 and that two ei -- 2. Then we may assume without loss of generality that a~ has i~ffinite order, U = a~ with s _> 2, (e~, ea) --(2, 2) and V ~- (a2a3)t with t _> 1. If t _> 2 then G has as a factor group the free product t -. s= 1, a~2 = a~ G ~ (a2a3) 1 > 1 =~ al,a2~ a3~ a which has a normal subgroup of index two iso~norphic to the free product of cyclics < x,y,z;x t = yS = zs = 1 > . Hence G and therefore also G has a subgroup of finite index mapping onto a free group of ra~k 2 if t _> 2. If t --- 1 then G has the presentation G ---< al, a2, a3; sa22 = a~ = a1a2a3 ~ 1 ~( al~ a2; a22 = I, ~as la2) ~2

=1>.

Since G is not solvable we have s _> 3. Then G has as a factor group the free product of cyclics ~ al, a2~a1 ---- a = 1 > .


237

Since s > 3 this has a subgroup of finite index mapping onto a free group of rank 2 and therefore G also has a subgroup of finite index mappingonto a free group of rank 2. Finally suppose n = 2. Since U and V are assumed to have infinite order and G is not solvable it follows that G must have a presentation < x,y;xSy ~ = 1 > with s k 2 and t _> 3. Then G has as a factor group the free product of cyclics G =< x,y;x ~ = yt = 1 > . Since (s, t) ~ (2, 2), G and hence G also has a subgroup of finite mapping onto a free group of rank 2. This completes the proof.

index

One of the most powerful techniques in the study of Fuchsian groups is the Riemann-Hurwitzformula relating the Euler characteristic of the whole group to that of a subgroup of finite index. As we remarked in section 7.6 Bass [Ba 3], Brown [Brn], Chiswell [Ch], Wall [Wa], and others have extended the concept of a rational Euler characteristic to more general finitely presented groups and have shown that in many cases these can be computed very easily from given finite presentations or given amalgam decompositions. Further these general rational Euler characteristics satisfy the Riemann-Hurwitz formula. In section 7.6 we showed how to compute the rational Euler characteristic for certain torsion one-relator products of cyclics. Weclose this section using results of C.T.C. Wall [Wa] and K.Brown[Brn] to extend this Euler characteristic to groups of F-type and give a Riemann-Hurwitz type formula. THEOREM 8.3.5. Let G be a group ofF-type. Euler characteristic x(G) given

Then G has a rational

where ai = -1 ire~ = 0 and ai = -1 + (l/e/) fie/k If IV: HI< oo then x(H) is defined and x(H) = IG HIx(G) Hurwitz formula). In addition G is of finite homological type and vcd(G) --< V > is the amalgamated subgroup. Since A is infinite cyclic, we have x(A) = and th erefore x( G) = x( GI) + x(G2). Fu rther G~ and are free products of cyclic groups, so we can apply the computation rules of Wall above to obtain n

=2 + i~l

ai = -1 if e~ = 0 and ai = -1 + (1/e~) if ei _> 2. Wecan also write this as

x(G)

= 2- n+

where f~s = 0 ifei = 0 and f~i = 1/ei ifei k 2. This second form is somewhat closer to the form usually written for F-groups. The RiemannHurwitz formula follows directly from Wall. Wenote further that the Euler characteristic of a group of F-type can be zero. In fact x(G) = if n =2. Since a group of F-type G is virtually torsion-free, it follows from K. Brown, as in section 7.6, that G is of finite homological type with vcd(G) _-potent if for any m _> 2 there exists a finite quotient of H where the image of h has order exactly m. This is clearly a strong form of residual finiteness. From F. Tang [Tan 2] and Allenby and Tang [A1-T 3] we have the following result. Let G = A *H B where H =< h > is infinite cyclic. If both A and B are < h > - potent and < h >-separable then G is residually finite. Groups of F-type have exactly this form where A and B are free products of cyclics. The theorem then follows since in any free product of cyclics K, K is < g >-separable and < g >-potent for any element g of infinite order in K. The < g >-separability follows from a result of P. Stebe [St] while the < g >-potency can be deduced in the following manner. Let K be a free product of cyclics with presentation Suppose g is an element of infinite order in K and m _> 2 a natural number. If g is a power of a generator of infinite order, the < g >-potency follows directly, so suppose that g has free product length _> 2. Let K map onto K=. K is then a torsion one-relator product of cyclics, so there exists an essential representation of K onto K in PSL2 (C). Then from Selberg there exists a torsion-free normal subgroup N of finite index in K. The sequence K---~K---*K---~K/N then gives a homomorphismof K onto a finite group where g has order m. Therefore K is < g >- potent. It follows then from the Theoremof Allenby and Tang that a group of F-type is residually finite and from the result of Malcev that it is Hopfian. Recall that a group G is subgroup separable or LERF(Locally extended residually finite) if given any subgroup H of G and any element g not in there exists a subgroup H* of finite index in G such that H is a subgroup of H* and g is not in H*. Againthis is a very strong form of residual finiteness. Brunner, Burns and Solitar [Br-B-S] proved that cyclically pinched onerelator groups are subgroup separable (Theorem3.5.10) while G. Scott [Sco] proved the same for finitely generated Fuchsian groups (Theorem 4.3.11). From two results, one of Allenby and Tang [A1-T 2,3] and one of Niblo [Ni 2], it can be deduced that groups of F-type are subgroup separable.

240


THEOREM 8.4.2.

A group ofF-type

is subgroup separable.

PROOF.Allenby and Tang’s method [A1-T 2,3] is purely group theoretic and proceeds by showing that the amalgamated free product of two virtually free groups amalgamating a cyclic group is subgroup separable. Since free products of cyclics are virtually free, this result can be applied both to groups of F-type, and also to the case of a group G with a presentation a=< ax,. .... , an; a~~ .........

a~~ = (UV) m = 1 >

with 1 # U = U(al,...,ap), 1 ~ V = V(av+l~..,an)~ 1 _ 2. This case will be considered in the next result, due to Allenby on conjugacy separability. Niblo [Ni 2] using some topological arguments shows that amalgamated free products of Fuchsian groups, amalgamating a cyclic subgroup are subgroup separable. Since free products of cyclics can be faithfully represented as Fuchsian group his result applies to groups of F-type. Wemention that M.Aab[Aa] has established the subgroup separability of a more general class of fundamental groups of graphs of groups. Recall that a group G is conjugacy separable if given any two dements of g, h of G, either g is conjugate to h, {g ,-, h} or there exists a finite quotient G of G where the images of g and h are non-conjugate. In Chapter 4 it was proved that Fuchsian groups are conjugacy separable. Allenby [A] has extended this to groups of F-type. THEOREM 8.4.3.

A group ofF-type

is conjugacy separable.

Allenby actually proved the following more general result from which the second part is just an addendumto his result¯ Wereproduce Allenby’s proof which is quite lengthy.. THEOREM 8¯4.4. (8.4.1)

Suppose

G=

where n >_ 2, ei = 0 or ei >_ 2, 1 _ 1. PROOF.The second part of the statement of the theorem, that is when neither U nor V has finite order clearly implies the conjugacy separability of groups of F-type.

8.4 ADDITIONAL RESULTS If V has infinite amalgamtion of

241

order and m >_ 2 then G is the free product with

1A, a;a al,..,a s =< ~

...

B =< 1ap+l, ...,

a~p = am = 1 > and

~P+I an; ap+

...

a~~ = 1 >

with the cyclic subgroup < h >=< U-la >=< V > a~nalgamated. If V has finite order and U has infinite order swap the roles of U and V. If both U and V have finite order, then both are conjugates of some generator and we may assume that G has the form G =< a~, .....

, a,~; a~~ .........

a~~ = 1, (a~a~)"~ = 1 >

withi 7& j. Assume further that rn_> 2. ThenG = H*RwhereHis ej ¯. i e a~ = a~ = the free product of cyclics on the ak, k ~ i,j and R =< a~, a~, (aiaj) = 1 Let k = e~/gcd(e~,c~), ~ = ce/gcd(e~,c~), 1 = ej/gcd(ej,~), f3/gcd(ej, ~). Then R is obtained from Ro =< x, y; xk = yt = (x~yS) "~ = 1 > by two successive free products with amalgamation. From a result of Dyer [Dy] these will be conjugacy separable if R0 is. However Ro ~-< x,y;x k = yt = (xy),~ = 1 > which, since m >_ 2, is an ordinary triangle group. From Theorem 4.3.10 Ro is conjugacy separable. Therefore R is conjugacy separable. A free product of cyclics is conjugacy separable and so in this case G = F * R is conj~acy separable. Therefore for the remainder of the proof we will assume that V has infinite order. Let first m _> 2. Wework with the above decomposition and use the fact that G now is actually a group of F-type. The proof with V of inifnite order and m > 2 then follows from the following series of lemmas. Following Allenby we use the notation A, h etc. to denote homomorphic images. LEMMA 8.4.1. Suppose G has form (8.4.1) ht ~ h-t -~. if and only ifh ~ h

and h = V as above. Then

PrtOOF. (Lemma8.4.1) Let A,B be the factors of G. Then if c = h~ ,-~c -t h -~ d then from the free product with amalgamation decomposition there exists a sequence h~,..., h~ of elements of < h > such that ((1))

ht ~Ahi~"~Bhi~ "~A .... . B h~ = h-t = d.

A and B are free products of cyclics and h has infinite order, so inside A,B we can only have ht ,~ ht or ht ~ h-t (see the arguments in 8.2.5 and 8.2.6). Thus each ik is a t or a -t. Further if say ht = b-lh±tb, then h is conjugate to h±~ using the same element b. Hence (1) implies that h ~ -1.

242


LEMMA 8.4.2. Let A be a free product of t~nitely many cyclic groups and h E A of infinite order. Then there exists a finite homomorphicimage ~ of A in which -~ has prescribed order t >_ 2. PROOF.(of Lemma8.4.2) (See also the proof of Theorem 8.4.1.) Without loss of generality we can assume that all the generators of A are involved in h. Consider the one-relator product of cyclics H = A/N(h~). Then there exists an essential representation of G in PSL2(C) and hence there is a homomorphic image ~ in PSL2 (C) where ~ has order t. ~ is finitely generated subgroup of PSL2 (C) and hence residually finite so there exists a finite homomorphicimage of A and therefore also of A where h has order t. LEMMA 8.4.3. Let A and h be as in Lemma 8.4.2. and let F be a normal free subgroup of finite index in A contained in the kernel of the natural map from A onto the direct product of its (finitely many) cyclic factors. Let h~ -1. ~ F, w >_ 1 and suppose h is not conjugate in A to h Then ~here exists a finite homomorphicimage ~ of A in which -~ 7~ 1 and notation {g}~ to denote the set of conjugates of g in G. PROOF.(of Lemma8.4.3) Let = {o ~1 ~- 1, o~2 , ... , at} be c oset representatives for A mod D where D = FC ~ with C the centralizer of h in A. Then every eleinent of A is of the form yfa where y ~ C,f ~ F, a ~ S. Let {~}~ be the set of conjugates of hTM in A. Hence {hw}A -~ {a~f-~h~fa~}k#~ U {f-~h~f} where f runs over all the elements of F. Now{f-~h~f : f e F}Q < hw >-- {hW}. (This is a singleton set since h~ ~ h~’ is ruled out by hypothesis if w~ = -w and by order considertaions if [w’[ ~ Iw[.) Similarly {a;~f-lh~fak}C~ < h~ >C {h~}. But, if k ¢ 1, a~f-~h~fa~ -- h~ implies that fak ~ C C D which implies that a~ ~ D, which is impossible by the choice of S. Hence {a~f-lh~fak}n < h~ >= ~ if k ~ 1. Consequently {f-~h~f} ~ {akh~a-~ ~} = O (if k ~ 1) - an empty intersection holding in a free group F. From results of Dyer [Dy] there exists a finite nilpotent homomorphicimage FIX of F in which {~--1]-~]~-}~ < ~-~ >= ~ for all ak ~ 1. Since X has finite index in A, we can assume without loss of generality that X is normal in A. To modify X further so that also {y--l~vy}~ ( ~-w )} = {~--w} in A/X, take Y = F~(F) the sth term of the lower central series for F where s is such that h~ ~ F~_~(F) \ F~(F). Let E be a characteristic subgroup of F ~ that F~(F) C E C F~_~(F) and EV~ < h >=< >where v is the orde r of ~ in A/X. Then ~ has order v in the residually finite group AlE and therefore A has a normal subgroup Z such that [A/Z[ < oc and in which

8.4 ADDITIONAL RESULTS

243

hZ has exact order vw. Replace X by X V~ Z = U. Then A = A/U satisfies the conclusions of the lemma. LEMMA 8.4.4. Let < g >, < k > be cyclic subgroups of a free group F and let s ¯ F\ < g >~ k >. Then there exists a finite homomorphic image P of F in which -~ ¯ -~\ < y > < ~ >. PROOF.(of Lemma8.4.4) Suppose [g, k] ~ 1 in F and suppose s-7 y~fl’ expresses the image of an element of < ~,~ >C F/F~(F) = F~.. Suppose[g,k] ¯ F~_I(F)/F~(F).. If for i _> t all the a.i are equal, then for somej _> t we have ~3j # f)t since a free group is residually nilpotent, that is V~IF~(F) =< 1 >. But this implies that ~ has order {f)j - ~t] in Ft which is a contradiction. Thus a~ ~ at for some r > t and then ~-~-~ = ~ in Ft for some f~ _> 0. f) = 0 leads to a similar contradiction to the one just noted. If/~ >_ 1 then ~,~]~ = [~,~] = 1 in F~ which contradicts ~,~] ~ 1 in Ft. Thus there is a finitely generated torsion-free nilpotent homomorphic image ~ of F in which ~ ~< y >< ~ >. From results of Stebe [St] then there is a finite homomorphicimage of F in which ~ @< ~ >. If [g, k] = 1 then < g >< k >=< g, k >=< gl > with g = g~, k = g~ for some g~ ¯ F and s, t ¯ Z. The desired result again follows from results of Stebe [St]. LEMMA 8.4.5. Let A and h be as in Lemma8.4.2. and let k,g ¯ A be such that k q~ HH~ where H =< h > and H~ = g-lHg. Then there exists ~. a finite homomorphicimage -~ of A in which k ~ HIf PROOF.Let F be the kernel of the natural map from A to the direct product of its finite cyclic factors and let (h~) a denote g-lhf~g etc. If k q~ HF factor out the normal sub’group HF. Otherwise suppose that k ¯ HF. Write H = Ho U hHo U ... t~ h~-iHo where r is the least power of h to lie in F, ho = h~ and H0 ---< h0 >. Then ~ _ HH~ {h~g-lh~g: a,f~ ¯ Z}={h ho(hoh) ~ ~ ~ ~ :O-- 1. LEMMA 8.4.7. Let A and h be as in Lemma 8.4.3, and let F be the kernel of the natural mapfrom A onto the direct product of its finite cyclic factors. Let hTM be the least power of h in F. Suppose for all r such that 1 < r < s that uT, vT E A and that ur = h*rvTh~ where i~,j~ are unique. Then there exists a finite homomorphicimage A of A in which h has order wp with p a prime and if for 1 < r < s, -fk~,-~lr is a solution oft ---- x~y then kr =- ir and IT =- j~ modulop. PROOF.(of Lemma 8.4.7) As before let H --< h > and ~ - - g-iHg ~ implies that for Note first that the uniqueness of the solution ur = h*~v~M ~ =< 1 >. This in turn implies that wr = [v~lh~vr, h~] 7~ 1 each r, HNH or else v~lhwvr, hTM generate an infinite cyclic subgroup of F which is a contradiction. Note that each w, lies in F which is free of rank > 1. If [wl, w2] -- 1 set w~ = [w2, z] for suitable z so that [Wl,W~] ~ 1. If [wl, wu] ~ 1 set w~ --- w2. Repeat this with wa, w4 .... in turn and consider q = [Wl, w~, ..., w~] ~ 1 in F. Nowchoose a finite homomorphicimage of F of exponent p where p is arbitrarily large and coprime to ]A/F[. Now pass to a homomorphic image A of A in which ~ has order p. None of the~7~ is trivial in ~since~ ~ 1. But this means that < ~’ > ~ < ~’ >~-7=< 1 >. Therefore suppose that, in ~, which is an extension of a finite group of exponent p by a group of coprime order ]A/F[, we kr -it,

ar = jT-mr. Assuming p does not divide p~ we have that p does not

These together imply that completing this lemma.

~r-1 --wh W ~
which is a contradiction

Wenow return to the main proof of Theorem 8.4.4. Wesuppose G has form (1) with m_> 2, V has infinite order and set a = UV. As in the proof of Lemma8.4.1, let A, B be the factors of G and < h >--< V >=< U-la >. Let c, d be elements of G of minimumlength in their conjugacy classes and such that c is not conjugate to d in G. Wehave to show that there is a finite quotient G of G where the images of c and d are not conjugate. If c -- I or d -- 1, then this follows from the residual finiteness of G considered as a group of F-type (see Theorem 8.4.1). Nowsuppose c ~ 1 and d ¢ Let Ix[ denote the length of x written as a normal form in ’the free product with amalgamation (see Chapter 2). Then Ix[ -- 0 if and only if x E< h and Ix[ = 1 if and only ifx ~ A\ < h > or x ~ B\ < h Case (1): Icl -- Idl -- 0. Then c hi ,d = h~wit h c ~ a d. In particular

8.4 ADDITIONAL RESULTS

245

i ~ j. If [i I ¢ IJl from Lemma8.4.2 we can find a homomorphicimage of G which is a free product with amalgamation G of two finite groups with j. So we may assume that I ~] = [illjl so that l~i[ ¢ I~Jl and hence ~i ~U~ i,d= -i i -~i c =h h and h ~G h In p~ticular h ~A h-~ and h~ -i. ~s h A being a ~ product of cyclics contai~ a fi~tely generated ~ group FA as a normal subgroup of finite index and contained in the kernel of the natural map ~omA onto the direct product of its fi~te cyclic factors. Simil~ly B contains such an FB. Let < h~ >= FA ~ F~. By Lemma8.4.2 there exists normal sub~oups X~, Y~ of fi~te index in A and B such that X~ < h >: Y~ < h >:< h~ >. Set A1 : FA ~X1,B~ : FB ~Y1. Then A~, B~ are normal of fiaite index in A, B respectively. Nowlet ~1 = 1, a~, ..., ~, fll = 1, ~2, ..., ~ be coset representatives in A, B respectively, chosen as in Lemma8.4.3 modulo sub~oups DA ALVA, DB = B1CB where CA,C~ are the centralizers in A, B of < h~ >. Use Lemma8.4.3 to find normal ~ subgroups X~, ~ of finite index such that (in A/X2) {~-

S h S~}N

y-

>= @ and in B/Y2

=

where f ~ A~,~ ~ B~ and 1 < k ~ r, 1 < t ~ s. Using Lemma8.4.2 we c~n modify X2, Y2 to assume without loss of generality that X2N< h >= Y2N h >=< hz~ >. As in the proof of Lemma8.4.3 choose normal sub~oups EA, E~ of fiMte index in A and B su~ that h~*~ > and such that

{VW }n
= E~N < h

>=

in A/EA and B/EB respectively. Finally set A3 : EA n X2, B3 : EB ~ Y2. Nowlet G : A/A3 * B/Ba the amalgamated ~ product of two finite ~oups. If ~i ~ then h and there exists a sequence of conjugate powers. But by choice of Aa, B~ the only conj~ate of~i~ in A/A3 and B/B3 is ~ itself. Hence ~i ~ ~-i in G. G is conjugacy separable so there exists a fi~te homomorphicimage of G and hence of G where c ~ d completing case (1). Case (2): (i) ~c] = 0,~d~ = 1. Without loss of generality let d (the case d ~ A is handled simil~ly). Thus d e B~H. Let c = i.

246

ALGEBRAIC

GENERALIZATIONS

OF DISCRETE

GROUPS

Since d is assumed to be of minimal length in its conjugacy class we have {d}BA < h >= 9. Suppose w is such that d~,h~ E FB withFB as in case (1). Then {d~}gA < ~ >= Oor els e b-l d~b = h t fol lows. App lying the proof of the first part of Lemma8.4.3 analogously we get a finite homomorphic image B B/M of B such that {~ }~ < >= ~). Case (2): (ii) Icl = 1, Idl = 0. This case may be dealt with as in (2): (i) in a symmetric manner. Case (2): (iii) All other cases where Icl ~ Idl are dealt with by passing to a free product with amalgamation G with finite, factors A and B where I~l = Icl ~ Idl = I~1. This is easily achieved by using the IIc property of the free products A and B (see [St]) to keep images of elements of A\ < and B\ < h > out of < ~ >. Case (3): Nowassume that c ~a d with Icl = tdl _> 2. Let c = ulu2...ur and d = vlv2...vr. Consider the system of equations (I(i))

ui+~ = x~v~x~

¯Ui4-r ~ xr--l~lVrXO

A solution of I(i) is a set of elements ho, h~, ..., h~_~ of < h > which satisfy the system simultaneously. By a result of Dyer [Dy] c ,-~ d iff for some value of i, 0 J, then for no i can the equations 7(i) solved in (A/Ma) * (B/MB) with the proper identifications. A similar result holds for case (ii) if we assume that ¢ is even which possible from the construction of essential representations. Thus ~-Zo # 1 since fl~ - fl0 is odd and ~ has even order. Thus we may assume that for at least one, and hence for all, i that the equations ui+~ = x~,~l_lvdY~5,1 < j < r, have solutions xi,~_~,yi,d in < h > and yet no solution in I(1) and that for each i at least one -1 v the equations, say ui+k(i) = xi,k(i)_ 1 k(i)Yi,k(i) is solvable with unique x, y. For each i select one such equation, say the k(i) th, as above. Fixing i, consider in turn the k(i) 1st, k( i) + 2nd, et c. eq uations of thesyst em, arranging if possible that Yi,k(i) = xi,~(i) etc. Since c ~ d this matching -1 must eventually fail, say at equation ui+i(i) = xij(i)_ 1~)f(i)Yi,I(i). Wenow choose p as in Lemma8.4.7 so that p is larger than all ]si -flil (as i runs

248


over all the integers from 0 to r - 1) and where ~i i s t he u nique s olution for xi,l(i)-i in the above equation and h~i is the forced value taken Yi,f(i)-I in the preceding equation. This choice of p leads, as in Lemma 8.4.7 to an amalgamated free product G with finite group factors which is a homomorphicimage of G and in which ~ ~U ~ as required. This completes the proof for m_> 2 by the result of Dyer [Dy]. Nowlet m = 1. Then U and V both have infinite order and G = A*H B with A -=

-~ a~" -- 1 > and

H==. The result then ibllows analogously. 8.5 Decision

Problems in One-Relator

Products

of Cyclics

In Chapter 2 we introduced the basic decision problems in combinatorial group theory; the word problem, the conjugacy problem and the isomorphism problem. Two additional decision problems are the generalized word problem and the power conjugacy problem. Let G =< X; R > and let H be a subgroup of G. The generalized word problem, abbreviated GWP,for G with respect to H is to determine if it can be decided algorithmicatly in finitely many steps whether an arbitrary element of G, written in terms of the generators X is in H. Wesay that the group G has a solvable GWPif G has a solvable GWPwith respect to the subgroups generated by recursive subsets of X. If G =< X; R > then the power conjugacy problem, abbreviated PCP, is given two elements in G, written in terms of the generators X to determine algoritmically, if a power of one is conjugate to a power of the other. The situation ibr one-relator groups is as ibllows: (1) Magnus,based directly on the classical Freiheitssatz, proved that onerelator groups have solvable word problem and fi~rther from the analysis it can be shown that they have solvable GWPas described above. (2) B.B. Newman[Ne], using the spelling theorem, proved that onerelator groups with torsion have solvable conjugacy problem (Theorem 3.2.6). Lipschutz [Li], using standard small cancellation theory, showed that cyclically pinched one-relator groups have solvable conjugacy problem (Theorem3.5.5). Juhasz [J 2] has described that all one-relator groups have solvable conjugacy problem, however a complete proof is not, yet available.

8.5 DECISION PROBLEMSIN ONE-RELATORPRODUCTS His methods -use an extension of small cancellation W(6) diagrams (see sections 3.8 and 5.3).

249

theory, the theory of

(3) Various special cases of the isomorphism problem have been settled. Pride [P 4] proved that the isomorphism problem for two-generator, one-relator groups is solvable (Theorem 3.2.7)¯ Rosenberger [R 21], using Nielsen reduction methods, showed that the isomorphism problem for cyclically pinched one-relator groups is solvable. Using essentially the same techniques, together with the Whitehead algorithm, Fine, Rosenberger and Stille [F-R-S 3], solved a restricted version of the isomorphismproblem for a class of para-free one-relator groups introduced by G. Baumslag. This was discussed in Chapter 3 -Thcorcm 3¯6¯5¯ More generally it has been shown that a one-relator group with torsion is word-hyperbolic (see Chapter 3). Sela [Se] has proved that there is positive solution of the isomorphism problem for torsion-free hyperbolic groups. In this section we extend these results to groups of F-type. The first result due to F.Tang [Tan 2] is that groups of F-type have solvable GWP. THEOREM 8.5.2. Let G --< al,. .... ,an; a~1 ......... a~~ = 1, (UV)m = 1 > where n >_ 2, e~ = 0 or e~ >_ 2, 1 =< V-1 >. Let Wbe a word in al,...,a=. Then W = x~y~x2y2...xky~ where xi are words in a~, ..., av and yi are words in a~+~, ..., a=. Since G~ and G2 are ~ products of cyclics, there is an algorithm to deter~ne whether x~ ~ and y~ ~< V > for i = 1, ...,k. Th~ there exists an algorithm to express Was a reduced word in G~ *A G2 ~OIIl which it follows that G has solvable GWP.

250


Now suppose that m > 1 and that either U or V has i~dinite order. Suppose without loss of gnerality that U has infinite order. Let a = UV so that U = aV-1. Then G = G~ *A G2 where G1--4 al, ..... , ap; a~1 ......... G~ --< ap+l , ..... , a,~, a;ap+l ~P+I - ¯- ......

a~p = 1 >

-- a~~ -- a "~ -- 1 > and

A ==< aV-1 >. Then as in the case where m = 1, G has solvable GWP. Wemention also that in [Tan 2] Tang proved a spelling theorem for torsion one-relator products of finite direct products of free, groups which extended the spelling theorem of B.B. Newman. Tang also studied the concept of Magnus relators. These are defined in the following manner. If Wis a word in the generators X, we use the notation {W}to stand for the elements of X involved in W. Suppose Go =< X; R > where X is a set of generators and R a set of relators in X and suppose S is a cyclically rcduccd word in thc frcc group on X. Thcn S is a Magnus relator for G =< X;R,S > if for every subset J of X with J ~ {S} ~ {S} the subgroup < .1 > of G is isomorphic to the subgroup < ,1 > of Go under the canonical homomorphismof Go to G. The classical Freiheitssatz says that the relator is a Magnusrelator for a one-relator group. The Freiheitssatz Ibr one-relator products of cyclics G--_ 2 and p~,p~+~as well as pi,p~ are in distinct factors, then g can be obtained by cyclically permutingpi,...,p~ and then conjugatingby an element ot" H. Another solution to the conjugacy problem was provided by techniques of Juhasz and Rosenberger [J-R] whose method of proof led to the fact that groups of F-type satis .fy certain hyperbolicity properties. Juhasz and Roscnbcrgcr’s proof is b~cd on a dctailcd analysis of thc structurc of thc involved Van Kampendiagrams. Juhasz and Rosenberger’s proof has the following interesting consequence. THEOREM 8.5.5. Let G be a group ofF-type. Assume ~hat at least one of U or V is neither a proper power nor a product of two elements of order 2 in the free product on the generators they involve. Then G is hyperbolic. In fact [J-R] contains deeper results on the combinatorial curvature of groups of F-type and related one-relator products. The above theorem can be considered as a special case of a muchmore general results (see.. section 2.7). Bestvina and Feign [Be-F] have shown that an amalgam of two hyperbolic groups over a cyclic subgroup is still hyperbolic, while Kharlamapovich and Myasnikov [Kh-M 3] have a more general result that the amalgam of two hyperbolic groups is again hyperbolic whenever one of the amalgamated subgroups is quasiconvex and malnormal in its respective factor. Thcsc rcsults can bc wcrc summarizcd in Thcorcm 2.7.6 which wc restate here.

252


THEOREM 8.5.6. (Theorem 2.7.6) IT H1,H2 are hyperbolic and H1 A H~ = H is a qu~iconvex s~ibgroup, malnormal in either H1 or H2, then ~,he amalgamatedproduct HI *H H2 is hyperbolic. In par~,ic~lax iT W~, W2 are elements oT infinite order in HI and H2 respectively and neither is a proper power then HI *wl=w2H2 is hyperbolic. Since tinitely generated subgroups of free groups axe quasiconvex it follows that i~ A and B are free groups and A ~ B = H is a t~nitely generated subgroup malnormMin either A or B then the amalgamated product A *H B is hyperbolic.

CHAPTER IX RELATED

GENERALIZATIONS

9.1 Related Generalizations

OF DISCRETE

of Discrete

GROUPS

Groups

Wenowconsider two new general classes of groups which are also natural extensions, via presentations, of discrete groups. Further these classes can be studied using the same techniques as in handling one-relator products of cyclics. First we consider a class of groups which arise in the study of nonEuclidean crystallographic groups (NECgroups) {See Chapter 4}. Wecall the members of this class groups of SN-type for Special NEe-type. A precise formulation will be given in the next section but, concisely, they can be described as iterated amalgamsof one-relator products of cyclics. The name arises since they have presentations which are similar in form to those of NECgroups. Weshow that a natural Freheitssatz exists for this class and that these groups admit essential representations into PSL2(C). As a consequence of these representations we obtain results on linearity properties for groups of SN-type analogous to those on one-relator products of cyclics. In particular we show that any group in this class is either SQuniversal or infinite solvable. The class of groups of SN-type contains as a subclass a collection of special NECgroups studied by Zieschang and Kaufman [Z-K]. Our techniques allow us to generalize a result on the ranks of these groups. Second we introduce and study a related class of groups, called generalized tetrahedron groups. The terminology was introduced by Vinberg, who studied them independently IV1. These groups generalize the class of ordinary tetrahedron groups discussed by Coxeter (see [Co - M]). The ordinary tetrahedron groups arise in 3-dimensional geometry in an analogous manner to the way the ordinary triangle groups were constructed in 2-dimensional geometry. The ge~mralized tetrahedron groups will be formally defined in section 9.3 but concisely they can be described as triangular products of generalized triangle groups. As for the class of Special NEC groups, we will prove that each generalized tetrahedron group admits an essential representation into PSL2 (C), and from this deduce many linearity properties. In addition we will use the concept of a Gersten-Stallings angle to prove a version of the Freiheitssatz. In particular we give sufficient 253

254

ALGEBRAIC

GENERALIZATIONS

OF DISCRETE

GROUPS

conditions for the generalized triangle group factors to inject into the group (see section 9.3.3.) 9.2 Groups

of Special

NEC Type

Recall that a non-Euclidean crystallographic group or NEC group, G is any discontinuous group of isometrics of the hyperbolic plane (see section 4.4). The class of NECgroups contains the Fuchsian subgroups as a subclass. Recall further that the geometric analysis of the structure of NECgroups follows a similar pattern to that of the Fuchsian groups. Using this geometric analysis it can be proved that a finitely generated NEC group G has a presentation of the following form (Theorem 4.4.1) (a) Generators: X ~2 J where X and J arc disjoint sets and J itself the disjoint union of ordered sets Jj = (Xl,~l, ..., xj,~), nj _~ (b) Relators: R0 U R1 t.J R2, the union of three sets of relators

2 . z~,h J} (i) R0= { xj,h,

(ii) R~is the union of sets R~,j = (tj,~ , ...,tj,n~ ) wherem~,h are positive integers and tj,h -= Xj,haj,hXj,h+laj--,~ (indexing h cyclically modulo rtj) with aj,h words containing only generators from X. (iii) R2 is a set of wordsof the form km~ where mk are positive integers and sk are words containing only generators from X Finally in the set of all aj,h together with all s~ each generator from X occurs exactly twice. This presentation of G can be deduced from the structure of a surface B. In the presentation, if the surface is orientable we can think of the X generators as the Fuchsian group part and the J generators as reflections. If the surface is orientable and there are no reflections this reduces to the structure obtained via the Poincare presentation. Wenow consider groups which have presentations similar in form to certain of those above. In particular we define a group of SN-type or group of Special NEC-type to be a group having a presentation of the form (9.2.1) G=

wheren_> 1, k_> 2, e~ -- 0 or ej _> 2 for i = l, .., n, fi=0orfi_>2fori= 1, .., k, m~>_ 2 for i --- 1, .., k andfor eachi = 1, ..., k, Ri = Ri(al, .., a,~, bi) is a cyclically reduced word in the free product on a~, .., a~, bi, which involves b~ and at least one of the a~s and which is not a proper power. In this case we call the Ri the relators. Using our previons terminology we recall that a representation p of a group of SN-type is essential if p(ai) has infinite order if ei = 0 or exact

9.2.1 ESSENTIALREPRESENTATIONSFOR GROUPS OF SN-TYPE 255 order e~ if e~ >_ 2 for i = 1,..,n,p(b~) has infinite order if f~ = 0 or exact order f~ if f~ > 2 for i = 1, .., k and p(R~) has exact order m~for i = 1, .., k. In the next section we show that each group of SN-type admits an essential representation in PSL2(C) and satisfies a FHS. From this follow many of the linearity properties of one-relator products of cyclics. 9.2.1

Essential

Representations

for Groups of SN-Type

Wefirst establish the FHSand the existence of essential representations for groups of SNtype. THEOREM 9.2.1. Le~ G be a group of SN-type, so that G has a presentation of the form 9.2.1

wheren> 1, k > 2,ej =Oorej >_2fori= l,..,n, fi=Oor fi > 2fori= 1,..,k, mi > 2 for i -= 1,..,k and for each i -= 1,...,k , R~= R~(al, ..,a,~,b~) is a cyclically reduced word in the free product on al, .., am, b~ which involves bi, no other b~, and at 1cast one of ~he a~s and which is not a proper power. Then (1) < al, ...,an >c-~< a];a~’ > *...* < a,,;a~c > (2) For any subset {bi~, ..., b~, } we have < al,..,an,bi~,...,bi, ~ < ai,.., a=,b~,.., bit, a~

a~ ~ ~.q b

(3) If each Ri involves a// a~, ...,a,, {a~,..., a~, } of {al, ..., a,~}

= >G b( ’~ q= RT

R.m~* -= 1 >

then for any proper subset

< a~,..,a~t,b~ >~= *...* < a~t;a~’~,* > * < b~;b~~ > (4) There exists an essential representation of G in PSL2 (C) which is faithful on < al, ...,a,~ >~ PROOF.Wesuppose first form G=< al,..,a,~,b~,b2;a~

that

~ ..

k = 2. Therefore the group G has the

a~~ =b(’ =b2/~ = R7~ =/L~ ~ = 1 >

wheree~, f~ are as before, m~, m2> 2 and R~= R~(a~,.., a,~, b~), i = 1,

256


Let a 1 =~(

al,..

,

an,

bl;

a~ 1 ..

a~e~ = b{1 = R~TM = 1 > and

an, b2; a~~ .. a~~ = b2/~ = R~2 = 1 > .

G2 =< al, ..,

G1 and G2 are one-relator products of cyclics. Since both ~’?.1 and m~are at least 2 we can apply the FHSfor one-relator products of cyclics (Theorem 6.3.2) to say that < al, ...,an >~=< al, ...,an >a2=< a~, ..,an;a~ ~ .... a~~ = 1 >. It follows that identifying {a~,...,an} in G~ with {a~,...,an} in G2 gives a subgroup isomorphism and therefore G is the free product of G~ with G~ amalgamated over < a~, ..., a~ >. Prom the structure theorems on free products with amalgamation, G~, G2 and < a~, ..., a~ > inject into G. If {ai~,.., ai~} is any proper subset of {al, ..., an} then {ail,.. , ai~, bi} generate a subgroup of Gi. If Ri involves all al, ..., an then again from the FHSfor one-relator products of cyclics this must be the free product. This establishes (1),(2) and (3) for the case If k = 3 then G has the form at, .., an, bl, b2; ael~ .... an ~" = b{~ = b2h = ba/~= R~= R~~ = R~a =: l’- Using the result for k = 2 and the FHSfor one-relator products of cylcics G must now be the amalgamated free product of al =< al, ..,

an, bl, b2; a~~ .. a~’~ = b{~ = b~~ = R~1 = R~’~ = 1 > and an, b3; a~~ .. a~’~ = ba~ = R~~ = 1 >.

C~2 =< al,..,

amalgamated over < al,..., an >. Parts (1),(2) and (3) of the theorem follow as before. For general k the first three parts now follow easily by induction. Wenow establish the existence of an essential representation. Again first suppose that k = 2. Let G, G~ and G~ be as before with k = 2. From Theorem6.3.1 there exists an essential representation Pl of which is faithful on < al,...,an >. Let A~,..,A~, B~ be the images of a~, .., an, b~ under p~. Let B2 = ~=

(t,

* 2cos(~r/f~) -

B2 = :t:

2 - t

) iff2>2and -

9.2.1

ESSENTIALREPRESENTATIONS FOR GROUPSOF SN-TYPE 257

with t a variable to be determined. Since tr(B2) = 2cos(Tr/f2) if f2 _> 2 we then have B2/2 = 1 for every choice of t. Weshow there exists a choice of t so that the image of R2 has order m2giving the essential representation. SupposeR2 = Wlb~1 w2 ..... b~~ whereWl, w2,. .... are non-trivial words in ~ where Wi are the corresponding imel, ...,an. Let R-~2 = W1B~IW2...B~2 ages of wl, w2, ..., wn under the representation pl. Since Pl is faithful on < al, ...,an > it follows that W1,W2,... are non-identity projective matrices. Therefore the same arguments as in the proof of Theorem 6.3.1 and Theorem6.3.2 can be used to assert that there exists a choice {perhaps by conjugating the original A~, ..., Anif necessary} so that (d~ d3 d2) W1B2klW2...B 2k~= -td4 where dl + da is a non-constant polynomial in t. From the fundamental theorem of algebra there is a choice tl so that (dl + da)(t~) = 2cos(tim2) With this t~ in B2 we then have tr(R--~) 2cos(tim2) and he nce ~2 2 = 1 Let p be the representation which takes a~,...,an,bl,b2 to A1, ..., An, B1, B2(t~) {or whichever conjugates of A1, ..., A,~, B1 were necessary so that the trace polynoinial was non-constant} respectively. This representation p is then faithful on < a~, ..., an > and from the trace conditions p(b~){= B1},p(b2){= B2},p(R~) and p(R~) have exactly the correct orders - that is p(bi) has infinite order if f~ = 0 or order fi if fi _> 2 and p(Ri) has exact order m~ for i = 1, 2. Therefore the representation p is essential. Once the essential representation for the case k = 2 is established the same argument can be iterated for the general case. This completes the theorem. Theorem 6.3.1 proved that if A and B are groups which admit faithful representations in PSL~(C) and R is a non-trivial word of syllable length at least 2 in A* B then the one-relator product G = (A * B)/N(R’*) with m _> 2 admits a representation p into PSL2(C) such that PlA is faithful, PlB is faithful and p(R) has order m. Using this result and an analagous proof to that of Theorem9.2.1 we obtain: THEOREM 9.2.2. Let Go =< A; re/A > and, /:or Bi; rel B~ >. Suppose that G =< A, B~, ..,

Ba; re/A, re/B1, ..,

i=l,..,k,

re/B~, R~~ .....

let G~ =< R~~ = 1 >

wherek > 2 and, for each i = 1,..,k, R~ = R~(A, Bi) is a non-~rivial cyclically reduced word in the free product A * B~ of syllable length at least two and

258


each m~ >_ 2. Assume further that each G~ [or i = 0,1,..,k representation into PSL2(C). Then

admits a faithful

(1) Each G~ [or i = 0,1,...,k injects in G (2) For each subset {il, .., it} o[1,2, .... ,k we have >G=< i~ A,B~, Bi,; relA,..,

< A, Bi~ ..,B~,

tel

RT~ = 1 >

Bi,,RT

(3) There exists a representation p of G into PSL2(C) such that is faithful for i = 0,1..,k and p(R~) has order m~for i = 1,2,..,k. Wenote that the conclusions of Theorem9.2.1 are also valid for certain modifications of the presentation type (2.2). For example the same proof can be used for groups with presentations of the form G =< a~,

..,

a,~+~;

~~ R~ ..... a~n~+~ .. = ~+~

Ra = 1 >

where n _> 1, k _> 2, e~- = 0orej _> 2 for i = 1,...,n+k, mi _> 2 for i = 1,2,..,k and for each i = 1, ..,k, R~= R~(a~,...,a~+,~) is a non-trivial, cyclically reduced word in the flee product on ai, ..., ai+,~ which involves all ai ~ ... ~ ai+,~. In general it is not clear what happens relative to essential representations for general multi-relator groups. Below we present an example of a group G very similar in form to a group of SN-type but which has no essential representation in PSL~(C). Let G =< x,y,z;x 4 = y3 = z ~ = (xy[x,y])~ = (x3z(x[x,y])2z-~)5 1 >. Suppose X, Y, Z are the images of x, y, z under a supposedly essential representation in PSL~(C). Then without loss of generality let tr(X) a = v/~, since x4 = 1, and tr(Y) = b = 1 since y3 =1. Suppose tr(XY) then from a direct computation we have trX[X, Y] = a(a~+ b2+ 2- abe-3) and trZY[X, Y] = c(a~ +b2 +c~ -abc-3). Since (XY[X, y])2 = 1 we have, if the representation is essential, that tr(XY[X, Y]) = 0. Therefore either c = 0 or a~ + b2 + c ~ - abc- 3 = 0. Howeverif c = 0 since a = v~ and b = 1 we have that a2 + b~ + c ~ -abc - 3 = 0 also. Therefore tr(X[X, Y]) = and so (X[X, y])2 = 1. However then from the second relation it would follow that X~s = 1 which coupled with X4 = 1 gives X = 1 and hence the representation is not essential. 9.2.2

Linearity

Results

for Special

NECGroups

In this section we show that groups of SN-type satisfy muchthe same linearity properties as one-relator products of cyclics. Specifically we examine the Tits Alternative, SQ-universality and virtual torsion-freeness.

9.2.2

LINEARITYRESULTSFOR SPECIAL NEC GROUPS

THEOREM 9.2.3. Let G be a group of SN-type. universal or G is infinite solvable.

Then either

259

G is SQ-

PROOF.For a group of SN-type having form (9.2.1) let N = n ÷ k. first prove the result for k - 2 so that N = n ÷ 2. The result for general k then follows in a straightforward manner. Thus we suppose that our group G has the form G=< ax,..,an,

bl,b2; a~1 ....

a~’

:bl

fl

:b~

f2

:

R m -~-

S t:

1

>

where n >_ 1, ej = 0 or ej _> 2 for i = 1,...,n, fi = 0 or fi _> 2 for i = 1, 2, m_> 2, t _> 2 and R = R(al, .., am bl) is a cyclically reduced word in the free product on al, .., a,~, bl which involes b~ and at least one of the ai and S = S(al, .., am, b2) is a cyclically reduced word in the free product on al, .., an, b2 whichinvolves b2 and at least one of the ai. Since, if a quotient of a group is SQ-universal, the group is itself SQuniversal we can, without loss of generality, assume that ej ~> 2 for i = 1, ...,n and fi _> 2 for i = 1, 2, passing to a quotient if necessary. For N _> 5 or N = 4 and (el,e2,f~,f~,m,t) (2 ,2,2,2,2) th e pr oof follows the outline of the proof of Theorem 6.2.3. From Theorem 6.3.2, G adlnits an essential representation p into PSL~(C) which is faithful on the free product on al, .., a,~. Therefore from Selberg’s theorem on fi~fitely generated subgroups of linear groups there exists a normal subgroup H of finite index in p(G) such that p(ai) has order ei moduloH, p(bi) has order fi modulo H, p(R) has order m modulo H and p(S) has order t modulo H. Thus the composition of maps {where 77 is the canonical map} G -~ p(a) -~ p(G)/H gives an essential representation ¢ of G onto a finite group. ~ .. a~~ =bll ~ =b~h= 1 > be the free Let X=< a~,...,a,~,bt,bu;a~ product on {a~, .., a,~, bl, b2} ¯ There is a canonical epimorphismf/: X -~ G. Wetherefore have the sequence X --~ G ~ p(G)/H Let Y = ker(¢o/3). Then Y is a normal subgroup of finite index in X and Y is torsion-free. Since X is a free product of cyclics and Y is torsion-free, Y is a free group of finite rank r. Suppose [X : YI = J. Regard X as a Fuchsian group with finite hyperbolic area p(X). {Every fintely generated free product of cyclics which is not i~ffinite dihedral can be faithfully represented as such a Fuchsian group}. From the Riemann- Hurwitz formula : j#(X)

= #(Y)

260


where #(Y) = r and #(X)

= N- 1 - (1 +.. el

+ __ en

),

we obtain 1 1 r=l-j(~+..+--+en~7~2+

1

1 -N+I)

G is obtained from X by adjoining the relations R’* and St and so G = X/K where K is the normal closure of R"~ and St. Since K is contained in Y the quotient Y/K can be considered as a subgroup of finite index in G. Applying the Reidemeister- Schreier process Y/K can be defined on r generators subject to (j/m) + (j/t) relations. The deficiency d of this presentation for Y/K is then d=r

J j m t

_l_j(1

1 1 1 ~+"+--+~en -f~ -f~2

1 1 m t+ +--+--N+I)

If N_>5, or N = 4 and (el,e2, fl,f.2, m,t) ~ ( 2,2,2,2,2,2), t hen d> 2. From the work of B. Baumslag and S. Pride [B.B. - P 2] Y/K and hence G has a subgroup of finite index mapping onto a free group of rank 2. Therefore Y/K is SQ-universal and since this has finite index in G, G is also SQ-universal. Therefore if N _> 5, or N = 4 and (el, e2, f], f2, m, t) (2, 2, 2, 2, 2, 2) then G is SQ-universal. Next suppose that N = 4 and (e~,e2,f~,fu,m,t) = (2,2,2,2,2,2), now G has the presentation (7=< al,a2,

bl,b2;a~

=a~2 =b~ =b~ =R2---o¢2=

1>

where R = R(a~, a2, b~) and S = S(al, a2, b2). We may assume that R and S are not proper powers since otherwise it reverts back to the previous arguments where one of the exponents is not 2. Without loss of generality we must consider the following four cases: (1) R al b~ and S = a~b~ (2) R al b] and S = a2b~ (3) R = alb~ and S involves both al and (4) Both R and S involve a~ and In case (1) if R = a~bl and S = a]b~ then a

=
.

9.2.2


261

Setting al = 1, G can be mapped onto < a2,b~,b2;a~ = b~ = b~2 = 1 >= Z2 * Z2 * Z2 which is SQ-universal and therefore G is. {Recall that any non-trivial free product except the i~dinite dihedral group Z2 * Z2, is SQuniversal}. In case (2), where R al bl and S -- a2b2 the n by set ting bl -- 1, G can be mapped onto < al, a2, b2; a~ = a~ = b~ = (a2b2) ~ = 1 >= Z2 * D: {where D~ is the Klein 4-group }. Since this is a non-trivial free product and not infinite dihedral, G is SQ-universal. In case (3) where R = alb~ and S involves both al and a2 set al = 1, to obtain as an image of G, < a~,51,52;a~ = 55 = b2~ = (a2b~) ~w = 1 >. This is a non-trivial free product Z~ * T where T has the presentation < a2, b2; a~ = b~ = (a2b~)2~’ -- 1 >. For no value of w is this cyclic of order 2 and therefore Zu * T is SQ-universal, and hence G is. Finally suppose case (4) where both R and S involve a~ and a: . Let be the subgroup of G generated by a = a~a.2, b = alb~ and c = a~b2. Then H has index 2 in G and there are three, possibilities for R and S relative to H. (1) Both R and S are elements of (2) R is an element of H but S is not {or vice versa}. (3) Both R and S are not elements of If both R and S are elements of H then R = T(a, b) where T(a, b) is a freely reduced word in the free group on a, b and S = U(a, c) where U(a, c) is a freely reduced word in the free group on a, c. Then H has a presentation H =< a,b,c;T2(a,b)

= T2(a -~,b-~)

= U~(a,c) = U~(a-1,c-~)

=1 >.

Using the arguments in [L-S pg. 293] we may assume that one of the generators a, b has exponent sum zero in T(a, b). If b has exponent sum zero, let a = 1 to obtain the quotient < b, c; c 2w = 1 >= Z*Z2~if w >_ 1 or Z*Zif w = 0. Either is SQ-universal and therefore G is. Ira has exponent sum zero let b = 1 to obtain the quotient < a, c; UU(a,c) a theorem of M. Edjvet [E 3] on groups which have balanced presentations, this quotient has a subgroup of finite index mapping onto a free group of rank 2, and is therefore SQ-universal. Thus G is SQ-universal. This completes the situation where both R and S are elements of H. Nowsuppose that R is an element of H but S is not. Suppose R -- T(a, b) as above then S = a~U(a, c) where U(a, c) is a freely reduced word in the free group on {a, c}. The subgroup H now has the presentation H----< a,b,c;T2(a,b)

= T2(a-~,b-~) ---- U(a,c)U(a-~,c-~) = 1 > .

Let a = 1 to obtain the quotient < b, c; b2w ~- 1 > = Z * Z,~, which as above is SQ-universal and therefore G is.

262


Finally suppose both R and S ~re not in H. Then R = alT(a, c), S alU(a, c) and H has the presentation H = . Letting a = 1 we obtain the quotient < b, c; >-- Z*Zwhich is SQ-universal, and therefore G is. This completes the situation when N = 4 and all exponents are 2. Wemust now consider the case when N = 3. Thus G has the presentation G =< al,bl,b2;a~ ~l=b~1=b~22=Rm=St = 1> . From the proof given above for N >_ 4 a group with a presentation as above is SQ-universal if the deficiency d of the analogously defined subgroup Y/K is 2 or greater, or equivalently, if (1/e~) + (1/f~) + (1/f2) + (l/m) (l/t) < 2. By a lengthy argument on the possible values of (el, f~, f2, m, the SQ-universality or not of groups with presentations of form (2.4) can established directly. Howeverthis can be simplified using a result of Lossov [Los]. He proved that if G ---- A *H B with IHI < ~, IA : HI _> 2 and IB : HI _> 3 then G is SQ-universal. In (2.4) we have G = G1 *H G2 where ~1 ~___b]fI = .~D~ G1 --< al,bl, . a~ = 1 >,G2 =< 1a~,b~;a~ =bf22 = st = 1 > ~ and H =< a~;a~ = 1 > . Since we can assume that el ~ 2, H is finite cyclic and Lossov’s theorem applies unless both G1 and Gu are finite with IG~ : HI = 2 and IG~ : HI = 2. This would imply that both G~ and G~ are finite dihedral and G would have the presentation G 1 =< al, b~, b2; a ~ = b~ = b~ = (a~bl) =~

(alb2)

2-- 1

>.

This has the structure of a free product of two isomorphic finite dihedral groups amalgamated over their cyclic subgroup of index 2. This group is infinite solvable. Therefore if N = 3 either G is SQ-universal or G is infinite solvable. This completes the proof for k = 2. If k > 2 using the same argument as before, employing an essential representation of G onto a finite group we obtain an analogously defined subgroup Y/K of finite index in G having a presentation with deficiency d given by 1 1 +..+--+ J(e, e~

1 -~1

+..+

1 1 +--+..+--~k

1

(n+k)+l)

ml

1 1 1 =j [ ( n +k ) - l ] - j ( ~ +. . +-- +e,~ ~ ~- " -~

1 ~ ~---1 ~- " -~ -- ) 1 mk

9.2.2


263

As in the case when k = 2 the deficiency is 2 or greater mfless n = 1 or n = 2 together with all exponents 2. Thus G is SQ-universal except possibly in these latter two cases. If n = 1 , Lossov’s theorem applies to show that G is SQ-universal if k > 2. If n = 2 and all exponents are 2 then G has the presentation G=< al,a2,

bl,b2,...,bk;a~

=a~ =b21 .....

b~ = R~ .....

R~ = 1 >.

Setting al = a2 = i we get the quotient < bl, b2, ..., bk; b2~ .... = b2~ = 1 >. This is a free product of cyclic groups of order 2 and since k > 2 it has more than two factors and is thus SQ-universal. Therefore G is SQ-universal. This completes the proof. Notice that, if a group is SQ-universal, it must contain a free subgroup of ra~k 2. From this we obtain the Tits Alternative. COROLLARY 9.2.1. Tits Alternative.

Let G be a group of SN-type then G satisfies

the

COROLLARY 9.2.2. Let G be a group of SN-type then G is infinite solvable if and only if G has a presentation < al, bl, b2; a~1 -- b2~ -- b22 = (albl) 2 = (alb2) 2 --- 1 > . Otherwise G is SQ-universal. If G is a group of SN-type then G admits an essential representation. If we can classify the elements of finite order we can apply Theorem6.2.1 to get that G is virtually torsion-free.. THEOREM 9.2.4. Let G be a group of SN-type. Assume each relator is not a proper power and satisfies one of the following conditions:

R~

(1) R~ -- U~V~where U~ -- U~(a~,..,a~) and V~ = V~(a~+~,..,a,,b~), k _< n are non-trivial words in the free product on ai, .., a~ and a~+l,.., a,~ bi respectively. (2) R~ is not conjugate in the free product on a~,..,a,,b~ to a word of the form XY where X, Y are elements of orders p >_ 2, q >_ 2 respectively with (1/m~) + (l/p) + (I/q) and m~ >_ 4. (3) R~ is arbitrary but e~ = 0 for i = 1,...,n, f~= 0 for i = 1,...,k. Then G is virtually torsion-free. PROOF.Let H be a one-relator

product of cyclics with presentation

H =< al,..,a~,b~,a~. ~ ....

a~~ = b~

=1 >

withn_> 1, ei =0or e~_> 2 for i = l, .., n, fj =0or fj _> 2 forj = 1 or 2. If m _> 2 and any one of the three conditions in the statement of the theorem is satisfied then any element of finite order in H is conjugate to a

264


power of one of the generators ai or a power of the relator R (see Chapter 7). Nowlet G be a group of SN-type. Wemay assume that k = 2. From the proof of Theorem 9.2.1, G is an amalgamated fxee product of one-relator products of cyclics. Hence, from the torsion theorem for such amalgams any element of finite order in G is conjugate to an element of finite order in one of the factors. Combiningthis with tim statement above on one-relator products of cyclics we have that if each relator Ri satisfies one of the stated conditions then any element of finite order in G is conjugate to a power of a generator al, ...am bl, b2 or a power of a relator R1, R2. Hence an essential representation must be essentially faithful and applying Theorem6.2¯ 1 we get that G is virtually torsion-free. COROLLARY 9.2.3. Let G be a group of SN-type satisfying the conditions o[ theorem 9.2.4 then the conjugacy classes of torsion elements are precisely given by the powers of the generators and the powers of the relators. 9.2.3

Rank Conditions

for Certain

NEC Groups

Zieschang and Kaufman[Z-K] have recently considered the rank { minimal number of necessary generators } of groups with presentations of the form These groups are contained in the class of groups of SN-type and are NEC groups with reflections. Zieschang and Kaufmanproved that if 2 < h < k, then G has rank 3 if h > 2 or h = 2 and k is even. Otherwise G has rank 2. Wenow give some extensions of this. THEOREM9.2.5. Let G (aiaa) k = 1 > with ei = 0 or ei >_ 2 for i = 1,2,3, 2 ~_ h ~_ k. Then 2 _ 2 and k >_3, k odd. { In the case where G has rank 2, G, as above, is generated by x = a and y = cb and G is an epiinorplfic image of the Fuchsian group F k 81,82, 83, 84; 812 = 822 ~--- 832 : 84 = 81S28384 = 1 > with k _> 3 and k odd.} PROOF.Let G be as above. If el = e~ = e3 = 2 then the result follows from Zieschang-Kaufman. Assume that e; where al ¯ ~ = a~~ = (a~a2) h 1 >,G~ =< a~,a3;a~ ~ ~ = a~ ~ (a~aa) = i > and H =< a~; a~~ = 1 > . Clearly 2 _< rank G _< 3. Assume G =< z, y > so that rank G = 2. Weuse Nielsen reduction on {z, B}. In Chapter 2 we stated the following result which is crucial in the proof here.

9.2.3

RANKCONDITIONSFOR CERTAINNEC GROUPS

THEOREM 9.2.6. Let G = H1 *A H2. H {x 1,..., x,~} is a finite elements in G then there is a Nielsen transformation from {Z1, ..., system {Yl,-.., Ym}for which one of ~he following c~es hold:

(i) (ii)

265 system of Zm} tO a

= 1 .. ., Eac~ w ~< y~,...,y~

> ca~ be wri$~e~ as w = ~=~y~,~ =

"~ a ¢ 1, withy~ ~ A(i 1,..,q) and Thereisaproduc~a =~a~=~y,~, in one of the factors Hi ~here is an elemen~ x ~ A wi~h x-~ax e A; (iv) There is a g e G such Chat for some i e {1, ...,m} we have y~ ~ gAg-~, bu~ for n suitable natural number k we have y~ e gAg-~; (v) Of $he y~ ~here are p ~ 1 contained in a subgroup of G conjugate ~o H1 or H~ and a certain produc~ of them is conjugate ~o a non-~rivial elemen~ of A. The Nielsen trnnsformn$ion can be chosen so that {y~, ...,y~} is shorter (wi~h respec$ ~o ~he len~h and a suitable order) Shan {Xl, ..., x,~} or the len~hs of the elements of {x~, ..., x~ } are preserved. ~r~herif {x ~, ..., x~ } is a generating system of G $hen in case (v) we find p ~ 2 for in ~his case conjugations determine a Nielsen $ransformn~ion. g we are interesSed in $he combinn$orial description of < x~, ..., x~ > in $erms of genera$ors and relations we find again $haf p ~ 2 in ease (v), possibly a~er suieable conjugations. (iii)

Case (i) of Th~rem 9.2.6 c~nnot occur for {x,y} in G since G is noncyclic while case (ii) of Theorem 9.2.6 cannot occur for {x,y} in G for otherwise G would be a ff~ product of cyclics. Each G~ is an ordin~y triangle group, so there is no d ~ G~ ~ H with 1 ¢ dt ~ H for some t e N. Therefore case (iv) cannot occ~. If case (v) occ~s we may assume that are both in G~ or both in G2, which is a contradiction since G1 ¢ H ¢ G2. Therefore case (iii) of Theorem9.2.6 occurs. Without loss of generality assume that 1 ¢ x = a~ for some positive integer t and that there is an elemem v in G~ ~ H with v-~a~v = a~ ~ 1 for some u,w ¢ 0 and a[ ~< x >=< a~ > . Regard G~ as a subgroup of PSL2(C). Then a direct computation shows that u = -w if e~ = 0 and u ~ -w rood el if e~ ~ 2 and v2 = 1. Therefore v-~a~v = a~~ so (va~) 2 = 1. Hence e~ = h = 2 and twe may assume that x = al became G =< x, y >=< a ~,y >C< al,y > ¯ Assu~ne that either e~ = 0 and k even or both e~ and k are positive even. Adjoining the relations a~ = (a~a3) ~ = al = 1 we obtain the quotient < ae, au; a~ = a~ = 1 > which is not cyclic. It follows that ~t least one of ez or k is odd and therefore there is no relation v-~a~v = a~ ¢ 1 for some integers s, t and some v e G2 ~ H. Analogously as above if e~ ~ 2 then the gcd of e2 and k is 1. Nowwe come back to our generating system {x, y} with x = a~.

266


Using the above arguments and Nielsen reduction we may assume that y = hla2h2a2...hma2 with m _> 1 and the hi coset representatives of H in G2. Because there is no relation v-latl v = a~ we must have m = 1 for otherwise x and y cannot generate G. Hence y = hla2 for some hi E G2\H. Further x and y can generate G only if x and yxy -~ or, equivalently, al and h 1 a ~- 1 h ~- 1 generate G2.Fromthe classification of generatingpairs of triangle groups {G2is also an ordinary triangle group} we get that one of e3 or k is 2. Therefore without loss of generality we have that e3 = 2 and k >_ 3 and hence G has a presentation < a,b,c;a p = b2 = c 2 = (ab) 2 = (ac) k = 1 > with p = 0 or p _> 2 and k >_ 3, k odd. This completes the proof when el ~2. Note that, if G has a presentation as above, let u = ab so that G also has the presentation < u,b,c;u 2 = b~ = c 2 = (ub) ~ = (ubc) t~ = 1 > . Let x = a = ub and y -- cu -- cba-1. Then [x,y] = (ubc) ~ and therefore ubc, c, u, b are in < x, y > since k is odd. Therefore {x, y} generate G. From the second presentation it is clear that G is an image of the Fuchsian group . 2 = s~ = s~ = sk~ = sls2sas4 = 1 > Fwithpresentation < sl, 2,sa, s4~s 8 1 with k odd and k _> 3. Nowsuppose that el ---- 2 and e2 _< h and ea _< k. Let {x,y} be a generating pair for G. Again we use Theorem 9.2.6 and case (iii) of that Theorem must hold. Therefore we may assume that x -- al is in A and without loss of generality there is a v E G1 \H with valv -~ -- al. Regarding G1 as a subgroup of PSL2 (C), and using direct calculations we get that e2 = 2 and h = 2r _> 2 and (ala2) ~ commutes with al. As for the case when el ~ 2 we use the classification of generating pairs for triangle groups to obtain that at least one of e3 or k is odd and the gcd of e3 and k is 1. Hence we get that e3 = 2 and k odd since, if e3 _> 3, then G2 cannot be generated by two elements of order 2 (see the arguments for the case el ~ 2. From this we must have h = 2 for, if h >_ 4 then G has rank 3 from the Zieschang-Kaufman result. This completes the proof. Using modifications of the same type of cancellation arguments as in the proof above we can give the following generalization which we just state. THEOREM 9.2.6. Le~ G =< al, a2, a3, ..., am bl, b2; a~1 = a~~ ...... ~ a~ = b{’ = bY2~ = (ala2 .... anbl)h = (ala2 .... anb2) k = 1 > with e~ = 0 or e d >_2fori--1,2,...n, fi=Oorfi>2fori=l,2andh_>2, k_>2. Then n+l_< rankG

More generally we call a group an ordinary tetrahedron group if it has a presentation of form (9.3.1). Coxeter{see [Co-M] } gave necessary and sufficient conditions for an ordinary tetrahedron group to be finite. Vinberg observed that one can state these conditions as follows: A group G with a presentation of the form (9.3.1) is finite if and only the matrix 1 - cos(n/k) cos(~r/k) .

-cos(r/l) - cos(~r/r)

- cos(r/m) - cos(~r/q)

1 - ~os(r/O- cos(~-/~) - cos(~/p) cos(r/m) - cos(r/q) - cos(~r/p) has positive determinant (see [Cox 1],[Cox 2],[Cox 3]). In the analysis of the Tits Alternative for generalized triangle groups and for special NECgroups subgroups with presentations of the following form related to (9.3.1) often arose. < al,a2,aa,

a~_ = a2 = aa = R’~(al,a2)

= R~(al, aa) = R~(a2,a3)

Vinberg IV] studied these groups independently and called them generalized tetrahedron groups and we adopt this terminology. Specifically a generalized tetrahedron group is a group with a presentation of the form

(9.3.2) < al, a2, an; a~1 = a~2 = aa = R~(al, a2) = R~(a,, a3) -= Rq3(a2, a3) = 1 where ei = 0 or e~ _> 2 for i -= 1,2,3;2 < m,p,q ; R~(al,a2) is a cyclically reduced word in the free product on a~, a2 which involves both a~ and R2(a~, an) is a cyclically reduced word in the free product on a~, 33 which involves both a~ and aa and Ra(a2, a3) is a cyclically reduced word in the free product on 32, aa which involves both a~ and aa ¯ Further each Ri, i = 1, 2, 3 is not a proper power in the free product on the generators it involves. As before a representation p : G -+ Linear Group is essential if p(ai) has infinite order if ei = 0 or exact order e~ if e~ >_ 2 for i -= 1, 2, 3, has order m, p(R2) has order p, and p(R3) has order q. In particular the

268


existence of any essential representation implies that the group G is nontrivial. Tsaranov ITs 1],ITs 2] considered the special generalized tetrahedron groups where each Ri(u, v) is a word of the form u’~v b and gave a complete classification of the finite groups in this case, extending the results of Coxeter. Recall that a group G is a polygonal product (see Chapter 2) if it can be described in the following manner. There is a polygon P. Each vertex v corresponds to a group G.. Each edge y corresponds to a group Gy and adjacent vertices are amalgamated via relations along the edges. The group G is then the group formed by the free product of the G. modulo the amalgamating edge relations. The groups G. are called the vertex groups. As mentioned in Chapter 2, Karrass, Pietrowski and Solitar have developed a subgroup theory of polygonal products which is parallel to the theory for free products with amalgamation. If the polygon has four or more sides then the group G decomposes as a free product with amalgamation and it follows that each of the vertex groups inject into the group G. In the language introduced in Chapter 3 we say that the vertex groups are FHSfactors whenever the polygon has four or more sides. If the polygon is a triangle it is called a triangular product. Amongall products {see Chapter 2} the triangular products are the least well-behaved and the question arises as to when the vertex groups inject. For a generalized tetrahedron group G with presentation (9.3.2) let G1 ----< al, a2; a~1 = a~2 -= R~(al, a2) = 1 ¯ el .= a~ .= R~(a~, a3) = 1 G2 1 ~< al~a3~ a G~ --< a2, a3; a~2 = a~a = R~(a2, a3) -- 1 > Each of these groups is a generalized triangle group and the generalized tetrahedron group G is then a triangular product of these groups with edge amalgamations over the cyclic subgroups < a~ >, < a2 >, < a3 >. As mentioned earlier it is not clear when the vertex groups in a triangular product inject into the group. In section 9.3.5 we give conditions for this to occur relative to the generalized tetrahedron case. Wecall G~, G2, G3 the generalized triangle group factors of the generalized tetrahedron group G. As in the analysis of one-relator products of cyclics and groups of SN-type we show first that generalized tetrahedron groups admit essential representations into PSL2(C). In particular this shows that these groups are never trivial. Using the existence of essential representations we give sufficient

9.3.1

ESSENTIALREPRESENTATIONS

269

conditions for both the SQ-universality of generalized tetrahedron groups and the existence of non-abelian free subgroups. Using techniques of Gertsen and Stallings we also give conditions for the generalized triangle group factors to inject. 9.3.1

Essential

Representations

Wefirst prove the existence of essential that these groups are non-trivial.

representations

and thus show

THEOREM 9.3.1. Let G be a generalized tetrahedron group with presentation as in (9.3.2). Then G admits an essential representation into PSL2(C). PROOF.Let G1,G2,G3 be as given above relative to G. Then G1 --< al,a2, at = a2 = R~n(al,a2) = 1 > is a generalized triangle group. Choose an essential representation of G1 in PSL2(C) which is possible from Theorem6.3.2 a.nd let At, A2 be the respective images of at, a2 ¯ If A,B are projective matrices in PSL2(C) and RI(A,B) q~ with all Pi¢ 0 if A has imqnite order and 1 _< Pi < p if A AplBql ..... AP~B has order p and all qi ¢ 0 if B has infinite order and 1 < qi < q if B has order q, then tr(R1 (A, B)) is a polynomial of degree k in tr(AB) with coefi%ients from. Z[tr(A), tr(B)]. {see Chapter 7}. Nowsuppose tr([gl, A2]) ?g 2. In [R 9] Rosenberger proved the following lemmawhich is crucial for the remainder of the proof of Theorem9.3.1. LEMMA9.3.1.

[R 9] Let A,B,C,X

be four elements in SLy(C) at with AB = C. For A we write A = (a~) a3 a4and we use simT and r = ilar notation for B,C, and X. Let x = (xt,x2,x3,x4) (tr(X), tr(AX), tr(BX), T and 1 0 0 al a3 a2 a4 bt ba ba b4 " c1 c3 c2 c4 Then Mx = r and det M= tr([A,

B] - 2.

Then from this lemma we can construct a projective matrix A3 in PSL2(C)such that tr(A3) = 2 if e3 = 0 or tr(Aa) -- 2cos(~r/ea) if ea _> 2, tr(R2(A1, A3)) -- -2cos(trip) and tr(Ra(A2,Aa)) -- -2cos(~r/q). Since a projective matrix B in PSL2(C)has order t if its trace is +2cos(~r/t) the representation defined by at -+ At, a2 -~ A2, a3 -+ A3 gives the desired essential representation. The construction of Aa using lemma9.3.1 used

270


the fact that tr(R2(A1, A3)) and tr(R3(A2, Aa)), analogously as above, are non-constant polynomials in tr(A1Aa) and tr(A2A3) respectively and then employed the Fundamental Theorem of Algebra. Nowsuppose al --~ A1, a2 --+ A2 defines an essential representation of G1 with tr([A1, A2]) = 2. Then lemma 9.3.1 cannot be used directly but essentially the same argument goes through as follows. Since tr([A1, A2]) 2 we may assume without loss of generality that < A1, A2 > is an i~nite metabelian but not abelian ~oup. A~er a s~table conjugation we may ass~e that ~ and A2

0

~rther if el = 0 we may assmne that assume that ~ = 0. Hence: A~=~

0

~-~

~ 0

~

~ ~ ~1,~ = 0. Ife~ ~ 2 we may

andA2=

~ 0

~

with 5 ~ 0 since < A~, Au > is non-abelian. We may ass~e further that ~ ~ ~1, for if fl = !1 then e2 = 0 and we may find an essential representation p~ : G1 ~ PSL:(C),aI ~ + ~-~ = tr(A~),tr(A~) ¢ ~2. Therefore assume ~ ¢ ~1. Let A3=~( xix3

x~

with x~,x2,x3,x~ considered as variables and A3 to be considered as an element of PSL:(C). Now tr(R2(A1,Aa)) = -2cos(r/p) and tr( Ra ( A~ , A3 ) ) = -2cos(u/q) are non-co~tant polyno~als in tr( ( A ~ A3 ) and tr(A2A3) respectively. Choose zeros of these two polynomials and let tr(A3) = 2 if c3 = 0 or tr(A3) = 2cos(u/ea)if e3 ~ 2. Assumefu~her that (trA3, tr(A~A3), tr(A2A3)) # (0, 0,0). Nowconsider the system of linear equations: x~ + x~ = tr(A3) (9.3.3)

O~Xl + o~-Ix4 = tr(A1A3) = ~X1 -~- (~X3~- ~--lx4 = tr(A2A3) =

Since det

I 0 ~ 0 ~-~

=~(~-~-~)#0,

9.3.1


there exists a unique solution -i rl(2 -X 1

(2__

(2--1

(Xl,

r2 ~-

(2--1

X3, X4)

with

_(2~-1_~_ ~(2--1 rlH 6~-- ~---1) r2+6-~r3 and

’x3

(2rl X4 --

271

(2

r2

__ (2_1

(2 __ (2--1"

Further this fixes tr(A~A2A3) = (2flxi+ (26X3 + (2--1fl--lx4 automatically. Nowconsider xlx4 - x2x3 = 1. If x3 ~ 0 choose an x~ to satisfy this. This choice of x2 together with (xt,x3, x4) will give an A3 in PSL2(C) completing the desired essential representation. Suppose then that x3 = 0. Then if (tr(Au),tr(AiA3)) = (0,0) must also have (tr(A3),tr(A2Aa)) = (0,0). If q ~ 2 then we may replace tr(R~(A2, A3)) = -2cos(~r/q) by tr(Ra(A~, A~)) = +2cos(~r/q) and then we obtain an A3 = ±

x~

(

X3

X4

E PSLu(C)

as desired because the polynomials tr(R3(A2,A3))2cos(~r/q) and tr(Ra(A2, A~)) + 2cos(~r/q) have no commonzero. Therefore assume that q = 2. If tr(R3(A2, A3)) has two different zeros for tr(A2A3) then we may argue analogously to the above by replacing one zero by another one. Therefore if (tr(A3), tr(AiA3), tr(A2A3)) (0, 0, 0) we areleft with the c ase that q = 2 and tr(Ra(A2, A3)) has exactly one zero (possibly with multiplicity). In an analogous manner we may argue on p and tr(R~(A~, A3)) and hence we may assume that p = 2. Therefore if < At, A2 > is metabelian and (tr(A3), tr(AiA3), tr(A~A3)) # (0, 0, 0) we obtain an essential representation for G except possibly when p = q = 2 and tr(Ra(A~, A3)) has exactly one zero and tr(R2(A~, A3)) has exactly one zero. The above arguments were based on starting with an essential representation of G~. Wemay also start with an essential representation o£ G2 and argue as above if (tr(A2),tr(A1A2),tr(A2A3)) ¢ (0, 0,0) and necessary start with an essential representation of Ga and argue as above if (tr(A~),tr(A1A2),tr(AiA3)) (0,0,0). Th erefore if we ass ume tha (tr(Aa),tr(AtAa),tr(AuA3)) # (0,0, 0), (tr(A2),tr(AiA2),tr(A2A3)) (0, 0, 0) and (tr(A1), tr(AiA2), tr(A1A3)) # (0, 0, 0) we obtain essential representations for G except possibly in the following situation: m = p = q = 2, tr(R3(A2, An)) has exactly one zero,tr(R2(A1, A3)) has exactly one zero, tr(Rt (A1, A2)) has exactly one zero and pi(Gi) is metabelian for each essential representation p~ : Gi --~ PSL2(C), i = 1, 2, 3. Therefore assume that we have the above situation. Since the images of the triangle group factors

272


are all metabelian this completely determines the traces of AIA2, A~A3and A2A3. For example if all ei >_ 2 we must have tr(A~A~) = 2cos(r/el ~r/e2), tr(AiA3) = 2COS(TO/el:1: r/e3) and tr(A2A3) = 2cos(~r/e2 zr/ e3). If we now start with A~=+

(: 0)

a- 1 and

As==t=

fl 1

with a + a-~ -- 2cos(Tr/el), ~ ~ 0and/3 + 13-~ = 2cos(~r/e2) asabove, the n in the system of equations (9.3.3) we automatically get that x~x4 ~- 1. This gives A3 in PSL2 (C) again completing the desired essential representation. Finally suppose that one of the triples (tr(Aa), tr(A~Aa), tr(A2Aa)) (tr( A2), tr(A~ A~), tr(A~.A3) and (tr(A~), tr(AiAe), tr(A~Aa)) is (0,0,0). Without loss of generality then suppose that tr(A3) tr(A1A3) = tr(A2Aa) = 0. If at least one of e3,p,q is not equal to 2 then we may choose one of tr(A3) , tr(A~A3) , Tr(A~Aa) non-zero. If for example e3 = 2, p _~ 3 and tr(A~A3) = th en werep lace tr( R2(A1, A3)) = -2cos(~r/p) by tr(R2(A~, A3)) +2cos(~r/p) and th en tr (A~A3) = 0 is not a solution to the equation tr(Ru(A~, A3)) = +2cos(~r/p). Therefore suppose that also e3 --- p = q -- 2. In this case because, of the trace conditions, G has an epimorphism onto the group

which preserves the orders. Recall the trace conditions give us the orders. Let

If e~ ~ 2 using the same argmnents as before we determine first an A~, A~ with tr([A~, A~]) ~ 2. This is possible since G~ is a non-abelian dihedral group. This then gives an essential representation of G~. Wecan then find an A~ to give an essential representation of G* and thus of G. In a symmetric manner we can in G* start with {a~,a3} or {a~,a~} and will obtain essential representations with the possible exceptions when el = e~ = m = 2 and tr(A~A3) = O, tr(A~A~) = O, tr(A~A~)

9.3.1


273

In this final case, again because of the trace and order conditions, G has an epimorphism onto a~ =a~=ag~- (ala2)2-~-

G** =< al:a2,

(ala3)2 ~-(a2a3)2-~

which preserves the orders. This group has the essential given by al -~

10)

,a2-~A2=±

Al=+(_01

and thus G has also.

0

i

representation

,aa-~Aa=±

This completes the proof.

Wemention that Vinberg has given an independent proof of this result IV] when the generalized triangle group factors have only non-metabelian images. Note that included in the proof is a proof of the following corollary which we will need in the next section. COROLLARY 9.3.1. Let p~ : G~ ~ PSL2(C),a~ --~ A~,a2 ~ A2 an essential representation. Suppose that < A1, A2 > is non-abelian and suppose further that one of the following holds: (1) tr([A1, A2]) # (2) tr(A~) # ±2, tr(A2) # ±2 and (p, q) # (2, Then p~ can be extended to an essential representation p : G --~ PSL2 (C) PrtOOF. The proof is a direct consequence of the proof of Theorem9.3.1. If tr([A1,A~]) # 2 then lemma 9.3.1 can be applied directly as in the beginning of the proof of Theorem 9.3.1. For condition 2 above, none of the exceptional cases handled in the proof of theorem 9.3.1 arise so therefore starting with an essential representation of G1 either lemma9.3.1 can be used directly or equations (9.3.3) can be solved in a suitable manner. Weremark that in constructing essential representations p~ : G~ --~ PSL~(C) for the generalized triangle group factors if pi(Gi) abelian we may replace pi by pi* with pi* (i) G non-abelian. Further if tr(Ai) = +2 and not the Klein bottle group, then ei = 0 and we may replace pi by p~* with tr(p~*(a~)) ~ ±2 (see Chapter 7). An iteration of the above procedure yields the following corollary. COROLLARY 9.3.2.

=< al,...,a,~;a~

~

""

Let n > 3 and an

/~1 (al,a2)

Pt 2m:(a2,aa) .. ..

Rm~"(an,al) = 1

where ei = 0 or ei > 2 for i = 1,...,n; mi > 3 for i = 1,..,n and for i = 1, .., n, Ri(ai, ai+l) is a cyclically reduced word in the free product on

274


ai,ai+l which involves both ai and ai+l ( with an+l = aa. Then G admits a representation p : G -* PSL2(C) such that p(ai) has infinite order ei = 0 or order ei if ei >_2 and p(Ri) has order mi for i = 1, ..., n. ~rther by a straightforward extension of the proof we get the following generalization. : THEOREM9.3.2. Let G =< al, ..., an~. a~~1 .... a~~ = R~’ (al, a~) .... R~_~ (a~,a~) = R~ (a~,aa) ..... (a2 ,an) = 1 >, where 3 n~ 3, ei ~0 orei ~ 2 [ori= 1,..,n, mj ~ 3[orj = 1,..,2n-3 and k = 1, .., n - 1, Rk is a cyclically reduced word ~ ~he ~ee product o~ a~, a~ i~o1~i~g bo~h a~ and ak and Tot s = 0,.., n- 3, ~+~ is a cyc~cally reduced word i~ She free product o~ a~, a~+3 involving bo$h a~ a~d a~+3. Then G admits a represen$ation p : G ~ PSL~(C) s~ch ~hat p(ai) h~ in~n1~e order iTe~ = 0 or order ei iTe~ ~ 2 and p(Rj) h~ order m~ [or = 1, ... ,2n - 3 .

9.3.2 Some Linearity Properties for Generalized Tetrahedron Groups Groups which admit essential representations in manycases satisfy additional linearity properties - for example the Tits Alternative and the presence of non-abelian free subgroups, SQ-universality and being virtually torsion-free. Wenow consider these linearity properties for generalized tetrahedron groups. First we discuss the Tits alternative and SQuniversality. THEOREM 9.3.3. Let G be a generalized tetrahedron group with pre. el =a = ~ ~3 = R~(al, a2) = R~(al, a3) sentation (2.1) - < al, a2, a3, I 2 a3 R~(a~,a3) = 1 > . If e~ >_ 2 let/3~ = 1/e~ while if e~ = 0 (so that a~ has infinite order) let fl~ = O. If ~3~ + ~2 + fl3 + (l/m) + (l/p) + (l/q) then G contains a subgroup of finite index which maps onto a free group of rank 2. In particular G is SQ-universal. PROOF.Recall that ira group G maps onto a non-abelian free group it is SQ-universal - that is every countable group is embeddable into a quotient of G. Further G is SQ-universal if i~ contains a subgroup of finite index which is SQ-universal. Further since if a quotient of a group is SQ-universal the group is itself SQ-universal we can without loss of generality assume that ei _> 2 for j -- 1, 2, 3, passing to a quotient if pecessary. The proof follows the basic outline of the proof of Theorem6.2.2. From Theorem 9.3.1, G admits an essential representation p into PSL~(C)

9.3.2

LINEARITYFOR GENERALIZED TETRAHEDRON GROUPS 275

Therefore from Selberg’s theorem on finitely generated subgroups of linear groups there exists a normal subgroup H of finite index in p(G) such that p(a~) has order e~ modulo H for i = 1,2,3 and p(R1) has order m modulo H, p(R2) has order p modulo H, p(R3) has order q modulo H . Thus the composition of maps ( where 77 is the canonical map)

a Z o(a) gives an essential representation ¢ of G onto a finite group. Let X =< ¯ el e2 e3 al, a2, aa, al = a2 : a.~ = 1 > be the free product on al, a2, a3. There is a canonical epimorphism f~ from X onto G. Wetherefore have the sequence X 2 G 2 p(G)/H Let Y = ker(¢ o ~). Then Y is a normal subgroup of finite index in X and Y is torsion-free. Since X is a free product of cyclics and Y is torsionfree Y is a free group of finite rank r. Suppose IX : YI = J" Regard X as a Fuchsian group with finite hyperbolic area #(X) (every finitely generated free product of cyclics can be faithfully represented as such a Fuchsian group unless it is an infinite dihedral group). From the Riema~m-Hurwitz formula : j#(X) = #(Y) where

,(Y)= r and

1 1 +--+--)

~(X)=2-(--1 el

e2

e3

Equating these we obtain; r=l

j(

1

1 1 +--+---2)

el

e2

e3

G is obtained from X by adjoining the relations R~"~,R P 2 and R~ and p so G = X/K where K is the normal closure of R~, R~ and R~. Since K is contained in Y the quotient Y/K can be considered as a subgroup of finite index in G. Applying the Reidemeister- Schreier process or repeated applications of Corollary 3 in [B-M-S] Y/K can be defined on r generators subject to (j/m) + (j/p) + (j/q) relations. The deficiency d of this presentation for Y/K is then d=r -j

m

J

J-1 p q

.... j(

1 1 1 1 1 1 +--+--+--+-+--2) el e2 e3 m p q

276


By assumption (1/el) + (l/e2) + (l/e3) + (l/m) + (l/p) + (l/q) so d > 2. From the work of B. Baumslag and S. Pride [B.B.- P 2] Y/K and hence G has a subgroup of finite index mapping onto a free group of rank 2. Therefore Y/K is SQ-universal and since this has finite index in G, G is also SQ-universal. Weremark that G is infinite if, in the notation of Theorern 9.3.3, /~1 + ~]2 + ~3 + (l/m) + (l/p) + (l/q) If a group G is SQ-universal then it must contain a non-abelian free subgroup. This would then be the case for generalized tctrahedron groups under the conditions of the theorem above. Vinberg has also obtained a proof of this result. However non-abdian free subgroups for generalized tetrahedron groups can be obtained under other conditions as well. THEOREM 9.3.4. sentation


tetrahedron

group with pre-

~ ~ ~ 1 al~a2, a3,¯ a~ = a~ = a~ = R~(al,a2) = R~(a~,a3) = R~(a:,a3) -- 1 > ~ with k _> 1 and for Suppose that e I ~_ e 2 and R,(a~,a2) = I 0:1 au ~1 . ..% aO~k 2 ~ i=l,...,k, ai 7 0,~3i # 0 and 1 _2, 1 _2. ~rther suppose that either (1) k ~_ 2and one of the following holds: (i) e~ >_ 4 and m >_ (ii)

e~ >_3andre>_4

(iii)

el >_ 3 and > 3

(iv) e~ = 0 and m >_ or

(2) at least one of p, q is not equal to 2, k >_ 1 and one of the following holds : (i)2 _< e~ _< e~ and (1/e~) + (1/eu) + (l/m) (ii)el = Then G contains a non-abelian free group. Symme~ricMstatements made replacing RI by 1~2 or 1~3 .

can be

PROOF.The general techniques of the proof follow the methods of Chapter 7, in partciular the proofs in section 7.3. The proof of Theorem9.3.4 depends on the following three lemmas.

9.3.2

LINEARITY FOR GENERALIZEDTETRAHEDRON GROUPS

277

LEMMA 9.3.2. Let G be as in the statement of the theorem and G1 be as defined in section 2. If G1 admits an essential representation into PSL2 (C) such that the image group is non-elementary then G contains non-abelian free subgroup. LEMMA 9.3.3. Let G be as in the statement of the theorem. Ilk ~_ 2 and one of the following holds: (i) e2 ~_ 4 and m >_ (ii)

e: >_3andre>_4

(iii) el >_3 and m ~_ (iv) el = 0 and m >_ Then G contains a non-abelian free subgroup. LEMMA 9.3.4. Let G be as in the statement of the theorem. If k ~ 1, at least one of p ,q is not equal to 2 and one of the following holds: (i)2 _< el _< e2 and (1/e~) + (1/e2) ÷ (l/m) (ii)el -~ Then G contains a non-abelian Tree group. (Wesplit lemmas9.3.2 and 9.3.3 since the proofs are somewhatdifferent) PROOFS.( of lemmas) (1) ( Lemma9.3.2) From the statement after proof of theorem 1, given an essential representation pl of G1 into PSL2(C) with p~(G~) non-elementary there exists an essential representation p of G extending Pl. If pl(G1) is non-elementary then it must contain a nonabelian free subgroup and hence p(G) does also. Since G maps onto p(G), G must also contain a non-abelian free subgroup. The proofs of lemmas9.3.2 and 9.3.3 are technical but follow esentially by showing that under the stated conditions either G1 admits an essential representation with non-elementary image or if not G has a factor group which contains a non-abelian flee subgroup. This was the method used to analyze free subgroups of generalized triangle groups in section 7.3. (2) ( Lemma9.3.3 ) As before let G~ be the generalized triangle group G~--< a~, a2; a~1 = a~~ ---- r~ RI (al,

a 2)

1>.

Call an essential representation into PSL2(C) suitable if the image group is non-elementary. From Lemma9.3.2, if G~ admits a suitable representation then G will contain a non-abelian free subgroup. From the

278


analysis of free subgroups of generalized triangle groups (section 7.3) if the length of the relator is _> 4 and the conditions of Lemma9.3.3 hold then G1 will admit a suitable representation except in the following five cases. (i) (~1 = 2, ¢2 6,m --- - 3, k - -- - 2 a ndG1 h as a presentation G1 -=<x, y; x 2 = y6 _-_ (xyxy3)3 _~ 1 (ii) el -= 2, e2 = m=- 4, k = 2 and G1 has a presentation GI --< x, y; x ~ -= y4 = (xyxy2)4 = 1 (iii)

el = e2 = m = 3, k =- 2 and G~ has a presentation G~ =< x, y; x3 = y3 = (xyxy~)3 _= 1

(iv) el --= e: = m-- 4, k --= 2 and G1 has a presentation G1 :< x, y;

X4 -~-

y4 _= (xy2xy2)4 = 1

(V) el =- 2, e2 -= m = k =- 4 and G1 has a presentation G~--< x, y; x~ -- y4 -- (xyxyUxy2xy~)’~ -= 1 > . In each of the five cases G~ does contain a free subgroup of rank 2. Further we can without loss of generality assume that x --= a~ and y -- an. Suppose first that the generalized tetrahedron group G has as a factor group 2 G* --.

G* has the subgroup H* of index two where H* has the presentation ~1 __a~~ = RT"(~ H* I :~ al,a2;a ,a~) = R~r~(a-1 ~ ~ ~ , a-l~ ~ ~ =1 >. In each of the five cases above the relation R~(a-~~, a~-~) -- 1 follows from m a hereH*= therelations a~~=a~2= R1 (1,a~) =1 ~nd therefore Thus G has G~ as a sub~oup and therefore G has a ~ subgroup of ra~ two. Suppose then that G does not have G* as a factor group and co~ider first case (i). Here G1 has a representation p~ onto the fi~te dihedral ~oup D~ =< A1, A2; A~ = A~ = (A1A2)2 = I > and a second representation p12 onto the (2,3,6) triangle ~oup T1 =< A;, A~; A~ = A~ = (AIA~)3 = 1 >. T~ is i~nite and metabelian. Let G3 =< a2, a3; a~ = a~~ = R~(a~, an) 1 >. If there is ~n essential representation p3 of G3 with respective images

9.3.2

LINEARITYFOR GENERALIZED TETRAHEDRON GROUPS

279

A2, Aa such that tr([A2, Aa]) -- 2 then we may choose the image < A2, Aa to be infinite and metabelian with tr(Aa) ~ +2 if e3 -- 0. Because m = the representation P3 of (~3 may be extended to an essential representation of G with non-elementary image (see the proof of theorem 1~ and therefore G will contain a free subgroup of rank 2. Suppose then that there exists no essential representation of Ga with tr([A2, A3]) = 2. If there is a suitable representation of G3 then G has a free subgroup of rank 2. If there is no suitable representation Pa of (~3 then Ga has the factor group G;" --< a2, a3;a~ ~- a~ = (a2a3)2 ~- 1 > and hence G has the factor group G** --< al, a2, a3; a~ = a26 = a~ = (ala2ala~) a = (alan) p = (a2aa) 2) = 1. If p _> 3 then we extend p12 and we can obtain a free subgroup of G** of rank 2 and hence of G. If p--- 2 then G* is a factor group of G. This handles case (i). Case (ii) is handled analagously. G~ has a representation p~ onto the finite dihedral group D4 =< A1, A2; A2~ = A~ = (A1A2)2 = 1 > and a second representation p~ onto the (2,4,4) triangle group T~ =< A1, A2;A~ = A~ = (A1A~)~ = 1 >. If we argue as above we must in addition consider the possibility that the finite symmetric group $4 is an image of Ga. Hence, G has a free subgroup of rank 2 or, possibly after changing the generators of G3, G has the factor group G**= and a second essential representation P12 onto the (3,3,3) triangle group Tt =< A~,A2;A~ = A~ = (AIA2) 3 -- 1 >. Since (p,q) ¢ (2, 2) we may extend pt2 directly (tr(Aa), tr(AiA3), tr(A2A3) (0, 0, 0 ) t o a representation p : G -- ~PSL2(C) such that p(G) is non-elementary. If (tr(A3), tr(A~A3), tr(A2A3) = (0, 0, 0) then G* is a factor group of G. This handles case (iii).

280


In case (iv) introduce the relation (ala2~ala’~) 2 = 1 and in case (v) the 2 2 2 2 = 1. In both these cases the corresponding relation (ala2ala2ala2ala~) factor group of G1 has a suitable representation into PSL~(/2) and therefore G contains a non-abelian subgroup completing the proof of Lemma9.3.3. (3){ Lemma9.3.4} Suppose that the conditions of Lemma9.3.3 hold that at least one of p, q is not equal to 2 and one of the following holds: (i)2 _< e~ _< e~ and (1/e~) + (l/e2) + (l/m)

(ii)el = Without loss of generality let q ¢ 2. Suppose (71 is as before and suppose that A1, A~ , with < A1, A2 > non-abelian, are the respective images of al, a2 under an essential representation of G1. For both cases (i) and (ii) above we may find a projective matrix Aa such that the triple (tr(A3), tr(A~Aa), tr(AuA3)) ¢ (0, 0, 0) as in the proof of Theorem9.3.1. Consider first case (i) above and assume further that k = 1. If rn _> 7 m = 5 then G1 has a suitable representation and as before since q _> 3 this extends to an essential representation of G. Let 2 _< rn _< 4 or rn = 6 and let pl : (71 -~ PSL2(C) be an essential representation, pl can be chosen be suitable except in the following four exceptional cases: (a) rn = 6, e~ = 2, e2 = 6 and Rl = ala~ (b) rn = 6, et = 3, e2 = 6 and Rt a~a~ (c) m = 6, el = 6, e~ = 6 and R~ a~a232 orR1 = ala 232 (d) rn = 4, e~ = 4, e~ = 4 and R1 at a~ orR1 = a~a2 In cases (a),(b) and (c) introduce the relation R~to obtain a factor G~ which has a suitable representation into PSL2(12). The corresponding factor group G* of G will then have a non-abelian free subgroup and hence so will G. In case (d) we have G2 =< a~, aa; 4 = a~~ = R~(a~, a3) = 1 >. Suppose G2 admits a suitable representation. Since m = 4 ¢ 2 this representation can be extended to an essential representation p of G such that p(G) is non-elementary. Therefore p(G) and hence G will contain a nonabelian free subgroup. Suppose then that Ge has an essential representation p~ with respective images A~, A3 such that tr([A~, A3]) = 2. Then as before we may choose p2 so that< A~, A3 > is infinite and metabelian with

9.3.2


281

tr(A3) ~ -1-2 if e3 -- 0. In G introduce the relation R~ -- 1 to obtain a factor group G*. Then p2 {considering G~. with respect to G*} may be extended to an essential representation p* : G* -+ PSL2 (C) such that p* (G*) is non-elementary. Therefore G* and hence G contains a non-abelian free subgroup. This completes case (i) when k = 1. Nowsuppose that k _> Choose an essential representation pl : GI -* PSL~(C) with respective images A~, A2 and assume that < A~, A2 > is elementary otherwise G would automatically have a non-abelian free subgroup from Lemma9.3.2 If < Aa, A2 > = As, the alternating group, then we may change the generators of Ga to get an image group of G1 in PSL2 (C) which is infinite. Hence we may assume as before that < A~, A2 > is infinite and metabelian and tr([A1, A2]) = 2. Construct A~ to get an essential representation of If either tr([A1, A3]) ¢ 2 or tr([A~, A3]) ~ 2 then < A1, A~, A3 > is nonelementary and G then contains a non-abelian free subgroup. Therefore we may assume that tr([A~, A3]) --- 2 and tr([A2, A3]) = 2. Using Lemma9.3.3 and the results above for k = 1 we can reduce to the case where m = 2. The condition tr([A~, A~]) --- 2 gives very strong restrictions on the exponents a~, fl,-..,~,fk occurring in R~(a~,a2). There would be analagous restrictions for the exponents occurring in R~(al, a3) and R3(a2, a3) since tr([Aa, A3]) = tr([A2, A~]) = 2. Despite these restrictions and the fact there are manyreductions from Lemma9.3.3 and the results for k = 1 the proof for this case still splits into several subcases to consider. Recall that we have the conditions 2 _< el ~ e2 and (1/O) + (l/e2) + (l/m) typical subcase is then e~ = e2 = 6, e3 = 2, m= 2, p = q = 3 and k

k

k

E o~, =-- O(mod6), Eft’ ~ O(raod6),2 ---- - 0(ro od6), i-~ l

i= l

R2(al, a3) al a3 and R3(a2, a3 ) = a2a3. Inthi s sub case we int roduce the relation a2a = 1 to obtain a factor group G* which has an image group in PSL2(C) which is non-elementary. Hence G* has a free subgroup of rank 2 and therefore G does also. The other subcases follow in a similar but sometimes lengthier manner. The details of the arguments and statements which are involved but not covered by the above example follow in the samer manner as in Chapter 7. Finally we consider case (ii) where e~ = If e2 = 0 then G~ has a suitable representation for any m> 2 so assume then that e2 > 2. Consider first e2 > 3. If m > 3 then from our previous results G contains a non-abelian free subgroup. An analagous statement holds if p > 3. Therefore if e2 > 3 we may reduce to the case where m = p = 2. By the same arguments we can reduce to e3 > 2 and we consider first e3 > 3. From Theorem9.3.2 if (l/e2) ÷ (1/e3) + (1/2) + (1/2) + (l/q) < 2 SQ-universal and must contain a non-abelian free subgroup. Therefore we

282


may assume (l/e2) + (l/e3) + (1/2) + (1/2) + (l/q) _> 2. Then e2 = e3 = q = 3 because e2, e3 >_ 3 and q ~ 2. Again from the techniques used in Chapter 7 and using the arguments in a symmetric fashion we see that G has a free subgroup of rank 2 via a suitable representation into PSL2(C). Nowlet el = 0, e2 _> 3, m= p --- 2 and e3 -- 2. If G1has a suitable representation into PSL2 (C) then G has a non-abelian free subgroup as before so assume that G1 has no such suitable representation. Using the analysis as in Chapter 7 we get that e2 = 2, k~ _> 4 and R~(a~,a2) = ak22T~(a1,a2) with T~(a~, a2) in the commutator subgroup of the free product on a~ and an. Nowwe argue in a symmetric fashion. If R2(al, a3) is conjugate in the free product on a~,a3 to some a~a3,n ~ 0 ,then we may extend each essential representation of G~ into PSL:(C) with tr(p~(al)) and infinite, non-abelian image to a suitable representation of G. If G2 has a suitable representation then G also has a free subgroup of rank 2. Hence we are left with the situation R2(al,a3) = a3T2(a~,a3) with T2(al,a3) in the commutator subgroup of the free product on al and a3. If (1/ee) ÷ (1/2) ÷ (1/2) ÷ (1/2) ÷ (l/q) < 2 then G is SQ-universal and must contain a non-abelian free subgroup. Therefore we may assume (l/e2) + (1/2) ÷ (1/2) ÷ (1/2) + (l/q) _> 2. Altogether then we al, a2, an; a~~ -~ a~3-- (ak2~Tl(al, a2))2 ---- (a3T2(al, a~))2 --~ R~(a~,an) -- 1 with q -- 4, e2 -- 4, k2 -- 2 or q -- 3, e2 -- 6, k2 : 3 or q -- 3, e2 -- 4, k2 ~- 2. If q = 4, e2 ~- 4, k2 -- 2 we introduce the relation a3 : 1 to obtain the factor group G* ~-< al, a2; a~ --- (a~T~(a~, a2)) 2 -- 1 > which has a free subgroup of rank 2. If q --- 3, e2 = 6, k2 -- 3 introduce the relations aa -- 1 and a23 -- 1 to obtain the factor group G* =< a~,a2; aa~ = (T~(a~,au)) 2 ~- 1 > which hasa free subgroup of rank 2. Nowlet q-- 3, e2 = 4, k~ = 2. We have tr([A~, A3]) ?~ 2 for any essential representation of G3 into PSL~(C). Hence if we choose an essential representation of G1 with tr(p~(a~)) ~ ±2 and infinite non-abelian image this may be extended to a suitable representation of G. Nowlet e~ ---- 2. By symmetrical arguments to the above we can reduce finally to the case where e2 ----- e3 --~ m ---- p = 2. Choose an essential representation of G~ with respective images A~,A~ and tr(A1) ~ ±2 as before. If tr([A~, A2]) -- 2 then we may assume as before that < A~, A2 is infinite metabelian. Since q ~ 2 the representation of G~ extends to G and G must contain a non-abelian free subgroup as before. If tr([A~, A3]) ¢ the only possibility where < A~, A: > does not contain a non-abelian free subgroup is if < A1, A~ > -- Z2 * Z2 the infinite dihedral group. In this case G has the factor group G*=

9.3.2


283

with 3 _< q _< qlWorking with G*, G* will contain a non-abelian free subgroup as above or G* will have the factor group G** G**=< a,, a2, a3; a~ = a] = (ala~.) 2 = (a,a3) 2 = (a2aa)q’ = 1 >. Now G** does contain a free subgroup of rank two. Starting with < AI,A: >= Z: * Z~ and tr(A~) ¢ +2 the infinite dihedral group we may construct Aa so that < A1, A2, Aa > is non-elementary. This can be done since 3 _< q _< ql and Z2 * Z2 has no element of finite order _> 3. Therefore G** and hence G contains a non-abelian free subgroup completing the proof of lemma9.3.3. In Theorem 9.3.4(2) we assumed that at least one of p, q is not equal to 2. This assmnption allowed us to extend essential representations Pi : G~ -~ PSLe(C) with pi(Gi) non-abelian. If p = q = 2 we also have some results. Using the same type of arguments as in Theorem9.3.4 we have the following. THEOREM 9.3.5. Let G be a generaBzed tetrahedron group wi~h p = q = 2. Suppose that either of the following two conditions hoM (1) el = and (e:, m) 7~ (2, 2) (2) 2 _< el _< e2 and (1/el) -t- (l/e2) -t- (l/m) Then G has a free subgroup of rank two with the possible exceptions ea = 2 and(el, e2, m)= (3, 8, 2), (3, 10, 2), (4, 5, 2), (4, 6, 2), (4, 8, 2) or PROOF.The proof is done in the same manner as the proofs of the lemmas for Theorem 9.3.4. We give a sketch and show how the possible exceptions can arise. If there is an essential representation p~ : G1 --~ PSL2(C) with tr ([pl (a~), p2 (a~)]) ¢ 2 then we mayalways find an extension to an essential representation of G. Hence if ea ¢ 2 and p = q = 2 we argue essentially as before, using the argument also in a symmetric fashion, to get a free subgroup of rank two, possibly passing to a factor group of G if necessary. Especially if et ¢ e2 we obtain that G has a free subgroup of rank 2. If e~ = e2 then as before we may reduce the problem to the group 2 G=_2, l _ 2 or that m _> 4 and R1 is not conjuga$e in the Tree product on al, a~ to a product X1Y1 for some X1 and of orders xl >_ 2, yl >_ 2 respec~i~ly where (l/m) ÷ (l/x1) ÷ (l/y1) R2 = a~a~,with w ~ O,u ¢ O,l _< W < el if e~ >_2, l _2 or that p >_ 4 and R~ is not conjugate in the Tree product on al, a3 to a product X2Y2 for some X2 and Y~ of orders x~ >_ 2, y2 >_ 2 respectively where (l/p) + (1/xu) + (l/y2) > 1; R3 = a~a~,with v ?~ O, z ¢ O, 1 _ 4 and R3 is not conjugate in the Tree product on a2, a3 to a product X3Y3 for some X3 and Y3 of orders x3 _> 2, y3 _> 2 respectively where (l/q) + (l/x3) + (1/y3) Then: order in G is conjugate to a power of (1) Any element of finite al,a2,a3,R1,R2 or R3. (2) G is virtually torsion-free. ( G contains a torsion-Tree normal subgroup of finite index.}

PROOF.From Theorem9.3.6 since (l/m) + (l/p) ~- (l/q) _< I each triangle group factors G~ inject into G and any finite subgroup is conjugate to a finite subgroup in one of the factors. The conditions on the relators guarantee that any element of finite order in a factor is conjugate to a power of either R~ or a powerof one of the two generators it involves. This handles

286


part (i). Nowfrom theorem 9.3.1 G admits an essential representation into PSL2 (C). However form the preceding statement this implies that an essential representation is essentially faithful. Therefore the remainder of the result follows from Theorem6.2.1. Wecan also extend Corollary 9.3.3 to the case where 3 _< m, p, q and no letter in Ri, i = 1, 2, 3, has order two, especially if all ei are odd. THEOREM 9.3.7. Let G be a generalized tetrahedron group and G1,G2,G3 its generalized triangle group factors. If Gi , i -- 1 or 2 or 3, is non-metabelian and admits a faithful representation into PSL2(C) then Gi injects into G. PROOF.Given a faithful representation ~ of G1 into PSL~(C) there exists an essential representation p of G which extends 7r since G1 is assumed non-metabelian. Since p maps G into PSL2(C) and p is faithful on G1, G1 must inject into G. Wenote that Theorem 9.3.7 also holds if Gi is metabelian and admits a faithful representation into PSL2 (C). There are some difficulties with the case whenG1 is the (3, 3, 3) triangle group (~1 ----( a~, a~; a3~ 3 --~ (ala2) a ---- 1 > . In this case the representation techniques reduces the problem of extending the faithful representation to the case where each Gi is a (3, 3, 3) triangle group. The calcuations are quite lengthy and we omit them. Wenext give a result as to when a generalized tetrahedron group actually admits a splitting as a non-trivial free product with amalgamation. THEOREM 9.3.8. Let G be a generalized tetrahedron group and Gi its first generalized triangle group factor. Suppose e~ -~ 0 and at least one of p, q is not equal to 2. Suppose further that t~ ~ a~a~T(a~, a2) with s ~ O, 0 ~_ ~ ~ e2 ire2 ~_ 2 and T(ai, a2) in the commutator subgroup in the free produc~ on ai, a2. Then G admits a splitting as a non-trivial free product with amalgamation. PROOF.Passing to a suitable factor group if necessary we may assume that e2 _~ 2 and e3 >_ 2. Given an essential representation pl : G1 --~ PSL2 (C) with tr(p~ (al)) ~ ±2 and pl (G~) non-abelian, since at least of p, q is not equal to 2 this can be extended to an essential representation p : G --~ PSL2 (C). NowH. Bass in [Ba 1] proved that if K is a finitely generated subgroup of GL2(C) then one of the following cases must occur. (1) There is an epimorphism f : K -~ Z such that f(U) -- fo r al l unipotent elements U in K (2) K is an amalgamated free product K = Ko *~ Ki with Ko ~ H ~ K1 and such that every finitely generated unipotent subgroup of K is contained in a conjugate of Ko or K1

9.3.3

FREIHEITSSATZANDVIRTUALTORSION-FREEPROPERTY 287

and d roots of unity (4) K is conjugate to a subgroup of GLe(A) where A is a ring of algebraic integers. Under the conditions on el and R1 the essential representation pl of G1 can be constructed so that the image < A1, Au > does not satisfy conditions (1),(3) or (4) of Bass’ theorem. Nowp: G-~ PSL~(C) be th e e ssential representation of G extending pl. The image p(G) cannot satisfy conditions (1), (3) or (4) of Bass’ theorem since then pl(G1) would also. It that p(G) must satisfy condition (2) and therefore p(G) is a non-trivial free product with amalgamation. However from a result of Zieschang [Z] groups which have non-trivial amalgamated free products as epimorphic images axe themselves non-trivial amalgamated free products and so G is a non-trivial free product with amalgamation. Finally we close by giving an extension of a representation results and applications to an extended class of three generator groups which includes the generalized tetrahedron groups. THEOREM 9.3.9. Let G =< a,b,c;a r = b R~(a,b) = Sm(a,c) = Tn(b,c) Oorr >_ 2, s = Oors >_ 2, t = Oort _> 2 and RI,...,Rk, S, T axe nontrivial cyclically reduced words in the free products on the generators they involve. Assume further that Ri(a,b), i -- 1,... ,k involves both a and b, S(a, c) involves both a and c and T(b, c) involves both b and c. G1 --< a,b;a r = bs = Rl(a,b) .... = Ra(a,b) = (1) If there exists a representation Pl : G1 --~ PSL2(C),a -~ A,b --~ with < A, B > non-metabelian then G ~ {1} and in fact G is non-metabelian. If there exists a representation Pl : G1 -~ PSL2(C), a -~ A, b --~ (2) with < A, B > non-elementary then G has a free subgroup of rank 2 (3) If GI is non-metabelian and if there is a faithful representation pi : G1 -~ PSL2(C) then GI embeds into PROOF.(1) Let Pl : G~ -~ PSL2(C),a --~ A,b --~ with < A, B > nonmetabelian. First suppose that G has the factor group G*--< a, b, c; a~" -b~ ~- c 2 = R~(a,b) ..... Rk(a,b) = ~ = (b e) ~ = 1 >. < A,B > embeds into a factor group G** of G* since there is a D in PSL2(C) with DAD-~ -~ A-~,DBD-1 = B-1. Therefore G ~ (1) because A, B > is non-~netabelian. Nowsuppose that, G* is not a factor group of G. Then we may extend

288


Pl to a representation p : G -~ PSL2(C) as before because < A, B > is non-metabelian. Therefore in this case also G ?~ {1}. (2)Let Pl : G1 -~ PSL2(C),a -~ A,b --~ with < A, B > nonelementary . Then tr([A, B]) ¢ 2 and we may extend pl to a representation p : G -~ PSL2 (C). Therefore G has a non-abelian free subgroup since < A, B > has one. The proof of (3) is entirely analagous to that for generalized tetrahedron groups (see Theorem9.3.7). 9.3.4

Euler Characteristic

for Generalized

Tetrahedron Groups

In section 7.6 the construction of a rational Euler characteristic for the generalized triangle groups was presented. A similar construction of Euler characteristic was given in section 8.3 for groups of F-type. M.Stille [St] has constructed a similar rational Euler characteristic for the generalized tetrahedron groups. In this final section we briefly review his results. If G =< a, b; ap = bq = Rm(a, b) = 1 > is a generalized triangle group, where as usual, R is a cyclically reduced word in the free product on a, b involving both a and b and m _> 2, then we say that the relator R has property E if either m >_ 4 and R is conjugate within the free product on a, b to a product of the form XY for some elements X, Y of respective orders x >_ 2, y >_ 2 where (l/m) + (l/x) + (l/y) > 1 or m -- 3 and there ~re letters of order involved in R. Wethen obtain. THEOREM 9.3.10. Let G be a generalized sentation of ~he form (9.3.2) so tha$

tetrahedron

group with pre-

1G ~-( al,a2,a3;a with the standard conditions on exponents and relators and let G~, G2, G3 be its generalized triangle group factors. Suppose that (l/m) + (l/p) (l/q) 2 ~ x(G) =I+~=~(X()-x(A~)) G where A~ =< a~ > ~or i =1,2,3 are reg~ded as sub~oups of G. vcd(G) ~

PROOF.Under the given conditio~ it follows ~om Theorem 9.3.6 and Corollary 9.3.3 that G is virtually torsion-~. As in the construction of the Euler ch~acteristic for generalized triangle groups (s~ the proof of Theorem 7.6.1) construct the Cayley complex for the given presentation.

9.3.5

FINITE GENERALIZEDTETRAHEDRON GROUPS

289

From results of Pride [Pr 2,3] the relation module splits as a direct sum of cyclic submodules. Therefore the complex X is a two-dimensional contractible G-complexsuch that every isotropy group is trivial or finite cyclic and X has only finitely many cells mod G. This implies then that X has an equivariant Euler characteristic xa(X). Because X is contractible and G is virtually torsion-free we have that x(G) = xa(X). Because of the decomposition of the relation modulethe cellular chain complex of X is a free resolution of Z over ZH for every torsion-free subgroup H of G. Hence G is of homological type WFL(see [Brn]) and vcd(G) _< 2. The computations follow as in the proof of Theorem7.6.1. COROLLARY 9.3.4. Let G be a generalized tetrahedron group as in Theorem 9.3.10. If G has a faithful representation p : G -~ PSL2(C) such that p(G) is a Kleinian group of finite covolume then: 3 I+I 1 with ~ = O if e~ = O or a~ = 1~ if e~ _> 2. PROOF.In these cases G has a rational Euler characteristic as in Theorem 9.3.10. If p(G) has finite eovolume then from work of Ratcliffe [Ra] X(G)_> 9.3.5

Finite

Generalized

Tetrahedron Groups

Recall that a complete classification of the finite generalized triangle groups has been given (see section 7.3.4). Analogous work has been started on the classification of the finite generalized tetrahedron groups. Recall that Coxeter gave a criterion for an ordinary tetrahedron group to be finite while this was extended by Tsaranov to generalized tetrahedron groups where each relator is of the form Ri(u, v) = uavb. Nowlet G be a general generalized tetrahedron group with a presentation of the form G ----< 1al,a2,a3, . a~1 = a;~ = a~ = RF(~I, ~) = R~(~, a~) = R~(a~, a~) >. FromTheorem9.3.3 it follows directly that G is infinite if 1 1 1 1 1 1 --+--+--+--+-+-

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