Elements of asymptotic geometry

M S E M E S EMS Monographs in Mathematics Edited by Ivar Ekeland (Pacific Institute, Vancouver, Canada) Gerard van d...

Author: Buyalo S. | Schroeder V.

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S E M E S

EMS Monographs in Mathematics Edited by Ivar Ekeland (Pacific Institute, Vancouver, Canada) Gerard van der Geer (University of Amsterdam, The Netherlands) Helmut Hofer (Courant Institute, New York, USA) Thomas Kappeler (University of Zürich, Switzerland) EMS Monographs in Mathematicsis a book series aimed at mathematicians and scientists. It publishes research monographs and graduate level textbooks from all fields of mathematics. The individual volumes are intended to give a reasonably comprehensive and selfcontained account of their particular subject. They present mathematical results that are new or have not been accessible previously in the literature.

Previously published in this series: Richard Arratia, A.D. Barbour, Simon Tavaré, Logarithmic combinatorial structures: a probabilistic approach Demetrios Christodoulou , The Formation of Shocks in 3-Dimensional Fluids

Sergei Buyalo Viktor Schroeder

Elements of Asymptotic Geometry

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S E M E S

EuropeanM athematicalSociety

Authors: Sergei Buyalo Steklov Institute of Mathematics at St. Petersburg 27 Fontanka St. Petersburg 191023 Russia Viktor Schroeder Institut für Mathematik Universität Zürich Winterthurerstrasse 190 8057 Zürich Switzerland

2000 Mathematical Subject Classification (primary; secondary): 51F99, 53C23, 55M10; 53C23, 54C20, 20F67, 20F69

ISBN 978-3-03719-036-4 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2007 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author's ETX files: I. Zimmermann, Freiburg Printed in Germany 987654321

To Tania and Cornelia

Contents

Preface

xi

1

Hyperbolic geodesic spaces 1.1 Geodesic metric spaces . . . . . . 1.2 Hyperbolic geodesic spaces . . . . 1.3 Stability of geodesics . . . . . . . 1.4 Supplementary results and remarks

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1 1 2 4 6

2 The boundary at infinity 2.1 ı-inequality and hyperbolic spaces . . . . . . 2.2 The boundary at infinity of hyperbolic spaces 2.3 Local self-similarity of the boundary . . . . . 2.4 Supplementary results and remarks . . . . . .

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Busemann functions on hyperbolic spaces 3.1 Busemann functions . . . . . . . . . . 3.2 Gromov products based at infinity . . 3.3 Visual metrics based at infinity . . . . 3.4 Supplementary results and remarks . .

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23 23 26 29 30

Morphisms of hyperbolic spaces 4.1 Morphisms of metric spaces and hyperbolicity . . . . 4.2 Cross-difference triples and cross-differences . . . . 4.3 PQ-isometric maps . . . . . . . . . . . . . . . . . . 4.4 Quasi-isometric maps of hyperbolic geodesic spaces 4.5 Supplementary results and remarks . . . . . . . . . .

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49 49 50 56 63

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Quasi-Möbius and quasi-symmetric maps 5.1 Cross-ratios . . . . . . . . . . . . . . . . 5.2 Quasi-Möbius and quasi-symmetric maps 5.3 Supplementary results and remarks . . . . 5.4 Summary . . . . . . . . . . . . . . . . .

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viii 6

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Contents

Hyperbolic approximation of metric spaces 6.1 Construction . . . . . . . . . . . . . . . . . . . . . . . 6.2 Geodesics in a hyperbolic approximation . . . . . . . . 6.3 The boundary at infinity of a hyperbolic approximation 6.4 Supplementary results and remarks . . . . . . . . . . .

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69 69 70 74 77

Extension theorems 7.1 Extension theorem for bilipschitz maps . . . 7.2 Extension theorem for quasi-symmetric maps 7.3 Extension theorem for quasi-Möbius maps . . 7.4 Supplementary results and remarks . . . . . .

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81 81 84 87 95

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Embedding theorems 97 8.1 Assouad embedding theorem . . . . . . . . . . . . . . . . . . . . . 97 8.2 Bonk–Schramm embedding theorem . . . . . . . . . . . . . . . . . 100 8.3 Supplementary results and remarks . . . . . . . . . . . . . . . . . . 102

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Basics of dimension theory 9.1 Various dimensions . . . . . . . . 9.2 Constructions . . . . . . . . . . . 9.3 P-dimensions . . . . . . . . . . . 9.4 The monotonicity theorem . . . . 9.5 The product theorem . . . . . . . 9.6 The saturation of families . . . . . 9.7 The finite union theorem . . . . . 9.8 Sperner lemma . . . . . . . . . . 9.9 Supplementary results and remarks

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107 107 111 117 121 122 122 123 124 126

10 Asymptotic dimension 10.1 Estimates from below . . . . . . . . . . . . . 10.2 Estimates from above . . . . . . . . . . . . . 10.3 Embedding of H2 into a product of two trees . 10.4 Supplementary results and remarks . . . . . .

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129 129 130 132 134

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11 Linearly controlled metric dimension: Basic properties 137 11.1 Separated sequences of colored coverings . . . . . . . . . . . . . . 138 11.2 Quasi-symmetry invariance of `-dim . . . . . . . . . . . . . . . . . 141 11.3 Supplementary results and remarks . . . . . . . . . . . . . . . . . . 145 12 Linearly controlled metric dimension: Applications 147 12.1 Embedding into the product of trees . . . . . . . . . . . . . . . . . 147 12.2 `-dimension of locally self-similar spaces . . . . . . . . . . . . . . 154 12.3 Applications to hyperbolic spaces . . . . . . . . . . . . . . . . . . 156

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12.4 Supplementary results and remarks . . . . . . . . . . . . . . . . . . 157 13 Hyperbolic dimension 13.1 Large scale doubling sets . . . . . . . . . . 13.2 Definition of the hyperbolic dimension . . . 13.3 Hyperbolic dimension of hyperbolic spaces 13.4 Applications to nonembedding results . . . 13.5 Supplementary results and remarks . . . . .

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159 159 160 162 164 166

14 Hyperbolic rank and subexponential corank 14.1 Hyperbolic rank . . . . . . . . . . . . . . . . . . . . 14.2 Subexponential corank . . . . . . . . . . . . . . . . 14.3 Applications to nonembedding results . . . . . . . . 14.4 Subexponential corank versus hyperbolic dimension . 14.5 Supplementary results and remarks . . . . . . . . . .

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167 167 169 175 175 177

Appendix. Models of the hyperbolic space Hn A.1 The pseudo-spherical model . . . . . . . A.2 The unit disc model . . . . . . . . . . . . A.3 The upper half-plane model . . . . . . . . A.4 The solvable group model . . . . . . . . . A.5 Generalizations to an arbitrary dimension A.6 Möbius transformations . . . . . . . . . . A.7 Cross-ratio . . . . . . . . . . . . . . . . .

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181 181 182 183 185 186 187 188

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Bibliography

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Index

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Preface

Asymptotic geometry is the study of metric spaces from a large scale point of view, where the local geometry does not come into play. An important class of spaces to be studied are the hyperbolic spaces (in the sense of Gromov), for which it turns out that the asymptotic geometry is almost completely encoded in the boundary at infinity. The basic example of these spaces is the classical hyperbolic space Hn . A main feature of this classical space is the deep relation between the geometry of Hn and the Möbius geometry of its boundary @1 Hn . For example the isometries of Hn correspond to Möbius transformations of @1 Hn . The classical space itself has different realizations, but there are natural isomorphisms between these models, which induce Möbius transformations between the boundaries at infinity. Mikhael Gromov realized that the essential asymptotic properties of Hn can be encoded in a simple condition for quadruples of points. A metric space X is called (Gromov) hyperbolic, if there exists some ı 0 such that every quadruple Q D fx; y; z; wg X satisfies the following inequality only involving the six distances between the four points: jxzj C jyuj maxfjxyj C jzuj; jxuj C jyzjg C 2ı: It is a remarkable fact that this inequality suffices to build up a theory of general hyperbolic spaces, which is very similar to the classical theory of the classical hyperbolic space but which allows much more flexibility and can be applied to many situations. We develop the basics of this theory of general hyperbolic spaces in the first eight chapters of the book. In our account we stress the analogy between a Gromov hyperbolic space X and the classical hyperbolic space Hn . We describe the boundary at infinity @1 X in different realizations as a bounded and as an unbounded metric space in analogy to the unit disc and the upper half-space model of Hn . We introduce a quasi-Möbius structure on @1 X and discuss in detail the relation between the morphisms of X and the quasi-Möbius transformations of the boundary. In the second part of the book we focus on several aspects of the asymptotic geometry of arbitrary metric spaces. It turns out that the simple philosophy to study “a boundary at infinity” does not work in this general situation. Instead we introduce various dimension type asymptotic invariants and give several interesting applications in particular for embedding and non-embedding results.

xii

Preface

In this book we only discuss a few elements of asymptotic geometry and our viewpoint is in no way exhaustive. For example, our book has little intersection with the recent book of John Roe [Ro] about the same subject and can be considered as a complement. Almost all of the results in this book are in the literature, but the presentation and some of the proofs are new. This book grew out of lectures which we gave at the Steklov Institute in St. Petersburg and the University of Zürich. We want to thank the audience of these lectures, in particular Kathrin Haltiner, Alina Rull and Deborah Ruoss, for their questions and suggestions. In particular we want to thank our friends and colleagues Yuri Burago, Thomas Foertsch, Urs Lang, Nina Lebedeva and Kolya Kosovskii for many interesting discussions about asymptotic geometry.

Chapter 1

Hyperbolic geodesic spaces

Here we recall basic notions related to metric spaces, define hyperbolic geodesic metric spaces and prove the fundamental theorem about the stability of geodesics in hyperbolic spaces.

1.1 Geodesic metric spaces A metric on a set X is a function d W X X ! R which (1) is positive: d.x; x 0 / 0 for every x; x 0 2 X and d.x; x 0 / D 0 if and only if x D x0; (2) is symmetric: d.x; x 0 / D d.x 0 ; x/ for every x; x 0 2 X ; (3) satisfies the triangle inequality: d.x; x 00 / d.x; x 0 / C d.x 0 ; x 00 / for every x; x 0 ; x 00 2 X . Given a metric d , the value d.x; x 0 / is called distance between the points x, x 0 . We often use the notation jxx 0 j for the distance between x, x 0 in a given metric space X , and X for the metric space obtained from X by multiplying all distances by the factor > 0. A map f W X ! Y between metric spaces is said to be isometric if it preserves the distances, i.e. jf .x/f .x 0 /j D jxx 0 j for each x; x 0 2 X . Clearly, every isometric map is injective. If f is in addition surjective, it is called an isometry. A geodesic in a metric space X is any isometric map W I ! X , where I R is an interval (open, closed or half-open, bounded or unbounded). The image .I / of such a map is also called a geodesic. A metric space X is said to be geodesic if any two points in X can be connected by a geodesic. We use the notation xx 0 for a geodesic in X between x, x 0 , calling it a segment (even in the case when there are possibly several such segments). Remark. In many theories where the local geometry plays an essential role as e.g. in Riemannian geometry, a geodesic means a curve W I ! X which is only locally isometric, while on large scales the length of a segment might be larger than the distance between its end points. However, we always consider geodesics in the sense of the definition above.

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Chapter 1. Hyperbolic geodesic spaces

1.2 Hyperbolic geodesic spaces A triangle xyz in a geodesic space X is the union of segments xy, yz, zx, called the sides, connecting pairwise its vertices x; y; z 2 X . More generally an n-gon x1 : : : xn in X is the union of segments x1 x2 ; : : : ; xn x1 . The property of a geodesic space to be hyperbolic is defined in terms of triangles and the Gromov product, which is a useful notion in many circumstances.

1.2.1 Gromov product Let X be a metric space. Fix a base point o 2 X and for x; x 0 2 X put .xjx 0 /o D 1 .jxoj C jx 0 oj jxx 0 j/. The number .xjx 0 /o is nonnegative by the triangle inequality, 2 and it is called the Gromov product of x, x 0 with respect to o. Geometrically, the product can be interpreted as follows. Lemma 1.2.1. Let X be a geodesic space and xyz a triangle in X . There is a unique collection of points u 2 yz, v 2 xz, w 2 xy such that jxvj D jxwj, jyuj D jywj, jzvj D jzuj. z v

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Figure 1.1. Gromov product and equiradial points.

Proof. The equation system a Cb D jxyj a C c D jxzj b C c D jyzj has a unique solution and a, b, c are nonnegative by the triangle inequality. Then the points u, v, w are uniquely determined by the conditions jxvj D a, jywj D b, jzuj D c. The points u 2 yz, v 2 xz, w 2 xy are called equiradial points. Note that a D 12 .jxyj C jxzj jyzj/ D .yjz/x and similarly b D .xjz/y , c D .xjy/z . For example, if a triangle xyz X is a tripod, i.e. the union wx [ wy [ wz with only one common point w 2 X , then .yjz/x D jxwj.

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1.2. Hyperbolic geodesic spaces

.z

y. w

.x Figure 1.2. Tripod.

Definition 1.2.2. A geodesic metric space is called ı-hyperbolic, ı 0, if for any triangle xyz X the following holds: If y 0 2 xy, z 0 2 xz are points with jxy 0 j D jxz 0 j .yjz/x , then jy 0 z 0 j ı. Roughly speaking, in a ı-hyperbolic geodesic space X two sides xy and xz of any triangle xyz coming out of the common vertex x run together within the distance ı up to the moment .yjz/x and after that they start to diverge with almost maximal possible speed. This point of view becomes effective at distances large compared to ı. The space is (Gromov) hyperbolic if it is ı-hyperbolic for some ı 0. The constant ı is called a hyperbolicity constant for X . Clearly, in a ı-hyperbolic space any side of any triangle lies in the ı-neighborhood of the two other sides. This is the case k D 1 of the following lemma. Lemma 1.2.3. Let x1 : : : xn be an n-gon with n 2k C 1 for some k 2 N, then every side is contained in the kı-neighborhood of the union of the other sides. Proof. We show that a point x 2 xn x1 has distance kı from x1 x2 [ [ xn1 xn . Choose the midpoint xm with m D Œn=2 C 1 where [ ] is the integer part and consider the triangle x1 xm xn . By ı-hyperbolicity there exists y 2 x1 xm [ xm xn with jxyj ı. In the case y 2 x1 xm (resp. y 2 xm xn ) the induction hypothesis for the polygon x1 : : : xm (resp. xm : : : xn ) implies that y has distance .k 1/ı from x1 x2 [ [ xm1 xm (resp. xm xmC1 [ [ xn1 xn ). The claim follows. Exercise 1.2.4. Show that if any side of any triangle in a geodesic space X lies in the ı-neighborhood of the union of the two other sides for some fixed ı 0, then X is hyperbolic (Rips’ definition of hyperbolicity). Estimate the hyperbolicity constant for X. Example 1.2.5. A metric tree is a geodesic space in which every triangle is a tripod (possibly degenerate). Clearly, every metric tree is a 0-hyperbolic space.

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1.3 Stability of geodesics In this section we show that geodesics in hyperbolic spaces are stable. This means that if we enlarge the class of geodesics to the larger class of quasi-geodesics, then still each quasi-geodesic stays in uniformly bounded distance to a geodesic. To make this concept precise we need the concept of quasi-isometric maps.

1.3.1 Quasi-isometric maps The notion of a quasi-isometric map is a rough version of a bilipschitz map; recall that a map f W X ! Y between metric spaces is bilipschitz if 1 jxx 0 j a

jf .x/f .x 0 /j ajxx 0 j

for some a 1 and all x; x 0 2 X (in this definition, we do not require that f .X / D Y ). A subset A Y in a metric space Y is called a net if the distances of all points y 2 Y to A are uniformly bounded. A map f W X ! Y between metric spaces is said to be quasi-isometric if there are a 1, b 0 such that 1 jxx 0 j a

b jf .x/f .x 0 /j ajxx 0 j C b

for all x; x 0 2 X . In other words, a map is quasi-isometric if it is bilipschitz on large scales. If, in addition, the image f .X / is a net in Y , then f is called a quasi-isometry, and the spaces X and Y are called quasi-isometric. We also say that f is .a; b/-quasiisometric and call a, b the quasi-isometricity constants. A quasi-geodesic in X is a quasi-isometric map W I ! X where I R is an interval. For general metric spaces a quasi-geodesic can be far from a geodesic. Consider, for example, in the Euclidean plane the spiral W .1; 1/ ! R2 , .t / D t.cos.ln t/; sin.ln t //. p Since j.t /j D t and j 0 .t /j D 2 for all t > 1, we easily see p1 j.t /.s/j 2

jt sj j.t /.s/j

which implies that is a quasi-geodesic. This curve is in no way close to any geodesic. In hyperbolic geodesic spaces the situation is completely different. We will show that in a geodesic hyperbolic space every quasi-geodesic will stay in uniform bounded distance to a honest geodesic. To start our argument we first show that, roughly speaking, in order to avoid a ball in a hyperbolic space one needs to go an exponentially long path. We use the notation Br .x/ for the open ball of radius r centered at x in a metric space X, Br .x/ D fx 0 2 X W jxx 0 j < rg. Furthermore, Bxr .x/ is the closed ball fx 0 2 X W jxx 0 j rg.

1.3. Stability of geodesics

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Figure 1.3. The spiral on logarithmic scale.

By an a-path, a > 0, in a metric space we mean a finite or infinite sequence of points fxi g with jxi xiC1 j a for each i . Lemma 1.3.1. Assume that an a-path f W f1; : : : ; N g ! X in a geodesic ı-hyperbolic space, ı > 0, lies outside of the ball Br .x/ centered at some point x 2 f .1/f .N /. Then N c 2 r=ı for some constant c > 0 depending only on a and ı. Proof. Let k be the smallest integer with N 2k C 1 (then N 2k1 ). By Lemma 1.2.3 there exists a point y 2 f .j /f .j C 1/ for some j 2 f1; : : : ; N 1g such that jxyj kı. Note that jxyj r a=2, and hence k r=ı a=.2ı/. Thus N 2k1 c 2r=ı with c D 2.a=2ıC1/ . We are now able to prove the stability of quasigeodesics. Theorem 1.3.2 (Stability of geodesics). Let X be a ı-hyperbolic geodesic space and a 1, b 0. There exists H D H.a; b; ı/ > 0 such that for every N 2 N the image im.f / of every .a; b/-quasi-isometric map f W f1; : : : ; N g ! X lies in the H -neighborhood of any geodesic c W Œ0; l ! X with c.0/ D f .1/, c.l/ D f .N /, and vice versa, c lies in the H -neighborhood of im.f /. Proof. We first show that c lies in the h-neighborhood of im.f /, where h D h.a; b; ı/ > 0 depends only on a, b and ı. Note that f is an .a C b/-path in X . Choose h maximal with the property that im.f / lies outside the ball Bh .x/ for some x 2 c. Take y 2 c.0/x, y 0 2 xc.l/ with jyxj D jxy 0 j D 2h (if the distance between x and one of the ends of c is less than 2h, we take as y or y 0 the corresponding end). There are i , i 0 2 A D f1; : : : ; N g with jf .i /yj, jf .i 0 /y 0 j h and the segments yf .i/, y 0 f .i 0 / lie outside the ball Bh .x/. By taking appropriate points on these segments together with f .i /; : : : ; f .i 0 /, we find an .a C b/-path between y and y 0 2h outside Bh .x/ which contains K ji i 0 j C 3 C aCb points.

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By quasi-isometricity of f , we have ji i 0 j a.jf .i /f .i 0 /j C b/ 6ah C ab: On the other hand, K c 2 h=ı by Lemma 1.3.1 where c D c.a; b; ı/. These estimates together give an effective upper bound h.a; b; ı/ for the radius h. To complete the proof, consider a maximal sub-interval fj; : : : ; j 0 g A such that f .fj; : : : ; j 0 g/ lies outside the h-neighborhood of c, h D h.a; b; ı/. Since c is contained in the h-neighborhood of im.f /, there are i 2 f1; : : : ; j g, i 0 2 fj 0 ; : : : ; N g and z 2 c so that jzf .i /j, jzf .i 0 /j h. Then jf .i /f .i 0 /j 2h, and ji i 0 j 2ah C ab by quasi-isometricity of f . Hence, im.f / is contained in the H -neighborhood of c, where H D h C a.2ah C ab/ C b, H D H.a; b; ı/. Exercise 1.3.3. Derive the following consequences of Theorem 1.3.2. Corollary 1.3.4. Let X be hyperbolic geodesic space. Then there is no quasiisometric map f W R2 ! X . (Hint: Assuming that such a map exists, consider images of larger and larger equilateral triangles to obtain a contradiction using the stability of geodesics in X). Corollary 1.3.5. If a geodesic space X is quasi-isometric to a hyperbolic geodesic space Y , then X is also hyperbolic. (Hint: Take any triangle in X and compare it with its image in Y to conclude using stability of geodesics in Y that the triangle satisfies a ı-hyperbolicity condition).

1.4 Supplementary results and remarks 1.4.1 The real hyperbolic space Hn The real hyperbolic space Hn is a simply connected, complete Riemannian manifold of dimension n 2 having the constant sectional curvature 1. Various models of Hn are discussed in the appendix. This is the basic example of Gromov hyperbolic spaces. Exercise 1.4.1. Using the parallelism angle formula (see Appendix, Lemma A.3.2), show that the space Hn is ı-hyperbolic with ı < ln 3 D 1:0986 …. Actually, ı D 2 ln D 0:9624 : : : where is the golden ratio, 2 D C 1.

1.4.2 Gromov hyperbolic groups An important class of hyperbolic spaces is the class of Gromov hyperbolic groups which are defined as follows.

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1.4. Supplementary results and remarks

Let G be a finitely generated group and S G a finite set generating G. We assume that S does not contain the unit element of G and is symmetric, i.e., g 2 S if and only if g 1 2 S. The Cayley graph of .G; S / is a graph D .G; S / with the vertex set G, and vertices g; g 0 2 G are connected by an edge if and only if g 1 g 0 2 S. The Cayley graph carries the path metric dS for which every edge has length one. Such a metric when viewed on G is called a word metric. Clearly, is a geodesic space. A finitely generated group G is said to be word hyperbolic or Gromov hyperbolic if its Cayley graph .G; S/ is a hyperbolic space for some generating system S . Exercise 1.4.2. Show (using Corollary 1.3.5) that the property of a finitely generated group G to be hyperbolic is independent of the choice of a generating system S .

1.4.3 CAT.1/-spaces Let xyz be a geodesic triangle in a geodesic metric space X . A comparison triangle xQ yQ zQ H2 is a triangle with the same side-lengths. Comparison points on the sides are obtained as follows. Let u be a point on one of the sides, say u 2 xy. Then the comparison point uQ is the unique point on the segment xQ yQ with juQ xj Q D juxj and juQ yj Q D juyj. A complete geodesic space X is a CAT.1/-space if for each triangle xyz X and each u 2 xy, v 2 xz, it holds that juvj juQ vj, Q where uQ 2 xQ y, Q vQ 2 xQ zQ are comparison points on the sides of xQ yQ zQ H2 . That is, any triangle in X is thinner than its comparison triangle in H2 . Thus by definition, every CAT.1/-space is ı-hyperbolic with ı ıH2 . The class of CAT.1/-spaces is very large. Recall that a Hadamard manifold is a complete simply connected Riemannian manifold with nonpositive sectional curvatures. Every Hadamard manifold with sectional curvatures K 1 is a CAT.1/space. Furthermore, any metric tree is a CAT./-space for each < 0, in particular, it is CAT.1/. The class of CAT.1/-spaces also includes various hyperbolic buildings. On the other hand, there are compact nonpositively curved (in Alexandrov sense) 2-polyhedra with word hyperbolic fundamental group that admit no metric with CAT.1/ universal covering; see e.g. [BB]. Taking comparison triangles in R2 , one similarly obtains the important class of CAT.0/ or Hadamard spaces, i.e. complete geodesic spaces with triangles thinner than the Euclidean comparison triangles. In any Hadamard space X , all points x; x 0 2 X are connected by a unique geodesic segment.

8


Bibliographical note. The stability of geodesics was discovered in the twenties of the last century by M. Morse, [Mo1], [Mo2]. There are several approaches to its proof. The proof presented in Section 1.3 is very close to Gromov’s proof, [Gr1]; see also [BrH].

Chapter 2

The boundary at infinity

We start this chapter with a discussion of further properties of the Gromov product with the aim of deriving the ı-inequality for hyperbolic geodesic spaces. This allows us to extend the notion of hyperbolicity to metric spaces which are not necessarily geodesic. An important point of this discussion is the Tetrahedron Lemma, which has various applications throughout the book. Next we define the boundary at infinity for any hyperbolic space and discuss various structures attached to it: Gromov product, quasi-metrics, visual metrics and topology. We also establish local self-similarity of the boundary at infinity of cocompact hyperbolic spaces.

2.1 ı-inequality and hyperbolic spaces The Gromov product is monotone in the following sense. Lemma 2.1.1. Assume that y 0 2 xy and z 0 2 xz in a geodesic space X . Then .y 0 jz 0 /x .yjz/x . Proof. Since jxzj D jxz 0 j C jz 0 zj; jy 0 zj jy 0 z 0 j C jz 0 zj; we have jxzj jy 0 zj jxz 0 j jy 0 z 0 j, and hence .y 0 jz 0 /x .y 0 jz/x . Similarly .y 0 jz/x .yjz/x . Proposition 2.1.2. If a geodesic space X is ı-hyperbolic, then .xjy/o minf.xjz/o ; .zjy/o g ı for any base point o 2 X and any x; y; z 2 X . Proof. Put t0 D minf.xjz/o ; .yjz/o g and assume that x 0 2 ox, y 0 2 oy and z 0 2 oz satisfy jox 0 j D joy 0 j D joz 0 j D t0 . Then jx 0 z 0 j, jy 0 z 0 j ı, thus jx 0 y 0 j 2ı. On the other hand, by Lemma 2.1.1, .xjy/o .x 0 jy 0 /o D t0 12 jx 0 y 0 j t0 ı:

10

Chapter 2. The boundary at infinity

The inequality from Proposition 2.1.2 is called ı-inequality. This inequality is characteristic for the property of a space to be hyperbolic. Proposition 2.1.3. Assume that a geodesic space X satisfies the ı-inequality for every base point o and every x; y; z 2 X. Then X is 4ı-hyperbolic. Proof. Assume that points x 0 2 ox, y 0 2 oy of a triangle oxy X satisfy the condition jox 0 j D joy 0 j D t .xjy/o . It suffices to show that then jx 0 y 0 j 4ı. By the ı-inequality we have .x 0 jy 0 /o minf.x 0 jy/o ; t g ı minfminf.xjy/o ; t g ı; t g ı D t 2ı; hence jx 0 y 0 j D 2t 2.x 0 jy 0 /o 4ı.

Finally, we show that the ı-inequality for some base point implies the 2ı-inequality for any other base point. The following terminology is useful. A ı-triple is a triple of real numbers a, b, c with the property that the two smallest of these numbers differ by at most ı. To rephrase the ı-inequality we can say that the numbers .xjy/o , .xjz/o , .yjz/o form a ı-triple. : : It is also convenient to write a D b up to an error c or a Dc b instead of ja bj c. The following important result, which has many applications in the sequel, is called Tetrahedron Lemma. Lemma 2.1.4. Let d12 , d13 , d14 , d23 , d24 , d34 be six numbers such that the four triples A1 D .d23 ; d24 ; d34 /, A2 D .d13 ; d14 ; d34 /, A3 D .d12 ; d14 ; d24 / and A4 D .d12 ; d13 ; d23 / are ı-triples. Then B D .d12 C d34 ; d13 C d24 ; d14 C d23 / is a 2ı-triple. Proof. Without loss of generality, we can assume that d34 is maximal among the listed : : numbers. Then d13 D d14 up to an error ı since A2 is a ı-triple, and d23 D d24 up to an error ı since A1 is a ı-triple. Adding these approximate equalities, we obtain : that d13 C d24 D d23 C d14 up to an error 2ı. Since d34 is maximal, this means, if we assume that B is not a 2ı-triple, that d12 < minfd13 ; d14 ; d23 ; d24 g 2ı. But this contradicts the fact that A3 and A4 are ı-triples. Thus B is a 2ı-triple. Lemma 2.1.5. Assume that a metric space X satisfies the ı-inequality for a base point o. Then for any other base point x 2 X , the 2ı-inequality is fulfilled.

11

2.1. ı-inequality and hyperbolic spaces

4

1

3

2 Figure 2.1. Tetrahedron Lemma.

Proof. Note that the expression A D .tjy/o C .xjz/o minf.t jz/o C .yjx/o ; .xjt /o C .yjz/o g does not depend on the base point o. Choosing x as the base point, we see A D .tjy/x minf.tjz/x ; .zjy/x g. Thus, we have to prove A 2ı. From the ıinequality for the base point o, it follows that the six numbers .t jx/o , .t jy/o , .t jz/o , .xjy/o , .xjz/o , .yjz/o satisfy the condition of the Tetrahedron Lemma, which implies A 2ı. Now we extend the notion of hyperbolicity to metric spaces which are not necessarily geodesic. Definition 2.1.6. A metric space X is (Gromov) hyperbolic if it satisfies the ıinequality .xjy/o minf.xjz/o ; .zjy/o g ı or, what is the same, the triple ..xjy/o ; .xjz/o ; .yjz/o / is a ı-triple for some ı 0, for every base point o 2 X and all x; y; z 2 X . For geodesic spaces this notion is equivalent to our initial definition by Propositions 2.1.2, 2.1.3. From now on, when speaking about a ı-hyperbolic space X we mean Definition 1.2.2 if X is geodesic, and Definition 2.1.6 otherwise. The same holds for hyperbolicity constants. This causes no ambiguity because of Proposition 2.1.2. Remark 2.1.7. By Lemma 2.1.5, to prove that a space X is hyperbolic, it suffices to check that the ı-inequality holds for some ı 0, some base point o 2 X and all x; y; z 2 X. We shall often use this remark.

12


2.2 The boundary at infinity of hyperbolic spaces There are several possibilities to define the boundary at infinity of a hyperbolic space, ranging from the most geometric one, geodesic boundary, see Section 2.4.2, to the most analytic one, called Higson corona, which is not discussed in this book. We choose the original Gromov definition, since it is well adapted to the basic property of hyperbolic geodesic spaces that quasi-isometric maps have a natural extension to boundary maps, and the definition appeals to the geometric intuition. Let X be a hyperbolic space and o 2 X a base point. A sequence of points fxi g X converges to infinity if lim .xi jxj /o D 1:

i;j !1

This property is independent of the choice of o since j.xjx 0 /o .xjx 0 /o0 j joo0 j for any x; x 0 ; o; o0 2 X . Two sequences fxi g, fxi0 g that converge to infinity are equivalent if lim .xi jxi0 /o D 1: i!1

Using the ı-inequality, we easily see that this defines an equivalence relation for sequences in X converging to infinity. The boundary at infinity @1 X of X is defined to be the set of equivalence classes of sequences converging to infinity. Remark 2.2.1. If fxi g is a sequence converging to infinity and fxi0 g a sequence equivalent to fxi g in the sense that lim.xi jxi0 /o D 1, then fxi0 g converges to infinity itself. This easily follows from the ı-inequality. Now we introduce natural metric structures on the boundary at infinity of a Gromov hyperbolic space X . This is done in three steps. In a first step, we extend the Gromov product to the boundary at infinity. More precisely, we define for a base point o 2 X and points , 2 @1 X the product .j /o . In a second step, we define the map

W @1 X @1 X ! Œ0; 1/ by .; / D a.j/o , where a > 1 is some parameter. The map turns out to be a quasi-metric. In a third step, we apply a standard procedure to obtain from a metric for parameters a > 1, a small enough.

2.2.1 Gromov product on the boundary Fix a base point o 2 X . For points , 0 2 @1 X , we define their Gromov product by .j 0 /o D inf lim inf .xi jxi0 /o ; i!1

where the infimum is taken over all sequences fxi g 2 , fxi0 g 2 0 . Note that .j 0 /o takes values in Œ0; 1, that .j 0 /o D 1 if and only if D 0 , and that

2.2. The boundary at infinity of hyperbolic spaces

13

j.j 0 /o .j 0 /o0 j joo0 j for any o, o0 2 X . Furthermore, we obtain the following properties. Lemma 2.2.2. Let o 2 X , let X satisfy the ı-inequality for o, and let , 0 , 00 2 @1 X . (1) For arbitrary sequences fyi g 2 , fyi0 g 2 0 , we have .j 0 /o lim inf .yi jyi0 /o lim sup .yi jyi0 /o .j 0 /o C 2ı: i!1

i!1

(2) .j 0 /o , . 0 j 00 /o , .j 00 /o is a ı-triple. Proof. (1) We only need to show that lim supi!1 .yi jyi0 /o .j 0 /o C 2ı. We can assume that ¤ 0 . Applying the standard diagonal procedure, we find sequences fxi g 2 , fxi0 g 2 0 with lim.xi jxi0 /o D .j 0 /o . Let fyi g 2 , fyi0 g 2 0 . For i : sufficiently large, we have .xi jxi0 /o D .yi jxi0 /o up to an error ı since .xi jxi0 /o , 0 .yi jxi /o , .xi jyi /o is a ı-triple, .xi jyi /o ! 1, and two other members are bounded : due to the assumption ¤ 0 . In the same way we see that .yi jxi0 /o D .yi jyi0 /o up to : an error ı for i large enough. Thus .xi jxi0 /o D .yi jyi0 /o up to an error 2ı, which implies the claim. (2) Without loss of generality, we have to show .j 00 /o minf.j 0 /o ; . 0 j 00 /o g ı: Choose fxi g 2 , fxi0 g 2 0 , fxi00 g 2 00 such that lim.xi jxi00 /o D .j 00 /o . Then .j 00 /o lim sup minf.xi jxi0 /o ; .xi0 jxi00 /o g ı minf.j 0 /o ; . 0 j 00 /o g ı:

i!1

Similarly, the Gromov product .xj/o D inf lim inf .xjxi /o i!1

is defined for any x 2 X , 2 @1 X , where the infimum is taken over all sequences fxi g 2 , and the ı-inequality holds for any three points from X [ @1 X .

2.2.2 Quasi-metric on the boundary A quasi-metric space is a set Z with a function W Z Z ! R which satisfies the conditions (1) .z; z 0 / 0 for every z; z 0 2 Z, and .z; z 0 / D 0 if and only if z D z 0 ; (2) .z; z 0 / D .z 0 ; z/ for every z; z 0 2 Z; (3) .z; z 00 / K maxf .z; z 0 /; .z 0 ; z 00 /g for every z; z 0 ; z 00 2 Z and some fixed K 1.

14


The function is then called a quasi-metric, or more specifically, a K-quasi-metric. The property (3) is a generalized version of the ultra-metric triangle inequality which is the case K D 1. Remark 2.2.3. If .Z; d / is a metric space, then d is a K-quasi-metric for K D 2. In general the p-th power d p of the distance d is not a metric on Z for p > 1. But d p is still a 2p -quasi-metric. Coming back to the Gromov hyperbolic space X , we fix a > 1 and consider the 0 function W @1 X @1 X ! R, .; 0 / D a.j /o . Then is a K-quasi-metric on ı @1 X with K D a : the properties (1), (2) are obvious, and (3) immediately follows from Lemma 2.2.2 (2). Remark 2.2.4. The quasi-metric defined on @1 X depends on the base point o 2 X and the chosen parameter a > 1. If we emphasize this dependence, we write o;a . Let o, o0 2 X. Since j.j 0 /o .j 0 /o0 j joo0 j we compute c 1

o;a .; 0 / c

o0 ;a .; 0 /

0

where c D ajoo j . If a, a0 > 1 are different parameters then we have ˛

o;a0 D o;a

where ˛ D

ln a0 . ln a

There is a standard procedure to construct a metric from a quasi-metric. Let .Z; / be a quasi-metric space. We are interested in obtaining a metric on Z. Since the only problem is the triangle inequality,P the following approach is natural. Define a map d W Z Z ! R, d.z; z 0 / D inf i .zi ; ziC1 /, where the infimum is taken over all sequences z D z0 ; : : : ; zkC1 D z 0 in Z. By definition, d is then symmetric and satisfies the triangle inequality. We call this construction of d the chain construction. The problem with the chain construction is that d.z; z 0 / could be 0 for different points z, z 0 and Axiom (1) would no longer be satisfied for .Z; d /. Lemma 2.2.5. Let be a K-quasi-metric on a set Z with K 2. Then the chain 1 construction applied to yields a metric d with 2K

d . Proof. Clearly, d is nonnegative, symmetric, satisfies the triangle inequality and d

. We prove by induction over the length of sequences D fz D z0 ; : : : ; zkC1 D z 0 g, j j D k C 2, that

.z; z 0 /

X

k1 X . / ´ K .z0 ; z1 / C 2

.zi ; ziC1 / C .zk ; zkC1 / : 1

(2.1)

15

2.2. The boundary at infinity of hyperbolic spaces

For jj D 3, this follows from the triangle inequality (3) for . Assume that (2.1) holds true for all sequences of length j j k C 1, and suppose that j j D k C 2. 0 00 Given p 2Pf1; : : : ; P k 1g, we Plet 00p D fz0 ; : : : ; zpC1 g, p D fzp ; : : : ; zkC1 g, 0 and note that . / D .p / C .p /. Because .z; z 0 / K maxf .z; zp /; .zp ; z 0 /g, there is a maximal p 2 f0; : : : ; kg with .z; z 0 / K .zp ; z 0 /. ThenP .z; z 0 / K .z; zpC1 /. Assume now that .z; z 0 / > . /. Then, in particular, .z; z 0 / > K .z; z1 / and

.z; z 0 / > K .zk ; z 0 /. It follows that p 2 f1; : : : ; k 1g and thus by the inductive assumption X X X

.z; zpC1 / C .zp ; z 0 / .p0 / C .p00 / D . / < .z; z 0 /: On the other hand,

.z; z 0 / K minf .z; zpC1 /; .zp ; z 0 /g .z; zpC1 / C .zp ; z 0 / because K 2. This is a contradiction. Now it follows from (2.1) that 2Kd . Hence d is a metric as required. Proposition 2.2.6. Let be a K-quasi-metric on a set Z. Then there exists "0 > 0 only depending on K such that " is bilipschitz equivalent to a metric for each 0 < " "0 . More precisely, there exists a metric d" on Z such that 1

" .z; z 0 / 2K "

d" .z; z 0 / " .z; z 0 /

for all z; z 0 2 Z. Proof. " is a K " -quasi-metric for every " > 0. If K " 2 then the chain construction applied to " yields a required metric d" by Lemma 2.2.5.

2.2.3 Visual metrics at infinity We now apply this construction to the quasi-metric on @1 X . A metric d on the boundary at infinity @1 X of X is said to be visual if there are o 2 X , a > 1 and positive constants c1 , c2 such that 0

0

c1 a.j /o d.; 0 / c2 a.j /o for all ; 0 2 @1 X . In this case, we say that d is a visual metric with respect to the base point o and the parameter a. The inequalities above are called the visual inequalities. Applying Proposition 2.2.6 we see: Theorem 2.2.7. Let X be a hyperbolic space. Then for any o 2 X , there is a0 > 1 such that for every a 2 .1; a0 there exists a metric d on @1 X which is visual with respect to o and a.

16


Now we consider what happens if the base point is changed. Proposition 2.2.8. Visual metrics d , d 0 on @1 X with respect to the same parameter a > 1 and base points o, o0 respectively are bilipschitz equivalent, c 1

d0 c d

for some constant c 1. Proof. This immediately follows from the visual inequalities for d , d 0 and from the fact that j.j 0 /o .j 0 /o0 j joo0 j for all ; 0 2 @1 X (see Remark 2.2.4). Next we consider the effect of the parameter change. Proposition 2.2.9. Visual metrics d , d 0 on @1 X with respect to the same base point o and parameters a, a0 > 1 respectively are Hölder equivalent, namely, there is a constant c 1 such that 1 ˛ d .; 0 / c

for all ; 0 2 @1 X , where ˛ D

d 0 .; 0 / cd ˛ .; 0 /

ln a0 . ln a

Proof. This immediately follows from the visual inequalities for the metrics d , d 0 and from the fact that a0 D a˛ (see Remark 2.2.4). We define the topology on the boundary at infinity @1 X for a hyperbolic space X as the metric topology for some visual metric on @1 X . It follows from Propositions 2.2.8 and 2.2.9 that this topology is independent of the choice of a visual metric. Exercise 2.2.10. Let X be a hyperbolic space. Show that @1 X is bounded and complete for any visual metric on @1 X .

2.3 Local self-similarity of the boundary Hyperbolic groups and more general cobounded hyperbolic spaces have a remarkable and useful property: their boundary at infinity are locally self-similar. A map f W Z ! Z 0 between metric spaces is called homothetic with coefficient R if jf .z/f .z 0 /j D Rjzz 0 j for all z; z 0 2 Z. Here we need a more flexible property. Let 1 and R > 0 be given. A map f W Z ! Z 0 between metric spaces is -quasi-homothetic with coefficient R if for all z; z 0 2 Z, we have Rjzz 0 j= jf .z/f .z 0 /j Rjzz 0 j:

17

2.3. Local self-similarity of the boundary

Note that f is also 0 -quasi-homothetic with coefficient R for every 0 . This property can be regarded as a perturbation of the property to be homothetic, and the coefficient describes the perturbation. We apply this notion usually to a family of quasi-homothetic maps with fixed when the coefficients R go to infinity. A metric space Z is locally similar to (subsets of) a metric space Y if there is 1 such that for every sufficiently large R > 1 and every A Z with diam A R1 there is a -quasi-homothetic map f W A ! Y with coefficient R. If a metric space Z is locally similar to itself then we say that Z is locally self-similar. Example 2.3.1. The standard ternary Cantor set X is locally self-similar. One can take D 3 in this case. Indeed, given R > 3 and A X with diam A 1=R, there is k 2 N with 3k < R 3kC1 . Then diam A < 1=3k . Hence, A is contained in the k-th step interval which it intersects. This interval is 3k -homothetic to Œ0; 1 and thus it is -quasi-homothetic to Œ0; 1 with coefficient R. The basic example of locally self-similar spaces is the boundary at infinity of a hyperbolic group. We consider a more general situation. A metric space X is cobounded if there is a bounded subset A X such that the orbit of A under the isometry group of X covers X . A metric space X is proper if every closed ball Bxr .x/ X is compact. Theorem 2.3.2. The boundary at infinity @1 X of every cobounded, hyperbolic, proper, geodesic space X is locally self-similar with respect to any visual metric. For the proof we need the following Lemma 2.3.3. Let o, g, x 0 , x 00 be points of a metric space X such that the Gromov products .x 0 jg/o , .x 00 jg/o jogj for some 0. Then .x 0 jx 00 /o .x 0 jx 00 /g C jogj .x 0 jx 00 /o C 2: Proof. The left-hand inequality immediately follows from the triangle inequality: since jox 0 j jogj C jgx 0 j and jox 00 j jogj C jgx 00 j, we have .x 0 jx 00 /o .x 0 jx 00 /g C jogj. Next we note that .x 0 jo/g D jogj .x 0 jg/o . This yields jx 0 oj D jogj C jgx 0 j 2.x 0 jo/g jogj C jgx 0 j 2 and similarly jx 00 oj jogj C jgx 00 j 2 . Now the right-hand inequality follows. Proof of Theorem 2.3.2. We can assume that the geodesic space X is ı-hyperbolic, ı 0, and that a visual metric d on @1 X satisfies 0

0

c 1 a.j /o d.; 0 / ca.j /o for some base point o 2 X , some constants c 1, a > 1 and all ; 0 2 @1 X . Note that then diam @1 X c.

18


There is > 0 such that the orbit of the ball B .o/ under the isometry group of X covers X. Now we put D c 3 aC4ı . Fix R > 1 and consider A @1 X with diam A 1=R. For each ; 0 2 A, we have .j 0 /o loga .R=c/: We fix 2 A. Since X is proper, there is a geodesic ray o X representing (see Exercise 2.4.3). We take g 2 o with ajogj D R=c. Then using the ı-inequality, we obtain for every 0 2 A . 0 jg/o minf. 0 j/o ; .jg/o g ı D jogj ı because .jg/o D jogj. For arbitrary 0 ; 00 2 A, consider sequences fxi0 g 2 0 , fxi00 g 2 00 such that 0 00 .xi jxi /g ! . 0 j 00 /g . We can assume without loss of generality that .xi0 jg/o , .xi00 jg/o jogj ı because possible errors in these estimates disappear while taking the limit; see below. Applying Lemma 2.3.3 to the points o; g; xi0 ; xi00 2 X and D ı, we obtain .xi0 jxi00 /o jogj .xi0 jxi00 /g .xi0 jxi00 /o jogj C 2ı: Passing to the limit, this yields . 0 j 00 /o jogj . 0 j 00 /g . 0 j 00 /o jogj C 4ı: There is an isometry f W X ! X with jof .g/j . Then . 0 j 00 /g .f . 0 /jf . 00 //o . 0 j 00 /g C

because the Gromov products with respect to different points differ one from another at most by the distance between the points. The last two double inequalities give . 0 j 00 /o jogj .f . 0 /jf . 00 //o . 0 j 00 /o jogj C C 4ı; and therefore c 3 a.C4ı/ Rd. 0 ; 00 / d.f . 0 /; f . 00 // ca Rd. 0 ; 00 /: This shows that f W A ! @1 X is -quasi-homothetic with coefficient R and hence @1 X is locally self-similar. We say that a metric space Z is doubling if there is a constant N 2 N such that for every r > 0 every ball in Z of radius 2r can be covered by N balls of radius r. If the property above holds for all sufficiently small r > 0 only, then we say that Z is doubling at small scales. Clearly, if a compact space is doubling at small scales then it is doubling.


19

Lemma 2.3.4. Assume that a metric space Z is locally similar to (subsets of ) a compact metric space Y . Then Z is doubling at small scales. Proof. There is 1 such that for every sufficiently large R > 1 and every A Z with diam A 1=R there is a -quasi-homothetic map f W A ! Y with coefficient R. We fix a positive 1=.4/. Since Y is compact, there is N 2 N such that any subset B Y can be covered by at most N balls of radius centered at points of B. Take r > 0 small enough so that R D =r satisfies the assumption above. Then for any ball B2r Z, we have diam B2r 4r 1=R; and thus there is a -quasi-homothetic map f W B2r ! Y with coefficient R. The image f .B2r / is covered by at most N balls of radius centered at points of f .B2r /. The preimage under f of any such ball is contained in a ball of radius =R D r centered at a point in B2r . Hence, B2r is covered by at most N balls of radius r, and Z is doubling at small scales. Example 2.3.5. The space Hn , n 2, is locally similar to a compact subspace, e.g. to any closed ball of radius 1. However, Hn is by no means doubling. Exercise 2.3.6. Let X be a proper geodesic hyperbolic space. Show that @1 X is compact. Corollary 2.3.7. Assume that a hyperbolic space X satisfies the condition of Theorem 2.3.2, e.g., X is a hyperbolic group. Then @1 X is doubling with respect to any visual metric.

2.4 Supplementary results and remarks 2.4.1 A quadruple condition for hyperbolicity Given a quadruple Q D .x; y; z; u/ of points in a metric space X with fixed base point o, we form the triple A D A.Q/ D ..xjy/o C.zju/o ; .xjz/o C.yju/o ; .xju/o C .yjz/o / as in the Tetrahedron Lemma and call it the cross-difference triple of Q. We define the small cross-difference of Q, scd.Q/, as the distance between the two smaller entries of the cross-difference triple A.Q/. Proposition 2.4.1. The metric space X is ı-hyperbolic, ı 0, if and only if scd.Q/ ı for every quadruple Q X . Proof. The condition scd.Q/ ı is a reformulation of the property of A.Q/ to be a ı-triple. Note that this property is independent of the choice of o and take as o any point of Q.

20


Explicitly written, the condition for A.Q/ to be a ı-triple for Q D .x; y; z; u/ is the inequality jxzj C jyuj maxfjxyj C jzuj; jxuj C jyzjg C 2ı: This formulation is more symmetric than the ı-inequality and has a geometric interpretation in the spirit of the Tetrahedron Lemma. Consider Q as an abstract tetrahedron. Adding the length of opposite edges of Q, we obtain three numbers which we can order as a b c. Then the inequality says c b 2ı.

2.4.2 Geodesic boundary Two geodesic rays ; 0 W Œa; 1/ ! X in a geodesic space X are called asymptotic if j.t/ 0 .t/j C < 1 for some constant C and all t a. To be asymptotic is an equivalence relation on the set of the rays in X , and the set of classes of asymptotic rays is sometimes called the geodesic boundary of X , @g X . In a geodesic hyperbolic space, asymptotic rays are at a uniformly bounded distance from each other. Moreover, we have Lemma 2.4.2. Let X be a geodesic ı-hyperbolic space. Assume that for some constant C > 0, geodesic rays , 0 in X with common vertex o contain points .t /, 0 .t 0 / with j.t / 0 .t 0 /j C for arbitrarily large t , t 0 . Then j. / 0 . /j ı for all 0; in particular, the rays , 0 are asymptotic. Proof. We have ..t /j 0 .t 0 //o D 12 .t C t 0 j.t / 0 .t 0 /j/ minft; t 0 g C =2: Thus for minft; t 0 g C =2 we have j./ 0 . /j ı by ı-hyperbolicity. Since t , t 0 can be chosen arbitrarily large, this inequality holds for all 0. If a geodesic space X is Gromov hyperbolic, then obviously @g X @1 X . In general, there is no reason that the geodesic boundary of a hyperbolic geodesic space coincides with the boundary at infinity. However, there are several important cases when @g X D @1 X . Exercise 2.4.3. Show that if X is a proper hyperbolic geodesic space, then @g X D @1 X. Another important case when @g X D @1 X is described in Chapter 6; see Proposition 6.4.3.

21


2.4.3 Meaning of the function o;e .1 ; 2 / D e .1 j2 /o for Hn For the unit ball model of the hyperbolic space Hn , n 2 (see Appendix, Sections A.2 and A.5), the quasi-metric o;e W S n1 S n1 ! R, o;e .1 ; 2 / D e .1 j2 /o , where the unit sphere S n1 Rn is identified with @1 Hn and o is the center of the ball, has a clear geometric interpretation: This function coincides with half of the chordal metric, e .1 j2 /o D 12 j1 2 j for every 1 ; 2 2 S n1 . This immediately follows from the next lemma which also implies that the angle metric on S n1 is a visual metric with respect to the center o and the parameter a D e. Lemma 2.4.4. For every 1 ; 2 2 @1 Hn D S n1 we have e .1 j2 /o D sin. =2/; where D ]o .1 ; 2 /. Proof. For the geodesic rays i W Œ0; 1/ ! Hn from o to i , i D 1; 2, we obviously have e .1 j2 /o D lim .e h t e 2t /1=2 ; t!1

where h t D d.1 .t /; 2 .t // is the distance in Hn . From the hyperbolic law of cosine cosh.h t / D cosh2 .t / sinh2 .t / cos and the trigonometric formula 1 cos D 2 sin2 . =2/, we easily obtain e h t e 2t sin2 . =2/ as t ! 1. Hence the claim.

2.4.4 The chain construction Lemma 2.2.5 and the idea of its proof is due to A. H. Frink, [Fr]. It provides a better constant than contemporary simpler arguments; see e.g. [He, Chapter 14]. Moreover, the condition of that lemma cannot be improved according to the following result: Example 2.4.5 ([Sch]). For every " > 0, there exists a .2 C "/-quasi-metric space .Z; / such that the chain construction applied to yields only a pseudo-metric d with d.z; z 0 / D 0 for some distinct z; z 0 2 Z.

22


Bibliographical note. It is proven in [Bou] that the function o .1 ; 2 / D e .1 j2 /o is a metric on the boundary at infinity, 1 ; 2 2 @1 X , of any CAT.1/-space X for every o 2 X (the only nontrivial point is to prove the triangle inequality). For further references we call this metric the Bourdon metric. Bourdon metrics associated with different o; o0 2 X are conformal to each other, and any isometry of X induces a conformal transformation of .@1 X; o / [Bou]. Local self-similarity of the boundary at infinity of hyperbolic groups and more general cocompact hyperbolic spaces is certainly well known to experts in the field. Explicitly, it is established in [BL] from where basic results of Section 2.3 are taken.

Chapter 3

Busemann functions on hyperbolic spaces

The boundary at infinity of a hyperbolic space X was defined in the previous chapter using a fixed basepoint o 2 X. Though the boundary is independent of the choice of o, the situation is different if we try to choose a basepoint at infinity. In this chapter we develop appropriate tools, the most important of which are Busemann functions. We also introduce and study properties of Gromov products and visual metrics based at infinity.

3.1 Busemann functions The notion of a Busemann function is very useful in many areas. Intuitively, a Busemann function on a metric space X is the distance function on X from a point ! at infinity. Since literally this makes no sense, one needs to normalize it subtracting the distance from ! to a fixed reference point o: one takes the difference bz;o .x/ D jzxj jzoj for z 2 X and looks at the limit as z ! !. Since jbz;o .x/j is bounded by jxoj by the triangle inequality, there is a good chance to get a well-defined function if one specifies how the limit limz!! bz;o .x/ is to be understood. Moreover the cost of normalizing is that one associates with ! a class of distance functions which differ from each other by a constant. This class has no canonical representative: for example, the function b W H2 ! R, defined in the upper half-plane model as b.u; v/ D c ln v, is a Busemann function based at ! D 1 for each constant c 2 R. In the case when X is Gromov hyperbolic, we avoid the problem how to define the limit limz!! bz;o .x/ by using the already defined Gromov product .!jx/o for any x; o 2 X, ! 2 @1 X . Geometrically, in the case when X is geodesic and ! is a point of the geodesic boundary, ! 2 @g X , the difference j!xj j!oj makes sense as jzo xj jzx oj, where zo 2 x!, zx 2 o! are equiradial points of an infinite triangle xo! X . Using equalities jzo xj D jz! xj D .!jo/x and jzx oj D jz! oj D .!jx/o , where z! 2 xo is the equiradial point, we arrive at the formula b!;o .x/ D jz! xj jz! oj D .!jo/x .!jx/o .

24

Chapter 3. Busemann functions on hyperbolic spaces

o

x

z! equiradial points

! Figure 3.1. Defining a Busemann function.

All of this motivates the following considerations. Let X be a ı-hyperbolic space. For every ! 2 @1 X , we have a well-defined function b! W X X ! R;

b! .x; y/ D .!jy/x .!jx/y :

The function b! is obviously skew symmetric in its variables. Busemann functions based at ! associated with different reference points o differ by a constant. We show that this property holds for b! up to an error depending only on the hyperbolicity constant of X . : We extend our agreement about D as follows. For two sequences ai , bi , we write : ai D bi up to an error c

or

: ai Dc bi

if lim supi!1 jai bi j c. Lemma 3.1.1. For every ! 2 @1 X , every sequence fzi g 2 ! and every x, o 2 X , we have : b! .x; o/ D .zi jo/x .zi jx/o up to an error 2ı. Proof. This follows from an adapted version of Lemma 2.2.2. Lemma 3.1.2. For every ! 2 @1 X , x; o; o0 2 X , we have : b! .x; o/ b! .x; o0 / D b! .o0 ; o/ up to an error 6ı.

25

3.1. Busemann functions

Proof. We start with the identity .zjo/x .zjx/o .zjo0 /x C .zjx/o0 D .zjo/o0 .zjo0 /o which is straightforward to check for every quadruple of points z; x; o; o0 2 X . When z is replaced by !, this identity transforms into the (approximate) equality we have to prove. Taking a sequence fzi g 2 ! and replacing z in the identity by zi , we apply Lemma 3.1.1 to complete the proof. We want the set B.!/ of Busemann functions based at ! 2 @1 X to have the following properties: – B.!/ contains all functions b!;o . / D b! . ; o/, o 2 X ; – for every b 2 B.!/ and every constant c 2 R, the function b C c is in B.!/; – any two functions b; b 0 2 B.!/ differ from each other by a constant up to an error with depending only on the hyperbolicity constant ı. We extend the set of canonical functions b!;o to achieve an additional flexibility which is sometimes helpful; see Example 3.1.4 and the proof of Propositions 3.1.5, 3.2.3. However, this inevitably leads to an artificial choice of the error bound . So consider the set B.!/ RX which consists of all functions b W X ! R for : each of which there are o 2 X and a constant c 2 R with b D b!;o C c up to an error 2ı. Then, by Lemma 3.1.2, any two functions b; b 0 2 B.!/ differ from each other by a constant up to an error 10ı. Definition 3.1.3. Any function from B.!/ is called a Busemann function based at ! 2 @1 X. Example 3.1.4. For every o 2 X , the function ˇ!;o .x/ D joxj 2.!jx/o is a Busemann function based at !, ˇ!;o 2 B.!/. Moreover, b!;o .x/ ˇ!;o .x/ b!;o .x/ C 2ı for every x 2 X . Proof. Given x 2 X , there is a sequence fzi g 2 ! with .!jx/o D lim.zi jx/o . Using the identity .zjo/x C .zjx/o D jxoj

(3.1)

which holds for every o; x; z 2 X , we obtain lim.zi jo/x D jxoj .!jx/o . Thus .!jo/x jxoj .!jx/o .!jo/x C 2ı by Lemma 2.2.2, and the claim follows.

Proposition 3.1.5. Every Busemann function b W X ! R based at ! 2 @1 X has the following properties:

26


(1) b is roughly 1-Lipschitz, that is jb.x/b.x 0 /j jxx 0 jC10ı for every x; x 0 2 X ; (2) b.yi / ! C1 for every fyi g 2 2 @1 X , ¤ !; (3) assume that b.xi / ! 1 for a sequence fxi g X ; then fxi g 2 !. Proof. (1) We obviously have jb! .x; x 0 /j .!jx 0 /x C .!jx/x 0 jxx 0 j for every x; x 0 2 X. Because b! is skew symmetric, we can apply Lemma 3.1.2 to its first variable to obtain jb!;o .x/ b!;o .x 0 /j jb! .x; x 0 /j C 6ı jxx 0 j C 6ı for every o 2 X . Now the claim follows easily from the definition of Busemann functions. (2) We can assume that b D ˇ!;o for some o 2 X , b.y/ D joyj 2.!jy/o ; see Example 3.1.4. The product .!jyi /o is uniformly bounded, since ¤ !. Thus b.yi / ! C1 together with joyi j because fyi g converges to infinity. (3) We can assume that b D b!;o . Since b! .xi ; o/ D .!jo/xi .!jxi /o ! 1, we have .!jxi /o ! 1. Hence fxi g 2 !. Remark 3.1.6. The converse to (2) and to (3) is not true: it is possible that b.xi / ! C1 for b 2 B.!/ and a sequence fxi g 2 !. For example, the sequence xi D .i; exp.i// 2 H2 in the upper half-plane model converges to infinity and fxi g 2 1, while b.xi / D i ! C1 for the Busemann function b.u; v/ D ln v based at 1.

3.2 Gromov products based at infinity Let X be a ı-hyperbolic space, ! 2 @1 X . Busemann functions allow to define a Gromov product based at !. We first define it on X .

3.2.1 Gromov products on X based at infinity We fix a Busemann function b 2 B.!/ and for x; y 2 X , we define their Gromov product based at b by .xjy/b D 12 .b.x/ C b.y/ jxyj/: Note that contrary to the standard case .xjy/o with o 2 X , the product .xjy/b might be negative. It immediately follows from the definition of Busemann functions that for different b; b 0 2 B.!/ there is a constant c 2 R so that : .xjy/b .xjy/b 0 D c

(3.2)

up to an error 10ı for every x; y 2 X . In other words, the choice of b 2 B.!/ does not play an essential role.

3.2. Gromov products based at infinity

27

Example 3.2.1. For the function b D ˇ!;o 2 B.!/, see Example 3.1.4, we have .xjy/b D .xjy/o .!jx/o .!jy/o for every x; y 2 X . Our next goal is to show that the Gromov product based at any Busemann function b 2 B.!/ satisfies the -inequality with 0 depending only on the hyperbolicity constant ı of X . We begin with : : Lemma 3.2.2. Assume that numbers a; b; c 2 R form a ı-triple, and a0 D a, b 0 D b, : c 0 D c up to an error . Then the numbers a0 , b 0 , c 0 form a .ı C 2 /-triple. Proof. We can assume that a maxfb; cg. Then jb cj ı. Thus jb 0 c 0 j ı C 2 because jb 0 c 0 j differs from jb cj by at most 2 . This implies the claim in the case a0 minfb 0 ; c 0 g. Suppose now that a0 < minfb 0 ; c 0 g. Since a0 a maxfb; cg minfb 0 ; c 0 g 2;

the claim follows.

Proposition 3.2.3. For every x; y; z 2 X , the numbers .xjy/b , .xjz/b , .yjz/b form a 2ı-triple for every function b D ˇ!;o 2 B.!/, and a 22ı-triple for an arbitrary b 2 B.!/. Proof. Six numbers .xjy/o , .yjz/o , .xjz/o , .!jx/o , .!jy/o , .!jz/o satisfy the condition of the Tetrahedron Lemma (Lemma 2.1.4) which implies that the numbers .xjy/o C .!jz/o , .yjz/o C .!jx/o , .xjz/o C .!jy/o form a 2ı-triple. We let a D .!jx/o C.!jy/o C.!jz/o and first assume that b D ˇ!;o . Then using Example 3.2.1, we obtain .xjy/b C a D .xjy/o C .!jz/o ; .yjz/b C a D .yjz/o C .!jx/o ; .xjz/b C a D .xjz/o C .!jy/o : Therefore, the numbers on the left-hand side form a 2ı-triple. Hence, the numbers .xjy/b , .yjz/b , .xjz/b form a 2ı-triple for b D ˇ!;o . In the general case, we apply : the approximate equality .xjy/b 0 D .xjy/b C c up to an error 10ı, where the constant c is independent of x; y 2 X , and Lemma 3.2.2.

3.2.2 Gromov products on X [ @1 X based at infinity There are several possibilities to introduce a Gromov product based at infinity on X [ @1 X. The simplest one, which allows to avoid any further limit procedure, is

28


motivated by the Gromov product on X based at the Busemann function b D ˇ!;o . We denote Z D X [ @1 X , and for every pair .; / 2 Z Z which is distinct from .!; !/, we put .j /!;o ´ .j /o .!j/o .!j /o : The so defined Gromov product takes values in Œ1; C1, .j /!;o D C1 if and only if D 2 @1 X n !, and .j /!;o D 1 if and only if one of the factors equals !. Certainly, .xjy/!;o D .xjy/b for every x; y 2 X . Just as in Proposition 3.2.3, we see that for every ; ; 2 Z distinct from !, the numbers .j /!;o , .j /!;o , . j /!;o form a 2ı-triple. It is unpleasant that this Gromov product depends not only on ! 2 @1 X but also on the reference point o 2 X, which is conceptually not right. The standard way to eliminate this dependence is to consider a family of Gromov products which depends then only on !. We proceed similarly to Section 2.2.1. For every Busemann function b 2 B.!/ and .; / 2 Z Z n .!; !/, we define the Gromov product based at b by .j /b D inf lim inf .xi jyi /b ; i!1

where the infimum is taken over all sequences fxi g 2 , fyi g 2 (here we assume that the sequence is constant for points in X ). It follows from equation (3.2) that for different b; b 0 2 B.!/, there is a constant c 2 R so that : .j /b .j /b 0 D c

(3.3)

up to an error 10ı for every ; 2 Z n !. As an exercise, one can check that for the Busemann function b D b!;o 2 B.!/ the approximate equality : .j /b D .j /!;o

(3.4)

holds up to an error 2ı. Furthermore, exactly as in Lemma 2.2.2, we obtain Lemma 3.2.4. Assume that the -inequality holds in X for the Gromov product based at b 2 B.!/, e.g., D 2ı for b D b!;o and D 22ı for an arbitrary b 2 B.!/. Let ; ; 2 Z n !. (1) For arbitrary sequences fxi g 2 , fyi g 2 , we have .j /b lim inf .xi jyi /b lim sup .xi jyi /b .j /b C 2 I i!1

(2) .j /b , . j /b , .j /b is a -triple.

i!1

29

3.3. Visual metrics based at infinity

3.3 Visual metrics based at infinity We recapitulate and slightly generalize the notion of a quasi-metric space. Definition 3.3.1. Let Z be a set, Z1 Z be a subset, D Z1 Z1 . A quasimetric on Z with infinitely remote set Z1 is any function W Z Z n ! Œ0; 1 with the following properties: (1) .a; b/ D 0 if and only if a D b; (2) .a; b/ D .b; a/; (3) there is K 1 with .a; b/ K maxf .a; c/; .c; b/g for all a; b; c 2 Z for which all members of the inequality are defined; (4) .a; b/ < 1 if and only if a, b 2 Z n Z1 . In this case, we also say that is a K-quasi-metric, and .Z; / is a quasi-metric space or shortly Q-metric space. This definition is a small modification of the definition in Chapter 2, Section 2.2.2. We allow the existence of an infinitely remote set Z1 Z such that .a; / D 1 for all a 2 Z n Z1 , 2 Z1 . However, .; / is not defined for ; 2 Z1 . In what follows, we always assume that the infinitely remote set contains at most one point, jZ1 j 1. In this case, the generalized ultra-metric triangle inequality (3) is fulfilled for all distinct a; b; c 2 Z. For example, if .Z; / is a Q-metric space with x D Z [ f1g (with the obvious extension of

.a; b/ < 1 for all a; b 2 Z, then Z

by .a; 1/ D 1 for all a 2 Z) is also a Q-metric space. For another interesting example see Section 3.4.1. The proof of Proposition 2.2.6 runs verbatim for modified Q-metrics, and we obtain Proposition 3.3.2. Let be a K-quasi-metric on a set Z with infinitely remote set Z1 . Then there exists "0 > 0 only depending on K such that " is bilipschitz equivalent to a metric on Z n Z1 for each 0 < " "0 . Coming back to our ı-hyperbolic space X , we fix ! 2 @1 X , a Busemann function b 2 B.!/ and for a parameter a > 1 we define the function

b W .@1 X /2 n .!; !/ ! Œ0; 1;

b .; / D a.;/b :

Then, similarly to the considerations in Section 2.2.2, b is a K-quasi-metric on Z D @1 X with the infinitely remote set Z1 D f!g and with K D a , where the -inequality holds in @1 X nf!g for the Gromov product based at b; see Lemma 3.2.4. Furthermore, b .; / D 1 if and only if one of the points , coincides with !. A metric d on @1 X n f!g is called visual if it is Lipschitz equivalent to b for some Busemann function b 2 B.!/ and some parameter a > 1: c1 a.j/b d.; / c2 a.j/b :

30


In this case we say that d is visual with respect to the Busemann function b and the parameter a. Because the Gromov products on @1 X n f!g based at different Busemann functions b, b 0 2 B.!/ differ from each other by a constant up to an error 10ı, see equation (3.3), the property of a metric to be visual is independent of the choice of b. Applying Proposition 3.3.2 we see Proposition 3.3.3. Let X be a hyperbolic space, ! 2 @1 X . Then for any Busemann function b 2 B.!/, there is a0 > 1 such that for every a 2 .1; a0 there exists a visual metric d on @1 X n f!g with respect to b and a. Example 3.3.4. For the upper half-space model of HnC1 , @1 HnC1 D Rn [ f1g, the Euclidean metric on Rn is visual with respect to the Busemann function b 2 B.1/, D Rn .0; 1/, and the parameter a D e. b.u; v/ D ln v for .u; v/ 2 RnC1 C Contrary to the case of visual metrics based at a point in X , the boundary at infinity of a hyperbolic space X equipped with a visual metric based at a Busemann function b 2 B.!/ is unbounded with infinitely remote point ! 2 @1 X . This is similar to the upper half-space model of Hn . Similarly to Propositions 2.2.8 and 2.2.9, we obtain that visual metrics with respect to one and the same parameter and different Busemann functions b; b 0 2 B.!/ are bilipschitz to each other, and the parameter change leads to Hölder equivalent visual metrics. Much more interesting is the effect which occurs when we change the point !. We study this effect in the next chapter.

3.4 Supplementary results and remarks 3.4.1 The boundary at infinity with respect to ! 2 @1 X Similarly to Section 2.2, we say that a sequence fxi g X converges to infinity with respect to ! 2 @1 X if lim .xi jxj /b D C1 i;j !1

for some and hence for any Busemann function b 2 B.!/. Two sequences fxi g, fxi0 g that converge to infinity with respect to ! are equivalent if lim .xi jxi0 /b D C1

i!1

for some and hence for every Busemann function b 2 B.!/. Using Proposition 3.2.3, we easily see that this defines an equivalence relation for sequences in X converging to infinity with respect to !. The boundary at infinity _.!X of X with respect to ! is defined as the set of equivalence classes of sequences converging to infinity with respect to !.


31

Proposition 3.4.1. For every ! 2 @1 X there are canonical inclusions @1 X n f!g ! _.!X ! @1 X; whose composition coincides with the inclusion @1 X n f!g @1 X . Proof. Fix a reference point o 2 X and recall that for the Busemann function b D ˇ!;o 2 B.!/, we have .xjy/b D .xjy/o .!jx/o .!jy/o for every x; y 2 X ; see Example 3.2.1. Also recall that the Gromov product based at o 2 X is nonnegative. Now if .xi jxj /b ! C1 for a sequence fxi g X , then .xi jxj /o ! 1, i.e., any sequence which converges to infinity with respect to ! converges to infinity in the standard sense. Similarly, two sequences equivalent to each other with respect to ! are equivalent to each other in the standard sense. This defines a map f W _.!X ! @1 X . Vice versa, fix 2 @1 X nf!g and consider a sequence fxi g 2 . Then .xi jxj /o ! 1, while .!jxi /o , .!jxj /o are uniformly bounded. Thus .xi jxj /b ! C1, and fxi g converges to infinity with respect to !. Similarly, any other fyi g 2 is equivalent to fxi g with respect to !. This defines a map g W @1 X n f!g ! _.!X . Obviously, the composition f B g coincides with inclusion @1 X n f!g @1 X . It remains to check that the map f is injective. Assume to the contrary that there is a sequence fxi g X with .xi jxj /b ! 1 and .!jxi /o ! 1. Then it follows from (3.5) .xi jxj /b D .xi jxj /o .!jxi /o .!jxj /o ! 1 and from the ı-inequality that j.!jxi /o .!jxj /o j ı and therefore, for the righthand side aij of equation (3.5), we obtain aij .xi jxj /o 2.!jxi /o C ı for all sufficiently large i , j . We fix such an i and look at the limit as j ! 1. This yields lim sup aij .xi j!/o C 3ı ! 1; j !1

as i ! 1. This contradicts the assumption .xi jxj /b ! 1.

3.4.2 Boundary continuous hyperbolic spaces Let X be a Gromov hyperbolic space which satisfies the ı-inequality for some ı 0. We have proven many estimates in terms of ı. In case ı D 0 these inequalities become equalities. But there are also other spaces, where we have sometimes exact equalities. This holds in particular for the classical hyperbolic space.

32


We call a Gromov hyperbolic space X boundary continuous if the Gromov product extends continuously onto the boundary at infinity in the following way: if fxi g; fyi g are sequences in X converging to points x; N yN in X or in @1 X , then .xi jyi /o ! .xj N y/ N o for every o 2 X . Gromov hyperbolic spaces satisfying the ı-inequality with ı D 0 are boundary continuous. The classical hyperbolic space Hn is boundary continuous. More generally, we have Proposition 3.4.2. Every proper CAT.1/-space X is a boundary continuous Gromov hyperbolic space. Proof. We fix o 2 X and consider distinct , 0 2 @1 X (the case when one of the points , 0 lies in X is even easier). Observe that there are geodesic rays o, o 0 X (see Exercise 2.4.3) which are uniquely determined because the space is CAT.1/. We use the notation D .t/ for the unit speed parametrization of o, .0/ D o. By monotonicity of the Gromov product (see Lemma 2.1.1), there exists a limit a D lim ..t /j 0 .t //o : t!1

Note that a < 1 because , 0 are distinct, and that a .j 0 /o by the definition of .j 0 /o . We shall prove that a D lim .xi jxi0 /o i!1

fxi0 g

0

for every fxi g 2 , 2. Since .xi j/o ! 1, we easily see that the segments oxi converge to the ray o as i ! 1 in the compact-open topology, that is, for every " > 0, R > 1 the segment oxi runs within the "-neighborhood of o during the time R for all sufficiently large i . To show our claim, we first prove that lim inf i .xi jxi0 / a. We fix " > 0 and take t large enough so that ..t /j 0 .t //o a ". Next, we fix xi .t / 2 oxi with joxi .t/j D jo.t /j D t. Then jxi .t /.t /j " for all sufficiently large i . Hence the lengths of the segments .t/ 0 .t / and xi .t /xi0 .t / differ from each other by at most 2". Therefore, the Gromov products ..t /j 0 .t //o and .xi .t /jxi0 .t //o differ from each other by at most ". Using monotonicity of the Gromov product, we obtain .xi jxi0 /o .xi .t /jxi0 .t //o a 2"; thus lim inf i .xi jxi0 / a. This shows in particular that a D .j 0 /o . To obtain the estimate from above, lim supi .xi jxi0 / a, we need the following fact: there exists a geodesic X with end points , 0 at infinity. (To prove this, we note that each geodesic segment .t/ 0 .t /, t 0, intersects a fixed ball BR .o/ with R 2.j 0 /o . Using that X is proper, we easily find as a sublimit of .t/ 0 .t / as t ! 1. We leave details to the reader.) We fix p 2 . Then the geodesic segments pxi , pxi0 converge to the subrays p, 0 p of respectively in the compact-open topology as i ! 1. It follows that given


33

" > 0, the angle ]p .xi ; xi0 / " for all sufficiently large i . Using comparison with H2 , we see that every point of xi xi0 is "-close to the union pxi [ pxi0 . By the CAT.1/ property, the distances dist..t /; /, dist. 0 .t /; / become arbitrarily small as t ! 1. Thus taking t large enough, we conclude that dist..t /; xi xi0 /, dist..t/; oxi / " and dist. 0 .t /; xi xi0 /, dist. 0 .t /; oxi0 / " for all sufficiently large i . It follows that .xi jxi0 /o ..t /j 0 .t //o C " for all sufficiently large i and hence lim supi .xi jxi0 / a.

For a boundary continuous hyperbolic space X , the notion of a Busemann function is simplified: the set of Busemann functions B.!/ RX for ! 2 @1 X consists of all functions b W X ! R for each of which there are o 2 X and a constant c 2 R with b D b!;o C c. The approximate equality of Lemma 3.1.2 becomes the precise equality and thus two Busemann functions b, b 0 2 B.!/ differ from each other by a constant. In particular, b!;o D ˇ!;o . Now Gromov products on X based at ! 2 @1 X differ from each other by a constant, .xjy/b .xjy/b 0 D c for different b, b 0 2 B.!/, some constant c and all x; y 2 X. Furthermore, we have the following refined version of Proposition 3.2.3. Proposition 3.4.3. Let X be a boundary continuous ı-hyperbolic space. Then for every x; y; z 2 X, ! 2 @1 X , the numbers .xjy/b , .xjz/b , .yjz/b form a ı-triple for every Busemann function b 2 B.!/. Proof. In view of the equality .!jx/o D limi!1 .wi jx/o for every fwi g 2 !, x 2 X , the numbers .xjy/o C.!jz/o , .yjz/o C.!jx/o , .xjz/o C.!jy/o form a ı-triple because the difference between any two of them is independent of o (compare the proof of Proposition 2.4.1), and we can take o D z for example. The rest of the proof runs exactly as in the proof of Proposition 3.2.3. The Gromov product on Z D X [ @1 X based at ! 2 @1 X is also simplified. Given fxi g 2 , fyi g 2 2 @1 X n f!g, we have .j /!;o D .j /o .!j/o .!j /o D lim Œ.xi jyi /o .!jxi /o .!jyi /o i!1

D lim .xi jyi /b D .j /b ; i!1

where b D ˇ!;o 2 B.!/. Then for any other Busemann function b 0 2 B.!/, the limit limi!1 .xi jyi /b 0 D limi!1 .xi jyi /b C c exists, and .j /b 0 D lim .xi jyi /b 0 D .j /b C c: i!1

In particular, Gromov products on Z based at ! 2 @1 X differ from each other by a constant: .j /b .j /b 0 D c for different b; b 0 2 B.!/, some constant c and all ; 2 Z.

34


Finally, we note that for every ; ; 2 Z distinct from ! and every b 2 B.!/, the numbers .j /b , . j /b , .j /b form a ı-triple by Proposition 3.4.3 because .j /b D limi!1 .xi jyi /b for fxi g 2 , fyi g 2 . Bibliographical note. A result close to Proposition 3.3.3 is obtained in [Ha] for negatively pinched Hadamard manifolds, where a family of functions on @1 X n f!g, defined via horospherical distances, is considered instead of the Gromov product based at a Busemann function b 2 B.!/. It is proven in [FS2] that the function b .1 ; 2 / D e .1 j2 /b is a metric on the boundary at infinity, 1 ; 2 2 @1 X n f!g, of any CAT.1/-space X for every ! 2 @1 X and every Busemann function b 2 B.!/. The proof is based on the properties of a Bourdon metric and the Ptolemy inequality.

Chapter 4

Morphisms of hyperbolic spaces

What is a natural class of morphisms between hyperbolic spaces? By natural we mean morphisms inducing maps of the boundaries at infinity in a way compatible with composition. There are different points of view on this question. The commonly accepted one is to consider quasi-isometric maps as natural, due to geometrical reasons. First of all, the universal covering of a reasonable compact metric space (by reasonable we mean spaces like manifolds and polyhedra for which the covering theory holds) is quasi-isometric to its fundamental group with a word metric which for a finitely generated group is only defined up to bilipschitz transformations related to changes of generating sets. Therefore, interesting geometric invariants should be quasi-isometry invariants. Secondly, the classical duality between the isometries of HnC1 and the Möbius transformations of the boundary sphere S n D @1 HnC1 extends to a much more general class of hyperbolic spaces. A well-known theorem of Efremovich–Tihomirova [ET] based on the stability of geodesics (which was discovered much earlier by M. Morse) says that any quasi-isometry of HnC1 has an extension to the boundary sphere S n . The argument works for quasi-isometric maps between any geodesic hyperbolic spaces, and with some additional effort one shows that the induced map between the boundaries at infinity is quasi-Möbius. However, the problem is that the condition for spaces to be geodesic is too restrictive, and without it, a quasi-isometric map between hyperbolic spaces has in general no extension to the boundaries at infinity. One can develop an approach which replaces geodesics by quasi-geodesics and therefore allows to recover the extension property of quasi-isometric maps between hyperbolic spaces which are more general than geodesic ones; see e.g. [BoS], [V2]. Instead, we narrow the class of quasi-isometric maps by putting the stronger condition that a map should have a bilipschitz type control over cross-differences, and we call such a map power quasi-isometric. A power quasi-isometric map between any metric spaces is automatically quasi-isometric, and in the case of arbitrary hyperbolic spaces it naturally induces a map between their boundaries at infinity (see Corollary 4.3.3 below), which is automatically (power) quasi-Möbius (see Proposition 5.2.10).

36

Chapter 4. Morphisms of hyperbolic spaces

From our point of view, power quasi-isometric maps constitute a natural class of morphisms between hyperbolic spaces. This is supported by the (nontrivial) fact that in the case of geodesic hyperbolic spaces any quasi-isometric map is power quasiisometric (see Theorem 4.4.1), and in that way, we recover the extension property of quasi-isometric maps. Moreover, we show in Chapter 7 that any hyperbolic space (with the mild natural restriction to be visual) is roughly isometric to a subspace of a geodesic hyperbolic space with the same boundary at infinity, and hence there is no need for quasification of geodesics.

4.1 Morphisms of metric spaces and hyperbolicity We consider in this section metric spaces X and X 0 and maps f W X ! X 0 . We classify maps f by considering the way these maps disturb the distances between points. We describe this by control functions. Given functions 1 , 2 W Œ0; 1/ ! R, we are considering the class of functions f such that for all x; y 2 X we have

1 .jxyj/ jx 0 y 0 j 2 .jxyj/; where x 0 D f .x/ and y 0 D f .y/ are the images of x, y respectively. In this way, one can define highly rigid classes of maps, e.g. isometric maps for 1 D 2 D id. On the other hand one can define much wider classes of maps. A map f is called coarse if there are control functions 1 , 2 for f with the property that lim t!1 i .t / D 1. For example, quasi-isometric or Q-isometric maps, introduced in Chapter 1, are characterized by affine control functions, 1 .t / D 1c t d , 2 .t / D ct C d , c 1, d 0. In Chapter 7, roughly homothetic (or R-homothetic) and roughly isometric (or R-isometric) maps, described by control functions 1 .t / D ct d , 2 .t / D ct C d , d 0 and c > 0, c D 1 respectively, play an important role. These characterizations of functions involve just the distance jxyj between two points and how this distance is disturbed by the map f . For the study of hyperbolic spaces it is necessary to consider more complicated expressions which involve the distances between three and four points. This is not surprising since hyperbolicity itself is a condition on quadruples of points. These expressions can be defined in a multiplicative or in an additive way. Well known and intensively studied is the cross-ratio, which involves the distances between four points and which is written in a multiplicative form. Given .x; y; z; u/ 2 X 4 the classical cross-ratio is given by Œx; y; z; u D

jxzjjyuj : jxyjjzuj

The additive version of the classical cross-ratio is the classical cross-difference hx; y; z; ui D 12 .jxzj C jyuj jxyj jzuj/:

4.1. Morphisms of metric spaces and hyperbolicity

37

The correspondence between the multiplicative and the additive version of the expression is more subtle than expected at first glance. A trivial calculation shows that hx; y; z; ui D .xjz/o .yju/o C .xjy/o C .zju/o ; where o 2 X is any chosen base point. One should think of .xjy/o to be the additive counterpart to the multiplicative jxyj (this explains the factor 1=2). This corresponds to the fact that the quasi-metric at the boundary at infinity of a hyperbolic space is given by the exponential a.xjy/o . It is easy to express hyperbolicity in terms of the classical cross-difference. Recall that by Proposition 2.4.1, a metric space X is hyperbolic if and only if the crossdifference triple A.Q/ of every quadruple Q D .x; y; z; u/ X is a ı-triple for some ı 0. Explicitly written, this condition is the inequality jxzj C jyuj maxfjxyj C jzuj; jxuj C jyzjg C 2ı: In turn, it can be rewritten via the classical cross-difference as minfhx; y; z; ui; hx; u; z; yig ı: One can now define classes of maps, which disturb the cross-difference in a similar way as above by using control functions,

1 .hx; y; z; ui/ hx 0 ; y 0 ; z 0 ; u0 i 2 .hx; y; z; ui/: We introduce a class of maps between metric spaces called (strongly) PQ-isometric maps. ‘PQ’ stands for ‘power quasi’ because every PQ-isometric map is quasiisometric, and in general this is stronger than the property to be quasi-isometric. Definition 4.1.1. We say that a map f W X ! X 0 between metric spaces is strongly PQ-isometric if there are constants c 1, d 0 such that for all quadruples .x; y; z; u/ 2 X 4 with hx; y; z; ui 0, 1 hx; y; z; ui c

d hx 0 ; y 0 ; z 0 ; u0 i chx; y; z; ui C d:

Remarks. (1) The term ‘PQ-isometric’ is reserved for a weaker condition which is introduced and used in the subsequent sections. (2) We can also estimate hx 0 ; y 0 ; z 0 ; u0 i in the case that this expression is negative. We use the antisymmetry in the second and third entry, i.e. the fact that hx; y; z; ui D hx; z; y; ui. We then obtain for hx; y; z; ui < 0 that chx; y; z; ui d hx 0 ; y 0 ; z 0 ; u0 i 1c hx; y; z; ui C d: Thus it is possible to write the condition to be a strongly PQ-isometric map as .hx; y; z; ui/ hx 0 ; y 0 ; z 0 ; u0 i .hx; y; z; ui/ where .x; y; z; u/ is now an arbitrary quadruple and W R ! R is the control function .t/ D maxfct; 1c t g C d .

38


One easily checks that the composition of strongly PQ-isometric maps is strongly PQ-isometric. It follows directly from the characterization of hyperbolic spaces by the quadruple condition of hyperbolicity that hyperbolicity is a strongly PQ-isometry invariant. More precisely we have the following. Proposition 4.1.2. Assume that a metric space X 0 is hyperbolic and that a map f W X ! X 0 is strongly PQ-isometric. Then X is also hyperbolic. Furthermore, the image of a hyperbolic space under a strongly PQ-isometric map is hyperbolic. Proof. There are c 1, d 0 so that for every quadruple Q D .x; y; z; u/ X , we have .hx; y; z; ui/ hx 0 ; y 0 ; z 0 ; u0 i .hx; y; z; ui/ and

.hx; u; z; yi/ hx 0 ; u0 ; z 0 ; y 0 i .hx; u; z; yi/

where .t/ D maxfct; 1c t g C d . Then minfhx 0 ; y 0 ; z 0 ; u0 i; hx 0 ; u0 ; z 0 ; y 0 ig ı for some ı 0 independent of Q0 D .x 0 ; y 0 ; z 0 ; u0 / D f .Q/ because X 0 is hyperbolic. This implies that minfhx; y; z; ui; hx; u; z; yig c.d C ı/ since we can assume that both hx; y; z; ui, hx; u; z; yi are nonnegative. Therefore, X is hyperbolic. A similar argument proves the second assertion of the proposition. Remark 4.1.3. (1) The proof shows that f preserves hyperbolicity if f only coarsely preserves the cross-difference in the sense that there are functions 1 , 2 W Œ0; 1/ ! R with lim t!1 i .t / D 1 such that

1 .hx; y; z; ui/ hx 0 ; y 0 ; z 0 ; u0 i 2 .hx; y; z; ui/ for all x; y; z; u 2 X with hx; y; z; ui 0. (2) In contrast it is not true in general that hyperbolicity is a quasi-isometry invariant. There are examples of quasi-isometric spaces X , X 0 such that one of them is hyperbolic, but the other is not. Consider the subset X D f0g [ fai ; bi ; ci 2 R2 W i 2 Ng R2 , where ai D .10i ; 0/, bi D .0; 10i / and ci D .10i ; 10i /. Then, considering the quadruples .0; ai ; bi ; ci /, one easily shows that X is not hyperbolic. However the map f W X ! R, defined by f .0/ D 0, f .ai / D 10i , f .bi / D 2 10i and f .ci / D 310i is a Q-isometric map whose image is (as a subset of R) hyperbolic. As another and simpler example, consider X D f.x; y/ 2 R2 W y D jxjg with the metric induced from R2 , X 0 D R. Then the projection f W X ! X 0 , f .x; y/ D x, is bilipschitz, X 0 is 0-hyperbolic, but X is not hyperbolic.

4.2. Cross-difference triples and cross-differences

39

4.2 Cross-difference triples and cross-differences Given a quadruple Q D .x; y; z; u/ of points in a metric space X , we have defined the classical cross-difference hx; y; z; ui of these points. However, there are drawbacks of this expression: it depends on a chosen order of the quadruple Q; as a consequence, there are six versions of the definition, and most of them (if not all) can be found in literature. Furthermore, it turns out that in a hyperbolic space X , essentially only one of those six cross-differences contains a significant geometric information encoded in the unordered Q – another one is obtained by reversing the sign and four others are inessential. In this chapter, we define a cross-difference in another way which does not depend on the ordering of the underlying four points. For hyperbolic spaces this new crossdifference contains essentially the same information as the classical one, but is often easier to handle. In the previous section, we have already observed the following. Given a quadruple Q D .x; y; z; u/ in a metric space X , the expression .xjy/o C.zju/o .xjz/o .yju/o is independent of the choice of a base point o 2 X , actually, it coincides with the classical cross-difference hx; y; z; ui. This expression has an interpretation in the spirit of the Tetrahedron Lemma. Consider the quadruple Q as an abstract tetrahedron with vertices x, y, z, u. Every edge of Q is labelled by the Gromov product of its vertices with respect to o. If we fix two different pairs of opposite edges of Q, then the expression above is the difference of the sums of the labels attached to the appropriate edges. To eliminate an ordering of the quadruple points in this expression, we proceed as follows. For a quadruple Q D .x; y; z; u/ of points in a metric space X , we form the cross-difference triple A D A.Q/ D ..xjy/o C .zju/o ; .xjz/o C .yju/o ; .xju/o C .yjz/o / as in the Tetrahedron Lemma. Every element a from A corresponds to a unique pair of opposite edges of Q. From the previous discussion, we obviously have the following fact which plays a fundamental role in the sequel. Theorem 4.2.1. Given a quadruple Q of points in a metric space X , the crossdifference triples Ao .Q/ and Ao0 .Q/ with respect to different base points o; o0 2 X differ from each other by the same constant in each of their entries. Every classical cross-difference of the unordered Q is a difference of two distinct members of A, and all together, we have six such differences, three nonnegative and three opposite to them. However, in a ı-hyperbolic space, A is a ı-triple, which means that only one out of the six differences, namely, the maximal one, contains a geometrically significant information. Hence, the definition of a strongly PQisometric map between hyperbolic spaces puts a lot of excessive conditions which are satisfied automatically. This is the reason for the following consideration.

40


We define the cross-difference of Q, .a a0 /: cd.Q/ D max 0 a;a 2A

As above, the cross-difference is independent of the choice of a base point o 2 X and it is an invariant of the unordered Q. Geometrically, the best way to understand the cross-difference in a hyperbolic space is to consider the case when X is a tree. Example 4.2.2. If Q is a quadruple of points in a metric tree X , then cd.Q/ is the maximal distance between opposite pairs of edges of the (degenerate) tetrahedron in X spanned by Q. x

u a

y

b

z

Figure 4.1. cd.Q/ D jabj D dist.xy; zu/.

In Section 2.4.1 we already defined the small cross-difference of Q, scd.Q/, as the distance between the two smaller entries of the cross-difference triple A.Q/. By definition, both entries are nonnegative, cd.Q/ scd.Q/ 0 for every quadruple Q X. Obviously, A.Q/ is, up to a constant, uniquely determined by the two numbers cd.Q/ and scd.Q/. Lemma 4.2.3. Let f W X ! Y be a strongly .c; d /-PQ-isometric map. Then 1 c

and 1 c

cd.Q/ d cd.Q0 / c cd.Q/ C d

scd.Q/ d scd.Q0 / c scd.Q/ C d

for every quadruple Q X with Q0 D f .Q/. Proof. Consider a quadruple Q D .x; y; z; u/ of points in X . We can assume that cd.Q/ D hx; y; z; ui 0. Then cd.Q0 / hx 0 ; y 0 ; z 0 ; u0 i 1c hx; y; z; ui d D

1 c

cd.Q/ d:

In order to prove the estimate from above for cd.Q0 /, we can assume that cd.Q0 / D hx 0 ; y 0 ; z 0 ; u0 i 0. In the case hx; y; z; ui 0 we obtain cd.Q0 / chx; y; z; ui C d c cd.Q/ C d:

41

4.3. PQ-isometric maps

In the case hx; y; z; ui < 0 the estimate is trivial cd.Q0 /

1 hx; y; z; ui C d < d c cd.Q/ C d: c

Similar argument proves the estimates for small cross-differences. We leave this as an exercise to the reader. Since hyperbolicity is equivalent to the condition ‘scd.Q/ is uniformly bounded for all quadruples Q’, it is now evident that strongly PQ-isometric maps preserve hyperbolicity, compare Proposition 4.1.2.

4.3 PQ-isometric maps In Section 4.1, we already introduced the notion of strongly PQ-isometric maps. We now take another look at this class of maps using the new notion of cross-difference. Maps between metric spaces are PQ-isometric if they perturb the cross-differences of quadruples in a bilipschitz manner. This is a weaker condition than to be strongly PQisometric. However, every PQ-isometric map is quasi-isometric, and this is in general stronger than the property of being quasi-isometric. Moreover, every PQ-isometric map between hyperbolic spaces naturally induces a map between their boundaries at infinity, which is not true necessarily for quasi-isometric maps between hyperbolic spaces. Definition 4.3.1. We say that a map f W X ! X 0 between metric spaces is PQisometric if there are constants c 1, d 0 such that 1 c

cd.Q/ d cd.Q0 / c cd.Q/ C d

for every quadruple Q of points in X , where Q0 D f .Q/. In this case we say that f is .c; d /-PQ-isometric. One easily checks that the composition of PQ-isometric maps is PQ-isometric. It follows from Lemma 4.2.3 that every strongly PQ-isometric map is PQ-isometric. Furthermore, we have Proposition 4.3.2. Let f W X ! X 0 be a .c; d /-PQ-isometric map. Then f is .c; d /quasi-isometric and moreover 1 .xjy/o c

d .x 0 jy 0 /o0 c.xjy/o C d

(4.1)

for every x; y; o 2 X , where ‘prime’ stands for the image under f . In particular, any .1; 0/-PQ-isometric map is isometric.

42


Proof. Taking the quadruple Q D .x; x; o; o/, we obtain that its cross-difference triple is A D .jxoj; 0; 0/. Thus 1 jxoj c

d jx 0 o0 j cjxoj C d

for every x; o 2 X , i.e., f is .c; d /-quasi-isometric. Similarly, the cross-difference triple of the quadruple Q D .x; y; o; o/ is A D ..xjy/o ; 0; 0/, and we obtain the inequalities (4.1). Corollary 4.3.3. Every PQ-isometric map f W X ! Y between hyperbolic spaces naturally induces a map @1 f W @1 X ! @1 Y between their boundaries at infinity. Proof. If 2 @1 X and a sequence fxi g 2 , then it follows from Proposition 4.3.2 that the sequence ff .xi /g converges to infinity. If fyi g 2 is another sequence, then similarly ff .xi /g and ff .yi /g are equivalent. Letting @1 f ./ be the class 2 @1 Y of ff .xi /g, we obtain a well-defined map @1 f W @1 X ! @1 Y . The map @1 f is natural in the sense that @1 .g B f / D @1 g B @1 f for PQ-isometric g W Y ! Z, where Z is a hyperbolic space. In Section 5.2.1 we give more information about the induced map @1 f ; see Proposition 5.2.10.

4.4 Quasi-isometric maps of hyperbolic geodesic spaces It is not clear whether a PQ-isometric map between general hyperbolic spaces is strongly PQ-isometric (see, however, Remark 4.5.6). But in the case of hyperbolic geodesic spaces we have a much stronger property. Theorem 4.4.1. Let f W X ! X 0 be a .c; b/-quasi-isometric map of hyperbolic geodesic spaces. Then there is a constant d 0 depending only on c, b and the hyperbolicity constants ı, ı 0 of X , X 0 such that f is strongly .c; d /-PQ-isometric and in particular .c; d /-PQ-isometric. As a consequence we obtain Corollary 4.4.2. Let f W X ! X 0 be a map between hyperbolic geodesic spaces. The following are equivalent: (a) f is quasi-isometric; (b) f is PQ-isometric; (c) f is strongly PQ-isometric.

The proof of Theorem 4.4.1 is based on the stability of geodesics in hyperbolic geodesic spaces. We first study the deviation from equiradial points. Let xyz be a geodesic triangle in a ı-hyperbolic geodesic space X . Then for the equiradial points

43

4.4. Quasi-isometric maps of hyperbolic geodesic spaces

u0 2 yz, v0 2 xz, w0 2 xy (see Lemma 1.2.1), we have ju0 v0 j, jv0 w0 j, ju0 w0 j ı by the definition of ı-hyperbolicity. We show that the converse is true in the following sense. Lemma 4.4.3. Let xyz be a geodesic triangle with equiradial points u0 2 yz, v0 2 xz, w0 2 xy in a ı-hyperbolic geodesic space X . Then for points u 2 yz, v 2 xz, w 2 xy with juvj; jvwj; juwj h, we have juu0 j, jvv0 j, jww0 j 2.ı C h/. y z u

u0

v

w0 w

v0 w0

x Figure 4.2. Estimating distances to equiradial points.

Proof. Without loss of generality, we can assume that w 2 xw0 and v 2 zv0 . Then for w 0 2 xv0 with jxw 0 j D jxwj we have jww0 j D jw 0 v0 j jw 0 vj jw 0 wj C jwvj ı C h: Thus jvv0 j jvw 0 j ı C h, and juu0 j juwj C jww0 j C jw0 u0 j 2.ı C h/. Hence the claim. Proof of Theorem 4.4.1. Fix o; x; y; z 2 X, and put o0 D f .o/, x 0 D f .x/, y 0 D f .y/, z 0 D f .z/. Taking o as a base point, we have hx; y; z; oi D .xjy/o .xjz/o D s and similarly hx 0 ; y 0 ; z 0 ; o0 i D .x 0 jy 0 /o0 .x 0 jz 0 /o0 D s 0 . It suffices to check that minfcs; s=cg d s 0 maxfcs; s=cg C d for an appropriate d 0. We first show that jf .u/u0 j d for the equiradial points u 2 xy, u0 2 x 0 y 0 of the triangles oxy, o0 x 0 y 0 respectively. We take the other equiradial points v 2 ox, w 2 oy and consider their images f .v/, f .w/. Then jf .u/f .v/j, jf .v/f .w/j, jf .u/f .v/j cı C b. By the stability x 0 2 o0 y 0 with juN 0 f .u/j, jvN 0 f .v/j, of geodesics, there are uN 0 2 x 0 y 0 , vN 0 2 o0 x 0 , w

44


jw x 0 f .w/j H , where H D H.c; b; ı 0 /. Thus juN 0 vN 0 j, jvN 0 w x 0 j, juN 0 w x 0 j h, where 0 0 0 h D cı C b C 2H . By Lemma 4.4.3, juN u j 2.ı C h/ and hence jf .u/u0 j d D 2.ı 0 C h/ C H . y x 1 0

11 00

v

11 00 00 11

11 00 00 11

w

11 00

o

11 00 11 00

z

Figure 4.3. Estimating the difference of Gromov products.

Consider the equiradial points v; w 2 ox for the triangles oxy, oxz respectively, and the equiradial points v 0 ; w 0 2 o0 x 0 for the triangles o0 x 0 y 0 , o0 x 0 z 0 respectively. Then jsj D jvwj and js 0 j D jv 0 w 0 j. The estimates jf .v/v 0 j, jf .w/w 0 j d imply 1 jsj c

.b C 2d / js 0 j cjsj C .b C 2d /:

(4.2)

To obtain the estimate for s 0 itself we assume first that s D jxvj jxwj 0. If s 0 0 then estimate (4.2) gives the desired estimate for s and s 0 . Assume now that s 0 < 0. We still have s 0 js 0 j cs C .b C 2d /. By the stability of geodesics, f .w/ lies within distance H from a geodesic o0 f .v/, which in particular implies jo0 f .w/j jo0 f .v/j C H and hence s 0 D jo0 v 0 j jo0 w 0 j .2d C H /. It follows that s 0 D js 0 j 2js 0 j 1c s .b C 2d / 2.2d C H /:

The case s < 0 is similar.

4.5 Supplementary results and remarks 4.5.1 Cross-difference in X based at infinity Assume that X is a ı-hyperbolic space, and let Q be a quadruple of points in X . Given ! 2 @1 X, for any Busemann function b 2 B.!/, we have the cross-difference triple A D Ab .Q/ based at b which is defined as above replacing the base point o by the Busemann function b. By the same argument as for Theorem 4.2.1 we obtain that two cross-difference triples Ab .Q/, Ab 0 .Q/, based either in X or at Busemann functions, differ from each other by a constant. In particular, the cross-difference based at b, .a a0 /; cdb .Q/ D max 0 a;a 2A

45


also depends neither on b 2 B.!/ nor on ! 2 @1 X , cdb .Q/ D cd.Q/:

4.5.2 Cross-differences at infinity There are several possibilities to extend the cross-difference to quadruples of points at infinity. We always assume that such quadruples consist of distinct points to ensure that cross-differences are well defined and finite. We first consider the case when a base point o 2 X is fixed. Given a quadruple Q D .˛; ˇ; ; / of distinct points in @1 X , we form the cross-difference triple A D ..˛jˇ/o C .j /o ; .˛j /o C .ˇj /o ; .˛j /o C .ˇj /o / as above and define the crossdifference cdo .Q/ D max .a a0 /: 0 a;a 2A

This cross-difference may depend on the choice of a base point. For another base point o0 2 X the cross-differences cdo .Q/ and cdo0 .Q/ differ from each other by at most 10ı. This follows from Lemma 2.2.2 (1). Next we fix ! 2 @1 X , o 2 X and for a quadruple Q of distinct points ˛, ˇ, ,

2 @1 X, we form the cross-difference triple A D ..˛jˇ/!;o C .j /!;o ; .˛j /!;o C .ˇj /!;o ; .˛j /!;o C .ˇj /!;o / with respect to the Gromov product .˛jˇ/!;o D .˛jˇ/o .!j˛/o .!jˇ/o ; see Section 3.2.2. Then the cross-difference .a a0 / cd!;o .Q/ D max 0 a;a 2A

is independent of ! 2 @1 X , cd!;o .Q/ D cdo .Q/. For different reference points o; o0 2 X, the corresponding cross-differences differ from each other by at most 10ı as above. We do not exclude the case when one of the quadruple points coincides with !, for example, D !. In this case, all members of the triple A contain a summand 1, and because the cross-difference depends only on differences of the members of A, we simply cancel out these (1)-summands. So we have cd!;o .Q/ D max .a a0 /; 0 a;a 2A

where A D ..˛jˇ/!;o ; .˛j /!;o ; .ˇj /!;o /. In this case, the cross-difference becomes the ordinary difference. Alternatively, fix ! 2 @1 X and a Busemann function b 2 B.!/. Given a quadruple Q of distinct points ˛, ˇ, , 2 @1 X , we form the cross-difference triple A D ..˛jˇ/b C . j /b ; .˛j /b C .ˇj /b ; .˛j /b C .ˇj /b / and define the crossdifference based at b of the quadruple by cdb .Q/ D max .a a0 /: 0 a; a 2A

46


It follows from Chapter 3, equation (3.4) that for b D b!;o 2 B.!/ the approximate equality : cdb .Q/ D cd!;o .Q/ holds up to an error 8ı for each quadruple Q of distinct points in @1 X . Proposition 4.5.1. Let X be a ı-hyperbolic space. For every quadruple Q of distinct points in @1 X , the cross-difference triples based at points o 2 X or at Busemann functions b 2 B.!/, where ! 2 @1 X , differ from each other by a constant up to a uniform error cı for some c > 0. In particular, the corresponding cross-differences differ from each other by at most cı for every quadruple of distinct points in @1 X . Proof. The proof is straightforward using Theorem 4.2.1, Lemma 2.2.2 and Lemma 3.2.4. We leave details as an exercise to the reader. In the case X is boundary continuous, all cross-differences at infinity defined above coincide. Example 4.5.2. Assume that X is a tree. Then for every quadruple Q of distinct points in @1 X , the cross-difference cd.Q/ is equal to the maximal distance between infinite geodesics in X representing opposite edges of the (abstract) tetrahedron with the quadruple vertices, cf. Example 4.2.2.

4.5.3 Cross-pairs Let Q be a quadruple of points in a metric space X , and let A D A.Q/ be its cross-difference triple (with respect to a base point o 2 X). Every member a 2 A corresponds to a pair of opposite edges of Q considered as an abstract tetrahedron. Such a pair is called an (additive) cross-pair of Q if a is maximal (this is independent of the base point o). This notion is motivated by the obvious fact that for any a; a0 2 A with cd.Q/ D a a0 , the element a defines a cross-pair of Q. In general, even a strongly PQ-isometric map may not preserve cross-pairs. Here is an example. Example 4.5.3. Consider a map f W R2 ! R2 which is the identity outside of a ball of radius " 2 .0; 1/ around the origin but moves the origin, f .0; 0/ D ."=2; 0/. That is, f is a perturbation of the identity map. The identity map is certainly strongly PQ-isometric. Since the property to be strongly PQ-isometric is a coarse property, f is also strongly PQ-isometric. Now take a large D, consider the quadruple Q0 D .o; x; y; z/ with o D .0; 0/, x D .D; 0/, y D .0; D/, z D .D; D/, and define Q to be the deformed quadruple, where we move the vertex x to x 0 D xpC . ; 0/ with ". One easily computes that the cross-difference, cd.Q0 / D . 2 1/D, is large; however, there are two cross-pairs of Q0 , namely, .xz; oy/ and .yz; ox/. The perturbation removes this degeneration, and there is only one cross-pair of Q, namely .x 0 z; oy/. The cross-pair

47


of f .Q/ is also unique; however, it is now .f .y/f .z/; f .x 0 /f .o// because ". Therefore, f does not preserve cross-pairs. This phenomenon is related to the fact that R2 is not hyperbolic. In the case the target space is hyperbolic, we have a remarkable addition to Lemma 4.2.3. Proposition 4.5.4. Assume that a map f W X ! X 0 is strongly .c; d /-PQ-isometric and the space X 0 is ı 0 -hyperbolic (we do not require that X is hyperbolic). Then f is a PQ-isometric map preserving cross-pairs of quadruples Q X with cd.Q/ > c.d C ı 0 /. Proof. It follows from Lemma 4.2.3 that f is PQ-isometric. Consider a quadruple Q D .x; y; z; u/ X . Assume that cd.Q/ > c.d C ı 0 / and cd.Q/ D hx; y; z; ui. Then hx 0 ; y 0 ; z 0 ; u0 i

1 c

cd.Q/ d > ı 0 :

Therefore, cd.Q0 / D hx 0 ; y 0 ; z 0 ; u0 i because the cross-difference triple A0 of Q0 D f .Q/ is a ı 0 -triple. We conclude that f preserves cross-pairs of Q. Combining this with Theorem 4.4.1, we obtain Corollary 4.5.5. Every quasi-isometric map f W X ! X 0 between hyperbolic geodesic spaces is PQ-isometric preserving cross-pairs of quadruples Q X with quantitatively large cross-difference cd.Q/. Remark 4.5.6. Vice versa, one can show that any PQ-isometric map f W X ! X 0 between hyperbolic spaces (not necessarily geodesic), which preserves cross-pairs of quadruples Q X with sufficiently large cd.Q/, is strongly PQ-isometric. We leave this as an exercise to the reader. Bibliographical note. It follows from Theorem 4.4.1 and Corollary 4.3.3 that every quasi-isometric map f W X ! Y between hyperbolic geodesic spaces naturally induces a map between their boundaries at infinity. This extension property was discovered by V. A. Efremovich and E. S. Tihomirova in the case X D Y D Hn ; see [ET]. Our approach to the extension property of quasi-isometric maps of hyperbolic geodesic spaces via strongly PQ-isometric maps, Theorem 4.4.1, is similar to that from [BoS, Proposition 5.5] and [V2, Theorem 3.21], where, however, neither strongly PQ-isometric, nor PQ-isometric maps are explicitly introduced. The second example of Remark 4.1.3 (2) is taken from [V2, Remark 3.19].

Chapter 5

Quasi-Möbius and quasi-symmetric maps

The goal of this chapter is to study properties of maps between boundaries at infinity of hyperbolic spaces induced by PQ-isometric maps between the spaces themselves. In other words, we generalize the classical result that every isometry of Hn induces a Möbius map of @1 Hn to arbitrary hyperbolic spaces.

5.1 Cross-ratios A cross-ratio is the multiplicative version of a cross-difference. Let X be a ıhyperbolic space. We fix a > 1 and define the cross-ratios cr o .Q/ D a cdo .Q/ for a base point o 2 X and cr b .Q/ D a cdb .Q/ for a Busemann function b 2 B.!/, ! 2 @1 X, where Q is a quadruple of distinct points in @1 X . These cross-ratios may depend on the choice of o or b respectively. However, by Proposition 4.5.1, such a dependence is completely controlled by the hyperbolicity constant ı. More generally, assume that .Z; / is a quasi-metric space with infinitely remote set Z1 Z, jZ1 j 1. Definition 5.1.1. Given a quadruple Q of distinct points a; b; c; d 2 Z, we call the triple M D . .a; b/ .c; d /; .a; c/ .b; d /; .a; d / .b; c//; formed by the products of distances attached to pairs of opposite edges of Q, the cross-ratio triple of Q, and define the cross-ratio cr .Q/ D

min m=m0 :

m;m0 2M

Speaking about the cross-ratio of a quadruple Q of points in a quasi-metric space .Z; / we always mean this notion with respect to and usually omit from its notation, cr.Q/ D cr .Q/. In the case when one of the quadruple points is at infinity, every member of M contains an infinite factor. In this case we cancel out every such factor, e.g., if d 2 Z1 then we put M D . .a; b/; .a; c/; .b; c// and define the cross-ratio cr.Q/ as above. The cross-ratio then becomes the ordinary ratio. The cross-ratio triple M from Definition 5.1.1 possesses a remarkable property which is the multiplicative version of the property to be a ı-triple. We say that a triple

50

Chapter 5. Quasi-Möbius and quasi-symmetric maps

M D .a; b; c/ of positive reals is a multiplicative K-triple, where K 1, if the two largest members of M , say a and b, coincide up to a multiplicative error K, 1 a K: K b As a shorthand for this, we use the notation a b up to a multiplicative error K, or a K b. For example, given distinct points a, b, c in a K-quasi-metric space .Z; /, the triple M D . .a; b/; .b; c/; .a; c// is a multiplicative K-triple. This follows from the K-ultra-metric triangle inequality for . In particular, if is a metric, then M is a multiplicative 2-triple, and if is an ultra-metric, then M is a multiplicative 1-triple. There is a multiplicative analog of the Tetrahedron Lemma. Lemma 5.1.2. Assume that is a K-quasi-metric on Z, K 1. Then for every quadruple Q of distinct points of Z, the cross-ratio triple M of Q is a multiplicative K 2 -triple. Proof. If one of the quadruple points is in Z1 , then M is a multiplicative K-triple (after we cancel out the infinite factors). Thus, we assume the quadruple contains no point at infinity. The numbers .ajb/ D ln .a; b/, a; b 2 Z, satisfy the ı-inequality with ı D ln K. Applying the Tetrahedron Lemma, we see that A D ln M is a 2ı-triple. Hence, M is a multiplicative K 2 -triple.

5.2 Quasi-Möbius and quasi-symmetric maps In classical hyperbolic geometry, a Möbius map is a composition of finitely many y n D Rn [ 1, R y n D @1 HnC1 . Such a inversions of the extended Euclidean space R map is characterized by the property to preserve the classical cross-ratio; see Theorems A.7.1 and A.7.2 in the appendix. We extend the notion of a Möbius map as follows.

5.2.1 Quasi-Möbius and PQ-Möbius maps Let Q be a quadruple of distinct points in a quasi-metric space Z and let M D M.Q/ be its cross-ratio triple. Recall that every member m 2 M corresponds to a pair of opposite edges of Q. A (multiplicative) cross-pair of Q is the pair of opposite edges of Q selected by a minimal element of M . We use the notation cp.Q/ for a cross-pair of Q even in the case that the cross-pair is not unique. Remark 5.2.1. In Chapter 4 we introduced the notion of an additive cross-pair; see Section 4.5.3. It is always clear from the context whether we speak about the additive or the multiplicative cross-pair. If we consider quadruples of points in a (hyperbolic)

5.2. Quasi-Möbius and quasi-symmetric maps

51

space, we take the additive viewpoint. If we consider quadruples in the boundary at infinity, then we take the multiplicative viewpoint. By Lemma 5.1.2, there is K 1 such that M.Q/ is a multiplicative K 2 -triple for every quadruple Q of distinct points in Z. Thus if cr.Q/ < K 2 then the cross-pair of Q is uniquely determined. Definition 5.2.2. An injective map f W .Z; / ! .Z 0 ; 0 / between quasi-metric spaces is said to be quasi-Möbius if there is a homeomorphism W Œ0; 1/ ! Œ0; 1/ such that 1 cr.f .Q// .cr.Q// .1= cr.Q// for every quadruple Q of distinct points in Z. In this case, we say that f is -quasi-Möbius, the function is called the control function of f . Definition 5.2.3. The map f is called strictly quasi-Möbius if in addition f eventually preserves cross-pairs, that is, there is a constant h 2 .0; 1/ such that cp.f .Q// D f .cp.Q// for every quadruple Q Z of distinct points with cr.Q/ h. In this case, we say that the constant h is a threshold constant. Remark 5.2.4. The cross-pair condition in Definition 5.2.3 is motivated by the fact that the boundary map of every strongly PQ-isometric map between hyperbolic spaces satisfies it; see Proposition 5.2.10. Furthermore, this condition is crucial for the proof of the extension theorems in Chapter 7, proving that every quasi-symmetric or quasi-Möbius map between uniformly perfect metric spaces is the boundary map of a quasi-isometric map between appropriate hyperbolic geodesic spaces. Remark 5.2.5. Consider the following example. Take a four-point space Z D fx; y; z; ug with distances jxyj D jzuj D l and jxzj D jxuj D jyzj D jyuj D L for some l, L with L > l > 0. This defines a metric on Z. Next, take Z 0 D fx 0 ; y 0 ; z 0 ; u0 g with distances jx 0 z 0 j D jy 0 u0 j D l, and all other distances equal to L. Now define f W Z ! Z 0 by f .x/ D x 0 , f .y/ D y 0 , f .z/ D z 0 , f .u/ D u0 . Then cr.Q0 / D cr.Q/ D .l=L/2 for the unique quadruple Q D Z of distinct points in Z, Q0 D f .Q/, while cp.Q/ D .xy; zu/ and cp.Q0 / D .x 0 z 0 ; y 0 u0 /. That is, f does not preserve the (unique) cross-pair of Q, see Figure 5.1. Nevertheless f is strictly quasi-Möbius (by a purely logical reason). Moreover, we prove in Section 5.3.3 that any quasi-Möbius map of a uniformly perfect space is strictly quasi-Möbius; see Proposition 5.3.7. Exercise 5.2.6. Show that every quasi-Möbius map f W .Z; / ! .Z 0 ; 0 / is continuous. Clearly, the inverse to a (strictly) quasi-Möbius map is (strictly) quasi-Möbius and the composition of (strictly) quasi-Möbius maps is (strictly) quasi-Möbius.

52


x0 z0 x

z

y

u y 0 u0

Figure 5.1. The cross-pair is not preserved. 1

Definition 5.2.7. For the control function .t / D q maxft p ; t p g with p, q 1, a (strictly) quasi-Möbius map is called (strictly) power quasi-Möbius or (strictly) PQMöbius. That is, an injective map f W .Z; / ! .Z 0 ; 0 / is (strictly) PQ-Möbius if 1 1 cr.Q/p cr.f .Q// q cr.Q/ p q for every quadruple Q of distinct points in Z (and if f eventually preserves crosspairs). In this case, we say that f is (strictly) .p; q/-PQ-Möbius. This definition is motivated by the fact that any (strongly) PQ-isometric map of hyperbolic spaces induces a (strictly) PQ-Möbius map of their boundaries at infinity; see Proposition 5.2.10. Let X be a hyperbolic space and let b be either a point in X or a Busemann function in B.!/, where ! 2 @1 X . Recall that to a parameter a > 1, we associate a quasi-metric b on @1 X , b .˛; ˇ/ D a.˛jˇ /b , based at b. We already know that quasi-metrics b , b 0 based at different points b; b 0 2 X are bilipschitz to each other, see Proposition 2.2.8. Now we are able to describe the effect which occurs in general when we change the base of b . Proposition 5.2.8. Let X be a ı-hyperbolic space. There exists a constant q 1 which depends only on ı such that for any two quasi-metrics b , b 0 on @1 X with one and the same parameter a > 1, based in X or at infinity, the identity map id W .@1 X; b / ! .@1 X; b 0 / is strictly .1; q/-PQ-Möbius. : Proof. By Proposition 4.5.1, we have cdb .Q/ D cdb 0 .Q/ up to an error cı for every quadruple Q of distinct points in @1 X . Thus 1 q

cr b .Q/ cr b 0 .Q/ q cr b .Q/;

where q D acı . Furthermore, it also follows from Proposition 4.5.1 that id eventually preserves cross-pairs. Corollary 5.2.9. Under the conditions of Proposition 5.2.8, the identity map id W .@1 X; d / ! .@1 X; d 0 / is strictly .1; q/-PQ-Möbius for any visual metrics d , d 0 on @1 X, where the constant q 1 depends on d , d 0 .

53


Proposition 5.2.10. For every (strongly) PQ-isometric map f W X ! X 0 between hyperbolic spaces, the induced map @1 f W @1 X ! @1 X 0 is (strictly) PQ-Möbius quantitatively with respect to any visual metrics on @1 X , @1 X 0 with base points in the corresponding spaces or in their boundaries at infinity. Proof. Fix o 2 X or a Busemann function b based at ! 2 @1 X . For the respective Gromov products, we use for brevity the same subscript !. Let Q D .˛; ˇ; ; / be a quadruple of distinct points in @1 X , fxi g 2 ˛, fyi g 2 ˇ, fzi g 2 , fui g 2 , Qi D .xi ; yi ; zi ; ui /. Using Lemmas 2.2.2 and 3.2.4, we obtain cd! .Q/ 4ı lim inf cd.Qi / lim sup cd.Qi / cd! .Q/ C 4ı i

i

for some ı 0 depending on the hyperbolicity constant of X (because the crossdifference of Qi is independent of a base point, we omit the subscript ! from the notations). Similar estimates hold for images Q0 , Qi0 of the quadruples. By Proposition 4.5.4, f eventually preserves cross-pairs (for cross-difference triples) if it is strongly PQ-isometric. This implies the claim. We leave details to the reader.

5.2.2 Quasi-symmetric and power quasi-symmetric maps Definition 5.2.11. A map f W X ! Y between metric spaces is called quasisymmetric if it is not constant and if there is a homeomorphism W Œ0; 1/ ! Œ0; 1/ such that from jxaj t jxbj it follows that jf .x/f .a/j .t /jf .x/f .b/j for any a, b, x 2 X and all t 0. In this case, we say that f is -quasi-symmetric. The function is called the control function of f . Exercise 5.2.12. Show that any quasi-symmetric map f W X ! Y is injective and continuous. Lemma 5.2.13. If f W X ! Y is -quasi-symmetric, then f 1 W f .X / ! X is 0 -quasi-symmetric, where 0 .t / D 1= 1 .t 1 / for t > 0. Moreover, if f W X ! Y and g W Y ! Z are f - and g -quasi-symmetric respectively, then g B f W X ! Z is . g B f /-quasi-symmetric. Proof. Assume that jxaj sjxbj. Then jxbj .1=s/jxaj, thus jx 0 b 0 j .1=s/jx 0 a0 j and jx 0 a0 j .1= .1=s//jx 0 b 0 j. Now if jx 0 a0 j < .1= .1=s//jx 0 b 0 j, then jxaj < sjxbj, since the opposite inequality would contradict the previous computation. We put t D 1= .1=s/. Then 1=s D 1 .1=t /, and s D 1= 1 .1=t /. By continuity, the inequality jx 0 a0 j t jx 0 b 0 j implies jxaj 0 .t /jxbj, where 0 .t / D 1= 1 .1=t /. The last assertion is obvious. A quasi-symmetric map is said to be power quasi-symmetric, or PQ-symmetric, if its control function is of the form .t / D q maxft p ; t 1=p g for some p, q 1.

54


Exercise 5.2.14. Show that for every strongly PQ-isometric map f W X ! X 0 between hyperbolic spaces, the induced map @1 f W @1 X ! @1 X 0 is PQ-symmetric quantitatively with respect to any visual metrics on @1 X , @1 X 0 with base points in the corresponding spaces. Quasi-symmetric and quasi-Möbius maps are closely related to each other, namely, every quasi-symmetric map is strictly quasi-Möbius (under the mild additional condition on spaces to be uniformly perfect) and every strictly quasi-Möbius map is quasi-symmetric if it preserves the infinitely remote set. In general, these classes of maps are different because quasi-symmetric maps preserve the property of subsets to be bounded, which is definitely not the case for quasi-Möbius maps. The described relation is not at all trivial, and we start with the easier implication; see the proposition below. The inverse implication is proven as a consequence of the extension theorems in Chapter 7; see Corollary 7.4.2. A remarkable fact is that quasi-symmetric and quasi-Möbius maps with general control functions turn out to be (for uniformly perfect spaces) power quasi-symmetric and power quasi-Möbius respectively. This is also a consequence of the mentioned extension theorems; see Corollaries 7.4.2 and 7.4.3. Proposition 5.2.15. Let f W Z ! Z 0 be a strictly quasi-Möbius map which preserves 0 the infinitely remote set, f .Z1 / D Z1 (we assume that Z1 is not empty). Then f is quasi-symmetric quantitatively, in particular, any strictly PQ-Möbius f is PQsymmetric if it preserves the infinitely remote set. Proof. The condition for f to be -quasi-symmetric can be written as 1 s 0 .s/ .1=s/ for all distinct x; a; b 2 Z n Z1 , where s D .x; a/= .x; b/, and the sign ‘prime’ stands for the image under f as usual. One easily checks that it suffices to consider the case s 1. The cross-ratio triple of the quadruple Q D .x; a; b; !/ is M D . .x; a/; .x; b/; .a; b//; where ! is the infinitely remote point of Z. Let h 2 .0; 1/ be a threshold constant of f . First, consider the case cr.Q/ h. Then s 0 1=cr.Q0 / .1= cr.Q// .1= h/ .s= h2 /; where we used the inequalities s 1, h < 1. On the other hand, s 1= cr.Q/ 1= h. Thus 1 1 1 : s 0 cr.Q0 / .1= cr.Q// .1= h/ .1=sh2 / Therefore, 1= 1 .1=s/ s 0 1 .s/ for 1 .s/ D .s= h2 /.


55

Now we consider the case cr.Q/ < h. We assume that Z and Z 0 are K- and K -quasi-metric spaces respectively. Suppose that .x; b/ is minimal in M , .x; b/

.x; a/; .a; b/. Then cr.Q/ 1=s K cr.Q/ (the right-hand inequality holds because M is a multiplicative K-triple). Furthermore, f preserves the cross-pair of Q, which means that 0 .x 0 ; b 0 / is minimal in the triple M 0 . Thus cr.Q0 / 1=s 0 K 0 cr.Q0 / because M 0 is a multiplicative K 0 -triple. We obtain 0

s 0 1= cr.Q0 / .1= cr.Q// .Ks/ 2 .s/ and s0

1 1 1 1 0 0 K 0 cr.Q0 / K .cr.Q// K .1=s/ 2 .1=s/

for 2 .s/ D K 0 .Ks/. It remains to consider the case that .a; b/ is minimal in M . Then 0 .a0 ; b 0 / is minimal in M 0 , and we have 1 s K, 1=K 0 s 0 K 0 . It follows that s 0 K 0 .s/= .1/ 3 .s/ and s0

1 .K=s/ .1/ 1 0 D 0 K .K=s/ K .K=s/ 3 .1=s/

for 3 .s/ D K 0 .Ks/= .1/. Putting D maxf 1 ; 2 ; 3 g, we obtain that f is -quasi-symmetric. Remark 5.2.16. This proposition does not cover the case that the spaces Z, Z 0 are bounded, even if we artificially extend the Q-metrics , 0 to Z [ f1g, Z 0 [ f1g respectively and put f .1/ D 1. The problem is that there is no obvious reason for the extended f to be still quasi-Möbius. We solve this problem in Chapter 7 for the case that , 0 are uniformly perfect metrics. The converse, that any quasi-symmetric homeomorphism between uniformly perfect metric spaces is strictly quasi-Möbius, is shown in Chapter 7, Corollary 7.4.2. Theorem 5.2.17. Let f W X ! Y be a quasi-isometric map of hyperbolic geodesic spaces. Then f naturally induces a well-defined map @1 f W @1 X ! @1 Y of their boundaries at infinity which is (1) strictly PQ-Möbius with respect to any visual metrics on @1 X , @1 Y with base points in X , Y or in @1 X , @1 Y respectively, and (2) PQ-symmetric with respect to any visual metrics with base points in X , Y or with base points ! 2 @1 X , @1 f .!/ 2 @1 Y respectively. Proof. By Theorem 4.4.1, f is strongly PQ-isometric (quantitatively). Now using Proposition 5.2.10, we obtain (1). The first part of (2) follows from Exercise 5.2.14. The second part of (2) follows from Proposition 5.2.15 because in this case the strictly PQ-Möbius @1 f preserves the infinitely remote points by the assumption.

56


Corollary 5.2.18. The hyperbolic spaces Hn , Hm , n, m 2, are not quasi-isometric for n 6D m. Proof. Indeed, any quasi-isometry between Hn and Hm would induce by Theorem 5.2.17 a homeomorphism between their boundary spheres S n1 and S m1 which is impossible for n 6D m.

5.3 Supplementary results and remarks 5.3.1 Möbius structure on a quasi-metric space An injective map f W Z ! Z 0 between quasi-metric spaces is called Möbius if it preserves the classical cross-ratio. y n which preserves the crossyn ! R Exercise 5.3.1. Show that any injective map f W R y n , is ratio, that is, cr.f .Q// D cr.Q/ for every quadruple of distinct points Q R Möbius, cf. Theorem A.7.2. Clearly, the inverse to a Möbius map is Möbius and the composition of Möbius maps is Möbius. The Möbius structure on a quasi-metric space .Z; / is the set of all quasi-metrics on Z which are Möbius equivalent to . As an example consider a boundary continuous hyperbolic space, see Section 3.4.2. Proposition 5.2.8 is then refined as follows. Proposition 5.3.2. Let X be a boundary continuous hyperbolic space. Then for two quasi-metrics b , b 0 on @1 X with one and the same parameter a > 1, based in X or at infinity, the identity map id W .@1 X; b / ! .@1 X; b 0 / is Möbius; in particular,

b , b 0 are Möbius equivalent to each other. Proof. For a boundary continuous hyperbolic space, one can take c D 0 in Proposition 4.5.1, that is, for every quadruple Q of distinct points in @1 X , the cross-difference triples based at points o 2 X or at Busemann functions b 2 B.!/, where ! 2 @1 X , differ from each other by the same constant in each of their entries. This follows from Theorem 4.2.1, Lemmas 2.2.2, 3.2.4 and from properties of boundary continuous hyperbolic spaces. Thus corresponding cross-differences coincide with each other irrespectively of their bases. Therefore, the identity map id W .@1 X; b / ! .@1 X; b 0 / is Möbius. Remark 5.3.3. In the case that X is a CAT.1/-space, it follows from the results [Bou] and [FS2] that any quasi-metric b .˛; ˇ/ D e .˛jˇ /b on @1 X based either at b 2 X or at a Busemann function b 2 B.!/ with ! 2 @1 X is actually a metric, i.e., it satisfies the triangle inequality. By Proposition 3.4.2, every proper CAT.1/space X is boundary continuous. Hence, there is a Möbius structure on @1 X whose members are honest metrics. This should be compared to [Bou] where it is proven that any b , b 0 as above with b; b 0 2 X are conformal to each other.

57


Assume that .Z; / is a K-quasi-metric space with infinitely remote set Z1 with jZ1 j 1. The most important example of a quasi-metric which is Möbius to is the inversion 0 of with radius r > 0 centered at o 2 Z, o 2 Z n Z1 . We put

0 .a; b/ D

r 2 .a; b/

.o; a/ .o; b/

for every a; b 2 Z, .a; b/ ¤ .o; o/, and 0 .!; !/ D 0. In the case b D !, this means 0 .a; !/ D 0 .!; a/ D r 2 = .o; a/, and in the case b D o, we have 0 .a; o/ D 0 y n, D fog. In the case Z D R

0 .o; a/ D 1, i.e., the infinitely remote set for 0 is Z1 0 n n y !R y is the inversion with respect to we have .a; b/ D .'.a/; '.b//, where ' W R the sphere of radius r centered at o 2 Rn ; see Appendix, Section A.6.1. This justifies our terminology. Furthermore, the inversion operation is involutive in the sense that the inversion 00 of 0 with the same radius r centered at ! coincides with , 00 D . Example 5.3.4. For a hyperbolic space X , we fix o 2 X , ! 2 @1 X and consider the Gromov product .; /!;o D .j /o .!j/o .!j /o on @1 X , see Section 3.2.2. Then the (unbounded) quasi-metric !;o D a. j /!;o , a > 1, on @1 X is the inversion of the bounded quasi-metric D a. j /o on @1 X centered at ! with radius r D 1,

!;o .; / D

.; /

.; !/ . ; !/

for each ; 2 @1 X:

We generalize the notion of the inversion of as follows. We call a function W Z ! Œ0; 1 admissible if Z1 D 1 .1/ and if there is r > 0 such that

.a; b/ K 0 maxfr.a/; r.b/g and r.a/ K 0 maxf .a; b/; r.b/g, i.e., . .a; b/; r.a/; r.b// is a multiplicative K 0 -triple for some K 0 K and all distinct a; b 2 Z. This is a projective invariant, i.e., if is admissible then s is admissible for every s > 0. We call r a coefficient of . Any admissible assumes the value 0 at most once. Furthermore, if .o/ D 0 for some o 2 Z then clearly r K 0 .o; / for some r > 0, K 0 K. Example 5.3.5. The following functions are admissible: (1) .a/ D .o; a/ for some o 2 Z n Z1 . Here, r D 1 and K 0 D K. (2) .a/ D maxf1; .o; a/g for some o 2 Z n Z1 . Here again r D 1 and K 0 D K. (3) .a/ D .1 C 2 .o; a//1=2 for some o 2 Z n Z1 . In the case that is a metric, the l2 -product metric O on Z R, O2 ..a; s/; .b; t // D 2 .a; b/ C .s t /2 is a 2-quasi-metric, and .a/ D ..a; O 0/; .o; 1// is admissible with r D 1, K 0 D K D 2. In the general case, i.e., for an arbitrary K-quasi-metric we only have that is admissible with r D 1 and K 0 D 2K. Indeed, this function equals at most two times the function of (2). In the last two examples, is uniformly separated from 0 on Z, .a/ 1 for all a 2 Z.

58


0 Given an admissible function with coefficient r > 0, we put Z1 D f1 .0/g 0 0 and define -inversion W .Z Z/ n .Z1 Z1 / ! Œ0; 1 of by

.a; b/ D

.a; b/ ; .a/.b/

where we set .a; !/ D .!; a/ D r=.a/ for ! 2 Z1 , in particular, .!; !/ D 0 0 0 0. Furthermore, for o 2 Z1 , we have .a; o/ D 1 for every a 2 Z n Z1 , i.e., Z1 is the infinitely remote set for . In the case that assumes the value 0, we clearly have K 0 2 0 , where 0 is the inversion of with radius r 2 centered at o, .o/ D 0. Proposition 5.3.6. Let .Z; / be a K-quasi-metric space with infinitely remote set Z1 D f!g, jZ1 j 1. Then for every admissible function on Z, the -inversion 0 of is a K 0 2 -quasi-metric on Z with infinitely remote set Z1 D f1 .0/g and some 0 K K, which is Möbius equivalent to . Furthermore, in the case .z/ 0 > 0 for every z 2 Z, the space .Z; / is bounded, diam.Z; / rK 0 =0 , where r is a coefficient of . Proof. The function obviously satisfies conditions (1), (2), (4) of Definition 3.3.1 0 with infinitely remote set Z1 . Let Q D .a; b; c; d / be a quadruple of distinct points in Z. If ! 62 Q, then the cross-ratio triple M of Q with respect to is proportional to the cross-ratio triple M of Q with respect to . Indeed .a/.b/.c/.d /M D M . If ! 2 Q, e.g. d D !, then M D

r r . .a; b/; .a; c/; .b; c// D M: .a/.b/.c/ .a/.b/.c/

It follows that the identity map id W .Z; / ! .Z; / is Möbius. 0 D f1 .0/g Now we check condition 3.3.1 (3). If assumes the value 0 then Z1 0 consists of one point which we denote by o, .o/ D 0, and K 0 2 , where 0 is an inversion of centered at o. Thus it suffices to check (3) for the inversion 0 of . Let a, b, c 2 Z be distinct points. If one of them coincides with o, then (3) is obvious for 0 with any K 0 1. Thus we assume that a, b, c are different from o. Then the triple M 0 D . 0 .a; b/; 0 .a; c/; 0 .b; c// is proportional to the cross-ratio triple M (with respect to ) of the quadruple Q D .a; b; c; o/, M D . .a; b/ .c; o/; .a; c/ .b; o/; .a; o/ .b; c//: It follows from Lemma 5.1.2 that M is a multiplicative K 2 -triple, i.e., condition (3) is fulfilled for 0 with K 0 D K 2 . y which is the Otherwise, if does not assume the value 0, consider the space Z y disjoint union of Z and a point o, Z D Z [ fog. Since is admissible with the

59


coefficient r > 0, . .a; b/; r.a/; r.b// is a multiplicative K 0 -triple, K 0 K, for y Z/ y n .!; !/ ! Œ0; 1 as all distinct a; b 2 Z. We define the function O W .Z follows: .o; O o/ D 0, .o; O a/ D .a; O o/ D r.a/ for every a 2 Z, and O restricted to y with infinitely .Z Z/ n .!; !/ coincides with . Then O is a K 0 -quasi-metric on Z y of any quadruple remote set Z1 . Hence, by Lemma 5.1.2, the cross-ratio triple M y is a multiplicative K 0 2 -triple with respect to . O Q D .a; b; c; o/ of distinct points in Z Now for distinct a; b; c 2 Z, we consider the triple M D . .a; b/; .a; c/; .b; c//: If one of the points is !, e.g. c D !, then M D

1 . .a; b/; r.b/; r.a// .a/.b/

y, is a multiplicative K 0 -triple. Otherwise, M is proportional to M y; .a/.b/.c/M D ..c/ .a; b/; .b/ .a; c/; .a/ .b; c// D M and thus M is a multiplicative K 0 2 -triple. Therefore, condition (3) of Definition 3.3.1 holds for , and is a K 0 2 -quasi-metric in Z. Finally, if 0 then .a; b/ rK 0 =0 for every a; b 2 Z, because is admissible. Hence, the space .Z; / is bounded.

5.3.2 Stereographic projections y n, As illustration, we look at the classical stereographic projection ' W S n ! R '.x/ D

1 .x1 ; : : : ; xn / 1 xnC1

for x D .x1 ; : : : ; xnC1 /:

Sr .i / i o

yn R

Sn

Figure 5.2. The stereographic projection as an inversion.

60


y nC1 ! R y nC1 of the extended R y nC1 D RnC1 [ f1g with The inversion 'y W R p nC1 , i D .0; : : : ; 0; 1/, r D 2 (see Section A.6.1), respect to the sphere Sr .i / R restricted to the standard unit sphere S n RnC1 , coincides with the stereographic y n ! S n are Möbius maps. projection, 'jS y n D '. Thus ' as well as its inverse W R nC1 We put o D .0; : : : ; 0/ 2 R and denote by the standard metric on RnC1 , y nC1 . We use the same notation for the

.x; y/ D jx yj, canonically extended to R n nC1 y n D fxnC1 D 0g [ induced metric on S R , and for the induced metric on R nC1 y . f1g R y n ; / (the choice 1 instead Consider the following two spaces: .S n ; 12 / and .R 2 1 n of for S is motivated by the fact that 2 is a visual metric for the unit disc model of Hn1 with parameter a D e, see Section 2.4.3). The function W S n ! R, 2.x/ D .x; i/, is admissible for .S n ; 12 /, see Example 5.3.5 (1), and the funcN y n ; /, see Examy n ! R, .x/ D .1 C 2 .x; o//1=2 , is admissible for .R tion N W R ple 5.3.5 (3). y nC1 , 'y .x; y/ D .'.x/; For the pull back metric 'y on R y '.y//, y we have (cf. proof of Theorem A.7.1, p. 189) 'y .x; y/ D

r 2 .x; y/

.x; i / .y; i /

for each x; y 2 RnC1 n fig. Thus ' D . 12 / for the pull back metric ' on S n . Since 'y is involutive, 'y2 D id, we also obtain

.x; y/

.x; y/ 1 .x; y/ D D

.x; i/ .y; i/ .1 C 2 .x; o//1=2 .1 C 2 .y; o//1=2 2 for each x; y 2 Rn . Then . 12 / D N for the pull back metric . 12 / which is y n. bounded on R In the general situation, if .Z; / is a quasi-metric space with at least two points, then using Proposition 5.3.6 and admissible functions as in Example 5.3.5, we easily see that there are bounded and unbounded Möbius equivalent quasi-metrics on Z, cf. Example 5.3.4.

5.3.3 Möbius maps of uniformly perfect spaces A (quasi) metric space Z is said to be uniformly perfect if there is a constant 2 .0; 1/ so that for every x 2 Z and every r > 0, for which the set Z n Br .x/ is nonempty, we have Br .x/ n Br .x/ ¤ ;. Proposition 5.3.7. If a quasi-metric space Z is uniformly perfect, then any quasiMöbius map f W Z ! Z 0 is strictly quasi-Möbius. Before we prove this proposition, we need some additional results about quadruples of points in a quasi-metric space. Let Z be a K-quasi-metric space, and

61


Q D .x; y; z; w/ a quadruple of four distinct points in Z. We use the following notation: Let l be the length of a smallest side and L the length of a largest side of Q. Note that cr.Q/ l 2 =L2 . Assume that jxyj D l. Lemma 5.3.8. Assume that M K and that cr.Q/ < M116 . Then the four distances jxzj, jxwj, jyzj, jywj are all > M 4 l. The cross-pair is the pair .xy; zw/. Proof. Assume that one of the four distances, say jxzj is M 4 l. By the quasi-metric property jyzj M 5 l. Hence we have l jxyj; jxzj; jyzj M 5 l: Since cr.Q/ l 2 =L2 we have L M 8 l. This implies that in any triangle containing the point w 2 Q, the two sides adjacent to w are the two largest sides. Using the Kand hence the M -quasi-metric property again, this implies that L=M jwxj; jwyj; jwzj L; and hence every entry of the cross-ratio triple is bounded from below by lL=M and above by M 5 lL. Thus cr.Q/ M16 , a contradiction to the assumption. It remains to prove that .xy; zw/ is the cross-pair. Therefore we prove jxyj jzwj < jzxj jywj. We can assume (w.l.o.g) that jzxj jywj. The quasi-metric inequality applied to the triangle xyw gives jxwj M jywj M jxzj. Thus jzwj M maxfjxwj; jzxjg M 2 jxzj and therefore jxyj jzwj lM 2 jxzj lM 4 jxzj < jywj jxzj; since jywj > M 4 l by the first part of the proof. Similarly we see that jxyj jzwj < jzyj jxwj. Lemma 5.3.9. Assume that there exists M K with cr.Q/ z 2 Q n fx; yg with l jxyj; jyzj; jxzj M 5 l.

1 . M3

Then there exists

Proof. If not, then the four distances jxzj, jxwj, jyzj, jywj are all > M 4 l. Note that by the quasi-metric property at least one of these distances, say jxzj, is L=M . We now have jxzj jywj > M 3 Ll, while jxyj jzwj Ll in contradiction to the assumption. The following is an easy consequence of the definition of uniform perfectness. Lemma 5.3.10. Let Z be uniformly perfect. Then there exists a constant C such that for any given differing points x; y 2 Z there exists v 2 Z with a jxyj; jxvj; jvyj C a; for some a > 0.

62


Proof of Proposition 5.3.7. Assume that Z is uniformly perfect and that f W Z ! Z 0 is quasi-Möbius. Firstly, choose M large enough such that M maxfK 4 ; K 04 ; C g, where C is the constant of Lemma 5.3.10. Secondly, choose M 0 M large enough such that the following holds: if Q Z is a quadruple with cr.Q/ M14 , then cr.Q0 / M103 . Finally choose h small enough such that (a) h < M116 and (b) cr.Q/ < h implies cr.Q0 / < M1068 . Now assume that Q D .x; y; z; w/ is a quadruple with cr.Q/ < h, and that xy is a shortest side. We show that f preserves the cross-pair. Since cr.Q/ < M116 we see by Lemma 5.3.8 that the cross-pair of Q is .xy; zw/. Thus we have to show that the cross-pair of Q0 is .x 0 y 0 ; z 0 w 0 /. Let us assume that this is not the case. Let l 0 be the length of the shortest side in Q0 . Then (since by our choices cr.Q0 / M1068 ) by Lemma 5.3.8, neither x 0 y 0 nor z 0 w 0 is a shortest side. We can thus (without loss of generality) assume that y 0 w 0 is a shortest side and hence jy 0 w 0 j D l 0 . By Lemma 5.3.8 (applied to Q0 ) and the fact that cr.Q0 / M1068 we see that the four lengths jx 0 y 0 j, jx 0 w 0 j, jz 0 y 0 j and jz 0 w 0 j are all > M 017 l 0 . Now we choose according to Lemma 5.3.10 a point v 2 Z with a jxvj, jxyj, jvyj M a, and consider the two quadruples Qz D .z; x; v; y/ and Qw D .w; x; v; y/. Let us take a closer look at Qz . Note that by Lemma 5.3.8 (applied to Q), we have jzxj; jzyj > M 4 jxyj and hence also jzvj > M 3 jxyj. Thus the smallest side is among the sides xv, vy, yx. Thus the smallest side has an adjacent side whose length is at most M times larger. By Lemma 5.3.8 (applied now to Qz with constant M 1=4 K) this implies, that cr.Qz / M14 . In the same way we see that cr.Qw / M14 . 0 / M103 . By the choices of our constants cr.Qz0 /, cr.Qw 0 0 Let lw be the length of the smallest side of Qw , in particular lw0 l 0 D jy 0 w 0 j. Recall jx 0 w 0 j; jx 0 y 0 j > M 017 l 0 > M 05 lw0 . This implies that x 0 cannot be among the 0 three distinguished points from Lemma 5.3.9 (applied to Qw ). Thus Lemma 5.3.9 0 0 0 0 0 0 0 05 0 implies that lw jv y j; jv w j; jy w j M lw . In particular jv 0 y 0 j M 05 l 0 . Now consider the quadruple Qz and its image Qz0 . Let lz0 be the length of a smallest side of this tetrahedron. By the above lz0 jv 0 y 0 j M 05 l 0 . Since jz 0 y 0 j > M 017 l 0 we obtain by the quasi-metric inequality that jz 0 v 0 j > M 016 l 0 . In the same way we have jx 0 y 0 j > M 017 l 0 and obtain also that jx 0 v 0 j > M 016 l 0 . This implies for all four distances jz 0 y 0 j; jz 0 v 0 j; jx 0 y 0 j; jx 0 v 0 j > M 016 l 0 M 05 lz0 : This is in contradiction to Lemma 5.3.9 applied to Qz0 .

Bibliographical note. Quasi-symmetric maps in metric spaces were introduced in [TV]. Quasi-Möbius maps were introduced in [V1] in a different form via classical

63

5.4. Summary

cross-ratio. Due to symmetry properties of the classical cross-ratio, it suffices to require Œx 0 ; y 0 ; z 0 ; u0 .Œx; y; z; u/ for every quadruple of distinct points x, y, z, u. As an exercise to the reader, we suggest to check that this definition is equivalent to Definition 5.2.3.

5.4 Summary A number of different types of morphisms between hyperbolic spaces and their boundaries at infinity are introduced in this and former chapters and various relations between them are discussed. To help the reader to get a general picture, we state here the basic definitions and results in a more systematic way.

5.4.1 Tuples of points and associated quantities For a metric space X, we have the distance jxyj between two points. For a triple x; y; z 2 X, it is useful to consider the ordinary difference hx; y; zio D .xjy/o .xjz/o ; which depends on a base point o 2 X . In the case that x, y, z are distinct, we also have the ordinary ratio, jxzj ; Œx; y; z D jxyj which is the multiplicative version of the ordinary difference and typically useful for studying the boundary maps of hyperbolic spaces. In the context of hyperbolic spaces it is most important to deal with quadruples of points and their cross-differences. Given Q D .x; y; z; u/, we have the classical cross-difference hx; y; z; ui D 12 .jxzj C jyuj jxyj jzuj/; which can also be expressed via Gromov products hx; y; z; ui D .xjz/o .yju/o C .xjy/o C .zju/o this time independently of the base point o. The best way to understand how these expressions are formed is to think of Q as an abstract tetrahedron, to label its edges by distances or Gromov products of vertices, and to take the difference of two sums corresponding to pairs of opposite edges. This leads to the notion of the cross-difference triple of Q, which is formed by three sums corresponding to the three pairs of opposite edges: A D ..xjy/o C .zju/o ; .xjz/o C .yju/o ; .xju/o C .yjz/o /:

64


It is remarkable that A is always a ı-triple if X is ı-hyperbolic. This follows from Theorem 4.2.1: take o D u for example. Thus out of six possible classical crossdifferences, which are associated with unordered Q, only one, namely a maximal one, has a geometrically significant meaning, and we call it the cross-difference of Q, cd.Q/ D max .a a0 /: 0 a;a 2A

To keep track of the corresponding pair, we have introduced the notion of the cross-pair of Q, cp.Q/, which is a pair of opposite edges of Q with maximal a 2 A. Given a quasi-metric space .Z; /, we associate the classical cross-ratio Œa; b; c; d D

.a; c/ .b; d /

.a; b/ .c; d /

to every quadruple Q D .a; b; c; d / Z of distinct points (in the case one of the points is infinitely remote, we cancel out the factors containing this point and the .a;c/ cross-ratio becomes an ordinary ratio, e.g., Œa; b; c; 1 D .a;b/ ). The cross-ratio is a multiplicative version of the cross-difference and the ordinary ratio is that of the ordinary difference. These multiplicative versions naturally occur on the boundary at infinity of a hyperbolic space. Since the triple . .a; b/; .b; c/; .a; c// is a multiplicative K-triple for a Kquasi-metric , the cross-ratio triple M D . .a; b/ .c; d /; .a; c/ .b; d /; .a; d / .b; c//; formed by the products of distances attached to pairs of opposite edges of a quadruple Q Z, is a multiplicative K 2 -triple by the multiplicative version of the Tetrahedron Lemma (Lemma 5.1.2). Thus again only one out of six classical cross-ratios associated with unordered Q, a minimal one, has a geometrically significant meaning, and we call it the cross-ratio of Q, cr.Q/ D

min m=m0 :

m;m0 2M

As in the additive case, to keep track of a corresponding pair, we call the pair of opposite edges of Q selected by a minimal element of M the cross-pair of Q, cp.Q/. An essential distinction from the additive case is that the cross-ratio cr.Q/ together with the cross-pair cp.Q/ contains information basically equivalent to that encoded in the six classical cross-ratios of an unordered Q for any quasi-metric space, while for cross-differences this is only true under the assumption that the space is hyperbolic.

5.4.2 Control functions for additive quantities Now using various control functions applied to the distance and to the (classical) cross-difference, we obtain two families of classes of maps f W X ! X 0 between metric spaces labelled by Q and PQ respectively:

65

5.4. Summary

(Q) 1 .jxyj/ jx 0 y 0 j 2 .jxyj/ for every x; y 2 X ; (PQ) 1 .hx; y; z; ui/ hx 0 ; y 0 ; z 0 ; u0 i 2 .hx; y; z; ui/ for every x; y; z; u 2 X (as usual, the sign ‘prime’ stands for the image under the map). Then PQ Q because jxyj D hx; x; y; yi for every x; y 2 X . Note that introducing similar conditions for the ordinary difference, we obtain nothing new because hx; y; zio D hx; y; z; oi. Why do we stick to affine control functions? An obvious reason is that these are the simplest ones in the coarse category. Deeper and mathematically more supported is the reason discussed in Chapter 7: quasi-Möbius maps with arbitrary control functions between the boundaries at infinity of hyperbolic spaces are automatically subordinated to very special control functions, namely power functions .t / D q maxft p ; t 1=p g, which correspond to affine control functions on the level of the spaces themselves. (This kind of rigidity holds under the assumption that the boundaries are uniformly perfect, however, this assumption is not too restrictive.) Maps with PQ affine control functions, we call (strongly) PQ-isometric. Together with general affine control functions 1 .t / D 1c t d , 2 .t / D ct C d , c 1, d 0, it is useful to consider a homothety or similarity type control 1 .t / D ct d , 2 .t/ D ct C d with c > 0, in particular, an isometry type control with c D 1. We discuss the last two classes in more detail and give important applications in Chapter 7. Note that for the homothety type control, the two families coincide, Q D PQ. Thus, we further discuss general affine control functions. For classical PQ affine classes, one should use a piecewise affine control, that is

1 .t/ D minft=c; ctg d , 2 .t / D maxfct; t =cg C d , because classical crossdifferences take also negative values. For maps between hyperbolic geodesic spaces, we have the coincidence Q D PQ of the two affine control classes, which in a sense completes the picture on the level of spaces. For general hyperbolic spaces, a quasi-isometric map between them may not induce any reasonable map between their boundaries at infinity, while any map in PQ affine classes automatically induces the boundary map. Moreover, the induced map is subordinated to strong control functions applied to the ordinary ratios and/or to the cross-ratios. In other words, we have a duality for general hyperbolic spaces which generalizes the classical duality isometries of HnC1

! Möbius maps of S n D @1 HnC1 :

More precisely, we recall that the canonical quasi-metrics on @1 X for a hyperbolic X , 0

.; 0 / D a.j /o , depend on two parameters, a > 0 and o 2 X . Changing the base point o results in a bilipschitz transformation of , and changing the constant a results in taking a power of ; see Remark 2.2.4. However, replacing o by a boundary point ! 2 @1 X already leads to a Möbius transformation, see Example 5.3.4 and Proposition 5.2.8. This should always be taken into account while discussing the duality.

66


5.4.3 Control functions for multiplicative quantities and duality Using various control functions applied to the ordinary ratio and to the cross-ratio, we obtain two families of classes of maps f W .Z; / ! .Z 0 ; 0 / between quasi-metric spaces labelled by QS and QM, respectively: (QS) there is a control function W Œ0; 1/ ! Œ0; 1/, which is a homeomorphism such that Œx 0 ; y 0 ; z 0 .Œx; y; z/ for all triples of distinct x; y; z 2 Z; (QM) there is a control function W Œ0; 1/ ! Œ0; 1/, which is a homeomorphism such that 1 cr.Q0 / .cr.Q// .1= cr.Q// for all quadruples of distinct points Q Z; moreover, if cr.Q/ < h for some h 2 .0; 1/ then cp.Q0 / D cp.Q/0 for the cross-pairs. We prove in Chapter 7 that QS QM under the assumption that the spaces are uniformly perfect, see Corollary 7.4.2. On the other hand, any quasi-Möbius map (QM-map) preserving infinitely remote points is quasi-symmetric (QS-map), see Proposition 5.2.15, and any quasi-Möbius map between uniformly perfect bounded spaces is quasi-symmetric, see Corollary 7.3.14. In general, these families are distinct, QS ¤ QM, because any quasi-symmetric map transforms bounded sets into bounded sets, which is not the case for quasi-Möbius maps. We further classify these families by choosing special control functions. The most important are of power type, .t / D .t / D q maxft p ; t 1=p g with p, q 1. The reason is that the power type control functions correspond via duality to general affine type control functions for additive quantities. Moreover, it turns out that there is basically no other class inside of QS or QM different from the power control type. This is proven in Chapter 7 under the assumption that the spaces are uniformly perfect; see Corollaries 7.4.2 and 7.4.3. QS-maps or QM-maps subordinated to the power type control are called PQ-symmetric or PQ-Möbius, respectively. Now the most general analog of the classical duality for hyperbolic spaces X is as follows: PQ-isometries of X

! PQ-Möbius maps of @1 X:

The right arrow, i.e., every PQ-isometric map between hyperbolic spaces induces a PQ-Möbius map of their boundaries at infinity, is explained in this chapter, see Proposition 5.2.10. The left arrow holds under the additional assumptions that X is geodesic and that @1 X is uniformly perfect, and this is explained in Chapter 7. One can further refine the classes of PQ-symmetric and PQ-Möbius maps by taking control functions .t / D .t / D qt p with p > 0, q 1. It follows from

5.4. Summary

67

the discussion in Section 5.3.2 that if .Z; / is a quasi-metric space, then there are always quasi-metrics 1 and 2 on Z, which are Möbius equivalent to such that 1 is bounded and 2 is unbounded. The most important example with p ¤ 1 is a snow-flake transformation 7! p of a quasi-metric, which occurs e.g. as the dual to roughly homothetic maps of a hyperbolic space into itself. Snow-flake transformations play an important role in the Assouad embedding theorem; see Chapter 8.

Chapter 6

Hyperbolic approximation of metric spaces

A hyperbolic cone is a hyperbolic space with prescribed boundary at infinity. More precisely, a hyperbolic cone X over a metric space Z is a hyperbolic space, usually geodesic, whose boundary at infinity is identified with Z, @1 X D Z, and the metric of Z coincides with a visual metric on @1 X . In this chapter we introduce a special kind of hyperbolic cones called hyperbolic approximations. The construction of a hyperbolic approximation is simple and transparent, and it has many applications. The main advantage of the construction is that it directly includes the combinatorics of coverings by balls of the space in the geometry of a hyperbolic approximation.

6.1 Construction A subset V of a metric space Z is called a-separated, a > 0, if d.v; v 0 / a for any distinct v; v 0 2 V . Note that if V is maximal with this property, then the union S v2V Ba .v/ covers Z. A hyperbolic approximation of a metric space Z is a graph X which is defined as follows. We fix a positive r 1=6 which is called the parameter of X . For every k 2 Z, let Vk Z be a maximal r k -separated set. We associate with every v 2 Vk the ball B.v/S Z of radius r.v/ ´ 2r k centered at v. We consider the disjoint union V D k2Z Vk , or better the set of balls B.v/, v 2 V , as the vertex set of a graph X . Vertices v; v 0 2 V are connected by an edge if and only if they either belong x x 0 / intersect, B.v/ x x 0 / ¤ ;, to the same level, Vk , and the closed balls B.v/, B.v \ B.v or they lie on neighboring levels Vk , VkC1 and the ball of the upper level, VkC1 , is contained in the ball of the lower level, Vk . An edge vv 0 X is called horizontal, if its vertices belong to the same level, v; v 0 2 Vk for some k 2 Z. Other edges are called radial. We consider the path metric on X for which every edge has length 1. We denote by jvv 0 j the distance between points v; v 0 2 V in X , and by d.v; v 0 / the distance between them in Z. The level function ` W V ! Z is defined by `.v/ D k for v 2 Vk .

70

Chapter 6. Hyperbolic approximation of metric spaces

We often use the following: Remark 6.1.1. For every z 2 Z and every k 2 Z, there is a vertex v 2 Vk with d.z; v/ < r k . This follows from the fact that Vk is a maximal r k -separated set in Z.

6.2 Geodesics in a hyperbolic approximation Here we study the behavior of geodesics in X . First, we note that any (finite or infinite) sequence fvk g V such that vk vkC1 is a radial edge for every k and the level function ` is monotone along fvk g, is the vertex sequence of a geodesic in X. Such a geodesic is called radial. Lemma 6.2.1. For every v 2 V there is a vertex w 2 V with `.w/ D `.v/ 1 connected with any v 0 2 V , `.v 0 / D `.v/, jvv 0 j 1, by a radial edge. Furthermore d.v; w/ r k where k D `.w/. We call the vertex w a central ancestor of v. In general, a central ancestor of v may not be unique. Proof. By Remark 6.1.1, there is a vertex w 2 Vk for which the distance in Z between v and w is at most r k , d.v; w/ r k . Thus for every vertex v 0 2 VkC1 adjacent to v in X we have d.v 0 ; w/ d.v 0 ; v/ C d.v; w/ < 4r kC1 C r k : For each z 2 B.v 0 / we have d.z; w/ d.z; v 0 / C d.v 0 ; w/ < 6r kC1 C r k 2r k ; since r 1=6. Hence B.v 0 / B.w/, and wv 0 is a radial edge.

Lemma 6.2.2. For every v; v 0 2 V there exists w 2 V with `.w/ `.v/; `.v 0 / such that v, v 0 can be connected to w by radial geodesics. In particular, the space X is geodesic. Proof. Let `.v/ D k and `.v 0 / D k 0 . Choose m < minfk; k 0 g small enough such that d.v; v 0 / r mC1 . Applying Lemma 6.2.1, we find radial geodesics D 0 in X connecting v D vk and v 0 D vk0 0 revk vk1 : : : vm and 0 D vk0 0 vk0 0 1 : : : vm spectively with m-th level. It follows from the definition of radial edges that v 2 B.u/, v 0 2 B.u0 / for every vertex u 2 , u0 2 0 . Then d.v 0 ; vm / d.v 0 ; v/ C d.v; vmC1 / C d.vmC1 ; vm / 3r mC1 C r m 2r m 0 0 x m x m / \ B.v /, and the vertices vm , vm are connected by since r 1=6. Thus v 0 2 B.v a horizontal edge. Applying Lemma 6.2.1 once again, we find w 2 Vm1 connected 0 with vm , vm by radial edges. Therefore v, v 0 are connected to w by radial geodesics, and X is connected. This implies that X is geodesic because distances between vertices take integer values.

6.2. Geodesics in a hyperbolic approximation

71

Lemma 6.2.3. Assume that jvv 0 j 1 for vertices v, v 0 of one and the same level, `.v/ D `.v 0 /. Then jww 0 j 1 for any vertices w, w 0 adjacent to v, v 0 respectively and sitting one level below. x x x 0 / intersect since they contain the intersecting balls B.v/, Proof. The balls B.w/, B.w 0 x B.v / respectively. From Lemma 6.2.3, we immediately obtain: Corollary 6.2.4. For any two radial geodesics , 0 X with common ends, the distance in X between vertices of and 0 of the same level is at most 1. Lemma 6.2.5. Any two vertices v; v 0 2 V can be joined by a geodesic D v0 : : : vnC1 such that `.vi / < maxf`.vi1 /; `.viC1 /g for all 1 i n. Proof. Let n D jvv 0 j 1. Consider Pn a geodesic D v0 : : : vnC1 from v0 D v 0 to vnC1 D v such that . / D iD1 `.vi / is minimal. We claim that has the desired properties. Let 1 i n, and let k D `.vi /. Consider the sequence .`.vi1 /; `.vi /; `.viC1 //. There are nine combinatorial possibilities for this sequence. To prove the result, it remains to show that the sequences .k 1; k; k 1/, .k; k; k/, .k 1; k; k/ and .k; k; k 1/ cannot occur. If the sequence is .k 1; k; k 1/, then jvi1 viC1 j 1 by Lemma 6.2.3, in contradiction to the fact that is a geodesic. In the case .k; k; k/ Lemma 6.2.1 implies the existence of w 2 Vk1 with jvi1 wj 1 and jviC1 wj 1. Replacing the string vi1 vi viC1 by vi1 wviC1 we obtain a new geodesic 0 between v, v 0 with . 0 / < . / in contradiction to the choice of . The two last cases are symmetric and we consider only the case .k 1; k; k/. Choose similar as above w 2 Vk1 with jviC1 wj 1. Then jvi1 wj 1 by Lemma 6.2.3. Again vi1 wviC1 defines a geodesic with smaller . From this we easily obtain the following Lemma 6.2.6. Any vertices v; v 0 2 V can be connected in X by a geodesic which contains at most one horizontal edge. If there is such an edge, then it lies on the lowest level of the geodesic. The following corollary is useful in many circumstances. Corollary 6.2.7. Assume that for some v; v 0 2 V the balls B.v/, B.v 0 / intersect. Then jvv 0 j j`.v/ `.v 0 /j C 1. Proof. We can assume that `.v/ `.v 0 /. For every vertex w 2 V of a radial geodesic descending from v we have B.v/ B.w/; in particular, if `.w/ D `.v 0 / then jwv 0 j 1. It follows that v 0 is the lowest vertex of a geodesic v 0 v X as in Lemma 6.2.6, hence the claim.

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We use the following terminology. Let V 0 V be a subset. A point u 2 V is called a cone point for V 0 if `.u/ inf v2V 0 `.v/ and every v 2 V 0 is connected to u by a radial geodesic. A cone point of maximal level is called a branch point of V 0 . By Lemma 6.2.2, for any two points v; v 0 2 V there is a cone point. Thus every finite V 0 possesses a cone point and hence a branch point. Corollary 6.2.8. Let v; v 0 2 V and let w be a branch point for fv; v 0 g. Then .vjv 0 /w 2 f0; 12 g, in particular jvv 0 j jvwj C jwv 0 j 1. Proof. Let u be any cone point of fv; v 0 g. Then jvuj D `.v/ `.u/ and jv 0 uj D `.v 0 / `.u/ and hence 2.vjv 0 /u D `.v/ C `.v 0 / 2`.u/ jvv 0 j: In particular for different branch points w1 ; w2 of fv; v 0 g the corresponding Gromov products coincide since `.w1 / D `.w2 /. Therefore it suffices to construct a branch point with this property. By Lemma 6.2.6 there is a geodesic between v and v 0 having at most one horizontal edge. If there are no horizontal edges, we pick w 2 V , which is the lowest level vertex of that geodesic. Clearly w is a branch point, and .vjv 0 /w D 0. If there is one horizontal edge, then by Lemma 6.2.1 there is w 2 V such that its level is one less than that of the vertices of the horizontal edge, and w is connected by radial edges with these vertices. Thus w is a cone point with .vjv 0 /w D 12 and therefore also a branch point. From Lemma 6.2.6 and Corollaries 6.2.4, 6.2.8 it is clear that the behavior of geodesics in X is similar to that in a tree (one should bear in mind that for every k 2 Z the subgraph Xk X spanned by Vk plays the role of a (horo)sphere in X ). Thus very likely X is hyperbolic. Proposition 6.2.9. Let v; v 0 ; v 00 2 V and let w, w 0 , w 00 be branch points for the pairs of vertices fv 0 ; v 00 g, fv; v 00 g and fv; v 0 g respectively. Let u be a cone point of fw; w 0 ; w 00 g. Then .vjv 0 /u minf.vjv 00 /u ; .v 0 jv 00 /u g ı with ı D 3=2. Proof. We will show that the numbers .vjv 0 /u , .vjv 00 /u and .v 0 jv 00 /u form a ı-triple with ı D 3=2. Note that .vjv 0 /u D .vjv 0 /w 00 C juw 00 j and corresponding equations hold for the other Gromov products. Since the terms .vjv 0 /w 00 are bounded by 1=2 due to Corollary 6.2.8, it remains to show that the numbers juwj, juw 0 j, juw 00 j form a ı-triple with ı D 1. We assume without loss of generality that juwj juw 0 j juw 00 j, and put ´ juw 0 j juwj. It suffices to show that 1. We pick vertices w1 and w10 on the radial geodesic uw 00 , for which juw1 j D juwj and juw10 j D juw 0 j. We also pick w2 on the radial geodesic uw 0 with juw2 j D juwj. Then D jw1 w10 j D jw 0 w2 j.

73

6.2. Geodesics in a hyperbolic approximation

The concatenation of the geodesics uw 0 and w 0 v is a radial geodesic from u to v. Also the concatenation of uw 00 and w 00 v is a (maybe different) radial geodesic from u to v. It follows from Corollary 6.2.4 that jw 0 w10 j 1. v v0

w 00

v 00

w10

w0

w1

w

w2

u Figure 6.1. ı-hyperbolicity.

The broken geodesic v 0 w 00 w10 w 0 v 00 has length L D jv 0 w10 j C jw10 w 0 j C jw 0 v 00 j jv 0 w10 j C 1 C jw 0 v 00 j: Furthermore we have and

jv 0 wj D jv 0 w1 j D jv 0 w10 j C jv 00 wj D jv 00 w2 j D jv 00 w 0 j C :

Putting everything together we estimate jv 0 v 00 j L jv 0 w10 j C 1 C jw 0 v 00 j D jv 0 wj C jwv 00 j C 1 2 jv 0 v 00 j C 2 2; where the last inequality comes from Corollary 6.2.8. Thus 1.

Proposition 6.2.10. A hyperbolic approximation of any metric space is a geodesic 2ı-hyperbolic space with 2ı D 3. Proof. Choose some base point x 2 V . We show that X satisfies the 2ı-inequality for the base point x. Let t , y and z be arbitrary points in V . (We use the notation as in Lemma 2.1.5 since we use similar arguments.) Choose a branch point for every

74


of the six pairs ft; xg, ft; yg, ft; zg, fx; yg, fx; zg, fy; zg, and choose a cone point u for the set of branch points. Proposition 6.2.9 implies now that the six numbers .tjx/u , .tjy/u , .tjz/u , .x; y/u , .xjz/u , .yjz/u satisfy the condition of the Tetrahedron Lemma 2.1.4 with ı D 3=2 which implies A 2ı for A D .tjy/ C .xjz/ minf.tjz/ C .yjx/; .xjt / C .yjz/g (recall that this expression is independent of a base point for the Gromov products involved). Since A D .t jy/x minf.t jz/x ; .zjy/x g (compare the proof of Lemma 2.1.4), we obtain the result.

6.3 The boundary at infinity of a hyperbolic approximation Let X be a hyperbolic approximation with parameter r 1=6 of a metric space Z. The main result of this section is that the metric d of Z is a visual metric for X under a natural identification @1 X D Z [ f1g. More precisely, we have the following result. Theorem 6.3.1. Given a complete metric space Z, its hyperbolic approximation X is a Gromov hyperbolic geodesic space, and there is a canonical identification @1 X D Z [ f1g such that the metric of Z is a visual metric on @1 X n f!g with respect to some and hence any Busemann function b 2 B.!/ and to the parameter a D 1=r, where ! 2 @1 X corresponds to the infinitely remote point 1. For the proof we need several lemmas. Lemma 6.3.2. For every z 2 Z, there is a sequence fk .z/ 2 Vk g such that d.z; k .z// < r k , in particular z 2 B.k .z//, and k .z/kC1 .z/ is a radial edge for every k 2 Z. Proof. By Remark 6.1.1, there is vk 2 Vk with d.z; vk / < r k . Then z 2 B.vk /, and for every z 0 2 B.vkC1 / we have d.z 0 ; vk / d.z 0 ; vkC1 / C d.vkC1 ; z/ C d.z; vk / < 2r kC1 C r kC1 C r k 2r k ; because r 1=6. Thus B.vkC1 / B.vk /, and vk vkC1 is a radial edge. It remains to define k .z/ ´ vk . Recall that every point 2 @1 X is represented by a sequence fxk g X which converges to infinity, see Section 2.2. Corollary 6.3.3. For every z 2 Z, .z/ D : : : 1 .z/0 .z/1 .z/ : : : is a bi-infinite geodesic line in X . Its ‘negative’ tails fk .z/ W k < 0g define one and the same point ! 2 @1 X for all z 2 Z.

6.3. The boundary at infinity of a hyperbolic approximation

75

Proof. Clearly, the ‘positive’ and ‘negative’ tails of .z/ converge to infinity. If z, z 0 2 Z then the balls of radius 2r k associated to the vertices k .z/, k .z 0 / intersect, B.k .z// \ B.k .z 0 // ¤ ;, for all k with d.z; z 0 / < r k , thus jk .z/k .z 0 /j 1. This shows that the sequences fk .z/ W k < 0g define the same point ! 2 @1 X for all z 2 Z. This point ! 2 @1 X will correspond to the point 1 under the identification @1 X D Z [ f1g. Now we fix a base point o 2 V0 which is also considered as a point of Z. The geodesic .o/ with 0 .o/ D o will serve as a reference geodesic. Lemma 6.3.4. Let v 2 V and let u be a branch point of the pair fo; vg. Then : jk .o/uj 1, where k D `.u/, k1 .o/ is a cone point of fo; vg and .vj!/o D jouj up to an error 3ı, where ı D 3=2. Proof. Since o 2 B.u/ \ B.k .o//, we have jk .o/uj 1. Since d.u; k1 .o// d.u; o/ C d.o; k1 .o// 2r k C r k1 ; we have B.u/ B.k1 .o// which implies that k1 .o/ is a cone point of fo; vg. Let n < k. It follows from the above that jn .o/vj D jn .o/uj C juvj, jn .o/oj D : jn .o/uj C juoj and jovj jouj C juvj jovj C 1. This implies .n .o/jv/o D jouj : up to an error 1=2 D ı=3. Since limn!1 .n .o/jv/o D .!jv/o up to error 2ı we obtain the result. Lemma 6.3.5. Every sequence fvn g V that converges to infinity in X and represents a point from @1 X different from ! has a limit in Z. For any other sequence fvn0 g V converging to infinity and equivalent to fvn g we have limn vn0 D limn vn . Proof. Note that if w 2 V is a branch point for v; v 0 2 V , then both balls B.v/, B.v 0 / are contained in the ball B.w/ by the definition of radial edges. Thus d.v; v 0 / < 2r.w/, where recall r.w/ is the radius of B.w/. The assumptions imply that .vn j!/o is bounded, hence by Lemma 6.3.4 also k D inff`.vn /g is bounded. Then this lemma implies that k1 .o/ is a cone point of the set fvn g. It follows from this and from .vn jvm /o ! 1 that the level of branch points wn;m for vn , vm increases to infinity and thus r.wn;m / ! 0 as n, m ! 1. Therefore, fvn g is a Cauchy sequence if considered as a sequence in Z. Thus there is a limit limn vn 2 Z because Z is complete. A similar argument shows that if fvn0 g is equivalent to fvn g then limn vn0 D limn vn . Lemma 6.3.6. Assume that a sequence fvn g V converges to infinity in X and represents a point 2 @1 X n f!g. Let z D limn vn .see Lemma 6.3.5). Then any ‘positive tail’ of the geodesic .z/ represents the same point , fk .z/gk2N 2 .

76


Proof. It follows from the condition that for every k 2 Z, a tail of fvn g is contained in B.k .z//. Passing to a subsequence, we can assume that vk 2 B.k .z// and `.vk / k for all k 2 N. Then B.vk /\B.k .z// 6D ; for all k 2 N. By Corollary 6.2.7, there is a geodesic k .z/vk X which is either radial or the level of its unique horizontal edge is k. Then .k .z/jvk /0 .z/ k 1 ! 1; as k ! 1. This shows that the sequences fk .z/g and fvk g, k 2 N, are equivalent. Lemma 6.3.7. The map W @1 X ! Z [ f1g that associates to every 2 @1 X n ! the limit point z D ./ of a sequence fvn g 2 , considered as the sequence in Z, and .!/ D 1, is a well-defined bijection. Proof. The map is well defined by Lemma 6.3.5. We define the map W Z [f1g ! @1 X letting .1/ D ! and .z/ 2 @1 X n ! for z 2 Z be the class of the ‘positive’ tail of the geodesic .z/; see Corollary 6.3.3. This class is well defined, since if 0 .z/ is another geodesic with z 2 B.k0 .z//, then jk .z/k0 .z/j 1 for all k 2 Z, and the sequences fk .z/g, fk0 .z/g, k 0, are equivalent. Clearly, limk k .z/ D z in Z, thus is a right inverse to , B D id. By Lemma 6.3.6, B D id, i.e., and are mutually inverse bijections. From now on we identify @1 X with Z [ f1g using the bijection

.

Proof of Theorem 6.3.1. To complete the proof of Theorem 6.3.1, we show that the metric d of Z D @1 X n f!g is visual with respect to any Busemann function b 2 B.!/ and the parameter a D 1=r. Because the functions b are bilipschitz to each other for all Busemann functions b 2 B.!/, it suffices to consider the case b D b!;o ; see Definition 3.1.3. Fix distinct ; 0 2 Z and consider corresponding bi-infinite geodesics ./, . 0 / in X . For brevity, we use the notations vk D k ./, vk0 D k . 0 /, k 2 Z. Let m D maxfk 2 Z W jvk vk0 j 1g. One easily estimates that r mC1 d.; 0 / 6r m r m1 :

(6.1)

0 By Lemma 6.2.1, there is a cone point o0 for fvm ; vm g of the level `.o0 / D m 1. 0 The idea is to compute the Gromov products .vk jvk /b as k ! 1 via .vk jvk0 /b 0 for the Busemann function b 0 D b!;o0 using the approximate equality

: b!;o .v/ D b!;o0 .v/ C b! .o0 ; o/

(6.2)

up to an error 12ı for all v 2 V ; see Lemma 3.1.2 (and take into account that X is 2ı-hyperbolic).

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6.4. Supplementary results and remarks 0

`.o/ D 0

`.o0 / D m 1

`.o00 / D n

!

The point o0 is obviously the branch point for the pairs fo0 ; vk g, fo0 ; vk0 g for all k m, thus by Lemma 6.3.4, we obtain j.!jvk /o0 j, j.!jvk0 /o0 j 3ı. Furthermore, : one easily computes .!jo0 /vk , .!jo0 /vk0 D2ı k m C 1. Using b 0 .vk / D .!jo0 /vk : : .!jvk /o0 , we obtain b 0 .vk /, b 0 .vk0 / D5ı k m C 1. Since jvk vk0 j D 2.k m C 1/ up 0 to an error 1 ı, we have j.vk jvk /b 0 j 6ı. Let o00 be a branch point for the pair fo; o0 g. Then n D `.o00 / `.o/ D 0, and by Lemma 6.3.4 we have : : .!jo/o0 D3ı jo0 o00 j D m n; .!jo0 /o D3ı joo00 j D n : and therefore b! .o0 ; o/ D .!jo/o0 .!jo0 /o D m up to an error 6ı. Using equation (6.2), we obtain : .vk jvk0 /b D 12 .b.vk / C b.vk0 / jvk vk0 j/ D12ı .vk jvk0 /b 0 C b! .o0 ; o/; : thus .vk jvk0 /b D m up to an error 24ı for all k m. Together with equation (6.1), this implies the existence of constants c1 , c2 > 0, depending only on r and ı, with 0

0

c1 a.j /b d.; 0 / c2 a.j /b for all ; 0 2 @1 X n f!g.

6.4 Supplementary results and remarks 6.4.1 Hyperbolic approximation of bounded spaces Assume now that the metric space Z is bounded and nontrivial, i.e. diam Z < 1, and contains at least two points. Then the largest integer k with diam Z < r k exists, and we denote it by k0 D k0 .diam Z; r/. Observe that if r < minfdiam Z; 1= diam Zg then k0 D 0 (the case diam Z < 1) or k0 D 1 (the case diam Z 1). Note that for every k k0 the vertex set Vk consists of one point, and therefore contains no essential information about Z. Thus we modify the graph X by

78


putting Vk D ; for every k < k0 , and call the modified graph the truncated hyperbolic approximation of Z. Clearly, all properties of the hyperbolic approximations discussed above hold as well for the truncated hyperbolic approximations. In particular, the level function ` has the unique minimum o, `.o/ D k0 , and the point o is naturally considered as the base point of X . Furthermore, o is obviously a cone point for V . The bounded version of Theorem 6.3.1 is the following: Theorem 6.4.1. Let X be a truncated hyperbolic approximation of a complete, bounded metric space Z. Then there is the canonical identification @1 X D Z under which the metric d of Z is a visual metric on @1 X with respect to the base point o of X and the parameter a D 1=r. Proof. Let Xy be the standard (non truncated) hyperbolic approximation of Z, from which X is obtained by truncation. We fix an auxiliary vertex oN of Xy with level zero, `.o/ N D 0, and consider the Busemann function b D b!;oN 2 B.!/. Now we only have to find a relation between the Gromov products .vjv 0 /b and .vjv 0 /o on the vertex set V of X. Since b!;o .v/ D .!jo/v .!jv/o D jovj for every v 2 V and b! .o; o/ N D .!jo/ N o .!jo/oN D k0 , we obtain from equation (6.2) : b.v/ D12ı b!;o .v/ C b! .o; o/ N D jovj k0 : : Thus .vjv 0 /b D .vjv 0 /o k0 up to an error 12ı for all v, v 0 2 V , which shows via Theorem 6.3.1 that 0 0 c1 a.j /o d.; 0 / c2 a.j /o for some positive constants c1 , c2 depending only on r, ı and k0 D k0 .diam Z; r/. Exercise 6.4.2. Show that the infinitely remote point 1 2 @1 X of a hyperbolic approximation of a metric space Z is isolated in @1 X if and only if Z is bounded.

6.4.2 Geodesic boundary of a hyperbolic approximation For the hyperbolic approximation of a complete bounded metric space (which is not necessarily compact), we have the following Proposition 6.4.3. Let X be a truncated hyperbolic approximation of a complete bounded space Z. Then @g X D @1 X , and for any two radial rays D o : : : vk : : : , 0 D o : : : vk0 : : : .vk , vk0 2 Vk / in X , representing the same point 2 @1 X , we have jvk vk0 j 1 for all k k0 . Proof. Recall that @g X @1 X for any hyperbolic geodesic space. For the truncated hyperbolic approximation X , the equality @g X D @1 X follows from Lemma 6.3.6. x k / \ B.v x 0 / for every For the rays , 0 , we have limk vk D z D limk vk0 and z 2 B.v k k k0 . Hence the claim.


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6.4.3 Hyperbolic approximation of a compact metric space Exercise 6.4.4. Show that a hyperbolic approximation X is proper if and only if @1 X is compact.

6.4.4 The hyperbolic cone of a bounded metric space There are several constructions of a hyperbolic cone over a bounded space besides a hyperbolic approximation. Here we discuss one of them, which is useful in some circumstances. Let Z be a bounded metric space. Assuming that diam Z > 0 we put D =diam Z and note that jzz 0 j 2 Œ0; for every z, z 0 2 Z. The hyperbolic cone Co.Z/ over Z is the space Z Œ0; 1/=Z f0g with metric defined as follows. Given x D .z; t/, x 0 D .z 0 ; t 0 / 2 Co.Z/ we consider a triangle oN xN xN 0 H2 with joN xj N D t, joN xN 0 j D t 0 and the angle ]oN .x; N xN 0 / D jzz 0 j. Now we put jxx 0 j ´ jxN xN 0 j. The point o D Z f0g 2 Co.Z/ is called the vertex of Co.Z/. In the case Z is isometric to the unit standard sphere S n1 Rn (with induced intrinsic metric) the cone Co.Z/ is isometric to Hn . In the general case, we have Proposition 6.4.5. Let Z be a bounded metric space. Then the hyperbolic cone Y D Co.Z/ is a hyperbolic space which satisfies the ı-inequality with respect to the vertex o with ı D ı.H2 /. Furthermore, there is a canonical inclusion Z @1 Y , and the metric of Z is visual with respect to the base point o and the parameter a D e. Proof. Assume that .yjy 0 /o .yjy 00 /o .y 00 jy 0 /o for y; y 0 ; y 00 2 Y . We have to show that .yjy 00 /o .yjy 0 /o C ı. To this end, consider triangles oN yN yN 00 and oN yN 00 yN 0 in H2 with common side oN yN 00 separating them such that joN yj N D joyj, joN yN 0 j D joy 0 j, 00 00 00 00 00 0 00 0 joN yN j D joyN j, and jyN yN j D jyy j, jyN yN j D jy y j. Then jyy 0 j jyN yN 0 j by the triangle inequality in Z. It follows that .yj N yN 00 /oN D .yjy 00 /o , .yN 00 jyN 0 /oN D .y 00 jy 0 /o and 0 0 00 .yj N yN /oN .yjy /o . Therefore, .yjy /o .yjy 0 /o .yj N yN 00 /oN .yj N yN 0 /oN ı since the 2 ı-inequality holds for H (see Exercise 1.4.1). For every z 2 Z the ray fzg Œ0; 1/ Y represents a point from @1 Y which we identify with z. This yields the inclusion Z @1 Y . It remains to check that the metric of Z is visual. Given z; z 0 2 Z, consider the geodesic rays .t / D .z; t /, 0 .t / D .z 0 ; t / in Co.Z/. Then 2 z, 0 2 z 0 viewed as points at infinity, and for .j 0 /o D lim t!1 ..t /j 0 .t //o we have (cf. Lemma 2.2.2) .zjz 0 /o . j 0 /o .zjz 0 /o C 2ı: For comparison geodesic rays x, x0 H2 with common vertex oN and ]oN .x ; x0 / D jzz 0 j (recall D =diam Z) we have .x jx 0 /oN D . j 0 /o and .x jx 0 /oN d .x jx 0 /oN C ı, 2 0 0 where d D dist.o; N zN zN / and zN zN H is the infinite geodesic with the end points at

80


infinity zN D x.1/, zN 0 D x0 .1/. By the parallelism angle formula (see Appendix, 0j D e d , therefore, we conclude that Lemma A.3.2), we have tan jzz 4 0

e 3ı e .zjz /o tan

jzz 0 j 0 e .zjz /o 4

is uniformly bounded and separated for every z; z 0 2 Z. The function s 7! 1s tan s 4 from 0 on Œ0; diam Z. It follows that the metric on Z @1 Y is visual with respect to the vertex o 2 Y and the parameter a D e. A disadvantage of this construction is that the space Co.Z/ is not necessarily geodesic even if Z is complete. However, Co.Z/ is roughly similar to any hyperbolic approximation of Z, see the next chapter. Exercise 6.4.6. Show that if Z is bounded and complete then the boundary at infinity of Co.Z/ coincides with Z. Bibliographical note. The construction of a hyperbolic approximation of a metric space is a further development of constructions in [El] for compact subspaces of a Euclidean space and in [BP] for arbitrary compact metric spaces. Our construction differs from that of [BP] by the definition of radial edges and radii of balls, which provides some technical advantages. Theorem 6.4.1 is similar to [BP], Proposition 2.1.

Chapter 7

Extension theorems

In this chapter we prove three extension results, each saying that given a map with certain properties between the boundaries at infinity of hyperbolic spaces, there is a map in an appropriate class between the spaces themselves which induces the given map of the boundaries. The simplest case is if the boundary map is bilipschitz (with respect to visual metrics). Then the extension map is roughly homothetic. This case is most important in view of a number of applications. In the other two cases, the extension map is quasi-isometric while the boundary map is quasi-symmetric or quasi-Möbius. These results show that all asymptotic properties of a hyperbolic space are encoded in its boundary at infinity.

7.1 Extension theorem for bilipschitz maps Here we discuss refined versions of quasi-isometric maps. A map f W X ! X 0 : between metric spaces is said to be roughly homothetic if jf .x/f .x 0 /j D ajxx 0 j up to an error b for some constants a > 0, b 0, and for all x; x 0 2 X (recall our : agreement to write A D B up to an error C instead of jABj C , see Chapter 2). If in addition the image f .X / is a net in X 0 , then f is called a rough similarity. In the case a D 1, the map f is called roughly isometric and a rough isometry respectively. If there is a rough similarity (isometry) between X and X 0 , then the spaces X and X 0 are called roughly similar (isometric) to each other, and these relations are obviously equivalence relations. A hyperbolic space Y is said to be visual if for some base point o 2 Y there is a positive constant D such that for every y 2 Y there is 2 @1 Y with joyj .yj/o C D (one easily sees that this property is independent of the choice of o). For hyperbolic geodesic spaces this property is a rough version of the property that every segment oy Y can be extended to a geodesic ray beyond the end point y. Exercise 7.1.1. Show that the property to be visual is a quasi-isometry invariant of hyperbolic geodesic spaces. Show that if @1 Y consists of one point then a visual hyperbolic Y is roughly isometric to a subspace of a ray. Theorem 7.1.2. Let X be a visual and X 0 a geodesic hyperbolic space. Assume that there is a bilipschitz embedding f W .@1 X; d / ! .@1 X 0 ; d 0 / where d; d 0 are visual

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metrics with respect to base points o 2 X , o0 2 X 0 and the same parameter a. Then there exists a roughly isometric map F W X ! X 0 such that f D @1 F . In the following arguments, all Gromov products are taken with respect to the base points o 2 X , o0 2 X 0 respectively. To simplify the notation, we omit the base point and denote jxj D joxj, jx 0 j D jo0 x 0 j for x 2 X , x 0 2 X 0 respectively. The idea of the proof is easily explained in the case that every point in X and X 0 lies on rays emanating from the origin. In this case for x 2 X let 2 @1 X such that x 2 o. Let 0 D f ./ and choose a ray o0 0 X 0 . Then define x 0 D F .x/ 2 o0 0 to be the point with jx 0 j D jxj. The bilipschitz property of f implies that for x1 ; x2 2 X and for the corresponding points 1 ; 2 2 @1 X the equality : .10 j20 / D .1 j2 / holds up to a uniformly bounded error. Using this one can check that F is roughly isometric. Under the more general assumptions of the theorem we have to modify the argument. Lemma 7.1.3. Let X be a Gromov hyperbolic space satisfying the ı-inequality with respect to the base point o 2 X . Assume that jxi j .xi jzi / C D for some D 0, xi 2 X, zi 2 X [ @1 X , i D 1; 2. Then : .x1 jx2 / D minfjx1 j; .z1 jz2 /; jx2 jg up to an error D C 2ı. Proof. Applying the ı-inequality twice and using the condition .xi jzi / jxi j D, i D 1; 2, we obtain .z1 jz2 / minf.z1 jx1 /; .x1 jx2 /; .x2 jz2 /g 2ı minfjx1 j; .x1 jx2 /; jx2 jg .D C 2ı/ D .x1 jx2 / .D C 2ı/; where the last equality follows from .x1 jx2 / minfjx1 j; jx2 jg. Thus minfjx1 j; .z1 jz2 /; jx2 jg .x1 jx2 / .D C 2ı/: : Similarly we have .x1 jx2 / minfjx1 j; .z1 jz2 /; jx2 jg .D C 2ı/. Hence .x1 jx2 / D minfjx1 j; .z1 jz2 /; jx2 jg up to an error D C 2ı. Proof of Theorem 7.1.2. By the assumption there exists D > 0 such that for every point x 2 X there is a point D .x/ 2 @1 X with .xj/ jxj D. Choose such a and let 0 D f ./ 2 @1 X 0 . Choose a point z 0 2 X 0 with .z 0 j 0 / jxj, in particular, jz 0 j jxj. Let o0 z 0 be a geodesic from o0 to z 0 . Then we define x 0 D F .x/ 2 o0 z 0 to be the point with jx 0 j D jxj. Note that F .o/ D o0 .

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7.1. Extension theorem for bilipschitz maps

Since X, X 0 satisfy the ı-, ı 0 -inequalities and d , d 0 are visual metrics with respect to the base points o 2 X , o0 2 X 0 and with respect to the same parameter a, we see that c1 a.1 j2 / d.1 ; 2 / c2 a.1 j2 / for every 1 , 2 2 @1 X , and 0

0

0

0

c10 a.1 j2 / d 0 .10 ; 20 / c20 a.1 j2 / for every 10 , 20 2 @1 X 0 . Using that 1 d 0 .f .1 /; f .2 // ƒ d.1 ; 2 / ƒ for some ƒ 1, we obtain : .f .1 /jf .2 // D .1 j2 / up to an error c for all 1 ; 2 2 @1 X , where the constant c depends only on a, ƒ and ci , ci0 , i D 1; 2. Now given xi 2 X consider i D .xi / 2 @1 X , zi0 2 X 0 with .zi0 jf .i // jxi j and xi0 D F .xi / 2 o0 zi0 , i D 1; 2. By Lemma 7.1.3 we have : .x1 jx2 / D minfjx1 j; .1 j2 /; jx2 jg up to an error D C 2ı. Since jxi j D jxi0 j D .xi0 jzi0 /, we obtain .xi0 jf .i // minf.xi0 jzi0 /; .zi0 jf .i //g ı 0 D jxi0 j ı 0 : : Then again by Lemma 7.1.3 we have .x10 jx20 / D minfjx10 j; .f .1 /jf .2 //; jx20 jg up to : : an error 3ı 0 . This implies .x10 jx20 / D .x1 jx2 / and hence jx10 x20 j D jx1 x2 j up to an error c C D C 2ı C 3ı 0 . This shows that F is roughly isometric, and it easily follows from the definition that @1 F D f . Corollary 7.1.4. Under the same conditions except that d 0 is now defined with respect to a different parameter a0 , the map F is roughly homothetic. Proof. Replace X 0 by the homothetic X 0 , where D ln a= ln a0 . This allows us to replace the parameter a0 of d 0 by a0 D a, while the metric d 0 on @1 X 0 D @1 X 0 remains unchanged. Then the result follows from Theorem 7.1.2. Corollary 7.1.5. Every visual hyperbolic space X is roughly similar to a subspace of a hyperbolic geodesic space X 0 with the same boundary at infinity, @1 X 0 D @1 X . Proof. Apply Corollary 7.1.4 to a hyperbolic approximation X 0 of @1 X and the identity map id W @1 X ! @1 X 0 .

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We say that hyperbolic spaces X, X 0 have bilipschitz equivalent boundaries at infinity if for some visual metrics d on @1 X , d 0 on @1 X 0 the metric spaces .@1 X; d / and .@1 X 0 ; d 0 / are bilipschitz equivalent. Corollary 7.1.6. Visual hyperbolic geodesic spaces X and X 0 with bilipschitz equivalent boundaries at infinity are roughly similar to each other. In particular, every visual hyperbolic geodesic space is roughly similar to any hyperbolic approximation of its boundary at infinity, and any two hyperbolic approximations of a complete bounded metric space Z are roughly similar to each other. Proof. Apply Corollary 7.1.4 to the spaces X , X 0 and then exchange their roles. For the last two assertions note that every hyperbolic approximation is a visual hyperbolic space.

7.2 Extension theorem for quasi-symmetric maps Recall that a metric space X is said to be uniformly perfect if there is a constant 2 .0; 1/ so that for every x 2 X and every r > 0 we have Br .x/ n Br .x/ ¤ ; unless X D Br .x/. Recall that sometimes we use the notation a b up to a multiplicative error c or a c b instead of 1 a c c b especially in situations when the error bound c is uniformly bounded over the range of variables a, b. Theorem 7.2.1. For every quasi-symmetric homeomorphism f W Z ! Z 0 of uniformly perfect, complete metric spaces, there is a quasi-isometry of their hyperbolic approximations F W X ! X 0 which induces f , @1 F .z/ D f .z/ for every z 2 Z. Remark 7.2.2. In the case Z and therefore Z 0 are bounded we consider truncated hyperbolic approximations in the theorem above. Proof. It suffices to define F as a map F W V ! V 0 of the corresponding vertex sets. Recall that X Sis a graph with the vertex set V whose edges have length 1. The vertices from V D k Vk are the balls B.v/, v 2 Vk , of radius r.v/ D 2r k , r 1=6, and their centers Vk form a maximal r k -separated set in Z. Furthermore, we assume (without loss of generality) that the hyperbolic approximation X 0 of Z 0 is defined with the same parameter r as X . We also use notation a D 1=r. For every v 2 V there is a vertex v 0 D F .v/ 2 V 0 of highest level for which the ball B.F .v// contains f .B.v// Z 0 . This defines a map F W V ! V 0 , v 7! v 0 . The basic idea is to show that the distance jv 0 w 0 j is uniformly bounded for any neighboring v; w 2 V . From the qualitative point of view this is almost obvious.

7.2. Extension theorem for quasi-symmetric maps

85

Assuming that this is not the case, we find a neighboring v; w 2 V with arbitrarily large distance jv 0 w 0 j. Since the balls B.v/, B.w/ and hence their images under f intersect, this means that the level difference j`.v 0 / `.w 0 /j is arbitrarily large; see Corollary 6.2.7. Therefore the ratio of radii r.v 0 /=r.w 0 / and hence the ratio diam f .B.v//= diam f .B.w// is arbitrarily large (assuming that `.v 0 / `.w 0 /). Since the balls B.v/ and B.w/ have comparable diameters due to the uniform perfection condition, this is certainly incompatible with the condition that f preserves the ratio of distances with common point (up to a uniformly bounded multiplicative error). Now F is Lipschitz because the space X is geodesic. By the same reason a similarly defined map G W V 0 ! V is Lipschitz, and similar arguments show that both compositions G B F and F B G are at a finite distance from the corresponding identities. We give details for the reader interested in a quantitative proof. We assume that the space Z is -uniformly perfect, and that f is -quasi-symmetric. Given neighboring v; w 2 V , jvwj 1, we estimate the distance jv 0 w 0 j as follows. The definition of F implies that the balls B.v 0 / and B.w 0 / X 0 intersect. Thus jv 0 w 0 j j`.v 0 / `.w 0 /j C 1 by Corollary 6.2.7, and we have to estimate the difference of levels j`.v 0 / `.w 0 /j. To this end, it suffices to show that r.v 0 / r.w 0 / up to a uniformly bounded multiplicative error. We do this in two steps. First, we show that r.v 0 / is comparable with diam f .B.v//, r.v 0 / diam f .B.v// quantitatively. Clearly, diam f .B.v// 2r.v 0 /. For the opposite estimate we choose k 2 Z with r kC1 diam f .B.v// < r k . There is u0 2 Vk0 with d.u0 ; f .v// r k and hence f .B.v// B.u0 /. It follows r.v 0 / r.u0 / D 2r k D 2ar kC1 2a diam f .B.v//:

(7.1)

Second, we check that the diameters diam f .B.v//, diam f .B.w// are comparable with each other. At this point we use both the uniform perfection property of X and the quasi-symmetry property of f . It follows from jvwj 1 that j`.v/ `.w/j 1 and x x B.v/ \ B.w/ ¤ ;. Thus r.w/ ar.v/ and d.v; w/ ar.v/. For any z 2 B.w/, we have d.z; v/ d.z; w/ C d.w; v/ r.w/ C ar.v/ 2ar.v/. On the other hand, we pick vO 2 B.v/ with d.v; v/ O 0 r.v/ which is going to play the role of a reference point, 2a O v/ d.z; v/ 0 d.v; for every z 2 B.w/ (the existence of vO follows from the -perfection condition if B.v/ ¤ Z, otherwise we take vO 2 B.v/ with d.v; v/ O 14 diam B.v/ and note that r.v/ r.o/ 2a diam Z by definition of the base vertex o 2 Vk0 ; now O f .v// for c D 2a=0 , 0 D minf; 1=8ag). Hence d.f .z/; f .v// .c/d.f .v/; which implies diam f .B.w// 2 .c/d.f .v/; O f .v// 2 .c/ diam f .B.v//:

(7.2)

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This completes the proof that the distance jv 0 w 0 j is uniformly bounded and thus F is Lipschitz. Consider the inverse homeomorphism g D f 1 W Z 0 ! Z. It also induces a map G W V 0 ! V , v 0 7! v 00 . Our next goal is to show that the composition G B F W V ! V is at a finite distance from the identity. Given v 2 V , we have B.v 0 / f .B.v// and thus B.v 00 / g.B.v 0 // B.v/; in particular, the balls B.v/ and B.v 00 / intersect. Thus as above to estimate the distance jvv 00 j it suffices to show that r.v/ r.v 00 / up to a uniformly bounded multiplicative error. The inclusion above implies r.v/ 2r.v 00 /. For the opposite estimate we note that as above r.v 00 / is comparable with diam g.B.v 0 //, and by (7.1) we have d.z 0 ; f .v// 2r.v 0 / 4a diam f .B.v// for every z 0 2 B.v 0 /. Applying (7.2) with w D v we see that d.f .v/; O f .v// O f .v// with t 8a .c/. Thus diam f .B.v//=2 .c/, hence d.z 0 ; f .v// t d.f .v/; d.g.z 0 /; v/ 0 .t /d.v; O v/ 0 .t /r.v/ for every z 0 2 B.v 0 / and therefore 0 0 diam g.B.v // 2 .t /r.v/, where 0 is the control function of g. This concludes the proof that G B F is at a finite distance from the identity, and hence that F is a quasi-isometry. It remains to check that F induces the initial homeomorphism f W Z ! Z 0 extended to the infinitely remote points by f .1/ D 1 in the case Z, Z 0 are unbounded (recall that @1 X D Z [ f1g and @1 X 0 D Z 0 [ f1g in that case). We consider only the unbounded case, the bounded case is even simpler. By what we have already proven, F W X ! X 0 is a quasi-isometry of hyperbolic spaces. Hence, by Theorem 5.2.17, F induces a homeomorphism @1 F W @1 X ! @1 X 0 . Every point z 2 Z represents a point of @1 X different from 1. By identification Z [ f1g D @1 X , we have vn ! z in Z for every sequence fvn g 2 z. We choose fvn g in a way that B.vnC1 / B.vn / for every n 0, see Lemma 6.3.2. The sequence fvn0 D F .vn /g V 0 converges to infinity and thus defines a point z 0 2 @1 X 0 D Z 0 [ f1g, z 0 D @1 F .z/. Because f .B.vn // B.F .v0 // for all n 0, z 0 is distinct from 1, z 0 ¤ 1. In particular, the levels `.n/ of vn0 tend to infinity as n ! 1. Furthermore d.f .vn /; vn0 / 2r`.n/ and since f .vn / ! f .z/, we have vn0 ! f .z/ in Z 0 . By identification Z 0 [ f1g D @1 X 0 , we have z 0 D f .z/ and thus @1 F .z/ D f .z/ for every z 2 Z. Hence, @1 F D f . Combining Corollary 7.1.6 and Theorem 7.2.1 we obtain Corollary 7.2.3. Let X , X 0 be visual hyperbolic geodesic spaces such that their boundaries at infinity @1 X , @1 X 0 are uniformly perfect. Then any quasi-symmetry f W @1 X ! @1 X 0 can be extended to a quasi-isometry F W X ! X 0 .

7.3. Extension theorem for quasi-Möbius maps

87

7.3 Extension theorem for quasi-Möbius maps It is convenient to use the following agreement. Let Z be a metric space. Using the notation Z [ f!g, we assume that f!g D ;, i.e. Z [ f!g D Z, if Z is bounded, and ! D 1 is an infinitely remote point, dist.z; !/ D 1 for every z 2 Z, otherwise. Theorem 7.3.1. Let Z, Z 0 be complete, uniformly perfect metric spaces, and let f W Z [f!g ! Z 0 [f! 0 g be a quasi-Möbius bijection. Then there is a quasi-isometry F W X ! X 0 between hyperbolic approximations X , X 0 of Z, Z 0 respectively, which induces f , @1 F D f (here as usual we use the canonical identifications @1 X D Z [ f!g, @1 X 0 D Z 0 [ f! 0 g and for bounded spaces, we consider truncated hyperbolic approximations). This theorem is obtained as a combination of the quasi-symmetry extension theorem and the following inversion extension theorem.

7.3.1 Extension theorem for inversions x we denote the metric completion of a metric space Z. Let Z 0 be an unbounded By Z metric space. One can think of the map f from the following theorem as a generalized inversion centered at z0 . Theorem 7.3.2. Let f W Z n z0 ! Z 0 be a quasi-Möbius map with control function S0 , so that .t/ D qt p , p > 0, q 1, and with dense image, f .Z n z0 / D Z f .z/ ! 1 as z ! z0 for z0 2 Z and Z is a uniformly perfect metric space. Then for every 0 < r 1=6, there is a rough similarity F W X ! X 0 with coefficient p between hyperbolic approximations of Z, Z 0 respectively both with parameter r such that @1 F .z/ D f .z/ for every z 2 Z n z0 , @1 F .z0 / D ! 0 , x of Z, where we use the identification of @1 X n ! with the metric completion Z x and the identification @1 X 0 n ! 0 D Z S0 , and where we assume that @1 X n ! D Z, X is truncated in the case Z is bounded. Corollary 7.3.3. Under the conditions of Theorem 7.3.2, f extends to a -quasix [f!g ! Z S0 [f1g with fN.z0 / D 1, where f!g D ; Möbius homeomorphism fN W Z if Z is bounded, and ! D 1 otherwise. Our main tool used in the proof is triples of points, called representative, associated with vertices of X which are balls B.v/ Z, v 2 V . We say that a triple of points T Bs .z/ is -representative for the ball Bs .z/ Z, 2 .0; 1/, if (1) T is s-separated, i.e. d.t; t 0 / s for each distinct t; t 0 2 T ; (2) cr.Q/ for the quadruple Q D .T; z0 /. We first establish the existence of -representative triples.

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Lemma 7.3.4. If the metric space Z is -uniformly perfect, then every ball Bs .z/ Z with nonempty complement, Z n Bs .z/ ¤ ;, contains a -representative triple for any positive 3 =16. Proof. We put D =4 and first consider the case d.z0 ; z/ s. There are x; y 2 Z with s=2 d.z; x/ s=2 and 2 s=4 d.z; y/ s=4. Clearly, the triple T D .x; y; z/ is contained in Bs .z/ and it is s-separated. Note that diam T .C2/s=4 and dist.z0 ; T / d.z0 ; z/s=2 s=2. Thus diam T = dist.z0 ; T / . C 2/=2 and using m ´ max t2T d.z0 ; t / dist.z0 ; T / C diam T , we obtain dist.z0 ; T /=m 2=. C 4/. It follows cr.Q/

dist.z0 ; T / s ; diam T m

that is, the triple T is -representative for the ball Bs .z/. Now consider the case d.z0 ; z/ < s. By uniform perfection, there is x 2 Z with s d.z0 ; x/ < s=4. Then d.z; x/ d.z; z0 / C d.z0 ; x/ < . C 1/s=4 and therefore B .x/ Bs .z/ for D s . C 1/s=4 D .3 /s=4. Since the upper estimate . C 2/s=4 for the diameter of the triple T constructed above is < , applying the same argument to B .x/ and z0 , we find a -representative triple for the ball Bs .z/ also in this case. Assume that Z is -uniformly perfect, and that for every vertex v 2 V of the hyperbolic approximation X a -representative triple Tv B.v/ is fixed with D 3 =16 according to Lemma 7.3.4. In a sense, the triple Tv replaces or represents the ball B.v/, considered as a vertex of X, and we use the construction of the extension F similar to that in the proof of Theorem 7.2.1. The distance between vertices of X can be expressed via special quadruples which are determined using representative triples, see equation (7.3) below. On the other hand, we have a good control over the cross-ratio of quadruples under the map f , see Lemma 7.3.9, which leads directly to desired properties of the extension we construct. The technical reason why pairs of points cannot be used instead of triples for this purpose is explained in the case `.v/ `.w/ C 1 in the proof of Proposition 7.3.5. Given v; v 0 2 V , a quadruple of distinct points Q Tv [ Tv0 is said to be admissible for the triples Tv , Tv0 if it contains pairs of points from both Tv and Tv0 which form a cross-pair for Q (in multiplicative setting with respect to the metric d of Z). Furthermore, we put a D 1=r. In the following proposition, we express the distance in X between distinct vertices via cross-ratios of admissible quadruples for the corresponding representative triples, which yields therefore a key ingredient of the proof of Theorem 7.3.2. Proposition 7.3.5. There are constants ƒ0 > 1, 0 2 .0; 1/ depending only on a and (one can take aƒ0 > maxf4a6 ; 36a3 =2 g and 0 D 2 a.ƒ0 C1/ =4) such that for given vertices v, v 0 2 V , the following holds:

89


(1) if jvv 0 j ƒ0 then there is an admissible quadruple Q for the triples Tv , Tv0 and 0 (7.3) ajvv j min cr.Q/ up to a multiplicative error at most 0 ; Q

where the minimum is taken over all admissible quadruples Q for Tv , Tv0 and 0 depends only on a and (one can take 0 D maxf4a6 ; .6a/2 =2 g); (2) if there is an admissible quadruple Q for Tv , Tv0 with cr.Q/ 0 then jvv 0 j ƒ0 . Proof. We fix a branch point w 2 V for the pair fv; v 0 g and note that jvwjCjwv 0 j1 jvv 0 j jvwj C jwv 0 j while jvwj C jwv 0 j D `.v/ C `.v 0 / 2`.w/. Therefore, jvv 0 j `.v/ C `.v 0 / 2`.w/ jvv 0 j C 1. Using that 2a`.v/ D r.v/ for every v 2 V (recall that r.v/ is the radius of the ball B.v/), we obtain the following estimates which are several times used in the proof: 0

a.jvv jC1/

r.v/r.v 0 / 0 ajvv j : r.w/2

(7.4)

Furthermore, since the triples Tv , Tv0 both are contained in B.w/, we have d.t; t 0 / 2r.w/ for any t 2 Tv , t 0 2 Tv0 . This yields 0

cr.Q/ 2 r.v/r.v 0 /=.2r.w//2 2 ajvv j =4a

(7.5)

for any quadruple Q admissible for Tv , Tv0 and hence 0

ajvv j 0 cr.Q/:

(7.6)

In what follows, we assume without loss of generality that `.v/ `.v 0 /. First, consider the case `.v/ `.w/ C 2. We show that then dist.B.v/; B.v 0 // a3 r.w/: Assume to the contrary that dist.v; Q B.v 0 // < a3 r.w/ for some vQ 2 B.v/. Since k the vertex set Vk is an r -net in Z for every k, there is a vertex u 2 V of the level `.u/ D `.w/ C 2 such that dist.u; v/ Q r.u/=2. By our assumption, Q C dist.v; Q B.v 0 // < r.u/=2 C a3 r.w/ r.u/ dist.u; B.v 0 // d.u; v/ because r.u/ D a2 r.w/ 2a3 r.w/. Thus the ball B.u/ intersects both balls B.v/ and B.v 0 /. Using Corollary 6.2.8, we obtain jvv 0 j jvuj C juv 0 j `.v/ C `.v 0 / 2`.u/ C 2 D `.v/ C `.v 0 / 2`.w/ 2 < jvv 0 j; a contradiction.

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Under assumption jvv 0 j ƒ0 , we show that every quadruple Q Tv [ Tv0 formed by any distinct t1 , t2 2 Tv and distinct t10 , t20 2 Tv0 is admissible. To this end, consider the cross-ratio triple M of Q, M D .d.t1 ; t2 /d.t10 ; t20 /; d.t1 ; t10 /d.t2 ; t20 /; d.t1 ; t20 /d.t2 ; t10 //: For its first member, we have d.t1 ; t2 /d.t10 ; t20 / 4r.v/r.v 0 /, while for any other m 2 M the estimate m dist.B.v/; B.v 0 //2 a6 r.w/2 holds. We have 4r.v/r.v 0 / < a6 r.w/2 because 2a`.v/ D r.v/ and `.v/ C `.v 0 / 2`.w/ jvv 0 j ƒ0 . Hence, the cross-pair of Q is determined by the first member of M , which means that Q is admissible. Furthermore, by (7.4), 0

cr.Q/ 4r.v/r.v 0 /= dist.B.v/; B.v 0 //2 4r.v/r.v 0 /=.a3 r.w//2 4a6 ajvv j and together with (7.6), we obtain (7.3).

Tv

Tv0

l.v/ l.w/ C 2

Tv0

Tv

l.v/ l.w/ C 1

Figure 7.1. Admissible quadruples (cross-pairs are in bold).

Now consider the case `.v/ `.w/ C 1. Then r.v/ r.w/=a and thus Tv is r.w/=a-separated. Furthermore, jvwj 1 and hence jvv 0 j jvwj C jwv 0 j 1 C `.v 0 / `.w/. The condition jvv 0 j ƒ0 implies `.v 0 / `.w/ C ƒ0 1 and therefore r.v 0 / a1ƒ0 r.w/. Thus diam Tv0 2r.v 0 / r.w/=3a. Now if dist.t0 ; Tv0 / r.w/=3a for some t0 2 Tv , then for any other t 2 Tv we have dist.t; Tv0 / dist.t; t0 / dist.t0 ; Tv0 / diam Tv0 r.w/=3a: Thus in any case, there are distinct t1 ; t2 2 Tv with dist.ti ; Tv0 / r.w/=3a, i D 1; 2. We show that every quadruple Q formed by t1 , t2 and any distinct t10 ; t20 2 Tv0 is admissible. Indeed, we obtain as above that d.t1 ; t2 /d.t10 ; t20 / 4r.v/r.v 0 / while m .=3a/2 r.w/2 for any other member m of the cross-ratio triple M of Q. We have chosen ƒ0 in a way that 4r.v/r.v 0 / < .=3a/2 r.w/2 , thus Q is admissible with cr.Q/ 4.3a=/2

r.v/r.v 0 / 0 4.3a=/2 ajvv j : r.w/2


91

Together with (7.6), this implies (7.3), and (1) is proved. Finally, assume that there is an admissible quadruple Q for Tv , Tv0 with 0 cr.Q/ 0 . Using (7.5), we obtain 0 2 a.jvv jC1/ =4 and thus jvv 0 j loga .2 =40 / 1 ƒ0 . Hence, (2) holds. Remark 7.3.6. In the proof above, we only used that triples Tv , v 2 V , are r.v/separated and did not use the condition cr.Tv ; z0 / . This last condition will be used when we consider what happens with triples Tv while applying the map f to them. We define a map F W V ! V 0 by taking for every v 2 V a vertex v 0 D F .v/ 2 V 0 of highest level for which the ball B.v 0 / Z 0 contains the triple f .Tv /. Here as usual, V 0 is the vertex set of the graph X 0 . The map F is well defined (up to the distance error 1) because the singular point z0 2 Z is a member of no triple Tv , v 2 V . One of the reasons why we change the definition of F comparing with that of Theorem 7.2.1 is that for vertices v 2 V with z0 2 B.v/ the image f .B.v// is unbounded. In the next step of the proof of Theorem 7.3.2 we show that F is roughly homothetic. For a triple of distinct points T Z, we let r.T / be the minimal ratio of their distances, d.t; t 00 / r.T / D min ; d.t; t 0 / where the minimum is taken over all permutations of the points of T . Lemma 7.3.7. Under the conditions of Theorem 7.3.2, assume that a triple T Bs .z/ is -representative. Then r.f .T // 1 , where 1 2 .0; 1/ depends only on , p and q (one can take 1 D p =q). Proof. Consider the quadruple Q D .T; z0 / Z. The condition f .z/ ! 1 as z ! z0 implies r.f .T // D cr.f .Q//. Since f is -quasi-Möbius, cr.f .Q// cr.Q/p =q. On the other hand, cr.Q/ by the assumption on T . Hence, the claim. Lemma 7.3.8. For every v 2 V , the triple Tv0 D f .Tv / B.v 0 / is 0 r.v 0 /-separated, v 0 D F .v/, with 0 2 .0; 1/ depending only on a, , p, q (one can take 0 D 1 =2a, see Lemma 7.3.7). Proof. There is k 2 Z with r kC1 diam Tv0 < r k and w 2 V 0 with `.w/ D k and dist.w; Tv0 / r k . Then d.w; t / dist.w; Tv0 / C diam Tv0 < r.w/ for every t 2 Tv0 , i.e. Tv0 B.w/. It follows from the definition of F that `.v 0 / `.w/ D k. Thus for s D r.v 0 /, we have s 2ar kC1 2a diam Tv0 . Since the minimal ratio r.Tv0 / 1 by Lemma 7.3.7 and thus the triple Tv0 is .1 diam Tv0 /-separated, we see that Tv0 is 0 s-separated with 0 D 1 =2a.

92


Since the space Z is uniformly perfect, the map f is strictly quasi-Möbius by Proposition 5.3.7. In particular, there is a threshold constant h 2 .0; 1/ such that cp.f .Q// D f .cp.Q// for every quadruple Q Z of distinct points with cr.Q/ h. Moreover, f 1 is strictly quasi-Möbius as well, and we easily see that any h0 2 .0; 1/ with h0 1= .1= h/ D hp =q is a threshold constant for f 1 . Lemma 7.3.9. There is 2 2 .0; 1/ depending only on h, p and q (one can take 2 h and qp2 hp =q) such that if v; v 0 2 V satisfy the condition m D min cr.Q/ 2 ; Q

where the minimum is taken over all admissible quadruples for the triples Tv , Tv0 , then m0 D min cr.Q0 / q mp ; 0 Q

where the minimum is taken over all admissible quadruples for the triples f .Tv /, f .Tv0 /. Proof. We fix admissible quadruples Q for Tv , Tv0 and Q0 for f .Tv /, f .Tv0 / with cr.Q/ D m and cr.Q0 / D m0 respectively. Since 2 h, the map f preserves the cross-pair of Q, f .cp.Q// D cp.f .Q//. Thus f .Q/ is admissible for f .Tv /, f .Tv0 / with m0 cr.f .Q// q cr.Q/p D mp ; in particular, cr.Q0 / cr.f .Q// qp2 . Since qp2 hp =q which is a threshold constant for f 1 , the quadruple Q00 Tv [ Tv0 with f .Q00 / D Q0 is admissible for Tv , Tv0 . We have m0 D cr.Q0 / q cr.Q00 /p mp ;

hence, the claim.

We denote by ƒ00 , 00 , 00 the constants of Proposition 7.3.5 obtained from ƒ0 , 0 , 0 respectively by replacing the separation constant by the constant 0 from Lemma 7.3.8. Proposition 7.3.10. There is ƒ > 1 depending only on a, h, p, q, such that for every v; v 0 2 V with jvv 0 j ƒ the equality : jF .v/F .v 0 /j D pjvv 0 j holds up to an error (one can take ƒ ƒ0 so that 0 aƒ 2 , q0p apƒ 00 and D loga .q00 0p /). Proof. Since jvv 0 j ƒ ƒ0 , we have by Proposition 7.3.5 (1) 0

m D min cr.Q/ ajvv j Q


93

up to a multiplicative error 0 , where the minimum is taken over all admissible quadruples Q for the triples Tv , Tv0 . We have m 0 aƒ 2 by the choice of ƒ, hence by Lemma 7.3.9 m0 D min cr.Q0 / q mp ; 0 Q

where the minimum is taken over all admissible quadruples for the triples f .Tv /, f .Tv0 /. 0 0 Since mp apjvv j up to a multiplicative error 0p , we obtain m0 apjvv j p p pƒ 0 0 up to a multiplicative error q0 . Therefore, m q0 a 0 by the choice of ƒ, and jF .v/F .v 0 /j ƒ00 by Proposition 7.3.5 (2) applied for F .v/, F .v 0 / (see Remark 7.3.6). Then again by Proposition 7.3.5 (1) we have 0

ajF .v/F .v /j m0 up to a multiplicative error at most 00 ; : hence jF .v/F .v 0 /j D pjvv 0 j with D loga .q00 0p /.

By Proposition 7.3.10 the map F W X ! X 0 is roughly homothetic and thus it induces a map @1 F W @1 X ! @1 X 0 . Corollary 7.3.11. We have @1 F .z/ D f .z/ for every z 2 Z nz0 and @1 F W @1 X ! @1 X 0 is a bijection with @1 F .z0 / D ! 0 , where ! 0 2 @1 X 0 corresponds to the S0 [ f1g. infinitely remote point 1 under the identification @1 X 0 D Z Proof. The argument is actually the same as at the end of the proof of Theorem 7.2.1. Namely, given z 2 Z n z0 , the point z0 misses a closed ball Bx in Z containing z. Then f .B/ Z 0 is bounded. Choosing a sequence of vertices fvn g 2 z with B.vnC1 / B.vn / B for every n 0, we find that the sequence fvn0 D F .vn /g S0 [ f1g. Note that for the corresponding V 0 determines a point z 0 D @1 F .z/ 2 Z representative triples, we have Tvn ! z and f .Tvn / ! f .z/ as n ! 1, because f is continuous. Since f .Tvn / f .B/ for all n 0, we see that z 0 ¤ 1 and the levels `.n/ of vn0 tend to infinity as n ! 1. Thus f .Tvn / ! z 0 which implies z 0 D f .z/ 2 Z 0 , i.e., @1 F .z/ D f .z/ for every z 2 Z n z0 . S0 , there is a Cauchy sequence zn0 2 f .Z n z0 / converging to For every z 0 2 Z 0 0 0 z , zn ! z . Then the singular point z0 2 Z cannot be an accumulation point for zn D f 1 .zn0 /, since otherwise zn0 ! 1. Using that f controls the cross-ratios, we x n z0 . This easily see that fzn g is Cauchy and thus z 0 D @1 F .z/ for some z 2 Z 0 0 S means @1 F .z0 / 62 Z , thus @1 F .z0 / D ! . Finally, the metric completion f .Z n z0 / is contained in the image @1 F .@1 X / S0 D because the last one is complete and contains f .Z n z0 /. Since f .Z n z0 / D Z 0 0 0 @1 X n ! , we see that @1 X coincides with the image of @1 F and thus @1 F is bijective. To complete the proof of Theorem 7.3.2, it remains to show that F .X / is a net in X 0 . This follows from a general fact:

94


Lemma 7.3.12. Let F W X ! X 0 be a quasi-isometric map between hyperbolic approximations with bijective induced map @1 F W @1 X ! @1 X 0 . Then F is a quasi-isometry, i.e., F .X / is a net in X 0 . Proof. Given v 0 2 V 0 , consider a bi-infinite radial geodesic 0 X 0 with 0 .0/ D v 0 . Then for its ends 0 , ! 0 2 @1 X 0 , there are distinct , 2 @1 X with @1 F ./ D 0 , @1 F . / D ! 0 . Consider a bi-infinite geodesic X with end points , (the existence of is obvious in the case or coincides with ! 2 @1 X ; otherwise, there is a branch point w 2 V for , , which yields ). Now by stability of geodesics, dist.v 0 ; F .// is bounded by a constant depending only on F . Now for the proof of Theorem 7.3.1, we use as the basic tool the following: Lemma 7.3.13. Given a metric space Z and a nonisolated point z0 2 Z, there is an unbounded metric space Z0 and a quasi-Möbius map ' W Z n z0 ! Z0 with dense S0 , and with control function .t / D qt p for some p > 0, image, '.Z n z0 / D Z q 1, such that '.z/ ! 1 as z ! z0 . Proof. Let d be the metric on Z. We first consider the inversion D d 0 of d with respect to z0 , .z; z 0 / D d.z; z 0 /=.d.z0 ; z/d.z0 ; z 0 // for every z, z 0 2 Z distinct from z0 , see Section 5.3.1. By Proposition 5.3.6, is a K-quasi-metric with K D 4, and .Z; / is an unbounded quasi-metric space with infinitely remote point ! D z0 . Moreover, the identity map ' W .Z n z0 ; d / ! .Z; /, '.z/ D z, is Möbius with dense image, and '.z/ ! 1 as z ! z0 . That is, all requirements of the lemma are satisfied for .Z; /, except that might not be a metric. To obtain a metric space Z0 , we take a power of , 0 D p with a sufficiently small p 2 .0; 1/. By Proposition 2.2.6, if p 1=4, then 0 is bilipschitz to a metric d0 . Now we take as Z0 the metric space .Z n z0 ; d0 /. Then the identity map ' W Z n z0 ! Z0 is quasi-Möbius with control function .t / D qt p for some q 1, and it satisfies all requirements of the lemma. Proof of Theorem 7.3.1. If the spaces Z, Z 0 both are unbounded and f preserves the infinitely remote points, f .!/ D ! 0 , then f is a quasi-symmetry by Proposition 5.2.15, and Theorem 7.2.1 can be applied. Assume that Z, Z 0 both are bounded. We fix z0 2 Z and put z00 D f .z0 / 2 Z 0 . Applying Lemma 7.3.13, we find quasi-Möbius maps ' W Z nz0 ! Z0 , ' 0 W Z 0 nz00 ! Z00 with dense images to which in turn Theorem 7.3.2 can be applied. In that way, we find rough similarities ˆ W X ! X0 , ˆ0 W X 0 ! X00 which induce ', ' 0 , where X0 , X00 are hyperbolic approximations of Z0 , Z00 respectively. Now the spaces Z0 , Z00 are unbounded, and the composition D ' 0 B f B ' 1 W Z0 ! Z00 is quasiMöbius preserving the infinitely remote points. Applying the first case above, we find a quasi-isometry ‰ W X0 ! X00 which induces . Therefore, the quasi-isometry ˆ01 B ‰ B ˆ W X ! X 0 induces ' 01 B B ' D f . The remaining cases are similar, and we leave them to the reader as an exercise.


95

Corollary 7.3.14. Every quasi-Möbius map f W Z ! Z 0 between bounded, uniformly perfect metric spaces is power quasi-symmetric. Proof. One can assume that the spaces are complete and f is bijective. Then, by Theorem 7.3.1, f is the boundary value of a quasi-isometry between (truncated) hyperbolic approximations X, X 0 of Z, Z 0 respectively. Since any hyperbolic approximation is geodesic, f is power quasi-symmetric by Theorem 5.2.17 (2), because the metrics of Z, Z 0 being bounded are visual with respect to points inside of X , X 0 respectively.

7.4 Supplementary results and remarks 7.4.1 The hyperbolic cone It follows from the proof of Theorem 7.1.2 that the condition on the target space X 0 to be geodesic can be relaxed to the existence of o0 2 X 0 such that every x 0 2 X 0 can be connected to o0 by a geodesic segment. This is useful e.g. for the hyperbolic cone construction; see Section 6.4.4. Thus we obtain Corollary 7.4.1. Every visual hyperbolic space X is roughly similar to a subspace of the cone Co.@1 X / over its boundary at infinity taken with any visual metric. Furthermore, if X is in addition geodesic then X and Co.@1 X / are roughly similar to each other.

7.4.2 Power quasi-symmetric and quasi-Möbius embeddings It is known that a quasi-symmetric embedding of a uniformly perfect space Z is power quasi-symmetric, see e.g. [He], Theorem 11.3. It is also known that such a map is quasi-Möbius (in classical sense and without uniform perfection assumption), [V1], Theorem 3.2. Theorem 7.2.1 combined with Theorem 5.2.17 gives another proof of these facts. Corollary 7.4.2. Any quasi-symmetric map f W Z [ f!g ! Z 0 [ f! 0 g, where Z is uniformly perfect, is power quasi-symmetric and power quasi-Möbius. Proof. We can assume without loss of generality that f is surjective, hence a homeomorphism, see Exercise 5.2.12, and that Z is complete. Then Z 0 is also uniformly perfect, complete, and, furthermore, f preserves the infinitely remote point, f .!/ D ! 0 (in the case ! D ;, i.e., Z is bounded, this means that the image Z 0 is also bounded). By Theorem 7.2.1, f is the boundary value of a quasi-isometry between hyperbolic approximations of Z, Z 0 . Since the metrics of Z, Z 0 are visual for X , X 0 with appropriately chosen base points, see Theorems 6.3.1 and 6.4.1, it follows from Theorem 5.2.17 (1) that f is PQ-Möbius and from Theorem 5.2.17 (2) that f is PQsymmetric.

96


Similarly, from Theorem 7.3.1 and Theorem 5.2.17 (1), we obtain Corollary 7.4.3. Let Z, Z 0 be complete, uniformly perfect metric spaces. Then any quasi-Möbius bijection f W Z [ f!g ! Z 0 [ f! 0 g is power quasi-Möbius. Bibliographical note. There are several approaches to extension results discussed in this chapter; see [Tu], [Pa], [BoS]. The idea of our approach via hyperbolic approximations was discussed in [BP] without giving however any detail. The important Lemma 7.1.3 is taken from [BoS], Lemma 5.1.

Chapter 8

Embedding theorems

In this chapter we prove two important embedding results which have a number of applications. The second one, the Bonk–Schramm embedding theorem (Theorem 8.2.1), is an application of the first one, the Assouad embedding theorem (Theorem 8.1.1).

8.1 Assouad embedding theorem A metric space Z is said to be doubling if there is a constant M 2 N such that every ball in Z can be covered by at most M balls of the half radius. For more about doubling spaces see Section 8.3.1. If d is a metric on a space Z then d p is also a metric on Z for every p 2 .0; 1/ (cf. Remark 2.2.3). The transformation d 7! d p is sometimes called a snow-flake transformation, because any curve in Z that is rectifiable with respect to d becomes nonrectifiable with respect to d p . Theorem 8.1.1. Let .Z; d / be a doubling metric space. Then for every p 2 .0; 1/ there is a bilipschitz embedding ' W .Z; d p / ! RN , where N 2 N depends only on p and the doubling constant of the metric d . For the proof we use a hyperbolic approximation of Z. Recall that for every k 2 Z a maximal r k -separated set Vk Z is fixed, where r 1=6. The graph structure X is not used in the proof and plays only a heuristic role. Proof. There are two important ingredients of the proof. The first one is a coloring S of the vertex set V D k2Z Vk . Since Z is doubling we can find a finite set A with cardinality jAj depending only on the doubling constant of d such that for every k 2 Z there is a coloring k W Vk ! A with k .v/ ¤ k .v 0 / for any distinct v; v 0 2 Vk with d.v; v 0 / 4r k1 (see Proposition 8.3.3). Furthermore, we take a finite set B, whose cardinality will depend only on p and which we will specify later in the proof. We fix a periodic coloring W Z ! B such that .k/ D .k 0 / if and only if k and k 0 are congruent modulo jBj. Now for C D A B we define the coloring W V ! C by .v/ D .k .v/; .k// for every v 2 Vk , k 2 Z. For every color c 2 C we put V c D 1 .c/, Vkc D V c \Vk .

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Chapter 8. Embedding theorems

Given an element z in Z let us define for a vertex v 2 V the number z.v/ D maxf0; 1 d.v; z/=r.v/g 2 Œ0; 1, where r.v/ D 2r k for v 2 Vk . The “decimal” decomposition of z for a color c 2 C is now defined as the set Dc .z/ D fz.v/ W v 2 V c g: Since z.v/ D 0 for d.v; z/ 2r k and 4r k < 4r k1 , there is, by the properties of the coloring , for every level k 2 Z at most one nonzero component z.v/ of Dc .z/. Therefore, for every color c 2 C every point z 2 Z determines a sequence c .z/ D fv 2 Vkc W z.v/ ¤ 0; k 2 Zg, which can be regarded as a “geodesic” in X and which converges to z as k ! C1. Note that for different z, z 0 2 Z the “geodesics” c .z/, c .z 0 / coincide on all sufficiently large negative levels. The second important ingredient of the proof is the scaling of Dc .z/ by the factor .2r k /p for every level k 2 Z in the following definition of the map ' W Z ! RC . We fix z0 2 Z and define ' by its coordinate functions 'c W Z ! R, X .z.v/ z0 .v// r.v/p ; c 2 C: 'c .z/ ´ v2V c

The reference point z0 is introduced to guarantee that the series converges on negative levels k < 0. Indeed, jz.v/ z0 .v/j r.v/p d.z; z0 /r.v/p1 , and since p < 1, the series converges on negative levels as a geometric series. On the other hand, jz.v/ z0 .v/j r.v/p 2r.v/p , and the series also converges on positive levels as a geometric series. Clearly, for the bilipschitz property the reference point z0 plays no role. The idea is that the labelling and the scaling together allow to locate precisely the place in the hyperbolic approximation X where the “geodesics” c .z/, c .z 0 / start to diverge by comparing the scaled “decimal” decompositions of z; z 0 2 Z for some color c 2 C . This leads directly to the required bilipschitz property of the map ' with respect to the metric d p . Given different z; z 0 2 Z, we define their critical level n D n.z; z 0 / 2 Z by the condition (8.1) 3r nC1 < d.z; z 0 / 3r n : In the following estimates we put d as a short for d.z; z 0 /. For c 2 C and k 2 Z we put X c;k D c;k .z; z 0 / D .z.v/ z 0 .v//r.v/p : v2Vkc

The following calculations are elementary. As tool we will only use the standard estimates for the geometric series: for 0 b < 1 we have X km

bk

bm 1b

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8.1. Assouad embedding theorem

and for 1 < b we have

X km

bk

b mC1 : b1

We will use this series for b D r < 1 respectively for b D r .p1/ > 1. We will also use the estimate r p.nC1/ 3p r p.nC1/ d p ; (8.2) p

which follows from (8.1). Since for every k there exists at most one v 2 Vkc with z.v/ ¤ 0 and one v 0 2 Vkc with z 0 .v 0 / ¤ 0, we have jc;k j 2.2r k /p D 2pC1 r pk and hence for all m n we obtain X 2pC1 pm jc;k j r c1 .r; p/r p.mn/ d p (8.3) 1 rp km

pC1

with c1 .r; p/ D r p2.1r p / , where we used in the last inequality the estimate (8.2). If k n and d.z; v/, d.z 0 ; v 0 / 2r k for v; v 0 2 Vk , then d.v; v 0 / 4r k C d.z; z 0 / 4r k C 3r n 4r k1 and hence v and v 0 have different colors or v D v 0 . This shows that for k n the set Vkc contains at most one vertex v for which z.v/ z 0 .v/ 6D 0. Thus we estimate jc;k j .2r k /p jz.v/ z 0 .v/j 2p1 d r .p1/k 3 2p1 r n r .p1/k for some v 2 Vkc , since 2r k jz.v/ z 0 .v/j d.z; z 0 / D d and d 3r n . We obtain therefore for m n X km

jc;k j

3 2p1 r n .p1/.mC1/ c2 .r; p/r .1p/.nm/ d p ; r r p1 1

(8.4)

where c2 .r; p/ D .3=2/1p =.r p r/. Here we used again (8.2) in the last inequality. Now ' is Lipschitz because X X j'c .z/ 'c .z 0 /j jc;k j C jc;k j c3 .r; p/d p .z; z 0 / kn

knC1

for every color c, where c3 .r; p/ D c1 .r; p/r p C c2 .r; p/. To prove the bilipschitz property we consider different z; z 0 2 Z. Since the balls Brk .v/, v 2 Vk , cover Z for every k 2 Z we note that there exists a color c 2 C c such that d.v; z/ r nC1 for some v 2 VnC1 , where n D n.z; z 0 / is the critical level. We decompose X X c;k C c;nC1 C c;k (8.5) 'c .z/ 'c .z 0 / D kn

knC2

100


and show that only the middle term essentially contributes to this decomposition. From d.v; z/ r nC1 we conclude that first z.v/ 1=2 and second z 0 is not in B2r nC1 .v/ since d.z; z 0 / > 3r nC1 . Thus z 0 .v/ D 0 and we obtain c;nC1 D .2r nC1 /p z.v/ 2p1 r p.nC1/ 2p1 .r=3/p d p by the right-hand side of estimates (8.1). Using the periodicity of the coloring, we obtain X X jc;k j D jc;k j c2 .r; p/r .1p/.jBj1/ d p kn

km

for m D n C 1 jBj by (8.4), and X X jc;k j D jc;k j c1 .r; p/r p.jBjC1/ d p knC2

km

for m D n C 1 C jBj by (8.3). If we choose jBj large enough (only depending on p and r), we see that the sum of the two boundary terms in the decomposition (8.5) is bounded by 2p2 .r=3/p d p which implies the bilipschitz property of '. Remark. For p D 1=2 one can check using the expressions for c1 .r; p/, c2 .r; p/ that four periodic colors, jBj D 4, suffice for the mapping ' to be bilipschitz if r is chosen sufficiently small. In general, jBj ! 1 as p ! 0 or p ! 1.

8.2 Bonk–Schramm embedding theorem Theorem 8.2.1. Let X be a visual Gromov hyperbolic geodesic space whose boundary at infinity is doubling for some visual metric. Then there exists n 2 such that X is roughly similar to a convex subset of Hn . Let Z be a compact subset of @1 Hn , n 2. Its convex hull in Hn is the intersection of all convex Y 0 Hn with @1 Y 0 Z. Lemma 8.2.2. Given the convex hull Y of a compact Z @1 Hn , n 2, containing more than one point, we have (1) @1 Y D Z; (2) Y is visual in Hn . Proof. First of all Y ¤ ; since Z contains more than one point. (1) It follows from the definition that Z @1 Y . To prove the opposite inclusion, given z 2 @1 Hn nZ, we consider a geodesic W R ! Hn with .1/ D z. Then every hyperplane E Hn orthogonal to bounds a half-space EC Hn , whose boundary at infinity @1 EC is a neighborhood of z in @1 Hn and such neighborhoods

8.2. Bonk–Schramm embedding theorem

101

form a basis for z. Since Z is compact there is a hyperplane E D E.t / orthogonal to at some .t / with @1 EC \ Z D ;. Then for every t 0 > t the half-space E .t 0 / opposite to EC .t 0 / is a convex subset in Hn with Z @1 E .t 0 / and z 62 @1 E .t 0 /. Thus z 62 @1 Y . (2) Fix o 2 Y . It suffices to show that for every y 2 Y there is 2 Z D @1 Y with joyj .yj/o C ı, where ı is the hyperbolicity constant of Hn . Assume that this is not the case, and for some y 2 Y we have joyj > .yj/o C ı for all 2 Z. We show that there is ˛ 2 .0; =2/ such that the angle of the (infinite) triangle oy at y is at most ˛, ]y .o; / ˛, for all 2 Z. Indeed, otherwise since Z is compact there is 2 Z with ]y .o; / =2. Let z0 2 oy, y0 2 o, o0 2 y be the equiradial points of oy. Then jz0 yj jz0 o0 j ı and joyj D joz0 j C jz0 yj .yj/o C ı, which contradicts our assumption. Now we conclude that the hyperplane E Hn , orthogonal to the segment oy at some point y 0 2 oy close enough to y, bounds the half-space EC Hn whose boundary at infinity contains Z, @1 EC Z. Therefore, y 62 Y , a contradiction. Lemma 8.2.3. Given a compact Z Rn , the Euclidean metric on Z is bilipschitz to the restriction to Z of some visual metric on @1 HnC1 D Rn [ f1g, where HnC1 is considered in the upper half-space model. Proof. Let g W B nC1 ! C nC1 be the isomorphism of the unit ball and the upper halfspace models of HnC1 ; see Appendix, Section A.3 and A.5. Then g 1 restricted to any compact Z Rn D @1 C nC1 is bilipschitz with respect to the Euclidean metric on Rn and the spherical metric on S n D @1 B nC1 (the bilipschitz constant depends on Z). On the other hand, by Section 2.4.3, the spherical distance on S n is a visual metric on HnC1 with respect to the base point o and the parameter a D e. Proof of Theorem 8.2.1. We can assume that X is unbounded. Then since it is visual, @1 X ¤ ;. If @1 X consists of one point, then X is roughly isometric to a ray (see Exercise 7.1.1), and the claim is obvious. Thus we assume that @1 X contains more than one point. By Theorem 8.1.1, there is a bilipschitz embedding .@1 X; d 1=2 / ! Rn for some n 1, where d is a visual metric on @1 X . Note that taking the power p D 1=2 of d corresponds to the p-homothety of X , pX . Let Z Rn be the image of the embedding. We consider Rn as a part of the boundary at infinity Rn [ f1g of the hyperbolic space HnC1 (in the upper halfspace model). Then the convex hull Y HnC1 of Z is hyperbolic and geodesic with respect to the induced metric, and its boundary at infinity coincides with Z by Lemma 8.2.2 (1). By Lemma 8.2.2 (2), Y is visual. By Lemma 8.2.3, there is a visual metric on @1 Y bilipschitz to the metric of Z induced from Rn . Now applying Corollary 7.1.6, we see that the space 12 X and hence X is roughly similar to Y .

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Remark 8.2.4. The same argument as above (with Corollary 7.1.6 replaced by Corollary 7.1.4) shows that any visual hyperbolic (not necessarily geodesic) space X with doubling boundary at infinity is roughly similar to a subspace of a convex subset of Hn for an appropriate n 2.

8.3 Supplementary results and remarks 8.3.1 More about doubling spaces An equivalent definition of the property of Z to be doubling is that there is M 2 N such that for every r > 0 every ball of radius 2r in Z contains at most M points which are r-separated. Exercise 8.3.1. Show that this definition is equivalent to the initial one. Show that the property of a metric space Z to be doubling is equivalent to the fact that there is a function M W Œ1; 1/ ! N such that for every r > 0 every ball of radius r with

1 in Z contains at most M. / points which are r-separated. The property to be doubling is hereditary: if a metric space X is doubling, then every subspace X 0 X is doubling. The basic example is Rn and its subsets. On the other hand, any tree with uniformly bounded length of edges and degree of every vertex 3 is not doubling, as well as Hn for any n 2 and any Hadamard manifold with sectional curvatures separated from 0. Recall that a Hadamard manifold X is a simply connected, complete Riemannian manifold with nonpositive sectional curvatures. Exercise 8.3.2. Show that the property to be doubling is quasi-symmetry invariant. The degree of a vertex v of a graph X is the number of edges in X adjacent to v. Proposition 8.3.3. A metric space Z is doubling if and only if the degree of vertices of a hyperbolic approximation X of Z is uniformly bounded. Proof. Assume that the degree of vertices of X is uniformly bounded, i.e., there is a constant M 2 N such that the number of edges adjacent to any vertex of X is at most M . Consider a ball B2s .z/ Z. There is k 2 Z with r kC1 < 2s r k , where we recall r 1=6 is the parameter of X (see Chapter 6). Since the vertex set Vk Z of level k is an r k -net in Z, there is v 2 Vk with d.z; v/ r k . Then B2s .z/ B.v/, where B.v/ D B2r k .v/. For the radius 2r kC2 of every ball B.w/ Z, w 2 VkC2 , we have 2r kC2 D 2r kC1 r < 4sr < s. Furthermore, recall that the balls Br j .w/, w 2 Vj , cover Z for every j 2 Z. Thus to show that Z is doubling, it suffices to estimate the number N of w 2 VkC2 with B.w/ \ B.v/ ¤ ;. By Corollary 6.2.7, the distance in X between v and w is jvwj j`.v/ `.w/j C 1 D 3. Hence N M 3 , and Z is doubling.


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Conversely, assume that Z is doubling, and let M W Œ1; 1/ ! N be an appropriate control function, see Exercise 8.3.1. Consider a vertex v of the graph X , v 2 Vk for some k 2 Z. Every horizontal edge of X adjacent to v corresponds to a vertex w 2 Vk x \ B.w/ x with B.v/ ¤ ;. Thus w 2 Bx4r k .v/ Z and the number of horizontal edges adjacent to v is bounded above by the number of points w 2 Vk sitting in the ball Bx4r k .v/. Since the set Vk is r k -separated, the last number is at most M.4/. By a similar argument, the number of radial edges connecting v with upper level vertices is bounded above by M.2=r/. Finally, consider radial edges connecting v with lower level vertices. Every such edge corresponds to a ball B.w/ Z with w 2 Vk1 , containing the ball B.v/. Hence w 2 B2r k1 .v/. Since Vk1 is r k1 -separated, we obtain that there are at most M.2/ radial edges connecting v with lower level vertices.

8.3.2 Spaces with bounded geometry Here we discuss the question how to decide whether a boundary at infinity of a hyperbolic space X is doubling by looking at the space X itself. This is important in view of the Bonk–Schramm theorem (Theorem 8.2.1). There are a number of definitions reflecting a property we are looking for. We prefer to use the following one basically for the reason that the corresponding property is quasi-isometry invariant. A metric space X has bounded geometry if there are a constant r > 0 and a function M W Œ1; 1/ ! N such that every ball of radius r, 1, in X can be covered by M. / balls of radius r. This is equivalent to the property that every ball of radius r 0 in X contains at most M 0 . / points which are r 0 -separated for some constant r 0 > 0 and a function M 0 W Œ1; 1/ ! N. The property to have bounded geometry is, obviously, hereditary. Moreover, it is straightforward to check the following. Lemma 8.3.4. If a metric space X has bounded geometry and f W X 0 ! X is quasi isometric, then X 0 also has bounded geometry. Consequently, the property to have bounded geometry is a quasi-isometry invariant. However, it is not at all clear how to check that a given metric space has bounded geometry. Thus we consider the following property which is typically easy to check. A metric space X is doubling at some scale if there are constants r > 0 and M 2 N such that every ball of radius 2r in X contains at most M points which are r-separated. For example, every doubling space is doubling at some scale. Clearly, spaces with bounded geometry are doubling at some scale. Exercise 8.3.5. Show that if a geodesic metric space X is doubling at some scale, then X has bounded geometry. This is not true in general without assumption that the space is geodesic.

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Examples 8.3.6. (i) Every Hadamard manifold X with bounded sectional curvature has bounded geometry. Indeed, X is geodesic and every ball Br .x/ X is bilipschitz to an open Euclidean ball of radius r, where the bilipschitz constant is arbitrarily close to 1 for sufficiently small r which is separated from 0 independent of x, since the sectional curvatures are uniformly bounded. Hence, X is doubling at some scale and has bounded geometry by Exercise 8.3.5. (ii) Every graph X with length of edges uniformly separated from zero and uniformly bounded degree of vertices is, obviously, doubling at some scale and therefore it has bounded geometry by Exercise 8.3.5. In particular, every word hyperbolic group (see Section 1.4.2) has bounded geometry and by Proposition 8.3.3, any hyperbolic approximation of a doubling metric space has bounded geometry. The converse to the last example is also true. Proposition 8.3.7. Assume that a hyperbolic approximation X of a metric space Z has bounded geometry. Then Z is doubling. In what follows, we use our standard notations Vk for the vertex set of X of level k 2 Z. For the proof we need the following lemma. Lemma 8.3.8. Given k 2 Z, for every w 2 Vk , there is a radial ray w0 w1 : : : in X starting at w0 D w such that for distinct w; w 0 2 Vk we have jwn wn0 j 2.n 1/ for every n 1. Proof. Assume that the vertex wn 2 VkCn , n 0, is already defined. Then we take as wnC1 a vertex from VkCnC1 with d.wn ; wnC1 / r kCnC1 (recall that we use notation d.v; w/ for the distance in Z between vertices v, w of X , and that Vj is an r j -net in Z for every j 2 Z). Then B.wnC1 / B.wn / since r 1=6, and this determines a radial ray in P X. We have d.w; wn / j 1 r kCj D r kC1 =.1 r/ for every n 1. For distinct w; w 0 2 Vk we have d.w; w 0 / r k , thus we obtain d.wn ; wn0 / r k 2r kC1 =.1 r/ D r k .1 3r/=.1 r/ for every n 1. On the other hand, let vj ; vj0 2 Vj be lowest level vertices of a shortest segment wn wn0 in X , jvj vj0 j 1 (see Lemma 6.2.6). Then B.wn / B.vj /, B.wn0 / B.vj0 /, and we obtain d.wn ; wn0 / 2r j Cd.vj ; vj0 /C2r j 8r j . Together with the former estimate, this yields 8r j k .1 3r/=.1 r/ and thus j k 1 since r 1=6. Therefore, jwn wn0 j 2.n C k j / 2.n 1/. Proof of Proposition 8.3.7. By Proposition 8.3.3, it suffices to show that the degree of vertices of X is uniformly bounded. Let v 2 Vk be a vertex of X . Applying Lemma 8.3.8 to the vertices of the same level j D k 1; k; k C 1 adjacent to v, we see that for every n 2, there are at least


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Nj points in the ball of radius n C 1 in X centered at v which are 2.n 1/-separated. Here Nj is the number of vertices of level j adjacent to v, so N D Nk1 CNk CNkC1 is the degree of v. On the other hand, there is s > 0 and a function M W Œ1; 1/ ! N such that every ball of radius s in X contains at most M. / points which are s-separated, since X has bounded geometry. Choosing n sufficiently large with 2.n 1/ s and taking

D .n C 1/=s, we obtain N 3M. / independently of v. Hence, the claim. Theorem 8.3.9. Assume that a hyperbolic geodesic space X has bounded geometry. Then its boundary at infinity @1 X is doubling with respect to any visual metric. Proof. By Theorem 7.1.2 and Corollary 7.1.4, any hyperbolic approximation of @1 X is roughly similar to a subset of X and hence it has bounded geometry. Then, by Proposition 8.3.7, @1 X is doubling. Corollary 8.3.10. The boundary at infinity of every Hadamard manifold with pinched negative sectional curvature, b 2 K a2 < 0, is doubling. The boundary at infinity of every word hyperbolic group is doubling. In particular, every such space is roughly similar to a convex subset of Hn for an appropriate n 2. Remark 8.3.11. The last corollary for hyperbolic groups has already been proved in Chapter 2 using local self-similarity of the boundary at infinity; see Corollary 2.3.7. Corollary 8.3.12. The boundary at infinity of every word hyperbolic group has finite topological dimension. Bibliographical note. Theorem 8.1.1 is due to P. Assouad [As2] and is optimal in the sense that in general p D 1 cannot be taken for the value of the parameter p: the Carnot–Caratheodory metric on S 3 which naturally occurs a boundary metric of the complex hyperbolic plane C H2 is doubling, but admits no bilipschitz embedding into any Euclidean space RN ; see [He], Chapter 12. Our proof of Theorem 8.1.1 follows the original ideas from [As2] but technically it is somewhat different due to the explicit use of a hyperbolic approximation. Theorem 8.1.1 is the main ingredient of the proof of Theorem 8.2.1, the Bonk– Schramm embedding result [BoS]. Theorem 8.3.9 has appeared in [BoS, Theorem 9.2] in a slightly different form. Our proof is based on different ideas and uses different techniques.

Chapter 9

Basics of dimension theory

Important tools in studying metric spaces are various coverings, and basic dimension type invariants are defined via multiplicity of coverings. In this chapter, we discuss a number of dimensions all of which are close relatives of the classical topological or covering dimension. Similarly as the topological dimension is invariant under homeomorphisms, other dimensions we consider are invariant under different types of morphisms, i.e., they are quasi-isometry, or bilipschitz, or even quasi-symmetry invariants. All those dimensions were proven to be useful for a large spectrum of problems, in particular, for embedding and nonembedding problems.

9.1 Various dimensions Let X be a metric space. For U , U 0 X we denote by dist.U; U 0 / the distance between U and U 0 , dist.U; U 0 / D inffjuu0 j W u 2 U; u0 2 U 0 g where juu0 j is the distance between u, u0 . For r > 0 we denote by Br .U / the open r-neighborhood of U , Br .U / D fx 2 X W dist.x; U / < rg, and by Bxr .U / the closed r-neighborhood of U , Bxr .U / D fx 2 X W dist.x; U / rg. We extend the notation Bxr .U / for all real r putting Br .U / D U for r D 0, and defining Br .U / for r < 0 as the complement of the closed jrj-neighborhood of X n U , Br .U / D X n Bxjrj .X n U /. Given a family U of subsets in X we define mesh.U/ D supfdiam U W U 2 Ug. The multiplicity of U, m.U/, is the maximal number ofSmembers of U with nonempty intersection. A family U is called a covering of X if fU W U 2 Ug D X.

9.1.1 Topological dimension The topological dimension of X is the minimal integer dim X D n such that for every " > 0 there is an open covering U of X with m.U/ n C 1 and mesh.U/ ". In this and all cases below, if the appropriate number n does not exist, we say that the corresponding dimension of X is infinite. The topological dimension is obviously invariant under uniformly continuous homeomorphisms, and it was extensively studied during the last century. The definitions of other dimensions involve control over the Lebesgue number of coverings, which – in contrast to the topological dimension – makes them depending

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on a chosen metric in a more crucial way. This feature is highly useful for applications. Let U be an open covering of a metric space X . Given x 2 X , we let L.U; x/ D supfdist.x; X n U / W U 2 Ug be the Lebesgue number of U at x and L.U/ D inf x2X L.U; x/ the Lebesgue number of U. For every x 2 X , the open ball Br .x/ of radius r D L.U/ centered at x is contained in some member of the covering U. Note that L.U/ might be larger than mesh.U/ and even infinite, e.g., if U contains X as a covering element for a bounded X. Though this definition can be applied to any covering, sometimes it yields inappropriate results, e.g. for the covering U D fŒi; i C 1 W i D 0; : : : ; n 1g of the segment X D Œ0; n, we have L.U/ D 0. Remark 9.1.1. We shall often use the following obvious fact. If the Lebesgue number of an (open) covering U of a metric space X is positive, r D L.U/ > 0, then the family Bs .U/ D fBs .U / W U 2 Ug is still a covering of X for every s < r.

9.1.2 Asymptotic dimension The asymptotic dimension of X is the minimal integer asdim X D n such that for every positive d there is an open covering U of X with m.U/ nC1, mesh.U/ < 1 and L.U/ d . Clearly, asdim X D 0 for every bounded X (take the covering with only one member, X ). However, for unbounded spaces, the asymptotic dimension is highly interesting and useful. Due to the large Lebesgue numbers, the condition for coverings to be open is not essential. We easily see that asdim X is a quasi-isometry invariant of X.

9.1.3 Assouad–Nagata dimension The Assouad–Nagata dimension of X is the minimal integer ANdim X D n with the following property: there exists ı 2 .0; 1/ such that for every positive there is an open covering U of X with m.U/ n C 1, mesh.U/ and L.U/ ı . This dimension is obviously a bilipschitz invariant. Surprisingly, the Assouad– Nagata dimension as well as the following dimension are in fact quasi-symmetry invariants, which is not at all obvious. The Assouad–Nagata dimension is an example of linearly controlled dimensions, which means that the Lebesgue number of coverings involved in the definition of a dimension is at least a fixed linear function of their mesh.

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9.1. Various dimensions

9.1.4 Linearly controlled metric dimension The linearly controlled metric dimension or `-dimension of a metric space X is the minimal integer `-dim X D n with the following property: there exists ı 2 .0; 1/ such that for every sufficiently small r > 0 there is an open covering U of X with m.U/ n C 1, mesh.U/ r and L.U/ ır.

9.1.5 Linearly controlled asymptotic dimension The linearly controlled asymptotic dimension or asymptotic `-dimension of X is the minimal integer `-asdim X D n with the following property: there exists ı 2 .0; 1/ such that for every sufficiently large R > 1 there is an open covering U of X with m.U/ n C 1, mesh.U/ R and L.U/ ıR. Exercise 9.1.2. Check that the linearly controlled asymptotic dimension is a quasiisometry invariant. The distinction between the last three dimensions is that while theAssouad–Nagata dimension takes into account all scales, the `-dimension reflects only features of the space at arbitrarily small scales, and the asymptotic `-dimension takes into account the large scales only. We obviously have dim X `-dim X ANdim X

and

asdim X `-asdim X ANdim X;

and we easily see that ANdim X D maxf`-dim X; `-asdim X g for every metric space X , in particular, `-dim X D ANdim X for every bounded space X. One can continue this list indefinitely, e.g., we discuss another useful dimension, the hyperbolic dimension, in Chapter 13. How to work with these definitions? For example, how to compute the topological dimension of the Euclidean space Rn ? Everybody knows that dim Rn D n, however, the argument is not at all on the surface. Even to observe the easier estimate dim Rn n, the best way is to use an alternative definition of the topological dimension, called the colored definition. Namely, we say that a family U of subsets in a metric space X is disjoint if its multiplicity equals 1, m.U/ D 1. S A covering U is said to be colored if it is the union of m 1 disjoint families, U D a2A Ua , jAj D m. In this case, we also say that U is m-colored. Clearly, the multiplicity of an m-colored covering is at most m.

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9.1.6 Colored definition of a dimension We define dimcol X as a minimal integer n such that for every " > 0 there is an open .n C 1/-colored covering U of X with mesh.U/ ". Similarly, there are colored definitions of every dimension from the list above. Clearly, dimcol X dim X and it turns out that dimcol X D dim X for every metric space X, see Proposition 9.3.7, and a similar equality is true for every dimension on the list. Example 9.1.3. We show that dimcol Rn n. As a color set, we take the cyclic group A D Z=.n C 1/Z of order n C 1, which we identify with A D f0; : : : ; ng. Now we define the family U0 as follows. Take the middle open subsegment J Œ0; 1 of length l 2 .0; 1/ and consider the cube J n Rn . The integer lattice Zn Rn acts on Rn by translations, and we put U0 D fJ n W 2 Zn g: For every color a 2 A, we take the cube Jan D J n C a, where the vector 2 Rn is of the form D .n C 1/1 f1; : : : ; 1g, and define Ua D fJan W 2 Zn g: Clearly, every Ua , a 2 A, is a disjoint familySin Rn . We easily see that if l is sufficiently close to 1, then the family U D a2A Ua is a covering of Rn (see Figure 9.1 for the case n D 2). Moreover, its Lebesgue number L.U/ ın , where the constant ın > 0 depends only on n.

Figure 9.1. Three shifted copies of J 2 (grey) cover the torus R2 =Z2 .

Taking a homothety h W Rn ! Rn , h .x/ Dp x, with coefficient > 0, we obtain a covering U D h .U/ with mesh.U / n and L.U / ın . This shows that the colored dimension of Rn of each type from the list above (i.e. topological, asymptotic and all three linearly controlled ones) is at most n. We explain how to get the estimate dim Rn n in Section 9.8.

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9.1.7 Polyhedral definition of a dimension There is another useful characterization of dimensions via Lipschitz maps into simplicial polyhedra. For example, one can define dimpol X as a minimal integer n such that for every " > 0 there is a Lipschitz map f W X ! K into a simplicial polyhedron K of combinatorial dimension n with diam f 1 .y/ " for every y 2 K. Similarly, there is a polyhedral definition of every dimension from the list above, and for every dimension, the different definitions are equivalent. The advantage of this is an additional flexibility in working with dimensions. Now to proceed further, we need to introduce some standard constructions.

9.2 Constructions 9.2.1 Uniform simplicial polyhedra Given an index set J , we let RJ be the Euclidean space of functions J ! R with finite support, i.e., x 2 RJ if and only if only finitely many coordinates xj D x.j / are non-zero. The distance jxx 0 j is well defined by X jxx 0 j2 D .xj xj0 /2 : j 2J J J Let J R P be the standard simplex, i.e.,J x 2 if and only if Jxj 0 for all j 2 J and j 2J xj D 1. The metric of R induces a metric on and on every subcomplex K J , i.e., the distance between two points in K is the distance between them in RJ . The topology of K is the metric topology. A metric in a simplicial polyhedron K is said to be uniform if K is isometric to a subcomplex of J for some index set J . Every simplex K is then isometric to the standard k-simplex k RkC1 , k D dim . So, for a finite J , dim J D jJ j 1, and this equality serves as the definition of the combinatorial dimension: the combinatorial dimension of a simplicial polyhedron is the maximal dimension of its simplices. Speaking about dimension of a simplicial polyhedron, we always mean the combinatorial dimension. For every simplicial polyhedron K, there is the canonical embedding u W K ! J , where J is the vertex set of K, which is affine on every simplex. Its image K 0 D u.K/ is called the uniformization of K, and u is the uniformization map. A subpolyhedron K 0 of a simplicial polyhedron K is said to be complete if every simplex of K whose vertices are in K 0 is also a simplex of K 0 . A simplicial polyhedron is locally finite if every vertex is the member of only finitely many simplices. The standard simplex J is not locally finite if jJ j D 1. Given a simplicial polyhedron K and a vertex v 2 K, its star st v consists of all simplices of K, containing v, and the open star stv is the star st v without faces opposite to v. For each vertex v of every standard simplex J , the open star stv is

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open in J . Indeed, any point x 2 stv is characterized by the condition xv > 0. Then for every " 2 .0; xv /, the intersection B" .x/ \ J is contained in stv because for y 2 B" .x/ we have jxyj < ", hence jxv yv j < " and thus yv > 0. As a consequence, we obtain that the open star of every vertex of a uniform simplicial polyhedron K is open in K. Remark 9.2.1. The open stars of the vertices cover any simplicial polyhedron K, and if dim K is finite for a uniform K then clearly the Lebesgue number of this covering is bounded from below by a positive constant depending only on dim K.

9.2.2 The nerve of a covering and barycentric maps With every covering U D fUj gj 2J of a space X , one associates a simplicial polyhedron N D N .U/ called the nerve of U. The vertex set of N is identified with the set J representing the covering members, and a subset J 0 J spans a simplex if and only if all Uj with j 2 J 0 have a common point. The nerve N can always be considered as a subcomplex of J , N J , and therefore as a uniform polyhedron. Furthermore, we have dim N D m.U/ 1. A covering U of a space X is said to be locally finite if any x 2 X has a neighborhood which meets only finitely many members of U. Let U D fUj gj 2J be a locally finite open covering of a metric space X and N D N .U/ J its nerve. One defines a barycentric map pW X ! N associated with U as follows. Note that the Lebesgue number at x, L.U; x/, is positive for every x 2 X though it might be infinite. We fix a positive function d W X ! R with d.x/ L.U; x/ for every x 2 X called a cut function. Given jP2 J , we put qj W X ! R, qj P .x/ D minfd.x/; dist.x; X n Uj /g. Then we have q .x/ d.x/ > 0, and j 2J j j 2J qj .x/ < 1 for every x 2 X because U is locally finite. NowPthe map p W X ! J is well defined by its coordinate functions pj .x/ D qj .x/= j 2J qj .x/, j 2 J , which are nothing else than the barycentric coordinates. Clearly, the image of a point lands in the nerve, p.X / N . Note that for each vertex v 2 N , the preimage of its open star, p 1 .st v / X , coincides with that member of the covering U associated with v. Lemma 9.2.2. Assume that the multiplicity of an open locally finite covering U is finite, m.U/ D m < 1, and L.U/ d > 0. Then there is a Lipschitz barycentric map p W X ! N with Lipschitz constant Lip.p/ .m C 1/2 =d . Proof. We take the barycentric map p W X !P N determined via the constant cut function, d.x/ D d for every x 2 X. Then j 2J qj .x/ d for every x 2 X .

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Furthermore, qj .x/ qj .x 0 / C jxx 0 j and X X qj .x/ qj .x 0 / C mjxx 0 j; j 2J

j 2J

because there are at most m nonzero summands in each sum. Using this, we obtain 1 mjxx 0 j 1 P P C : 0 d j 2J qj .x/ j 2J qj .x / j 2J qj .x/ P P Then abbreviating D j 2J qj .x/, 0 D j 2J qj .x 0 /, we obtain qj .x 0 / qj .x/ 1 1 mC1 0 C qj .x/ jxx 0 j: pj .x / pj .x/ D 0 0 d P

Finally, for p D fpj gj 2J we have jp.x 0 /p.x/j2 D

X .m C 1/2 .2m/ .pj .x 0 / pj .x//2 jxx 0 j2 ; d2

j 2J

hence, Lip.p/ .m C 1/2 =d .

9.2.3 Barycentric subdivision The barycentric subdivision of the standard simplex J is a subcomplex ba J of the standard simplex J where J is the set of all finite nonempty subsets of J called the finite power set. The subcomplex ba J is defined as follows: a finite collection ˛ of vertices of J spans a simplex of ba J if and only if for any two members of ˛ considered as subsets in J , one of them is contained in the other. There is the canonical bijection baJ W J ! ba J called the barycentric subdivision map. The inverse map ba1 J sends every vertex a 2 J into the barycenter ba 1

1 Figure 9.2. The barycentric subdivision of 1 .

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of the corresponding face a J , and it is affine on every face of ba J . Given a uniform polyhedron K J , its barycentric subdivision is by definition the subcomplex ba K D baJ .K/ ba J . Therefore, ba K is also a uniform simplicial polyhedron. Often we consider ba K as the barycentric triangulation of K, i.e., we identify ba K with K via baJ if there is no danger of confusion. Lemma 9.2.3. Assume that an index set J is finite, jJ j < 1. Then for each y; y 0 2 ba J , there is y 00 2 ba J such that the pairs y, y 00 and y 00 , y 0 lie in simplices of ba J and jyy 00 j C jy 00 y 0 j C jyy 0 j for some constant depending only on jJ j. Proof. There are simplices , 0 ba J with a common face, \ 0 ¤ ;, containing y, y 0 respectively. Take 3 y, 0 3 y 0 of minimal dimension. If \ 0 D ; then the subset I of J which is the union of the subsets representing all vertices of , 0 is the common vertex of the simplices spanned by , I and 0 , I . Now we take y 00 2 \ 0 such that the broken geodesic path yy 00 y 0 is a shortest path in [ 0 between y, y 0 . It suffices to estimate from below the angle ˛ between the segments y 00 y and y 00 y 0 at y 00 by a positive constant depending only on jJ j. We can assume that ¤ 0 , and thus \ 0 is a proper face of , 0 . Let A RJ be the affine subspace spanned by the face \ 0 , and let B, B 0 RJ be the affine half subspaces bounded by A, @B D A D @B 0 , containing y, y 0 respectively. Then ˛ is bounded from below by the angle between B, B 0 , that is, by the angle between the rays with a common vertex in A, which are orthogonal to A and lie in B, B 0 respectively. When y, y 0 run over , 0 respectively, the corresponding subspaces B, B 0 are parameterized by points of the faces ı , ı 0 0 opposite to \ 0 , i.e., we can assume that y 2 ı, y 0 2 ı 0 . The worst case occurs when we take y 2 ı, y 2 ı 0 as the barycenters and dim \ 0 D 0. In this case, ˛ > 0 depends only on the distances jyj, jy 0 j to the origin of RJ , which are bounded from below by a constant depending only on jJ j. Lemma 9.2.4. For every uniform, simplicial, finite dimensional polyhedron K J , the map baJ W K ! ba K is bilipschitz, jxx 0 j=C j baJ .x/ baJ .x 0 /j C jxx 0 j for each x; x 0 2 K, where the constant C 1 depends only on dim K. Proof. The property is obvious for x, x 0 whose images baJ .x/, baJ .x 0 / lie in one and J the same simplex of ba J because the map ba1 J is affine on every face of ba . 0 J In the general case, we note that x, x are contained in a simplex whose dimension depends only on dim K, and we consider the restriction of baJ to . Then for any y; y 0 2 ba according to Lemma 9.2.3, there is y 00 2 ba such that each of the pairs y, y 00 and y 00 , y 0 lie in simplices of ba and jyy 00 j C jy 00 y 0 j C jyy 0 j for some constant C depending only on dim . From this, we easily obtain the bilipschitz property of baJ for x, x 0 .

9.2. Constructions

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Remark 9.2.5. One important feature of the barycentric subdivision is that the covering U of any finite dimensional polyhedron K J by the open stars of the vertices of its barycentric subdivision ba K is open and m-colored with m D dim K C 1. The color of a star stv ba K for v 2 J is jvj, the number of elements from J ,

Figure 9.3. Three disjoint families of open stars cover 2 .

jvj dim K C 1. Furthermore, it follows from Remark 9.2.1 and Lemma 9.2.4 that the Lebesgue number of U is bounded from below by a positive number depending only on dim K.

9.2.4 The barycentric triangulation of a product Here, we describe the canonical triangulation of the product of two simplicial complexes which is based on the barycentric subdivision. We start with the case of two simplices. Let J1 , J2 be index sets, J D J1 [ J2 the disjoint union, and J1 , J2 , J the power sets of J1 , J2 , J respectively. Recall that the vertex set of the simplicial complex ba J is J. Every member of J1 J2 being a pair with members from J1 , J2 is canonically identified with a member of J, that is, it is a vertex of ba J . By definition, the barycentric triangulation of J1 J2 is the complete simplicial subcomplex J1 s J2 of ba J spanned by J1 J2 . This means that a finite collection ˛ of members of J1 J2 spans a simplex of J1 s J2 if and only if for any two members of ˛ considered as subsets in J , one of them is contained in the other, see Section 9.2.3.

Figure 9.4. The barycentric triangulation of 1 1 (left) and a part of the barycentric triangulation of 2 1 (right).

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There is the canonical bijection ' W J1 J2 ! J1 s J2 also called the barycentric triangulation map, for which the inverse map ' 1 sends every vertex a D .a1 ; a2 / 2 J1 J2 of J1 s J2 into .b1 ; b2 / 2 J1 J2 , where bi is the barycenter of the corresponding face ai Ji , i D 1; 2, and it is affine on every simplex of J1 s J2 . Now let K1 , K2 be uniform simplicial polyhedra which we identify with subpolyhedra of J1 , J2 respectively, where Ji is the vertex set of Ki , i D 1; 2. Then K1 K2 J1 J2 and '.K1 K2 / J1 s J2 is clearly a subcomplex, where ' W J1 J2 ! J1 s J2 is the barycentric triangulation map. We define the barycentric triangulation of the product K1 K2 as K1 s K2 D '.K1 K2 /. Lemma 9.2.6. Given finite dimensional simplicial polyhedra K1 J1 , K2 J2 , the barycentric triangulation map ' W K1 K2 ! K1 s K2 is bilipschitz with bilipschitz constant depending only on the dimensions of K1 , K2 . Proof. Any pair of points in either of the factors K1 , K2 is contained in a simplex with dimension at most two times the dimension of the factor. Thus we can assume that Ki D Ji with finite Ji , and we are looking for an estimate depending only on both jJi j, i D 1; 2. Now the proof is similar to that of Lemma 9.2.4. If y D '.x/, y 0 D '.x 0 / lie in one and the same simplex of K D K1 s K2 for some x; x 0 2 K1 K2 then the property is obvious because ' 1 is affine on every simplex. Otherwise, the argument of Lemma 9.2.3 can be applied to K as well, and we find 00 y 2 K such that the pairs y, y 00 and y 00 , y 0 lie in simplices of K and jyy 00 jCjy 00 y 0 j C jyy 0 j for some constant depending only on jJ1 j C jJ2 j. Hence, the claim. Remark 9.2.7. Since the open stars of the vertices of K1 s K2 form a covering, their preimages under the barycentric triangulation map cover K1 K2 . In the case that dim K1 , dim K2 are finite, this covering is open, and it follows from Remark 9.2.1 and Lemma 9.2.6 that its Lebesgue number is bounded from below by a positive constant depending only on dim K1 , dim K2 . We need the following: Lemma 9.2.8. Let K1 , K2 be uniform simplicial polyhedra, K D K1 s K2 . For every vertex v 2 K, there are vertices v1 2 K1 , v2 2 K2 such that ' 1 .st v / stv1 stv2 , where ' W K1 K2 ! K is the barycentric triangulation map. Proof. Consider first the case of two simplices. Let J1 , J2 be index sets and J1 , J2 the power sets of J1 , J2 respectively. Every vertex v of J1 s J2 is a member of J1 J2 , v D .a1 ; a2 /, and therefore, there are v1 2 J1 , v2 2 J2 entering v; v1 2 a1 ,

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v2 2 a2 . Then ' 1 .v/ D .b1 ; b2 /, where bi is the barycenter of ai . Because vi is a vertex of ai , i D 1; 2, we have ' 1 .st v/ stv1 stv2 . In the general case, we can assume that Ki Ji , where Ji is the vertex set of Ki , i D 1; 2. Then the open star of any vertex in Ki , K is the part of the open star in Ji , J1 s J2 of the same vertex sitting in Ki , K respectively, i D 1; 2. The claim follows because K D '.K1 K2 /.

9.3 P-dimensions The dimensions listed at the beginning of the chapter have three types of equivalent definitions and possess the same basic properties like monotonicity and a product theorem. It would be awkward to discuss these things for every dimension separately. Thus we introduce the general concept of P-dimension (P stands for Property, indicating the characterizing property of an appropriate dimension) with the aim to provide a device, which allows us to present a large number of similar results in a unified way.

9.3.1 Property spaces Recall that a filter F on a set P is a collection of subsets of P with the following properties: (1) ; 62 F ; (2) if A 2 F and B A then B 2 F ; (3) if A, A0 2 F then A \ A0 2 F . Every P-dimension of metric spaces is characterized by a property space. A property space P associated with a dimension is a set together with a filter F of subsets called characteristic. Every point p 2 P represents a property of open coverings of a metric space X or a property of Lipschitz maps of X into uniform simplicial polyhedra. We write U 2 p if a covering U of X has the property (represented by) p. Similarly, f 2 p for a Lipschitz map f W X ! K into a uniform simplicial polyhedron K, if the covering U D ff 1 .stv / W v 2 Kg of X by the preimages of the open stars stv of the vertices of K has the property p. Speaking more formally, every point p 2 P is a function on the set of all open coverings of X or of all Lipschitz maps of X into uniform simplicial polyhedra with values in f0; 1g. Its value p.U/ on a covering U equals 1 if and only if the covering U has the property p. Now we give examples of property spaces. For the topological dimension, the property space P is identified with positive reals .0; 1/, and the filter F consists of all subsets each of which contains some subinterval .0; t / .0; 1/, that is, F is generated by the intervals .0; t /, t > 0. We say that a covering U of a metric space X has the property t 2 .0; 1/, U 2 t , if and only if U is open and mesh.U/ t.

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For the asymptotic dimension, the property space P is identified with the interval .0; 1/, and the filter F is generated by the intervals .t; 1/ .0; 1/, t > 0. We say that a covering U of a metric space X has the property t 2 .0; 1/, U 2 t , if and only if U is open, mesh.U/ < 1 and L.U/ t . For the Assouad–Nagata dimension, the property space P is identified with P D .0; 1/ .0; 1/, and the filter F is generated by the sets .0; 1/ .0; ı/, ı 2 .0; 1/. We say that a covering U of a metric space X has the property p D .; ı/ 2 P , U 2 p, if and only if U is open, L.U/ ı and mesh.U/ . For the `-dimension of X , the property space P is identified with P D .0; 1/ .0; 1/, and the filter F is generated by the sets .0; / .0; ı/ P , ı 2 .0; 1/, > 0. We say that a covering U of a metric space X has the property p D .; ı/ 2 P , U 2 p, if and only if U is open, L.U/ ı and mesh.U/ .

9.3.2 Axioms of property spaces We introduce a set of axioms which allow us to prove a number of basic properties of the P-dimension. Let X be a metric space, U its locally finite open covering and pU W X ! N a barycentric map; see Section 9.2.2. We denote by ba.pU / the covering of X by the 1 .st v / W v 2 preimages of the open stars in ba N with respect to pU , ba.pU / D fpU ba N g. Given two barycentric maps fi W Xi ! Ni associated with open, locally finite coverings Ui of Xi , fi D pUi , i D 1; 2, we denote by Uf1 ;f2 the covering of the product X1 X2 by the preimages of the open stars in K D N1 s N2 with respect to the map ' B .f1 f2 / W X1 X2 ! K, where ' W N1 N2 ! K is the barycentric triangulation map; see Section 9.2.4 and Lemma 9.2.6. Note that if m.Ui / ni C 1 then dim Ni ni , i D 1; 2, therefore dim K n1 Cn2 and m.Uf1 ;f2 / n1 Cn2 C1. We assume that any property space P we consider satisfies the following axioms. Axioms 9.3.1. (1) For every natural number m, there exists a map bam W P ! P such that bam .F / F for the characteristic filter F , and ba.pU / 2 bam .p/ for all p 2 P , every open covering U 2 p of a metric space X with multiplicity m and some barycentric map pU W X ! N . This axiom is responsible for the equivalence of the three definitions of P-dimension given below. (2) If X 0 X and a covering U of X has the property p 2 P , U 2 p, then the restriction of U to X 0 has the same property p, UjX 0 2 p. This axiom is responsible for monotonicity of the P-dimension. (3) For every natural number m, there exists a map prodm W P P ! P such that prodm .F F / F for the characteristic filter F , and such that Uf1 ;f2 2 prodm .p1 ; p2 / for every pi 2 P , each open covering Ui 2 pi of a metric space Xi with multiplicity m, and some barycentric map fi D pUi W Xi ! Ni , i D 1; 2. This axiom is responsible for the product theorem.

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Example 9.3.2. For the topological dimension, recall that P D .0; 1/ and the filter F is generated by the intervals .0; t /, t > 0. We take the identity map as the map bam W P ! P for each m 0. Axiom 9.3.1 (1) is satisfied because the open star of every vertex of the barycentric subdivision is contained in the open star of an appropriate vertex of the polyhedron, and the covering of the finite dimensional nerve N by the open stars of ba N is open; see Remark 9.2.5. Axiom 9.3.1 (2) is obvious. Axiom 9.3.1 (3) is also satisfied, if we define prodm W P P ! P by prodm .t1 ; t2 / D 2 maxft1 ; t2 g for every natural m. Indeed, given ti 2 P and an open covering Ui 2 ti of a metric space Xi with multiplicity m, there is a Lipschitz barycentric map fi D pUi W Xi ! Ni , i D 1; 2 by Lemma 9.2.2. For any vertex v 2 K D '.N1 N2 /, there are by Lemma 9.2.8 vertices v1 2 N1 , v2 2 N2 with ' 1 .stv / stv1 stv2 , where ' W N1 N2 ! K is the barycentric triangulation map. Then for U D .f1 f2 /1 B ' 1 .stv / we have that U U1 U2 , where Ui D fi1 .st vi / 2 Ui . Since diam.U1 U2 / 2 maxi diam Ui 2 maxi ti , and ' 1 .st v / N1 N2 is open (see Remark 9.2.7), we have Uf1 ;f2 2 prodm .t1 ; t2 /. Finally, the inclusion prodm .F F / F follows from the definition of prodm and properties of filters. Example 9.3.3. For the asymptotic dimension, P D .0; 1/, and the filter F is generated by the intervals .t; 1/, t > 0. Recall that a covering U of X has the property t 2 P , U 2 t , if and only if U is open, mesh U < 1 and L.U/ t . By Remark 9.2.5, there is for every natural number m a lower bound lm 2 .0; 1/ for the Lebesgue number of the covering of any uniform polyhedron K, dim K C 1 m, by the open stars of ba K. We put m D lm =.m C 1/2 and define bam W P ! P as bam .t/ D m t for every t > 0. Axiom 9.3.1 (1) is satisfied because, for any covering U 2 t 2 P with multiplicity m, the covering U0 D ba.pU / is open, mesh.U0 / mesh.U/ < 1 and L.U0 / lm = Lip.pU / m t , where we used the estimate Lip.pU / .m C 1/2 =t of Lemma 9.2.2. The property bam .F / F is evident. Axiom 9.3.1 (2) is obvious because L.UjX 0 / L.U/ for every open covering U of X and every subspace X 0 X . For every natural number m, there is by Remark 9.2.7 a constant cm 2 .0; 1/ with the property: given uniform polyhedra K1 , K2 , dim Ki C 1 m, the Lebesgue number of the covering of K1 K2 by ' 1 .st v /, v 2 K1 s K2 , is bounded from below by cm , where ' W K1 K2 ! K1 s K2 is the barycentric triangulation map. We put m D cm =.m C 1/2 and define prodm W P P ! P by prodm .t1 ; t2 / D m minft1 ; t2 g for every t1 , t2 > 0. Then clearly prodm .F F / F . Moreover, for every covering Ui 2 ti 2 P with multiplicity m, there is by Lemma 9.2.2 a barycentric map fi D pUi W Xi ! Ni with Lip.fi / .m C 1/2 =ti , i D 1; 2. Then the covering Uf1 ;f2 of X1 X2 is open and L.Uf1 ;f2 / cm = Lip.f1 f2 / m minft1 ; t2 g;

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because Lip.f1 f2 / .m C 1/2 maxf1=t1 ; 1=t2 g. That is, Uf1 ;f2 2 prodm .t1 ; t2 /, and Axiom 9.3.1 (3) is satisfied. Example 9.3.4. For the `-dimension, P D .0; 1/ .0; 1/, and the filter F is generated by .0; / .0; ı/ for all > 0, ı 2 .0; 1/. Recall that a covering U of X has the property p D .; ı/ 2 P , U 2 p, if and only if U is open, mesh.U/ and L.U/ ı. Given a natural number m, we define bam W P ! P by bam .; ı/ D .; m ı/ and prodm W P P ! P by m minfı1 1 ; ı2 2 g prodm ..1 ; ı1 /; .2 ; ı2 // D 2 maxf1 ; 2 g; ; 2 maxf1 ; 2 g where the constants m ; m 2 .0; 1/ are defined as in Example 9.3.3. Combining the arguments of the two examples above, one easily checks the Axioms 9.3.1 for the `-dimension. We leave details to the reader. Exercise 9.3.5. Check the axioms for the Assouad–Nagata dimension. Exercise 9.3.6. Describe the property space and check the axioms for the asymptotic `-dimension. In the following let X be a metric space.

9.3.3 Colored definition of the P-dimension For every integer n 0, the space X is represented in P by a subset col D col.X; n/ of P : namely, p 2 col if and only if there is an .n C 1/-colored covering U of X with the property p. Now the colored P-dimension of X is Pdimcol X D minfn W col.X; n/ contains some set from F g:

9.3.4 Covering definition of the P-dimension For every integer n 0, the space X is represented in P by a subset cov D cov.X; n/ P : namely, p 2 cov if and only if there is a covering U of X with the multiplicity m.U/ n C 1 and the property p. Now the covering P-dimension of X is Pdimcov X D minfn W cov.X; n/ contains some set from F g: For the property spaces of the topological, asymptotic, Assouad–Nagata, `- and asymptotic `-dimensions, this gives back the definitions of these dimensions at the beginning of the chapter.

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9.3.5 Polyhedral definition of the P-dimension For every integer n 0, the space X is represented in P by a subset pol D pol.X; n/ of P : namely, p 2 pol if and only if there is a Lipschitz map f W X ! K having the property p, where K is a uniform simplicial complex of the (combinatorial) dimension n. Now the polyhedral P-dimension of X is Pdimpol X D minfn W pol.X; n/ contains some set from F g: Note that col.X; n/ cov.X; n/ pol.X; n/: the first inclusion follows from the fact that the multiplicity of an m-colored covering is at most m, and the second one follows from the existence of a barycentric map associated with a covering; see Section 9.2.2. Hence, Pdimcol X Pdimcov X Pdimpol X for every metric space X . Proposition 9.3.7. For every metric space X, the dimensions Pdimcol X , Pdimcov X , Pdimpol X coincide, Pdimcol X D Pdimcov X D Pdimpol X: Proof. It suffices to check that col.X; n/ contains a set F 0 2 F for some n if pol.X; n/ contains a set F 2 F , because this implies the inequality Pdimpol X Pdimcol X completing the proof. Now if p 2 F pol.X; n/, then there is a Lipschitz map f W X ! K with f 2 p, where K is a uniform polyhedron of dimension n. Consider the covering U of X by preimages of open stars of K, U D ff 1 .st v / W v 2 Kg. Note that the nerve of U coincides with K. By Axiom 9.3.1 (1), there is a barycentric map pU W X ! K with ba.pU / 2 banC1 .p/, and the covering ba.pU / is an .n C 1/colored covering of X, see Remark 9.2.5. Thus banC1 .p/ 2 col.X; n/. This shows that banC1 .F / col.X; n/. Finally, we note that F 0 D banC1 .F / 2 F again by Axiom 9.3.1 (1). From now on, we use the notation Pdim X for the common value of all three P-dimensions.

9.4 The monotonicity theorem Theorem 9.4.1. Given a property space P , for a metric space X and any subspace X 0 X, we have Pdim X 0 Pdim X . Proof. It follows from Axiom 9.3.1 (2), that cov.X 0 ; n/ cov.X; n/ for every n. Hence, the claim.

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9.5 The product theorem Theorem 9.5.1. For metric spaces X1 , X2 and a given dimension of type P , we have Pdim.X1 X2 / Pdim X1 C Pdim X2 . Proof. Let .P; F / be the property space associated with the given dimension. Assume that Fi cov.Xi ; ni / P for some characteristic set Fi 2 F and some integer ni , i D 1; 2. It suffices to show that the set cov.X1 X2 ; n1 C n2 / P contains a characteristic set F 2 F , because this would imply Pdim.X1 X2 / n1 C n2 and therefore the required inequality. It follows from the assumption that for every property pi 2 Fi there is a covering Ui 2 pi with multiplicity ni C 1. By Axiom 9.3.1 (3), there is a barycentric map fi D pUi W Xi ! Ni , i D 1; 2, such that Uf1 ;f2 2 prodm .p1 ; p2 / with m D maxfn1 ; n2 g C 1. Since Uf1 ;f2 is a covering of X1 X2 with multiplicity n1 C n2 C 1, we obtain prodm .p1 ; p2 / 2 cov.X1 X2 ; n1 C n2 /. This shows that prodm .F1 ; F2 / cov.X1 X2 ; n1 C n2 /. Finally, F D prodm .F1 ; F2 / 2 F again by Axiom 9.3.1 (3).

9.6 The saturation of families Here, we discuss a powerful and flexible construction called the saturation of a given family U of subsets of a space X by another such family V . The construction has important applications in a number of questions including estimations of various types of dimensions and embedding questions. The saturation of U 2 U by the family V is the union U V of U and all members V 2 V with U \ V ¤ ;. Now the saturation of U by V is the family U V D fU V W U 2 Ug: Note that f;g V D ;, U f;g D U. We slightly modify the notion of the Lebesgue number to adapt it to open coverings of a subset A in a metric space X . Let U be a family of open subsets in a metric space X which cover A X . Then we put L.U/ D inf x2A L.U; x/, the Lebesgue number of the covering U of A, where as usual L.U; x/ D supfdist.x; X n U / W U 2 Ug. For every x 2 A, the open ball Br .x/ X of radius r D L.U/ centered at x is contained in some member of the covering U. Proposition 9.6.1. Suppose that X is a metric space and A, B X . Let [ [ Uc and V D Vc UD c2C

c2C

be coverings of A and B, respectively, which are both open in X and S m-colored with m D jC j 1. If mesh.V/ L.U/=2 then the family W D c2C W c is

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the open m-colored covering of A [ B with L.W / minfL.U/=2; L.V /g and mesh.W / maxfmesh.U/; mesh.V/g where W c D Br .Uc / V c [ fV 2 V c W Br .U / \ V D ; for all U 2 Uc g; for r D L.U/=2 Proof. We can assume that L.U/ < 1, i.e. no member of U covers X , because otherwise there is nothing to prove. The family Br .U/ D fBr .U / W U 2 Ug still covers A because r < L.U/. Thus, the family W covers A [ B. Clearly, for the Lebesgue number of W , we have L.W / minfL.Br .U//; L.V/g minfL.U/=2; L.V /g: Since diam V r for every V 2 V , we have Br .U / V c U for every U 2 Uc , c 2 C . Hence, mesh W maxfmesh.U/; mesh V g. Now since V c is disjoint, it easily follows from the definition that W c is disjoint for every c 2 C . Hence, W is m-colored.

9.7 The finite union theorem We apply the saturation construction to prove the following theorem. Theorem 9.7.1. Assume that a metric space X is the union of two subsets, X D A[B. Then Pdim X D maxfPdim A; Pdim Bg for each Pdim from the list: asdim, ANdim, `-dim, `-asdim. Proof. Because of the monotonicity theorem, it suffices to show that Pdim X m D maxfPdim A; Pdim Bg. We can assume that m is finite. The case Pdim D asdim. Given d > 0, there is an open .m C 1/-colored covering V of B with mesh.V/ < 1 and L.V / d . For d 0 2 maxfd; mesh.V /g, there is an open .m C 1/-colored covering U of A with mesh.U/ < 1 and L.U/ d 0 . Then mesh.V/ L.U/=2. By Proposition 9.6.1, there is an open .m C 1/-colored covering W of X with mesh.W / < 1 and L.W / minfL.U/=2; L.V /g d . Hence, asdim X m. The case Pdim D ANdim: We can assume that the constants ıA , ıB 2 .0; 1/ from the definition of ANdim for A, B coincide, ıA D ıB D ı, taking the smaller one if necessary. Given > 0, there is an open .m C 1/-colored covering U of A with mesh.U/ and L.U/ ı. For 0 D ı=2, there is an open .m C 1/colored covering V of B with mesh.V / 0 and L.V / ı 0 . Then mesh.V / L.U/=2, and, by Proposition 9.6.1, there is an open .m C 1/-colored covering W of 2 X with mesh.W / maxf; 0 g D and L.W / minfı=2; ı 0 g D ı2 . Hence, ANdim X m. The cases Pdim D `-dim and Pdim D `-asdim are similar.

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9.8 Sperner lemma The following nice combinatorial result definitely has a homological nature, and it can be used for elementary proofs of a number of classical topological facts like the invariance of the dimension and interior points for manifolds, the existence of fixed points of continuous maps, etc. Nowadays, these facts are usually proven by homological means. It seems, however, that the potential of the Sperner lemma is not yet completely exhausted. Lemma 9.8.1. Let be a triangulation of the standard simplex n . Assume that with every vertex t 2 one associates a vertex '.t / of n such that the following holds: if t 2 n , where is a face of n , then '.t / 2 . Then there is an n-dimensional simplex D Œt0 : : : tn such that the vertices '.t0 /; : : : ; '.tn / are pairwise distinct. Proof. Let be the set of all n-dimensional simplices of the triangulation . A simplex from is said to be normal, if the vertices of n associated with the vertices of are pairwise distinct. We claim that the number of the normal simplices is odd. For n D 0 this is obvious. Assume that the assertion is proven for all .n 1/simplices, and prove it for n . A face of dimension n 1 of a simplex 2 is called distinguished if its vertices are mapped onto the vertices e1 ; : : : ; en of n D Œe0 e1 : : : en . Note that the number of distinguished faces of is 0, or 1, or 2. Indeed, if is normal, then the number of its distinguished faces is 1; if is not normal and contains a distinguished face, then for the remaining vertex t0 2 , we have '.t0 / 2 fe1 ; : : : ; en g. Thus has another distinguished face which contains the vertex t0 , and no other distinguished face apart from these two. Let a be the number of distinguished faces of . We put aD

X

a :

2

Then the number of normal simplices has the same parity as a. Thus it suffices to show that a is odd. We compute a in another way: if one of n 1 faces of a simplex 2 lies inside of n , then it has exactly 2 adjacent simplices from that list. Thus its contribution to a is even; if such a face lies on some proper face of n different from Œe1 : : : en , then, by the condition on ', it cannot be distinguished, and its contribution to a is zero. The remaining case is that such a face lies in Œe1 : : : en . Clearly, in this case its contribution to a is 0 or 1. By the inductive assumption, the number of faces with contribution 1 in a is odd. Thus a is odd. Let n RnC1 be the standard n-simplex, and let U be a covering of a metric space X. A map f W n ! X is said to be coherent with U if no element of U

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intersects the image of every .n 1/-face of n . A covering V of n is called coherent if the identity map id W n ! n is coherent with V . Lemma 9.8.2. Let V be a coherent covering of n . Then for any triangulation of n , there is an n-simplex whose vertices lie in n C 1 different elements of the covering V. Proof. Any set containing two vertices of n intersects all .n 1/-faces of n . Thus there are n C 1 different elements V0 ; : : : ; Vn 2 V such that every vertex of n is contained in one and only one of those elements, ei 2 Vi , i D 0; : : : ; n. Assume that there is an element V 2 V different from V0 ; : : : ; Vn . Since V is coherent, there is an .n1/-face of n disjoint with V . Let ei 2 n be the opposite vertex, i 2 f0; : : : ; ng. We modify V by taking the union Vi [V . This does not change the coherent condition. Therefore, from the beginning we can assume that V D fV0 ; : : : ; Vn g. Now with every vertex t 2 , we associate a vertex '.t / D ei 2 n so that t 2 Vi . Note that any face Œe0 : : : er n is contained in V0 [ [ Vr , because if i 62 f0; : : : ; rg then Vi misses an .n 1/-face containing Œe0 : : : er and hence their intersection. Therefore, the condition of Lemma 9.8.1 is satisfied, and the claim follows. Corollary 9.8.3. The multiplicity of every open coherent covering V of n is at least n C 1, m.V/ n C 1. Proof. Since n is compact, we can assume that V is finite. Then the Lebesgue number L.V / D 2r > 0, and the family V 0 D Br .V / still covers n . By Lemma 9.8.2, for any triangulation of n , there is an n-simplex whose vertices lie in n C 1 different elements of the covering V 0 . Assuming that mesh < r, we find a simplex which is contained in n C 1 different elements of the covering V . Then m.V / n C 1. As an application, we obtain the following theorem. Theorem 9.8.4. For every n 0, we have dim Rn D asdim Rn D ANdim Rn D `-dim Rn D `-asdim Rn D n: Proof. We have already shown that all dimensions, dim, asdim, ANdim, `-dim and `-asdim, of Rn , are at most n, see Example 9.1.3. Now we first show that dim Rn n. By the monotonicity theorem, it suffices to prove that dim n n. We fix " > 0 and consider an open covering U of n with mesh.U/ ". If " is sufficiently small, then U is obviously coherent. By Corollary 9.8.3, m.U/ n C 1 and thus dim n n. Since ANdim `-dim dim, we also obtain the required estimate for the `- and Assouad–Nagata dimensions.

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It remains to consider the asymptotic dimension. Given any covering U of Rn with mesh.U/ < 1, we have mesh.h.U// mesh.U/ for the covering h.U/, where h W Rn ! Rn , h.x/ D x, is the homothety with coefficient > 0. Choosing sufficiently small, we produce a covering h.U/ with arbitrarily small mesh. Thus, dim Rn asdim Rn `-asdim Rn .

9.9 Supplementary results and remarks 9.9.1 The finite union theorem for the topological dimension For the topological dimension, the classical Menger–Urysohn theorem states: dim.A [ B/ dim A C dim B C 1 with equality in some cases. The difference to the finite union theorem for the asymptotic and the linearly controlled dimensions is rooted in the control of the Lebesgue number of coverings. However, for a compact X and closed A, B X , one has dim.A [ B/ D maxfdim A; dim Bg.

9.9.2 Qualified multiplicity and separation There is an alternative approach to definitions of dimensions where, instead of the Lebesgue number, one uses a qualified multiplicity or separation condition. For d > 0, the d -multiplicity of a family U of subsets in a metric space X , md .U/, is the multiplicity of the family Bd .U/ obtained by taking open d -neighborhoods of the members of U. So md .U/ D m.Bd .U//. Then the covering definition of the asymptotic dimension can be formulated as follows. The asymptotic dimension of a metric space X is at most n, if for every d > 0 there is a covering U of X with mesh.U/ < 1 and md .U/ n C 1. A family U is called d -disjoint if m.Bd .U// D 1. Then the colored definition of the asymptotic dimension can be formulated as follows. The asymptotic dimension of a metric space X is at most S n if for every d > 0 there is an .n C 1/-colored uniformly bounded covering U D a2A Ua , jAj D n C 1 of X , where each family Ua , a 2 A, is d -disjoint. Exercise 9.9.1. Formulate similar colored, covering and polyhedral definitions for asdim, ANdim, `-dim and `-asdim using qualified multiplicity and separation conditions instead of the Lebesgue number and prove their equivalence.

9.9.3 Asymptotic dimension and coarse maps Recall that a map f W X ! Y between metric spaces is coarse if there are control functions 1 , 2 for f , that is,

1 .jxx 0 j/ jf .x/f .x 0 /j 2 .jxx 0 j/


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for all x; x 0 2 X , with the property that lim t!1 i .t / D 1. The spaces X , Y are coarsely equivalent if there exist coarse maps f W X ! Y and g W Y ! X such that g B f and f B g are at a finite distance from the identity maps on X and on Y respectively. Exercise 9.9.2. Show that the asymptotic dimension is a coarse invariant, i.e. it is invariant under coarse equivalence.

9.9.4 Mapping simplices into a metric space There is another application of the Sperner lemma in the spirit of the Borsuk–Ulam theorem. Theorem 9.9.3. Let X be a metric space with dim X n. Then for any continuous f W nC1 ! X , there is x0 2 X such that f 1 .x0 / meets every n-face of nC1 . Proof. Assume to the contrary that for every x 2 X , f 1 .x/ misses an n-face of nC1 . Then f is coherent with every open covering U of X with sufficiently small mesh. Choosing U with multiplicity m.U/ n C 1, we obtain the open coherent covering V D f 1 .U/ of nC1 with m.V/ nC1. This contradicts Corollary 9.8.3. Recall that the boundary @nC1 is homeomorphic to the sphere S n and that by the classical Borsuk–Ulam theorem for any continuous map f W S n ! Rn , there is x0 2 Rn such that f 1 .x0 / contains a pair of antipodal points of S n . Corollary 9.9.4. For any continuous f W @nC1 ! Rn , there is x0 2 Rn such that f 1 .x0 / meets every n-face of @nC1 . Proof. The map f can be extended to a continuous map nC1 ! Rn .

9.9.5 An alternative approach to the product theorem Here we briefly discuss an alternative approach to the product theorem based on the so called Kolmogorov trick and the following theorem due to Ostrand [Os1]. Theorem 9.9.5 (P. A. Ostrand). A metric space X is of dimension n if and only if for each open cover C of X and each integer k n C 1 there exist k disjoint families of open sets U1 ; : : : ; Uk such that the union of any n C 1 of them is a covering of X which refines C . Here, we say that a covering U refines C or is inscribed in C if every member of U is contained in some member of C . The existence of additional k .n C 1/ families can be used in the proof of the product theorem as follows. Assume that dim X D m, dim Y D n, and let k D m C n C 1. Then we can find k disjoint families

128


fUi g in X and k disjoint families fVi g in Y as in Theorem 9.9.5 with arbitrarily small mesh. Then the family fUi Vi gkiD1 covers X Y , as for any point .x; y/ 2 X Y , x is contained in sets from at least k m D n C 1 families from fUi g and y is contained in sets from at least k n D m C 1 families from fVi g. Thus there is at least one index i such that x is covered by Ui and y is covered by Vi . The covering fUi Vi gkiD1 is disjoint and its mesh is as small as required. A similar argument works for any dimension we discussed above, see [BDLM].

9.9.6 Coarse structures and dimensions In this book, we took a moderate approach using a metric to define various asymptotic dimensions. There is a more advanced approach based on the notion of coarse structures that does not rely on metrics. For that, we refer to [Ro], [Gra], [DH], [Dr2]. Historical note. The covering definition of the topological dimension originates from A. Lebesgue. The colored definition goes back to the paper [Os1] by P. A. Ostrand which in turn is based on Kolmogorov’s idea used in the solution to Hilbert’s 13th problem, [Ko], [Ar]. The notion of the asymptotic dimension is introduced in [Gr2], the notion of the Assouad–Nagata dimension is due to P. Assouad, see [As1] where it is called the Nagata dimension. The idea of the saturation of one family of subsets by another was used in [BD1] to prove a countable union theorem for the asymptotic dimension which includes the finite union theorem. Basic results of Section 9.8 (Lemmas 9.8.1, 9.8.2 and Corollary 9.8.3) are due E. Sperner, [Sp].

Chapter 10

Asymptotic dimension

10.1 Estimates from below We prove two theorems which give optimal estimates from below for the asymptotic dimension of different classes of metric spaces. Recall that a Hadamard manifold is a complete simply connected Riemannian manifold X with nonpositive sectional curvature. For every point x 2 X , there is the exponential map expx W Tx X ! X from the tangent space Tx X which is a noncontracting diffeomorphism, jexpx v expx wj jvwj for each v; w 2 Tx X . Theorem 10.1.1. For every Hadamard manifold X , we have asdim X dim X: Proof. The proof is based on the Sperner lemma, see Section 9.8. Let U be a uniformly bounded covering of X , mesh.U/ < 1. Then for every sufficiently large R > 1, we obtain an n-dimensional, n D dim X , simplex f W n ! X coherent with U as follows. Consider an isometric copy of n in some tangent space Tx X with the barycenter at the origin and identify n with the closed ball BxR Tx X of the radius R via a radial homotopy. Composing with expx W BxR ! X , we obtain a map f W n ! X. Since expx is noncontracting, no member of U meets all .n 1/-faces of f .n / if R is chosen sufficiently large compared to mesh.U/. Thus f is coherent with U. Then m.U/ n C 1 by Corollary 9.8.3. It follows that asdim X n. For Hadamard manifolds that are also hyperbolic spaces, the following argument can be applied, which also works for arbitrary proper geodesic hyperbolic spaces. That argument is based on the well-known fact from dimension theory that dim.Z I / D dim Z C 1 for every compact space Z, where I D Œ0; 1. For the proof of this fact we refer to [Dr1]. Theorem 10.1.2. For every proper geodesic hyperbolic space X , we have asdim X dim @1 X C 1:

130

Chapter 10. Asymptotic dimension

Proof. We can assume that @1 X ¤ ;. By Corollary 7.1.5, the hyperbolic cone Co.Z/ over Z D @1 X , taken with some visual metric, can be roughly similarly embedded in X because X is geodesic. Thus asdim X asdim Co.Z/, and we show that asdim Co.Z/ dim Z C 1. The annulus An.Z/ Co.Z/ consists of all x 2 Co.Z/ with 1 jxoj 2, where o 2 Co.Z/ is the vertex. Clearly, An.Z/ is homeomorphic to Z Œ0; 1. Since X is proper, Z is compact. Then dim An.Z/ D dim Z C 1. Consider the sequence of contracting homeomorphisms Fk W Co.Z/ ! Co.Z/ given by Fk .z; t / D .z; k1 t /, .z; t / 2 Co.Z/, k 2 N. Given a uniformly bounded covering U of Co.Z/, the coverings Uk D Fk .U/ \ An.Z/ of the annulus An.Z/ have arbitrarily small mesh as k ! 1. Therefore, asdim Co.Z/ dim An.Z/, and the estimate follows. Remark 10.1.3. For some important classes of metric spaces, we obtain an optimal estimate from below for the asymptotic dimension via the monotonicity theorem. For example, let X be a Euclidean building whose apartments are isometric to Rn , n 1. Then asdim X asdim Rn D n (see Theorem 9.8.4). In particular, for every metric tree T with nonempty boundary at infinity, we have asdim T 1. Similarly, let X be a hyperbolic building whose apartments are isometric to Hn , n 2. Then asdim X n because asdim Hn n by Theorem 10.1.1. Actually, in all cases discussed in this remark, the equality holds (we show this below for the case n D 1).

10.2 Estimates from above To estimate the asymptotic dimension from above is both, more important and more difficult. Estimates from above have interesting applications. By a result of G.Yu [Yu], for every finitely generated group G with asdim G < 1 and finite classifying space, the Novikov higher signature conjecture holds. As a corollary, the Gromov–Lawson– Rosenberg conjecture, saying that there is no Riemannian metric with positive scalar curvature on closed K.; 1/-manifolds, holds when the fundamental group has finite asymptotic dimension. Furthermore, the extension property for Lipschitz maps is closely related to finiteness of the Assouad–Nagata dimension [LS]. On the other hand, even the question whether every cocompact Hadamard manifold has finite asymptotic dimension is open. For hyperbolic spaces the situation is better understood, and we give in Chapter 12 some general optimal estimates. In part, they are based on the following estimate for trees. Proposition 10.2.1. The asymptotic dimension as well as the asymptotic `-dimension of every metric tree T is at most 1, asdim T `-asdim T 1. Proof. Fix a base point o 2 T and consider Gromov products with respect to o,

131

10.2. Estimates from above

.xjy/ D .xjy/o . Since T is 0-hyperbolic, the triple ..xjy/; .yjz/; .xjz// is a 0-triple, i.e., the two smallest members coincide for every x; y; z 2 T . We use the notation jxj D joxj for x 2 T . Given R > 1, we consider the annuli Ak D fx 2 T W kR jxj < .k C 1/Rg for every integer k 0. The relation x x 0 if and only if .xjx 0 / .k 1=2/R is an equivalence relation on Ak because T is 0-hyperbolic. If x x 0 then jxx 0 j D jxj C jx 0 j 2.xjx 0 / < 2.k C 1/R 2.k 12 /R D 3R; i.e., each equivalence class has diameter 3R. Furthermore, for nonequivalent x; x 0 2 Ak , we have jxx 0 j D jxj C jx 0 j 2.xjx 0 / > 2kR 2.k 12 /R D R; i.e., different equivalence classes are R=2-disjoint, see Section 9.9.2. The covering U of T by all equivalence classes has mesh.U/ 3R. Now the coloring of the annuli by even and odd k makes this covering 2-colored such that members of the same color are R=2-disjoint. Thus asdim T `-asdim T 1, cf. Exercise 9.9.1. There is a general idea behind this proof that can be applied in many other cases though the estimates from above it produces are often not optimal. Let fX˛ g be a family of metric spaces. The uniform asymptotic dimension of fX˛ g is the minimal n such that for every positive d there is R > 1 and an (open) .n C 1/-colored covering U˛ of each X˛ with L.U˛ / d and mesh.U˛ / R. Lemma 10.2.2. Let f W X ! Y be a Lipschitz map between metric spaces. Suppose that asdim Y n and for each R > 1 the inverse image f 1 .BR .y// has asymptotic dimension k uniformly in y 2 Y . Then asdim X .n C 1/.k C 1/ 1. Proof. Let D Lip.f / be the Lipschitz S constant of f . Given a positive d , there is an open .n C 1/-colored covering U D a2A Ua , jAj D n C 1, of Y with L.U/ d z D f 1 .U/ of X , and mesh.U/ R for some R > 1. Note that for the covering U z we have L.U/ L.U/= d . Then there is D > 1 such that for each a 2 A, the inverse image V D f 1 .U / of every member U 2 Ua can be covered by a .k C 1/-colored family WV of open sets with L.WV / d and mesh.WV / D. Here the condition L.WV / d means that BdS.x/ \ V is contained in some member of WV for S every x 2 V . Therefore, the union U 2Ua f 1 .U / is covered by the open family V WV that is .k C 1/-colored and uniformly bounded by D with Lebesgue number d . Taking the union over all a 2 A, we obtain the open .nC1/.kC1/-colored covering W of X with mesh.W / D. Since each ball Bd X of radius d is contained in some z we see that L.W / d . Hence, asdim X .n C 1/.k C 1/ 1. member V 2 U,

132


As an application, we obtain: Proposition 10.2.3. For every m 1, we have asdim HmC1 2m C 1. D Rm .0; 1/, see Proof. Consider HmC1 in the upper half-space model RmC1 C Appendix. The Busemann function b W HmC1 ! R, b.u; v/ D ln v for .u; v/ 2 is 1-Lipschitz and the inverse images b 1 .BR / of balls BR R are isometric to RnC1 C each other for every fixed radius R > 0. In particular, b 1 .BR / have one and the same asymptotic dimension uniformly. Since the asymptotic dimension is a quasi-isometry invariant, asdim b 1 .BR / coincides with the asymptotic dimension of any horosphere S D b 1 .t/, t > 0. The horosphere S is isometric to Rm with respect to the induced intrinsic metric and thus it has open, uniformly bounded, .m C 1/-colored coverings with arbitrarily large Lebesgue number with respect to , see Example 9.1.3. To apply Lemma 10.2.2, it remains to note that jxx 0 j ln .x; x 0 / for sufficiently separated x; x 0 2 S, see Exercise A.3.3, where jxx 0 j is the distance in HmC1 . This estimate is not optimal: we show below that asdim H2 D 2 by constructing a quasi-isometric embedding into the product of two trees. The embedding approach into products of trees works in the general case and we use it to show that asdim Hm D m for every m 2, see Theorem 12.3.3. However, Proposition 10.2.3 suffices to prove that the asymptotic dimension of every hyperbolic group is finite. Moreover, we have: Corollary 10.2.4. Let X be a visual Gromov hyperbolic space whose boundary at infinity is doubling for some visual metric. Then asdim X < 1. Proof. Indeed, by the Bonk–Schramm embedding theorem, Theorem 8.2.1 and Remark 8.2.4, X is roughly similar to a subset of Hm for some m 2 N. Hence asdim X asdim Hm 2m 1.

10.3 Embedding of H2 into a product of two trees We describe a quasi-isometric embedding of H2 into a product of two trees using a tessellation of H2 by regular right-angled hexagons. The trees have infinite valence at every vertex and the embedding is equivariant with respect to the action of the Coxeter group generated by reflections on the sides of a hexagon. Consider a right-angled hexagon F in H2 . Let x0 2 F be the symmetry center, and let A D fa1 ; a2 ; a3 g and B D fb 1 ; b 2 ; b 3 g denote the alternating lines that contain the sides of the hexagon. Thus, ai \ aj D ; and b i \ b j D ; for i ¤ j . We denote by G the Coxeter group generated by reflections with respect to the lines ai , b j , i; j D 1; 2; 3 (we use the same notation for the reflections as for the corresponding lines). We then have the tessellation [ g.F / H2 D g2G

10.3. Embedding of H2 into a product of two trees

133

of the hyperbolic plane H2 by regular right-angled hexagons. An element r 2 G is called a-reflection (b-reflection) if r D gag 1 (r D gbg 1 ), a 2 A (b 2 B), g 2 G. The fixed point set of an a-reflection r is called a-mirror Mr D fx 2 H2 W r.x/ D xg. Let La denote the union of all a-mirrors. Let Ta be a the graph whose vertices are components of H2 nLa , and whose edges correspond to common boundary a-mirrors. Clearly, the graph Ta is a tree. We consider it as a metric tree assuming that the length of each edge of Ta is 1. The metric tree Tb is defined similarly.

Figure 10.1. a-mirrors and the tree Ta .

Proposition 10.3.1. There exists a quasi-isometric embedding H2 ! Ta Tb . Proof. To define a quasi-isometric map, it suffices to define it on a net. In our case, we take as a net the orbit G.x0 / D fg.x0 / W g 2 Gg H2 . We define a map fa W G.x0 / ! Ta by the rule: fa .x/ is the vertex in Ta that corresponds to the component of x 2 G.x0 /. Similarly, we define fb W G.x0 / ! Tb . The geodesic segment g.x0 /g 0 .x0 / runs inside of a tessellation hexagon during a time not longer than diam F . Furthermore, any two disjoint hexagons are separated : at least by the distance l, the edge length of F . It follows that jg.x0 /g 0 .x0 /j D m up to uniformly bounded error, where m is the number of hexagons which are met by g.x0 /g 0 .x0 /. Since every geodesic segment of the form g.x0 /g 0 .x0 / intersects any mirror at most once, we see that the product map .fa ; fb / W G.x0 / ! Ta Tb is quasi-isometric where the product of trees is considered with the l1 -metric. It is easy to prove the following corollary by directly constructing appropriate coverings (see [Gr1], 1.E). However, for higher dimensions, this does not work properly while embeddings into products of trees are still effective.

134


Corollary 10.3.2. asdim H2 D `-asdim H2 D 2. Proof. By Theorem 10.1.1, we have asdim H2 2. Using Proposition 10.3.1 and Proposition 10.2.1, we obtain `-asdim H2 2.

10.4 Supplementary results and remarks 10.4.1 Estimates from above for the asymptotic dimension Proposition 10.2.1 is well known, see e.g. [DJ], Proposition 4. The proof shown above was taken from [Ro], where it is generalized to the case of hyperbolic groups, avoiding the Bonk–Schramm embedding theorem, see also [Ro1]. The above Lemma 10.2.2 is [Ro], Lemma 9.16, which generalizes arguments from [BD1], Theorem 2. In [DJ], it was shown that Coxeter groups have finite asymptotic dimension. The following optimal estimate for a finite graph of groups is obtained in [BD2]. Theorem 10.4.1. Let be the fundamental group of a finite graph of groups with finitely generated vertex groups Gv having asdim Gv n for all vertices v. Then asdim n C 1. This result is generalized in [Be] to the case of groups acting on complexes. The last two results are further generalized and sharpened in [BD3], where the following Hurewicz-type theorem for the asymptotic dimension is proven. Theorem 10.4.2. Let f W X ! Y be a Lipschitz map of a geodesic metric space X to a metric space Y . Suppose that for every R > 0, ff 1 .BR .y// W y 2 Y g satisfies the inequality asdim n uniformly in R. Then asdim X asdim Y C n. A further progress is achieved in [BDLM], where a Hurewicz-type theorem is proven for theAssouad–Nagata dimension as well as for the linearly controlled asymptotic dimension. Refined versions of Lemma 10.2.2 have been used in [LS] to prove the following: 1. ANdim X D n for any Euclidean building X of rank n 1. The argument actually gives asdim X D `-asdim X D n; 2. Let X be a homogeneous Hadamard manifold, i.e. a Hadamard manifold with transitive isometry group. Then ANdim X < 1. Since ANdim X D maxf`-dim X; `-asdim X g and `-dim X D dim X , this yields asdim X `-asdim X < 1. Let M D G=K be a homogeneous space with a G-invariant metric, where G is a connected Lie group, and K its maximal compact subgroup. It is proven in [CG] that asdim M D dim M . In particular, this equality holds for each symmetric space of noncompact type.


135

Let S be a compact orientable surface. The curve graph X of S is a graph whose vertices are free homotopy classes of essential simple closed curves on S . Two distinct vertices are joined by an edge if the corresponding classes can be realized by disjoint curves. Let g be the genus of S, and p the number of the boundary components. If 3g 3 C p > 1 then, by a result of H. Masur and Y. Minsky [MM], the curve graph X.S / is hyperbolic with respect to the intrinsic metric with length 1 edges. The graph X D X.S / plays an important role in the study of the mapping class group of S, which naturally acts on X . It is proven in [BeFu] that if 3g 3 C p > 1 then the asymptotic dimension of X is finite, asdim X < 1.

Chapter 11

Linearly controlled metric dimension: Basic properties

Let U be an open covering of a metric space Z with mesh.U/ < 1. We define the capacity of U by cap.U/ D supfı W L.U/ ı mesh.U/g: The basic motivation of linearly controlled dimensions is that in some circumstances we need control over the Lebesgue number L.U/ of coverings involved in the definition of a dimension, e.g., that the capacity cap.U/ stays separated from zero for appropriately chosen U’s. In general, there is no reason for that. However, if we allow coverings with larger multiplicity, we can typically gain control over L.U/. Examples. (1) The topological dimension of the space Z D f0g [ f1=m W m 2 Ng Œ0; 1 is zero, dim Z D 0, because it obviously admits open coverings of multiplicity 1 with arbitrarily small mesh. However, `-dim Z > dim Z by the following argument. Given > 0, let U be an open covering of Z with m.U/ D 1 and mesh.U/ . Take the member U 2 U containing 0 and consider the maximal z D 1=m 2 U . Then 1=m D diam U , and we have L.U/ dist.z; Z n U /

1 1 2 : m1 m 1

Hence, there is no way to find a multiplicity one open covering U of Z with mesh.U/ , L.U/ ı for some fixed ı > 0 and arbitrarily small . This implies `-dim Z > 0, and in fact, `-dim Z D 1 by the monotonicity theorem because `-dim R D 1. Actually, the `-dimension of a space might be arbitrarily larger than the topological dimension; see Section 11.3.1. (2) The asymptotic dimension of the space X D fn2 W n 2 Ng with metric induced from R is zero, asdim X D 0, while `-asdim X D 1. We leave the proof to the reader as an exercise. The basic result of the chapter is the existence of separated sequences of colored coverings related to the linearly controlled metric dimension, `-dim, see Theorem 11.1.1. This theorem has a number of applications in the sequel. The proof of

138

Chapter 11. Linearly controlled metric dimension: Basic properties

quasi-symmetry invariance of the linearly controlled dimensions, which is the main result of Section 11.2, is based on Theorem 11.1.1. In the next chapter, we prove a general and sharp embedding result of hyperbolic spaces into the product of metric trees. This result uses the whole power of Theorem 11.1.1.

11.1 Separated sequences of colored coverings The main feature of linearly controlled dimensions is that the coverings involved in their definitions have the Lebesgue number at the same scale as their mesh. This is the source of astonishing flexibility in manipulating with coverings, which allows to achieve many useful properties. Theorem 11.1.1. Let Z be a metric space with finite `-dimension, `-dim Z n. Then there are constants ı, 2 .0; 1/ such that for every sufficiently small r 2 .0; 1/ there exists a sequence Uj , j 2 N, of .n C 1/-colored (by a set A) open coverings of Z with the following properties: (i) mesh Uj < r j and L.Uj / ır j for every j 2 N; (ii) for every a 2 A and any different members U 2 Uja , U 0 2 Uja0 with j 0 j , we have either Bs .U / \ U 0 D ; or Bs .U / U 0 with s D r j . The construction of fUj g is based on the saturation construction, see Section 9.6, and consists of infinitely many steps. The idea of an elementary step can be explained as follows. Recall that a family U is r-disjoint, r 0, if the multiplicity m.Br .U// D 1. Assume we have a separated family V of subsets in X , that is, any two members of V are either disjoint or one of them is contained in the other. Now given an r-disjoint family U of subsets in X with r > mesh.V /, the saturated union .U V/ [ V is a separated family, where we recall U V D fU V W U 2 Ug is the saturation of U by V, i.e., U V is the union of U and all members V 2 V with U \ V ¤ ;. This is because V Br .U / whenever U \ V ¤ ;. We begin with the following fact used in Lemma 11.1.5. Lemma 11.1.2. Given U Z, for every 0 < s < t we have B ts .U / Bs .B t .U //: Proof. If z 62 Bs .B t .U // then there is z 0 2 Z n B t .U / with d.z; z 0 / s. For every u 2 U , we have d.z 0 ; u/ t and thus d.z; u/ d.z 0 ; u/ d.z; z 0 / t s. Therefore, z 62 B ts .U /. Next we prepare a sequence of colored coverings modifying which we construct a required separated sequence.

139

11.1. Separated sequences of colored coverings

Lemma 11.1.3. Under the condition of Theorem 11.1.1, there are constants ı, r0 2 .0; 1/ such that for every r 2 .0; r0 /, there exists a sequence of open, .n C 1/yj , j 2 N, of Z such that mesh.U yj / < r j , colored (by a color set A) coverings U yj / ır j and for every a 2 A, the family U y a is ır j -disjoint for every j 2 N. L.U j Proof. It follows from the colored definition of the `-dimension that there are constants ı 0 , r0 2 .0; 1/, with the following property. Given r 2 .0; r0 /, for every j 2 N, there is an open .n C 1/-colored covering Uj of Z with mesh.Uj / < r j and yj D Bs .Uj /. L.Uj / ı 0 r j . Fix j 2 N and for s D ı 0 r j =2 consider the family U y yj / Then Uj is an open covering of Z (see Remark 9.1.1), and we have mesh.U j j 0 yj / ır with ı D ı =2. Furthermore, for every color a 2 A, mesh.Uj / < r , L.U a j y the family Uj is ır -disjoint. We need a qualified version of the saturation construction. Given s 0, U Z and the family V of subsets in Z, the s-saturation of U by V is the union U s V of U and all members V 2 V with Bs .U / \ Bs .V / ¤ ;. Lemma 11.1.4. Assume that a family of sets V is ıs-disjoint for ı 2 .0; 2=3, s > 0, and mesh.V / < 2s. Then the operation U 7! U D Bıs .B4s .U / ıs V / does not increase any set U Z, U U , and for every V 2 V it holds that either Bıs .V / \ U D ; or Bıs .V / U .

V B4s .U /

Figure 11.1. Operation U 7! U .

Proof. For every V 2 V , we have that diam Bıs .V / < 2s C 2ıs and therefore ıs C diam Bıs .V / < 2s.1 C 3ı=2/ 4s. Hence if Bıs .V / intersects Bıs .B4s .U // then Bıs .B4s .U / [ V / U , and U U , in particular, in that case Bıs .V / U . Otherwise, Bıs .V / misses the ıs-neighborhood of B4s .U / as well as the one of every other member of V , and thus Bıs .V / \ U D ;.

140


This sophisticated version U 7! U of the saturation is chosen to provide also the following important property which allows to apply the operation inductively infinitely many times leaving invariant quantitative separation conditions. Lemma 11.1.5. Under the conditions of Lemma 11.1.4 assume that B t .U1 / U2 ; t > 4s; for some sets U1 , U2 Z. Then B t 0 .U1 / U2 for t 0 D t 4s. Proof. By Lemma 11.1.4, we have B t 0 .U1 / B t 0 .U1 /. By Lemma 11.1.2, B t4s .U1 / B4s .B t .U1 //. Finally, we have B4s .B t .U1 // B4s .U2 / U2 . yj , j 2 N, Proof of Theorem 11.1.1. We take the sequence of the colored coverings U constructed in Lemma 11.1.3 for r 2 .0; r0 /, and modify it to obtain a separated 2r sequence. We further assume that 1r ı=4 1=6. a y a . Then the family Ua is Fix a color a 2 A and define V1 D Ua1;1 ´ U 1 1;1 a ır-disjoint and mesh.U1;1 / < r. Assume that for k 1 the family Vka is already defined and has the following properties: S a ; (i) Vka D jkD1 Uj;k a a (ii) the family Uj;k is ır j -disjoint and mesh.Uj;k / < r j for every 1 j k; a (iii) given 1 j 0 < j k, for every U 0 2 Uja0 ;k , U 2 Uj;k , we have either B t .U / \ U 0 D ; with t D ır j =2 or B t .U / U 0 with t D k;j r j , where k;j is defined recurrently by j;j D ı=2 and k;j D k1;j 2r kj for k > j .

We define

a ya / [ U ya VkC1 ´ Bıs .B4s .Vka / ıs U kC1 kC1 SkC1 a a kC1 with s D r =2. Then VkC1 D j D1 Uj;kC1 , where a a ya / D Bıs .B4s .Uj;k / ıs U Uj;kC1 kC1

y a . Since the family U y a is 2ıs-disjoint for 1 j k and UakC1;kC1 D U kC1 kC1 y a / < 2s, we can apply Lemma 11.1.4, by which every U 2 Ua and mesh.U j;kC1 kC1 a with 1 j k is contained in the appropriate U 2 Uj;k , in particular, the family a a a Uj;kC1 is ır j -disjoint and mesh.Uj;kC1 / mesh.Uj;k / < r j . Furthermore, for every Uy 2 Ua we have either Bıs .Uy / \ U D ; or Bıs .Uy / U . kC1;kC1

Now if U 0 2 Uja0 ;k with j 0 < j and B t .U / \ U 0 D ; with t D ır j =2 then B t .U / \ U 0 D ; since U U and U 0 U 0 . In the case when B t .U / U 0 with t D j;k r j , we have B t2r kC1 .U / U 0 by Lemma 11.1.5 and t 2r kC1 D

11.2. Quasi-symmetry invariance of `-dim

141

2r ı=4 kC1;j r j with kC1;j D k;j 2r kC1j . Note that limk!1 k;j D ı=2 1r for every j > 1. Therefore, for every color a 2 A, we have the sequence Vka , k 2 N, of families of sets in Z with properties (i)–(iii). It follows from the definition of the -operation a that every member U 2 VkC1 is contained in its well-defined predecessor U 2 Vka , a a if and only if U 2 Uj;k . In this sense, the sequence Vka is moreover, U 2 Uj;kC1 T a a a a monotone, Vk VkC1 , and we define Uj D Int Uj;k , where Int U is the kj S a interior of a subset U Z, Uj D a2A Uj for every j 2 N. P 2r j kC1 yj / and We put sOj D D 1r r . Then sOj ır j =4 < L.U kj 2r y a / Ua for every a 2 A, j 2 N. By Remark 9.1.1, the family Uj is BOs .U j

j

j

still an open .n C 1/-colored covering of Z with L.Uj / ır j sOj ır j =2. From (ii) we obtain mesh.Uj / < r j and the family Uja is ır j -disjoint for every a 2 A, j 2 N. Property (iii) implies that given 1 j 0 < j , for every U 0 2 Uja0 , U 2 Uja , we have either B t .U / \ U 0 D ; or B t .U / U 0 with t D r j , ı=4.

11.2 Quasi-symmetry invariance of `-dim By the definition, the `-dimension is a bilipschitz invariant. On the other hand it turns out that the `-dimension perfectly fits a number of questions related to the boundary at infinity of hyperbolic spaces taken with visual metrics. Recall that visual metrics (based at interior points of a hyperbolic space) are only defined up to a quasi-symmetry transformation. Surprisingly, the `-dimension is actually a quasi-symmetry invariant of uniformly perfect spaces. In that way the `-dimension of the boundary at infinity of a hyperbolic space is well defined independently of the choice of a visual metric. Theorem 11.2.1. The `-dimension is a quasi-symmetry invariant of uniformly perfect metric spaces, that is, if there is a quasi-symmetric homeomorphism f W X ! Y between uniformly perfect metric spaces then `-dim X D `-dim Y . The proof is based on the existence of a separated sequence of colored coverings, Theorem 11.1.1. The idea can be explained as follows. One easily realizes that there is no reasonable control over the capacity cap.U/ of a covering U of a metric space X under quasi-symmetry homeomorphisms, and thus there is no way for a straightforward argument. However, the local capacity of U defined by caploc .U/ D inf supfı W L.U; x/ ı mesh.U; x/g; x2X

where mesh.U; z/ D supfdiam U W x 2 U 2 Ug, has the advantage over the capacity that its positivity is preserved under quasi-symmetries quantitatively, see Lemma 11.2.2. Furthermore, if U is c-balanced, that is inffdiam.U / W U 2 Ug c mesh.U/ with c > 0, then cap.U/ c caploc .U/, see Lemma 11.2.3. We combine these two facts to obtain the desired estimate in the following way.

142


Taking a separated sequence of colored coverings Uj , j 2 N, of X , we construct out of it a covering V of X with local capacity uniformly separated from 0, see Lemma 11.2.6, such that the image f .V/ is balanced and has an arbitrarily small mesh, see Lemma 11.2.7. Then the capacity of the covering f .V / of Y is positive quantitatively, which implies `-dim Y `-dim X . Recall that a (nonconstant) map f W X ! Y between metric spaces is -quasisymmetric with control function W Œ0; 1/ ! Œ0; 1/ if from jxaj t jxbj, it follows that jf .x/f .a/j .t /jf .x/f .b/j for any a; b; x 2 X and all t 0; see Definition 5.2.11. Lemma 11.2.2. Let U be an open covering of a metric space Z with finite mesh, mesh.U/ < 1, let f W Z ! Z 0 be an -quasi-symmetry and let U0 D f .U/ be the image of U. Then for the local capacities of U and U0 , we have 2 1 16 : caploc .U0 / caploc .U/ Proof. We can assume that caploc .U/ > 0 and that no member of U coincides with Z, since otherwise caploc .U/ D caploc .U0 / D 1. We fix z 2 Z and consider U 2 U for which z 2 U and dist.z; Z n U / L.U; z/=2. For z 0 D f .z/ and U 0 D f .U / there is a0 2 Z 0 n U 0 with jz 0 a0 j 2 dist.z 0 ; Z 0 n U 0 /. Then jz 0 a0 j 2L.U0 ; z 0 /, and for a D f 1 .a0 / we have jzaj dist.z; Z n U / L.U; z/=2. Similarly, consider V 0 2 U0 with z 0 2 V 0 and diam V 0 mesh.U0 ; z 0 /=2. There is b 0 2 V 0 with jz 0 b 0 j diam V 0 =4. Then jz 0 b 0 j mesh.U0 ; z 0 /=8, and for b ´ f 1 .b 0 / we have jzbj mesh.U; z/. Therefore, we have caploc .U/

L.U; z/ 2jzaj mesh.U; z/ jzbj

and

jzbj t jzaj

with t D 2= caploc .U/. It follows that jz 0 b 0 j .t /jz 0 a0 j and L.U0 ; z 0 / jz 0 a0 j .16 .t //1 mesh.U0 ; z 0 / 16jz 0 b 0 j 1 . for every z 0 2 Z 0 . Then caploc .U0 / 16 cap 2.U/ loc

Lemma 11.2.3. If an open covering U of a metric space Z is c1 -balanced and its local capacity satisfies caploc .U/ c0 , then cap.U/ c0 c1 . Proof. Since U is c1 -balanced, we have mesh.U; z/ c1 mesh.U/ for every z 2 Z. Since caploc .U/ c0 , we have L.U; z/ c0 mesh.U; z/ for every z 2 Z. Therefore L.U/ c0 c1 mesh.U/. It suffices to prove that `-dim Y `-dim X , because then the opposite inequality is obtained be permuting X and Y . Thus we assume that n D `-dim X < 1.

11.2. Quasi-symmetry invariance of `-dim

143

By Theorem 11.1.1, for some ı 2 .0; 1/ and for every sufficiently small r > 0, there is a separated sequence of .nC1/-colored coverings Uj of X with mesh.Uj / < r j and L.Uj / ır j . We take r < ı. Then mesh.Uj C1 / < L.Uj / and, hence, the covering Uj C1 is inscribed in Uj for every j 2 N, that is, every member of Uj C1 is contained in some member of Uj . Since X is uniformly perfect, there is 2 .0; 1/ such that for every x 2 X and every r > 0 with X n Br .x/ ¤ ; we have Br .x/ n Br .x/ ¤ ;. We can additionally assume that diam U L.Uj / for every U 2 Uj , since if diam U < L.Uj / then U cannot coincide with the ball B .x/ of radius D L.Uj / centered at any point x 2 U because X is -uniformly perfect. Thus U is contained in another member U 0 2 Uj and hence it can be deleted from Uj without destroying any property of the sequence. This is the only place where we use the uniform perfection condition. S We put U D j 2N Uj and for s > 0 consider the family U.s/ D fU 2 U W diam f .U / sg. Lemma 11.2.4. For every s > 0, the family U.s/ is a covering of X . Proof. We fix x 2 X , consider an element x 0 2 X different from x and put y D f .x/, y 0 D f .x 0 /. For every j 2 N, there is Uj 2 Uj containing x. Take yj 2 f .Uj / with diam f .Uj / 4jyyj j and consider xj D f 1 .yj /. Then jxxj j tj jxx 0 j with tj ! 0 as j ! 1, since diam Uj r j ! 0. Therefore, diam f .Uj / 4jyyj j 4 .tj /jyy 0 j s for sufficiently large j . Hence, Uj 2 U.s/. A family V U.s/ is maximal if every U 2 U.s/ is contained in some V 2 V and neither of different V , V 0 2 V is contained in the other. Lemma 11.2.5. For every s > 0, there is a maximal family V U.s/. Every maximal family V U.s/ is an .n C 1/-colored covering of X . Proof. Given s > 0, we construct a family V U.s/ by deleting every U 2 U.s/ which is contained in some other U 0 2 U.s/. Now V is what remains. We need only to check that for every U 2 U.s/ there is a maximal U 0 2 U.s/ with U U 0 . Since the covering Uj is .n C 1/-colored, for every j 2 N there are only finitely many U 0 2 Uj containing U (because all of them must have different colors). Since mesh.Uj / ! 0 as j ! 1, there are only finitely many U 0 2 U.s/ containing U and hence there is a maximal U 0 2 U.s/ among them. Let V U.s/ be a maximal family. By Lemma 11.2.4, the family U.s/ is a covering of X , and it follows from the definition of a maximal family that V is also a covering of X . It follows from Theorem 11.1.1 (ii) that different V , V 0 2 V having one and the same color are disjoint. Thus V is .n C 1/-colored. Lemma 11.2.6. There is a constant > 0 depending only on ı, , r and such that for every s > 0, every maximal covering V U.s/ has the local capacity caploc .V/ .

144


Proof. Let V U.s/ be a maximal family. Given x 2 X we put j D j.x/ D minfi 2 N W x 2 V 2 V \ Ui g. Then mesh.V ; x/ < r j . We fix V 2 V \ Uj containing x; v 2 V with 4jxvj diam V and note that diam V L.Uj / ır j by our assumptions. Next we fix > 0 with 4 .4=ı/ 1. Now we check that for i 2 N with r ij every U 2 Ui containing x is a member of U.s/. There is u 2 U with 4jf .x/f .u/j diam f .U /. We have jxuj t jxvj for some t 4 diam U= diam V 4r ij =ı 4=ı. Then diam f .U / 4jf .x/f .u/j 4 .4=ı/jf .x/f .v/j diam f .V / s, thus U 2 U.s/. Therefore, L.V ; x/ L.Ui / ır i . Assuming that i is taken minimal with L.V;x/ r ij , we obtain L.V; x/ ı r j C1 . Thus mesh.V;x/ D ı r for every x 2 X and caploc .V/ . Lemma 11.2.7. Given a maximal family V U.s/, the .n C 1/-colored covering W D f .V/ of Y satisfies diam W s=4 .t/ for every W 2 W , where t D 4=ır. In particular, W is c-balanced with c 1=4 .t /. Proof. Note that mesh.W / s by the definition of U.s/. Take any W 2 W and consider V D f 1 .W /. We can assume that V 2 Uj for some j 2 N. Then diam V L.Uj / ır j by our assumption on the sequence fUj g. For any U 2 U with V U , we have diam f .U / > s, since the family V is maximal. The covering Uj is inscribed in Uj 1 , and thus there is U 2 Uj 1 containing V , in particular, diam f .U / > s. Take y 2 W f .U /. There is y 0 2 f .U / with jyy 0 j diam f .U /=4 > s=4. For x D f 1 .y/, x 0 D f 1 .y 0 / we have jxx 0 j diam U mesh.Uj 1 / r j 1 . There is v 2 V with jxvj diam V =4 ır j =4. Thus jxx 0 j r j 1 t jxvj for t D 4=ır. For w D f .v/ 2 W we obtain jyy 0 j .t /jywj .t / diam W . Hence, diam W s=4 .t /. Proof of Theorem 11.2.1. By Lemmas 11.2.5 and 11.2.7 for every s > 0, we have an open .n C 1/-colored covering W D f .V / of Y with cs mesh.W / s which is c-balanced, c 1=4 .t /, where t D 4=ır. Moreover, by Lemmas 11.2.6 and 11.2.2, its local capacity caploc .W / d , where the constant d > 0 depends only on , ı, , r. Then, by Lemma 11.2.3, we have cap.W / c d independently of s and thus L.W / c 2 d s. This shows that `-dim Y n. Remark 11.2.8. Separation property (ii) from Theorem 11.1.1 has only been used in the proof of Theorem 11.2.1 in a weak qualitative form; see Lemma 11.2.5. Combining with the monotonicity theorem, we obtain Corollary 11.2.9. Assume that there is a quasi-symmetric f W X ! Y between uniformly perfect metric spaces. Then `-dim X `-dim Y .


145

11.3 Supplementary results and remarks 11.3.1 The `-dimension versus the topological dimension The following proposition allows to construct examples of metric spaces with different topological and linearly controlled metric dimensions, and the distinction can be arbitrarily large. Proposition 11.3.1. Let X , Y be bounded metric spaces such that for every " > 0 there is A X and a homothety h" W A ! Y with "-dense image, dist.y; h" .A// < " for every y 2 Y . Then `-dim X dim Y . Proof. We can assume that dim Y > 0, in particular, diam Y > 0. Then we have ."/ 0 > 0 as " ! 0 for the coefficient ."/ of the homothety h" , because X is bounded. Assume that n D `-dim X < dim Y . There is ı > 0 such that for every sufficiently small > 0 there is an open covering U of X of multiplicity n C 1 with mesh.U / and L.U / ı. Using the estimate ."/ 0 , we can find D ."/ such that ."/ ."/ ! 0 as " ! 0 and ı."/ ."/ 4". Then for the covering V" D h" .U / of h" .A/, we have mesh.V" / ."/ ."/ and L.V" / ı."/ ."/. Furthermore, m.V" / n C 1. Therefore, the family V"0 D B2" .V" / still covers f .A/. Taking the "-neighborhood in Y of every V 2 V"0 , we obtain an open covering V of Y with mesh.V / mesh.V" / ! 0 as " ! 0. Let us estimate the multiplicity of V. Assume that y 2 Y is a common point of members Vj 2 V , j 2 J . By the definition of V , for every j 2 J , there is ai 2 A such that f .aj / 2 Vj0 2 V"0 and jf .aj /yj < ". Then the mutual distances of the points f .aj /, j 2 J , are < 2". Since Vj0 D B2" .Uj / for Uj 2 V" , we see that every point f .aj /, j 2 J , is contained in every Ui , i 2 J , and therefore jJ j n C 1 because the multiplicity of V" is at most n C 1. Hence, m.V / n C 1 and dim Y n, a contradiction. As an application, we obtain the following examples. Let Z D f0g [ f1=m W m 2 Ng be the space from the example (1) on p. 137. Then `-dim Z n D n for any n 1, while dim Z n D 0 (e.g. by the product theorem). Indeed, the spaces X D Z n and Y D Œ0; 1n , obviously, satisfy the condition of Proposition 11.3.1, thus `-dim Z n dim Y D n (we have equality here because Z n Y ). Further examples: Take any monotone sequence of positive "k ! 0, "1 D 1=3, and repeat the construction of the standard ternary Cantor set K Œ0; 1, only removing at every k-th step, k 1, instead of the .1=3/k -length segments, the middle segments of length sk D "k lk , l1 D 1, where the length lkC1 of the segments obtained after processing the k-th step is defined recurrently by 2lkC1 Csk D lk . The resulting compact space Ka Œ0; 1 is homeomorphic to K. We easily see that `-dim K D 0. However, X D Ka and Y D Œ0; 1 satisfy the condition of Proposition 11.3.1, thus

146


`-dim Ka D 1, while dim Ka D 0. Similarly, one can construct ‘exotic’ Sierpinski carpets, Menger curves etc with the capacity dimension strictly bigger than the topological dimension. Any of those compact metric spaces is not quasi-symmetric to any locally self-similar space, in particular, it is not quasi-symmetric to the boundary (viewed with a visual metric) of a hyperbolic group. To compare, it is well known that the boundary at infinity of a typical hyperbolic group is homeomorphic to the Menger curve.

11.3.2 Quasi-symmetry invariance of the Assouad–Nagata dimension It is proved in [LS] that the Assouad–Nagata dimension is a quasi-symmetry invariant of arbitrary metric spaces. Since for bounded metric spaces the Assouad–Nagata dimension coincides with the linearly controlled dimension, this strengthens Theorem 11.2.1 in the case of bounded spaces because then we can omit the uniform perfection condition. As the following example due to Nina Lebedeva shows, in the general case the linearly controlled dimension as well as the linearly controlled asymptotic dimension is not a quasi-symmetry invariant. Example 11.3.2 (N. Lebedeva). Let X D f2k C i W i; k 2 N; i kg and Y D f2k Ci=k W i; k 2 N; i kg be metric spaces with metrics induced from R. Then one easily checks that `-dim X D 0, `-asdim X D 1 and `-dim Y D 1, `-asdim Y D 0, in particular, ANdim X D ANdim Y D 1. On the other hand, the natural map f W X ! Y , f .2k C i / D 2k C i=k is a quasi-symmetry (the map is well defined because the representation x D 2k C i is unique for every x 2 X ). Bibliographical note. The construction of Theorem 11.1.1 fits a large array of similar constructions for various dimensions, e.g., see [Os2] for the topological dimension, [Dr2] for the asymptotic dimension, [LS] for the Assouad–Nagata dimension to name few. Proposition 11.3.1 and its application to examples of metric spaces with different topological and linearly controlled dimensions is due to N. Lebedeva; see [BL].

Chapter 12

Linearly controlled metric dimension: Applications

12.1 Embedding into the product of trees Here we use the whole power of Theorem 11.1.1 to prove the following embedding result. Theorem 12.1.1. Let X be a visual Gromov hyperbolic space whose boundary at infinity has finite `-dimension, `-dim.@1 X / < 1. Then there exists a quasi-isometric embedding f W X ! T1 Tn of X into an n-fold product of metric trees T1 ; : : : ; Tn with n D `-dim.@1 X / C 1. Remark 12.1.2. On a finite product of metric spaces we consider any of standard product metrics, l1 , l2 or l1 , which are the sum, the square root of the sum of squares, or the maximal of the coordinate distances respectively. These product metrics are bilipschitz equivalent. It is only important that the distance between any two points in the product is at least the distance between their projection to any factor. All metric trees Tk , k D 1; : : : ; n, constructed during the proof are simplicial with length 1 edges, that is, metric trees which admit a triangulation. We can speak about vertices and edges of a simplicial tree. The valence of a vertex is the number of edges adjacent to it. Vertices of Tk , k D 1; : : : ; n, typically have infinite valence. The estimate of the number of tree-factors, needed for a quasi-isometric embedding given by Theorem 12.1.1, is sharp: we show in Chapter 13 that this embedding theorem is optimal in a strong sense. The class of Gromov hyperbolic spaces to which Theorem 12.1.1 can be applied contains all visual Gromov hyperbolic spaces with doubling boundary at infinity; see Lemma 12.2.2 and Sections 2.3, 8.3.1. In particular, all hyperbolic geodesic spaces with bounded geometry as well as cobounded ones are in this class, see Corollary 2.3.7 and Theorem 8.3.9, that includes all Gromov hyperbolic groups and Hadamard manifolds with pinched negative curvature. The idea of the embedding can be explained as follows. First of all, replacing the space X by a hyperbolic approximation of its boundary at infinity, we assume that X is a hyperbolic approximation of a metric space Z with finite `-dimension. As usual,

148

Chapter 12. Linearly controlled metric dimension: Applications

we use notation V D fVj g for the vertex set of X and assume that X is truncated for a bounded Z, see Chapter 6. Theorem 11.1.1 now takes the following form. Theorem 12.1.3. Let Z be a bounded metric space with finite `-dimension, n D `-dim Z < 1. Then for every sufficiently small r, r 2 .0; r0 /, there exists a sequence Uj , j 0, of .n C 1/-colored (by a set A) open coverings of Z with Ua0 D fZg for all a 2 A and mesh Uj < r j for every j 2 N such that for any hyperbolic approximation X of Z with parameter r, we have the following. (1) For every v 2 Vj C1 , j 0, there is U 2 Uj such that B.v/ U . (2) For every a 2 A and for different members U 2 Uja , U 0 2 Uja0 with j 0 j S the following holds: let B.U / D fB.v/, v 2 Vj C1 ; B.v/ \ U ¤ ;g; then either B.U / U 0 or B.U / \ U 0 D ;. S Condition (2) means that the family Ua D j Uja has combinatorially the structure of a tree Ta for every color a 2 A: the vertices are the members of the family and the edges are defined by the inclusion relation. We formalize this in the notion of a levelled tree, see Section 12.1.1. It is now possible to define a map X ! Ta . The map is defined on the set V of vertices. A vertex v, which is just the ball B.v/ X , is mapped to theQ smallest a U 2 U such that B.v/ U . It turns out that the product map V ! a Ta is quasi-isometric. Proof of Theorem 12.1.3. Choosing r > 0 sufficiently small, we can assume that k0 2 f1; 0g where k0 D k0 .diam Z; r/ is the largest integer k with diam Z < r k , see Section 6.4.1. Since X is truncated, the vertex sets Vj are nonempty for j k0 only. By Theorem 11.1.1, there are constants ı, 2 .0; 1/ such that for every sufficiently small r > 0 there exists a sequence Uj , j 2 N, of .n C 1/-colored (by a set A) open coverings of Z with the following properties: (i) mesh Uj < r j and L.Uj / ır j for every j 2 N; (ii) for every a 2 A and for different members U 2 Uja , U 0 2 Uja0 with j 0 j , we have either Bs .U / \ U 0 D ;, or Bs .U / U 0 for s D r j . Furthermore, we add to the sequence Uj , j 2 N, the member U0 which consists of (copies of) Z for every color a 2 A, Ua0 D fZg. Because L.Uj / ır j , every ball B .z/ Z of radius ır j is contained in some member U 2 Uj . Assuming that r < ı=2, we obtain property (1) because recall that B.v/ D B2r j C1 .v/ for v 2 Vj C1 . Finally, assume additionally that r < =4. Now if a color a 2 A and different members U 2 Uja , U 0 2 Uja0 with j 0 j are given, we have B.U / Bs .U / with s D r j by the choice of r and the definition of B.U /. Hence, property (2).

12.1. Embedding into the product of trees

149

12.1.1 Levelled trees A poset (partially ordered set) V is called directed if for any u; v 2 V there is w 2 V with u w, v w. A levelled tree T is a directed poset V , called the vertex set of T , together with a level function ` W V ! Z which is strictly monotone in the following sense: If v; v 0 2 V are different elements and v v 0 , then `.v/ > `.v 0 /. In this case, v 0 is called an ancestor of v, and v is a descendant of v 0 . We require that the following condition is satisfied: .C/ if distinct elements v; v 0 2 V have a common descendant then one of them is an ancestor of the other. A collection E of two point subsets of V called the edge set of T is defined by the condition: A pair of vertices .v; v 0 / forms an edge, .v; v 0 / 2 E, if and only if one of its members, say v 0 , is an ancestor of the other and the level `.v 0 / is maximal with this property. If there is a vertex which has no ancestor, then such a vertex is unique by directedness, and it is called the root of T . Note that the root is an ancestor of every other vertex. It follows from .C/ that for every vertex v 2 V (except the root) there is exactly one edge .v; v 0 / in which v is the descendant. Hence, by the uniqueness part, T has no circuit. By the existence part (together with .C/), every vertex is connected with any of its ancestors by a sequence of edges in T . Now because V is directed, any two vertices in T are connected by a sequence of edges, i.e., T is connected. Therefore, T is a simplicial tree. Assuming that the length of every edge equals 1, we see that each levelled tree is a simplicial metric tree with length 1 edges.

12.1.2 Colored trees Let Z be a bounded metric space with finite `-dimension, n D `-dim Z < 1; let X be a hyperbolic approximation of Z with sufficiently small parameter r < minfdiam Z; 1= diam Zg satisfying the condition of Theorem 12.1.3; and let fUj g, j 0, be a sequence of .n C 1/-colored (by a set A) open coverings of Z with parameter r as in Theorem 12.1.3. Then recall that k0 D k0 .diam Z; r/ D 0 or 1. We use notation d.v; v 0 / for the distance in Z of vertices v; v 0 2 V considered as points of Z, and jvv 0 j for the distance in X. For every a 2 A, we defineSa rooted levelled tree Ta as follows. Its vertex set Ua is the disjoint unionUa D j 0 Uja with the root va D Z which is the unique member of Ua0 . The partial order of Ua is defined by the inclusion relation, U U 0 if and only if U U 0 , and this order is directed due to the existence of the root. We say that a vertex U 2 Uja has level j ; this defines the level function. It follows easily from separation property (2) that the level function is strictly monotone. Then

150


a vertex U 2 Uja is a descendant of U 0 2 Uja0 if j 0 < j and U U 0 . It follows from separation property (2) that .C/ is satisfied. Hence, Ta is a rooted levelled tree: a pair of vertices U 2 Uja , U 0 2 Uja0 forms an edge of Ta if and only if it is a pair (descendant, ancestor), and the level j 0 of the ancestor U 0 is maximal with this property. Typically, vertices of the tree Ta have infinite valence. We use notation jU U 0 j for the distance in Ta between its vertices U , U 0 , and jU j for the distance jU va j. Note that jva j D 0 and jU j j k0 if U 2 Uja . Furthermore, if U 0 2 Uja0 is an ancestor of U 2 Uja , then the level difference j j 0 might be arbitrarily large compared to the distance jU U 0 j even if .U; U 0 / is an edge of Ta .

12.1.3 A map into the product of colored trees We now define a map fa W V ! Ta , V is the vertex set of the hyperbolic approximation X of Z, as follows. The root of X is mapped into the root of Ta , fa .v/ D va for the unique member v 2 Vk0 . Given v 2 Vj , j > k0 , we let fa .v/ D U 2 Uja0 be the covering element containing the ball B.v/, B.v/ U , and j 0 j 1 is maximal with this property. By Theorem 12.1.3 (2), fa .v/ is well defined. Lemma 12.1.4. For every a 2 A, the map fa W V ! Ta is Lipschitz, jfa .v/fa .v 0 /j 2jvv 0 j for every v; v 0 2 V: Proof. Since the hyperbolic approximation X is geodesic, it suffices to estimate the distance jfa .v/fa .v 0 /j for neighbors, jvv 0 j D 1. Assume that the edge .v; v 0 / X is horizontal, i.e., v; v 0 2 Vj for some j k0 and the balls B.v/, B.v 0 / intersect. Thus the covering elements U D fa .v/, U 0 D fa .v 0 / also intersect. By separation property (2), either U D U 0 and so fa .v/ D fa .v 0 / or these elements have different levels and one of them is contained in the other, say U 2 Uai , U 0 2 Uai0 with i > i 0 and U U 0 . It follows from the definition of fa that i < j and from separation property (2) that any U 00 2 Uai00 , i 00 < i , intersecting U , also contains B.v 0 /. Thus .U; U 0 / Ta is an edge by the definition of fa , and jfa .v/fa .v 0 /j D 1 in this case. Assume now that the edge .v; v 0 / X is radial, say v 2 Vj C1 , v 0 2 Vj . Then B.v/ B.v 0 / and as in the previous case, U \ U 0 ¤ ;. Thus we can assume that these elements have different levels, U 2 Uai , U 0 2 Uai0 with i ¤ i 0 , and one of them is contained in the other. Moreover, i > i 0 because B.v/ U 0 , thus U U 0 . We have i j by definition of fa , and as above, any ancestor U 00 2 Uai00 , i 00 < i 1, of U separated from U by at least one generation, jU U 00 j 2, also contains B.v 0 /. Therefore, it follows from the definition of fa that at most one generation can separate U from its ancestor U 0 , and jfa .v/fa .v 0 /j 2.

151


To prove Theorem 12.1.1, it suffices to show that the map Y Y fa W V ! Ta f D a

a

is quasi-isometric. Indeed, by Corollaries 7.1.5 and 7.1.6, every visual hyperbolic space is roughly similar to a subspace of a hyperbolic approximation of its boundary at infinity. The map f defined by its coordinate maps fa W X ! Ta is Lipschitz by Lemma 12.1.4. To prove that f is roughly bilipschitz, we begin with the following lemma, which is the main ingredient of the proof. For i 0, we denote by Ta;i D Uai the vertex set of Ta of level i . Lemma 12.1.5. Given v 2 Vj C1 , j 0, for every integer i , 0 i j , there is a color a 2 A such that dist.fa .v/; Ta;i / M with M C 1 .j i C 1/=jAj. Furthermore, if for k i a vertex w 2 Ta;k is the lowest level vertex of the segment fa .v/w Ta , then jfa .v/wj M . Proof. Consider a radial geodesic viC1 : : : vj C1 X with vertices vm 2 Vm , where vj C1 D v. This means in particular that B.vmC1 / B.vm / for every m D i C 1; : : : ; j . By Theorem 12.1.3 (1) for every vertex vmC1 , there is a covering element Um 2 Um with B.vmC1 / Um , m D i; : : : ; j . There is a color a 2 A such that the set fUi ; : : : ; Uj g contains M C 1 .j i C 1/=jAj members having the color a, i.e. each of those Um 2 Uam . Since B.v/ Um for every m j , we have U D fa .v/ Um for every Um having the color a by the definition of fa and separation property (2). Using again the separation property, we obtain that any path in Ta between fa .v/ and the set Ta;i must contain at least M C1 vertices and hence dist.fa .v/; Ta;i / M . Finally, let W 2 Uak be the set corresponding to the vertex w of Ta . By the assumption on w, the set W contains U and every set from the list fUi ; : : : ; Uj g having the color a. Hence, jfa .v/wj M . We say that distinct points v 2 Vj , v 0 2 Vj 0 , j j 0 0 are horizontally close to 0 each other if d.v; v 0 / < r j . This terminology is motivated by the fact that there is a geodesic segment vv 0 X which is almost radial. More precisely, we have Lemma 12.1.6. Assume that the distinct points v; v 0 2 V are horizontally close to each other. Then their levels are different and the upper level ball is contained in the lower level ball, say `.v/ > `.v 0 /, B.v/ B.v 0 /. In particular, jvv 0 j j`.v/ `.v 0 /j C 1: Proof. We can assume that v 2 Vj , v 0 2 Vj 0 , j j 0 0. Then j > j 0 because v, v 0 are distinct and because Vj Z is r j -separated for every j 0. Furthermore, 0 0 B.v/ B.v 0 / because 2r j C r j < 2r j . By Corollary 6.2.7, we have jvv 0 j .j j 0 / C 1.

152


Lemma 12.1.7. Given v; v 0 2 V horizontally close to each other, we have that fa .v/fa .v 0 / Ta is a radial segment for every color a 2 A, i.e. its lowest level vertex is one of its ends, and there is a color a 2 A such that jvv 0 j jAjjfa .v/fa .v 0 /j C ; where D jAj C 1. Proof. We can assume that v, v 0 are distinct. Then, by Lemma 12.1.6, their levels are different, say `.v/ > `.v 0 /, and B.v/ B.v 0 /. Thus fa .v/ D fa .v 0 / or fa .v/ is a descendant of fa .v 0 / for every color a 2 A. In any case, fa .v 0 / is the lowest level vertex of the segment fa .v/fa .v 0 / Ta . On the other hand, fa .v 0 / 2 Ta;ma with ma `.v 0 / for all a 2 A by the definition of fa . By Lemma 12.1.5, we have jfa .v/fa .v 0 /j D dist.fa .v/; Ta;ma / M with M C 1 .`.v/ `.v 0 //=jAj for some color a 2 A. Therefore, using again Lemma 12.1.6, we obtain jvv 0 j j`.v/ `.v 0 /j C 1 jAj.M C 1/ C 1 jAjjfa .v/fa .v 0 /j C :

Lemma 12.1.7 shows a rough bilipschitz property of the map f in the case v, v 0 are horizontally close to each other. The opposite case is more complicated. We say that vertices v; v 0 2 V are horizontally distinct if `.v/, `.v 0 / 0 and they 0 are not horizontally close to each other, d.v; v 0 / r minf`.v/;`.v /g . In this case, there is an integer l with r l d.v; v 0 / < r l1 : We call l the critical level of v, v 0 . Note that r l diam Z < r k0 and thus l > k0 , in particular, l 0. Furthermore, l minf`.v/; `.v 0 /g. Recall that a similar notion of a critical level has been used in the proof of the Assouad embedding theorem; see Chapter 8. Lemma 12.1.8. Let l be the critical level of horizontally distinct vertices v, v 0 2 V . Then jvv 0 j `.v/ C `.v 0 / 2l C 3. Proof. Consider a central ancestor radial geodesic v in X between the root of X and v, see Lemma 6.2.1. For the vertex u 2 v of the level l 1, u 2 Vl1 , we have jvuj D `.v/l C1 and d.v; u/ r l1 . Thus d.u; v 0 / d.u; v/Cd.v; v 0 / < 2r l1 . Therefore, v 0 2 B.u/, the balls B.u/, B.v 0 / intersect and hence, jv 0 uj `.v 0 / l C 2 by Corollary 6.2.7. We see that jvv 0 j jvujCjuv 0 j `.v/C`.v 0 /2l.v; v 0 /C3. Lemma 12.1.9. Let l be the critical level of horizontally distinct v; v 0 2 V . Assume that v 2 U , v 0 2 U 0 for some elements U , U 0 2 Ta and some color a 2 A. Then k < l for the lowest level vertex W 2 Ta;k of the segment U U 0 Ta .

153


Proof. The set W 2 Uak contains both U and U 0 , and we have by Theorem 12.1.3, d.v; v 0 / diam W < r k . It follows r l < r k and thus k < l. Lemma 12.1.10. Given horizontally distinct v; v 0 2 V , `.v/ `.v 0 /, there is a color a 2 A such that jvv 0 j 2jAjjfa .v/wj C , where w 2 fa .v/fa .v 0 / Ta is the lowest level vertex and D 2jAj C 1. Proof. By Lemma 12.1.9 for every color a 2 A, any path in Ta between fa .v/ and fa .v 0 / passes through a vertex of a level k < l, where l D l.v; v 0 / is the critical level of v, v 0 . By Lemma 12.1.5, there is a color a 2 A such that dist.fa .v/; Ta;l1 / M with M C 1 .`.v/ l C 1/=jAj. Let w 2 Ta;k be the lowest level vertex of the segment fa .v/fa .v 0 /. By Lemma 12.1.5, we have jfa .v/wj M because k l 1. Since `.v/ `.v 0 /, we have by Lemma 12.1.8 jvv 0 j 2.`.v/ l C 1/ C 1 2jAjjfa .v/wj C ;

hence the claim.

ProofQ of Theorem 12.1.1. We use also the notation jww 0 j for the distance between w, 0 w 2 c Tc . Since the map f D

Y a

fa W V !

Y

Ta

a

is Lipschitz, it suffices to show that there are constants ƒ > 0, 0 depending only on jAj such that jvv 0 j ƒjf .v/f .v 0 /j C for all v; v 0 2 V . If vertices v, v 0 are horizontally close to each other, then the required estimate follows from Lemma 12.1.7. If they are horizontally distinct, then the estimate follows from Lemma 12.1.10 since jfa .v/wj jfa .v/fa .v 0 /j. Corollary 12.1.11. Let X be a visual Gromov hyperbolic space. Then asdim X `-asdim X `-dim @1 X C 1: Proof. We can assume that `-dim @1 X < 1. Then, by Theorem 12.1.1, there is a quasi-isometric embedding of X into the n-fold product of metric trees with n D `-dim @1 X C1. Using the product and monotonicity theorems for the asymptotic `-dimension and Proposition 10.2.1, we obtain `-asdim X n.

154


12.2 `-dimension of locally self-similar spaces The `-dimension of a metric space can be larger than the topological dimension, see Section 11.3.1. Thus in view of Theorem 12.1.1 it is important to know for which spaces these dimensions coincide. Here, we show that for locally self-similar spaces, see Section 2.3, the `-dimension coincides with the topological one. Recall that a metric space Z is locally similar to (subsets of) a metric space Y if there is 1 such that for every sufficiently large R > 1 and every A Z with diam A 1=R there is a -quasi-homothetic map f W A ! Y with coefficient R; see Section 2.3. Theorem 12.2.1. Assume that a metric space Z is locally similar to .subsets of / a compact metric space Y . Then `-dim Z < 1 and `-dim Z dim Y . Recall that a metric space Z which is locally similar to (subsets of) a compact metric space Y is doubling at small scales; see Lemma 2.3.4. Lemma 12.2.2. Assume that a metric space Z is doubling at small scales. Then `-dim Z < 1. Proof. By the assumption, there is n 2 N such that every ball B4r Z of radius 4r is covered by at most n C 1 balls Br=2 for all sufficiently small r > 0. We fix a maximal r-separated set Z 0 Z, i.e., jzz 0 j > r for each distinct z; z 0 2 Z 0 . Then the family U0 D fBr .z/ W z 2 Z 0 g is an open covering of Z. Since every ball Br=2 contains at most one point from Z 0 and B4r .z/ is covered by at most n C 1 balls Br=2 , the ball B4r .z/ contains at most n C 1 points from Z 0 for every z 2 Z 0 . Thus, there is a coloring W Z 0 ! A, jAj D n C 1, such that .z/ ¤ .z 0 / for each z; z 0 2 Z with jzz 0 j < 4r, cf. the proof of Theorem 8.1.1. For a 2 A, we let Za0 D 1 .a/ be the set of the color a. Then jzz 0 j 4r for 0 distinct z; z 0 2 Za0 . Putting S Ua D fB2r .z/ W z 2 Za g, we obtain an open .n C 1/colored covering U D a2A Ua of Z with mesh.U/ 4r and L.U/ r. This shows that `-dim Z n. We shall use the following facts implied by the definition of a quasi-homothetic map. Lemma 12.2.3. Let h W Z ! Z 0 be a -quasi-homothetic map with coefficient R. z Then we have: z be an open covering of h.V / and U D h1 .U/. Let V Z, U z R mesh.U/; (1) R mesh.U/= mesh.U/ z where L.U/ is the Lebesgue number of U as a covering (2) R L.U/ L.U/, of V . Proof of Theorem 12.2.1. It follows from Lemmas 2.3.4 and 12.2.2, that `-dim Z D N is finite. We can also assume that dim Y D n is finite. There is a constant ı 2 .0; 1/ such that for every sufficiently small > 0 there exists an .N C 1/-colored open

12.2. `-dimension of locally self-similar spaces

155

S covering V D a2A V a of Z with mesh.V / and L.V / ı . It is convenient to take A D f0; : : : ; N g as the color set. There is a constant 1 such that for every sufficiently large R > 1 and every V Z with diam V 1=R there is a -quasi-homothetic map hV W V ! Y with coefficient R. Using that Y is compact and dim Y D n,Swe find for every a 2 A a finite .n C 1/zc za D z a of Y , U colored open covering U c2C Ua , jC j D n C 1, such that the following holds: z 0/ ı ; (i) mesh.U 2

z aC1 / (ii) mesh.U

1 22

z a /; mesh.U z a /g for every a 2 A, a N 1. minfL.U

z a / W a 2 Ag > 0 and mesh.U z a / ı for every a 2 A. Then l ´ minf2aN L.U 2 0 D Bı =2 .V /. Then the For every V 2 V , consider the slightly smaller subset V S S sets Za D V 2V a V 0 Z, a 2 A, cover Z, Z D a2A Za because L.V / ı . Given V 2 V , we fix a -quasi-homothetic map hV W V ! Z with coefficient R D z a;V D 1= and put Vz D hV .V 0 /. Now for every a 2 A, V 2 V a consider the family U z z z z fU 2 Ua W V \ U ¤ ;g, which is obviously an .n C 1/-colored covering of Vz . Then z z z Ua;V D fh1 V .U / W U 2 Ua;V g is an open .n C 1/-colored covering of V 0 . z z z Note that U D h1 V .U / is contained in V for every U 2 Ua;V because z dist.v 0 ; Z n V / > ı=2 for every v 0 2 V 0 and diam U S diam U ı=2. Thus the family Ua;V is contained in V . Now the family Ua D V 2V a Ua;V covers the set Za of the color a, and has the following properties: (1) for every a 2 A, the family Ua is at most .n C 1/-colored (by C ); (2) mesh.UaC1 / 12 minfL.Ua /; mesh.Ua /g for every a 2 A, a N 1 (L.Ua / means the Lebesgue number of Ua as a covering of Za ); z a / and L.Ua / .=/L.U z a / for every a 2 A. (3) mesh.Ua / mesh.U Indeed, distinct V1 , V2 2 V a are disjoint and thus any U1 2 Ua;V1 , U2 2 Ua;V2 are disjoint because U1 V1 , U2 V2 . This proves (1). Furthermore, for every a 2 A, a N 1, and every U 2 UaC1 , we have z aC1 / diam U mesh.U z a /; mesh.U z a /g 1 minfL.U

2R 1 minfL.Ua /; mesh.Ua /g 2

by Lemma 12.2.3, hence (2). Finally, (3) also follows from Lemma 12.2.3. y 0 D U0 and assume that for some a 2 A, we y 1 D fZg, U Now we put U y a so that U y a is an .n C 1/-colored y 0; : : : ; U have already constructed families U

156


y a1 /, mesh.U y a / mesh.U0 /, covering of Z0 [ [ Za and mesh.Ua / 12 L.U y a1 /g. Then we have y a / minfL.Ua /; 1 L.U L.U 2 mesh.UaC1 /

1 2

y a1 /g 1 L.U y a /: minfL.Ua /; 12 L.U 2

y a , UaC1 , we obtain an open Applying Proposition 9.6.1 to the pair of families U y aC1 of Z0 [ [ ZaC1 with .n C 1/-colored covering U y aC1 / maxfmesh.U y a /; mesh.UaC1 /g mesh.U0 / mesh.U y aC1 / minfL.UaC1 /; 1 L.U y a /g. and L.U 2 yN Proceeding by induction, we obtain an open .n C 1/-colored covering U D U aN L.Ua / W of Z with mesh.U/ mesh.U0 / ı=2 and L.U/ minf2 a 2 Ag .l=/. Since we can choose > 0 arbitrarily small and the constants ı, , l are independent of , this shows that `-dim Z n. Corollary 12.2.4. The `-dimension of every compact, locally self-similar metric space Z is finite and coincides with its topological dimension, `-dim Z D dim Z. Proof. We have dim Z `-dim Z for every metric space Z. By Theorem 12.2.1, `-dim Z < 1 and `-dim Z dim Z, hence `-dim Z D dim Z is finite.

12.3 Applications to hyperbolic spaces Recall that the boundary at infinity of any proper hyperbolic space is compact, see e.g. Exercise 6.4.4. Now Theorem 2.3.2 and Corollary 12.2.4 give the following Theorem 12.3.1. The `-dimension of the boundary at infinity of every cobounded, hyperbolic, proper, geodesic space X is finite and coincides with the topological dimension, `-dim @1 X D dim @1 X . The class of spaces satisfying the condition of Theorem 12.3.1 is very large. It includes in particular all symmetric rank one spaces of noncompact type (i.e. the real, complex, quaternionic hyperbolic spaces and the Cayley hyperbolic plane), all cocompact Hadamard manifolds of negative sectional curvature, various hyperbolic buildings, etc. The most important among them is the class of Gromov hyperbolic groups; see Section 1.4.2. Every cobounded, hyperbolic, proper, geodesic space is certainly visual. Thus, combining Theorems 12.3.1 and 12.1.1, we obtain the following. Theorem 12.3.2. Let X be a cobounded, hyperbolic, proper, geodesic space. Then there exists a quasi-isometric embedding f W X ! T1 Tn of X into the n-fold product of metric trees T1 ; : : : ; Tn with n D dim.@1 X / C 1.


157

Now we are able to compute the asymptotic dimension and the asymptotic `dimension of any hyperbolic space from that class. Theorem 12.3.3. Let X be a cobounded, hyperbolic, proper, geodesic space. Then asdim X D `-asdim X D dim @1 X C 1: Proof. The estimate from below, asdim X dim @1 X C 1, holds for a larger class of spaces; see Theorem 10.1.2. The estimate from above, `-asdim X dim @1 X C 1, follows from Corollary 12.1.11 and Theorem 12.3.1.

12.4 Supplementary results and remarks 12.4.1 `-dimension as an obstacle to quasi-symmetry Let Z D f0g [ f1=m W m 2 Ng R be the (bounded) space from the first example on p. 137. Since ANdim Z n D `-dim Z n D n and dim Z n D 0 for every n 2 N (see Section 11.3.1), it follows from quasi-symmetry invariance of the Assouad–Nagata dimension and Corollary 12.2.4 that Z n is not quasi-symmetric to any locally selfsimilar space. Standard fractal spaces like the ternary Cantor set, the Sierpinski carpet or the Menger curve are self-similar and in particular locally self-similar (see Example 2.3.1). Therefore, their `-dimension coincides with the topological dimension. On the other hand, these spaces admit metrics with the `-dimension strictly larger than the topological one; see Section 11.3.1. As above, any such metric is not quasi-symmetric to any locally self-similar metric. Bibliographical note. A quasi-isometric embedding of Hn into the product of n 2 metric trees was constructed in [BS2]. Theorem 12.1.1 is obtained in [Bu] combining ideas from [BS2] with the notion of the linearly controlled dimension. The target trees of Theorem 12.1.1 typically have infinite valence at every vertex. Quasi-isometric embeddings of hyperbolic spaces into the product of binary trees with optimal number of factors are constructed in [BDS] combining Theorem 12.1.1 with a sophisticated combinatorial argument. The results of Sections 12.2 and 12.3 are taken from [BL].

Chapter 13

Hyperbolic dimension

The hyperbolic dimension of a metric space is a close relative of the asymptotic dimension. The main feature is that one allows coverings by unbounded sets. Of course, it would be useless to consider coverings by arbitrary unbounded sets. We do require certain conditions restricting the size of covering members. These are the large scale doubling condition and some uniformity condition. The hyperbolic dimension possesses usual properties of dimensions like the monotonicity and product theorems. However, unlike the asymptotic dimension, the hyperbolic dimension of Euclidean spaces as well as of all doubling spaces is zero, hypdim Rn D 0 for every n 0. What makes the hyperbolic dimension useful is that it behaves for hyperbolic spaces just like the asymptotic dimension: The main result of this chapter is the estimate hypdim Hn n for every n 2. This result has a number of applications to nonembedding results.

13.1 Large scale doubling sets A subset U of a metric space X is large scale doubling if there is a constant N 2 N such that, for every sufficiently large r > 1 and for every ball B2r X of radius 2r, the intersection B2r \U can be covered by at most N balls of radius r. More precisely, we say that U is .N; R/-large-scale doubling or .N; R/-ls-doubling if the condition above holds for all r R. We also say that U is N -ls-doubling if only N is of importance. An equivalent definition is that every intersection B2r \ U contains at most N points which are r-separated for all sufficiently large r. Exercise 13.1.1. Show that the covering and separation definitions of large scale doubling sets are equivalent. What is the relation between the implied constants? Examples 13.1.2. (1) Any Euclidean space Rn is .N; R/-ls-doubling for some N D N.n/ and R D 0. (2) Let B be a bounded metric space. Then the metric product X D B Rn is .N; R/-ls-doubling for some N D N.n/ and R 2 diam B. We emphasize that in this example the parameter N counting the number of separated points or of covering

160

Chapter 13. Hyperbolic dimension

balls is actually independent of B, while the parameter R describing the corresponding scales tends to infinity as diam B ! 1, e.g., if one takes as B an R-tree. Lemma 13.1.3. Assume that U X is .N; R/-ls-doubling. Then for any r R, we have: every intersection Br \U is covered by N n balls B with n D log2 .2r= /. In particular, Br \ U contains at most N n points which are 2 -separated. Proof. Every intersection Br \ U is covered by N n balls of radius r=2n . There is n 2 N with r r < n1 : n 2 2 Then n log2 2r , which proves the first assertion. For the second one note that any ball B contains at most one point of any 2 -separated set. The property to be large scale doubling is a quasi-isometry invariant. Proposition 13.1.4. Let f W X ! Y be a quasi-isometric map. Then for any large scale doubling U X , V Y , we have f .U / Y , f 1 .V / X are large scale doubling quantitatively. Proof. We can assume that f is .a; b/-quasi-isometric for some a 1, b 0, and that r 2b. The inverse image of any ball B2r Y is contained in a ball BR X of radius R D a.4r C b/. The image of any ball B X of radius D a1 .r b/ is contained in a ball Br Y . By Lemma 13.1.3, the intersection BR \ U is covered by N k balls B with k D log2

2R 2a2 .4r C b/ D log2 log2 18a2 :

r b

Thus the intersection B2r \ f .U / is covered by N k balls Br , where the upper bound N k is independent of r. Lemma 13.1.5. If U X , V Y are large scale doubling then U V X Y is large scale doubling quantitatively. Proof. By Proposition 13.1.4, we can consider the l1 -product metric on X Y . Then Br ..x; y// D Br .x/ Br .y/ for any .x; y/ 2 X Y and r > 0. The large scale doubling property of U V is now obvious.

13.2 Definition of the hyperbolic dimension Definition 13.2.1. A covering U of a metric space X is called uniformly large scale doubling or uniformly ls-doubling if there exists an N 2 N with: (1) there exists R 0 such that every element of the covering is .N; R/-ls-doubling;

13.2. Definition of the hyperbolic dimension

161

(2) any finite union of elements of the covering is N -ls-doubling. We also call such a covering uniformly N -ls-doubling or uniformly .N; R/-lsdoubling. Definition 13.2.2. The hyperbolic dimension of X is the minimal integer hypdim X D n such that for every d > 0 there is an open covering U of X with m.U/ n C 1 and L.U/ d which is uniformly large scale doubling. This is the covering definition. One can also introduce the colored and polyhedral definitions of the hyperbolic dimension, see Chapter 9. For the hyperbolic dimension, the property set P (see Chapter 9) is identified with the interval .0; 1/ and the filter F is generated by all subintervals .d; 1/ .0; 1/, d > 0. We say that a covering U of a metric space X has the property d 2 .0; 1/, U 2 d , if and only if U is open, L.U/ d and U being uniformly large scale doubling. Lemma 13.2.3. The property space P for the hyperbolic dimension satisfies the Axioms 9.3.1. Proof. The proof is similar to the one for the asymptotic dimension. We use the notations from Section 9.3. 1. For every natural number m, there is a lower bound lm 2 .0; 1/ for the Lebesgue number of the covering of any uniform polyhedron K, dim K C 1 m, by the open stars of ba K. We put m D lm =.m C 1/2 and define bam W P ! P as bam .t / D m t for every t > 0. Axiom 9.3.1 (1) is satisfied because for any covering U 2 t 2 P with multiplicity m, the covering U0 D ba.pU / is open, L.U0 / lm = Lip.pU / m t , and U0 being inscribed in U is large scale doubling. 2. Axiom 9.3.1 (2) is obvious because the property to be large scale doubling is hereditary and L.UjX 0 / L.U/ for every open covering U of X and every subspace X 0 X. 3. By Remark 9.2.7, for every natural number m, there is a constant cm 2 .0; 1/ with the following property: given uniform polyhedra K1 , K2 , dim Ki C 1 m, the Lebesgue number of the covering of K1 K2 by ' 1 .st v /, v 2 K1 s K2 , is bounded from below by cm , where ' W K1 K2 ! K1 s K2 is the barycentric triangulation map. We put m D cm =.m C 1/2 and define prodm W P P ! P as prodm .t1 ; t2 / D m minft1 ; t2 g for every t1 , t2 > 0. Then clearly prodm .F F / F . Furthermore, for each covering Ui 2 ti 2 P with multiplicity m, there is by Lemma 9.2.2 a barycentric map fi D pUi W Xi ! Ni with Lip.fi / .m C 1/2 =ti , i D 1; 2. Then the covering Uf1 ;f2 of X1 X2 is open and L.Uf1 ;f2 / cm = Lip.f1 f2 / m minft1 ; t2 g because Lip.f1 f2 / .mC1/2 maxf1=t1 ; 1=t2 g. Furthermore, the covering Uf1 ;f2 is inscribed in the product covering U1 U2 . Thus using Lemma 13.1.5, we easily check that Uf1 ;f2 is uniformly large scale doubling.

162


That is, Uf1 ;f2 2 prodm .t1 ; t2 /, and Axiom 9.3.1 (3) is satisfied.

Now using results of Section 9.3, we conclude that the three definitions of the hyperbolic dimension, the colored, covering and polyhedral ones, are equivalent, and that for the hyperbolic dimension the monotonicity and product theorems, Theorem 9.4.1 and Theorem 9.5.1, hold true. Furthermore, by the definition, we have hypdim X D 0 for every large scale doubling space X , and hypdim X asdim X for every metric space X because any uniformly bounded covering is certainly uniformly large scale doubling. It follows from Proposition 13.1.4 that the hyperbolic dimension is a quasiisometry invariant.

13.3 Hyperbolic dimension of hyperbolic spaces We study here the hyperbolic dimension of hyperbolic spaces. The main result of the section, Theorem 13.3.2, is based on the fact that large scale doubling sets in CAT.1/-spaces, when observed from distance c, look exponentially small in c if measured by the angle measure.

13.3.1 Large scale doubling sets in CAT.1/-spaces Let X be a CAT.1/-space. We fix a base point x0 2 X and define the angle metric ]1 in the geodesic boundary at infinity @g X as follows. Given ; 0 2 @g X , we consider the unit speed geodesic rays c , c 0 from x0 to , 0 respectively, and put ]1 .; 0 / D lim ].cN .s/oN cN 0 .s//; s!1

where ].cN .s/oN cN 0 .s// is the angle at oN of the comparison triangle in H2 for the triangle c .s/x0 c 0 .s/. By the parallelism angle formula, we have tan

1 4

N N N 0 / ]1 .; 0 / D e dist.o; ;

N N 0 / D ]1 .; 0 /, and N N 0 is the geodesic in H2 N N 0 2 @1 H2 satisfy ]oN .; where , N N N 0 / N N 0 . Thus ]1 .; 0 / 4e dist.o; with the end points at infinity , . The shadow of a set A X is a subset sh.A/ @g X which consists of the ends of all rays x0 intersecting A (so sh.x0 / D @g X ). Given ı > 0 we define the angle ı-measure of A, ]ı A, by X diam.sh.B//; ]ı A D inf C

B2C

where the infimum is taken over all coverings C of A by balls of radius ı in X .

163

13.3. Hyperbolic dimension of hyperbolic spaces

Lemma 13.3.1. Given N 2 N, R > 1, there is for every sufficiently large ı a positive constant C depending only on N and ı such that if a subset A X is .N; R/-lsdoubling and dist.x0 ; A/ c > ı, then ]ı A C e c=2 : Proof. We take ı=2 R. Then by Lemma 13.1.3, every ball Br X with r > ı=2 contains at most N n D d r k points of A which are ı-separated, where k D log2 N C 1/k for every c > ı. and d D .4=ı/k . Furthermore, we can assume that e c=2 .c S 0 Take a maximal ı-separated subset A A. Then A a2A0 Bı .a/. For any ball Bı .a/, a 2 A0 , consider , 0 2 sh.Bı .a// with ]1 .; 0 / arbitrarily close to diam.sh.Bı .a///. For simplicity, we assume that ]1 .; 0 / D diam.sh.Bı .a/// because possible errors disappear in the final conclusion. N N N 0 / Then diam.sh.Bı .a/// 4e dist.o; in the notations introduced above. We take x 2 x0 \ Bı .a/, x 0 2 x0 0 \ Bı .a/ and consider the piecewise geodesic curve in X which consists of the geodesic rays x, x 0 0 and the segment xx 0 . The curve connects in X the points , 0 , and dist.x0 ; / dist.x0 ; Bı .a// D jx0 aj ı. Furthermore, dist.x0 ; / dist.o; N N N 0 / by comparison with H2 . Thus diam.sh.Bı .a/// 4e ıjx0 aj and ]ı A

X

diam.sh.Bı .a///:

a2A0

Since c > ı, for every c C 1, the number of points from A0 whose distances to x0 lie in the interval Œ 1; / is d k . Thus we have ]ı A 4e ı 4de

X

e jx0 aj 4de ı

a2A0 1 X ı

1 X

.c C q C 1/k e q e c

qD0

.q C 1/k e q .c C 1/k e c C e c=2

qD0

by the choice of c.

13.3.2 Estimate from below Here we give a proof based on Sperner’s lemma that the hyperbolic dimension of any Hadamard manifold X with sectional curvatures K 1 is at least n D dim X . Moreover, we prove the following. Theorem 13.3.2. Let U be a uniformly large scale doubling open covering of X, where X is a Hadamard manifold. Then the multiplicity of U is at least n C 1.

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Proof. Recall that X is a CAT.1/-space. One can assume that U is locally finite, that U is uniformly .N; R0 /-ls-doubling for some N 2 N, R0 > 1, and that the elements of U are connected. By Corollary 9.8.3, it suffices to find a continuous simplex f W n ! X coherent with U. We fix a small positive constant ˛0 so that the maximal number of ˛0 -separated points in the unit sphere S n1 Rn with respect to the angle metric is > N . Let W S n1 be a ball of radius ˛0 =4 equipped with a structure of the .n 1/-dimensional standard simplex. Furthermore, let d > 0 be the minimal diameter of a subset B S n1 which meets all .n 2/-dimensional faces of W . This d depends only on ˛0 . We fix a sufficiently large ı as in Lemma 13.3.1 (one can take ı 2R0 ). Let C D C.N; ı/ be the constant from Lemma 13.3.1. Now we choose r > ı such that C e r=2 < d . Since U is locally finite, there are only finitely many elements of U which intersect the ball Br .o/ centered at a base point o 2 X . The union A of all those elements is N -ls-doubling by properties of the covering U. Thus for every sufficiently large R > r, the intersection A \ S2R contains at most N points which are R-separated, where S2R D @B2R .o/. Next, we fix a maximal ˛0 -separated subset in S n1 . Then the balls of radius ˛0 =4 in S n1 To X centered at its points are pairwise separated by an angle distance ˛0 =2. Radially projected to S2R by expo W To X ! X these balls are pairwise separated by the distance > R in X if R is sufficiently large (at this point, we use that X is a CAT.1/-space). Since the number of the balls is > N , there is at least one such ball W2R S2R which misses the closure of A. Using a standard .n 1/-simplex structure on the appropriate ball W S n1 , we introduce the induced simplex structure on W2R and consider the geodesic cone X over W2R with the vertex o as a continuous n-simplex in X . Let us check that is coherent with U. Any element of U which intersects the ball Br .o/ misses the .n 1/-face W2R by the construction. Any other element of U 2 U is at the distance r from o, and thus has the angle measure ]ı .U / C e r=2 < d by Lemma 13.3.1. Since X is CAT.1/ and U is connected, the angle diameter of U observed from o is at most ]ı .U /. Hence also U cannot intersect all .n 1/-faces of by the choice of d . Therefore, the simplex is coherent with U.

13.4 Applications to nonembedding results Theorem 13.4.1. Let X be a metric space with hypdim X p and let T1 ; : : : ; Tk be any metric trees. Then there is no quasi-isometric embedding X ! T1 Tk Rm for p > k and any m 0. Proof. Assume there is a quasi-isometric embedding f W X ! T1 Tk Rm :

13.4. Applications to nonembedding results

165

Since hypdim asdim, we have hypdim T 1 for any metric tree. Thus using the quasi-isometry invariance of the hyperbolic dimension and the monotonicity and product theorems, we obtain p hypdim X hypdim.T1 Tk Rm / k:

Using Theorem 13.3.2, we obtain: Corollary 13.4.2. Let X be a Hadamard manifold with sectional curvatures K 1. Then there is no quasi-isometric embedding X ! T1 Tk Rm into the product of any k < dim X metric trees stabilized by any Euclidean factor Rm . This corollary shows that the embedding result obtained in Theorem 12.3.2 is optimal in a strong sense with respect to the number of tree factors: each cobounded hyperbolic Hadamard manifold admits a quasi-isometric embedding into the n-fold product of metric trees, n D dim X . One can generalize Corollary 13.4.2 as follows. Let X n be a universal covering of a compact Riemannian n-dimensional manifold, n 2, with nonempty geodesic boundary and constant sectional curvature 1. Then X n satisfies the condition of Theorem 12.3.1, and hence dim @1 X n D `-dim @1 X n . Note that X n can be obtained from the real hyperbolic space Hn by removing a countable collection of disjoint open half-spaces, and @1 X n S n1 is a compact, nowhere dense subset obtained from S n1 by removing a countable collection of disjoint open balls. In particular, for n D 2, @1 X n S 1 is a Cantor set, for n D 3, @1 X n S 2 is a Sierpinski carpet, and for n 4, @1 X n S n1 is a higher dimensional version of a Sierpinski carpet. Thus dim @1 X n D n 2. The space X n contains isometrically embedded copies of Hn1 as boundary components. In the next chapter, we show that the k-fold product Hn1 Hn1 contains a quasi-isometrically embedded Hp with p D k.n2/C1, see Section 14.1. Theorem 13.4.3. Let X n be a space as above, and let Ykn D X n X n be the k-fold product, k 1. Then there is no quasi-isometric embedding Hp ! Ykn Rm for p > k.n 1/ and any m 0. Proof. Indeed, we have that hypdim X n asdim X n . By Theorem 12.3.3, asdim X n D `-asdim X n D dim @1 X n C 1 D n 1. Furthermore, hypdim Hp p by Theorem 13.3.2. Now applying the argument of Theorem 13.4.1, we obtain the result.

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Theorem 13.4.3 implies in particular that there is no way to embed H5 quasiisometrically into X 3 X 3 Rm for any m 0 though there is a quasi-isometric embedding H3 ! X 3 X 3 (and H5 ! H3 H3 /. In general, for an arbitrary n 2, there is a quasi-isometric Hp ! X n X n for p D 2n 3 and there is no quasi-isometric Hp ! X n X n Rm for p D 2n 1 and an arbitrary m 0. In the case n D 2, the space X 2 is quasi-isometric to the binary tree T whose edges all have length 1 because X 2 covers a compact hyperbolic surface with nonempty geodesic boundary. By [BDS], there is a quasi-isometric embedding H2 ! T T , and hence there is a quasi-isometric embedding Hp ! X n X n in the remaining case p D 2n 2 if n D 2. For n 3, the question whether there is a quasi-isometric embedding H2n2 ! X n X n remains open. Moreover, the same question is open for quasi-isometric Hk.n1/ ! Ykn ; n; k 3:

13.5 Supplementary results and remarks 13.5.1 Hyperbolic dimension of general hyperbolic spaces The proof of Theorem 13.3.2 generalizes the one of Theorem 10.1.1 for the asymptotic dimension. Recall that there is another approach to estimates from below of the asymptotic dimension of hyperbolic spaces, using the hyperbolic cone over the boundary at infinity, see Theorem 10.1.2. A more sophisticated version of that allows to prove the following, see [BS3]. Theorem 13.5.1. Let X be a geodesic hyperbolic space, whose boundary at infinity @1 X is infinite and doubling. Then hypdim X dim @1 X C 1:

In that general case, one needs to use the Lebesgue number condition from the definition of the hyperbolic dimension. Similarly to Section 13.4, the embedding result in Theorem 12.3.2 is optimal in a strong sense with respect to the number of tree factors. Corollary 13.5.2. Let X be a geodesic hyperbolic space, whose boundary at infinity @1 X is infinite and doubling. Then there is no quasi-isometric embedding X ! T1 Tk Rm into the product of any k < dim @1 X C 1 metric trees stabilized by any Euclidean factor Rm .

Chapter 14

Hyperbolic rank and subexponential corank

In this chapter we consider other quasi-isometry invariants which are useful to prove a number of important nonembedding results in asymptotic geometry.

14.1 Hyperbolic rank Let X be a metric space. Consider all proper geodesic hyperbolic spaces Y quasiisometrically embedded in X and define the hyperbolic rank of X as rank h X D sup dim @1 Y; Y

where the supremum is taken over all such Y . It follows from the definition that the hyperbolic rank is monotone, i.e., if there is a quasi-isometric embedding X ! X 0 , then rank h X rank h X 0 . Therefore, the hyperbolic rank is a quasi-isometry invariant. This invariant measures in a sense how much hyperbolicity there is in a space, i.e., how big a space is intrinsically. If X is a proper geodesic hyperbolic space, then rank h X D dim @1 X : the identity map X ! X shows that rank h X dim @1 X ; the opposite inequality follows from the stability of geodesics: any quasi-isometric embedding Y ! X of a geodesic hyperbolic space Y induces an embedding @1 Y ! @1 X and thus dim @1 X dim @1 Y ; see Theorem 5.2.17. As an illustration for nonhyperbolic spaces, we show that rank h .H2 H2 / 2: The Riemannian metric of HnC1 , n 1, in the solvable group model takes the form ds 2 D dt 2 C e 2t dx 2 ; see Section A.4, A.5, where .t; x/ 2 R Rn are horospherical coordinates in HnC1 , D HnC1 and dx 2 D dx12 C C dxn2 is the Euclidean metric. The space HnC1 nC1 by multiplying all distances by the factor > 0 has the constant obtained from H curvature 1=2 and its Riemannian metric in horospherical coordinates takes the form ds 2 D dt 2 C e 2t= dx 2 :

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Chapter 14. Hyperbolic rank and subexponential corank

Now take D

p 2 and consider the map

f W H3 ! H2 H2 ;

p p f .t; x1 ; x2 / D .t = 2; x1 ; t = 2; x2 /

in horospherical coordinates x D .t; x1 ; x2 / 2 H3 and x1 x2 D .t1 ; x1 / .t2 ; x2 / 2 H2 H2 . The embedding f is isometric with respect to the Riemannian metric p 2 2 2 t 2 .dx1 C dx22 / on H3 and the metric product ds12 C ds22 , dsi2 D ds D dt C e 2 2 2ti dti C e dxi , i D 1; 2, on H2 H2 , that is, jdf .v/j D jvj for every tangent to H3 . Proposition 14.1.1. The embedding f W H3 ! H2 H2 is quasi-isometric, in particular, rank h .H2 H2 / 2. Proof. Since f is Riemannian isometric, its leaves invariant the length of the curves. Thus jf .x/f .x 0 /j jxx 0 j for every x; x 0 2 H3 . To obtain the estimate from below, let us recall that for any x; x 0 2 HnC1 lying in a horosphere S HnC1 we have jxx 0 jS D 2 sinh.jxx 0 j=2/; where jxx 0 jS is the distance between x, x 0 along the horosphere S ; see Exercise A.3.3. D HnC1 , this gives Since HnC1 jxx 0 j 2 ln jxx 0 jS 2 ln . Thus for x; x 0 2 S HnC1 p p jxx 0 j 2 2 ln jxx 0 jS C c D 2 2 ln jf .x/f .x 0 /jf .S/ C c p 2 2 ln.2 max jfi .x/fi .x 0 /jSi / C c i p 2 jf .x/f .x/j C c for some universal constant c > 0 (which may differ from part to part of the computation above) and for every x; x 0 2 H3 lying in the same horosphere given by the equation t D t0 . In the general case, where x, x 0 are from different horospheres, the geodesic between them goes first from the upper one to the lower one almost radially, and along this part f is isometric. The image Y D f .H3 / H2 H2 can be described as Y D f.x1 ; x2 / 2 H2 H2 W b1 .x1 / D b2 .x2 /g for appropriately chosen Busemann functions b1 , b2 W H2 ! R. The straightforward generalization shows that there is a quasi-isometric embedding Hn ! Hn1 Hnk with n 1 D n1 C C nk k. In particular, rank h .Hn1 Hnk / n 1 D n1 C C nk k: This raises the question whether the obtained estimates of the hyperbolic rank are optimal. To obtain estimates from above, we introduce another quasi-isometry invariant in a sense complementary to the hyperbolic rank.

169

14.2. Subexponential corank

14.2 Subexponential corank A continuous map g W X ! T of topological spaces can be regarded as a continuous foliation [ XD g 1 .t / t2T

of X over T . We define its rank as rank.g/ D supK dim g.K/, where the supremum is taken over all compact K X . Roughly speaking, the subexponential corank of X is the minimal rank of continuous foliations g W X ! T all fibers of which have a subexponential growth rate. For example, for X D Hn Rm , the foliation g W X ! Hn given by the first factor projection is subexponential, and its rank is n D dim Hn . However, this is not the least rank of subexponential foliations of X . The factor Hn possesses a subexponential foliation g1 W Hn ! Rn1 which is the projection onto a fixed horosphere S Hn from its center (at infinity). The rank of this foliation is n 1, thus X has a continuous subexponential foliation of rank n 1. The precise definition of the subexponential corank is somewhat complicated since we have to make the continuity condition compatible with the quasi-isometry invariance. Assume that ı and that a maximal ı-separated set Xı X are fixed. We define the size of A X (with respect to Xı and ) as the number sizeXı ; .A/ 2 N [ f1g of points x 2 Xı for which the balls B .x/ intersect A. Now we can formulate the basic definition. A continuous foliation g W X ! T is said to be subexponential if, for any maximal ı-separated net Xı X (with sufficiently large ı), for any ı and any " > 0, there is a constant R0 D R0 .Xı ; ; "/ 1 such that for all R R0 and all t 2 T we have 1 ln sizeXı ; g 1 .t / \ BR .x0 / < " R for some fixed point x0 2 X ; clearly, this property is independent of the choice of x0 . The subexponential corank of a metric space X is defined as corank X D sup inf rank.g W Z ! T /; ZX

where the supremum is taken over all spaces Z quasi-isometric to X , and the infimum over all subexponential foliations of Z. By definition, corank X is a quasi-isometry invariant. Taking the supremum over all Z quasi-isometric to X in the definition of corank X is necessary because for any discrete space Z the trivial foliation id W Z ! Z is subexponential and has rank 0, i.e. minimal possible rank.

170


Remark 14.2.1. The property of a foliation g W X ! T to be subexponential means roughly speaking that every fiber g 1 .t / has a subexponential growth rate and a bounded distortion in X . The last condition is essential. For example, all fibers of the foliation given by a Busemann function g W Hn ! R are isometric to Rn1 in the induced Riemannian metric and hence they have a subexponential (in fact, polynomial) growth rate. However, this foliation is by no means subexponential because every horosphere g 1 .t /, t 2 R is exponentially distorted in Hn and for a fixed x0 2 Hn the balls BR .x0 / contain exponentially large pieces of it. Lemma 14.2.2. If f W X ! Z is a continuous quasi-isometric map and g W Z ! T is a continuous subexponential foliation, then g B f W X ! T is a continuous subexponential foliation. Proof. Assume that f is .a; b/-quasi-isometric. We fix x0 2 X and put z0 D f .x0 /. Let ı0 , 0 ı0 be the separation and the radius constants for the foliation g. We put ı D a.ı0 C b/ and take a maximal ı-separated set Xı X . Then jf .x/f .x 0 /j a1 jxx 0 j b a1 ı b ı0 for different x; x 0 2 Xı . Thus f .Xı / Z is ı0 -separated. We extend it to a maximal separated net Zı0 f .Xı /. Furthermore, we fix maxfı; a1 .0 b/g, and consider t 2 T . If the ball B .x/ intersects the set .g B f /1 .t / \ BR .x0 / for some x 2 Xı , then its image f .B .x// intersects the set g 1 .t / \ BaRCb .z0 / since f .BR .x0 // BaRCb .z0 /. Furthermore, f .B .x// Ba Cb .f .x//. Thus the ball BaCb .f .x// intersects the set g 1 .t/ \ BaRCb .z0 /, and we have sizeXı ; .g B f /1 .t / \ BR .x0 / sizeZı0 ;aCb g 1 .t / \ BaRCb .z0 / " for every R > 0, t 2 T . Fix " > 0. Then, for aR C b R0 .Zı0 ; a C b; aCb /; we get 1 " : ln sizeXı ; .g B f /1 .t / \ BR .x0 / < aCb aR C b " Put R0 .Xı ; ; "/ D maxf1; a1 ŒR0 .Zı0 ; a C b; aCb / bg. For R R0 .Xı ; ; "/ we have 1 1 ln size .t / \ B .x / < ": .g B f / X ; R 0 ı R

Thus the foliation g B f is subexponential.

14.2.1 QPC-spaces To apply the subexponential corank to specific situations, we need to study continuous subexponential foliations of spaces. An appropriate class to study is formed by QPC-spaces. A metric space Z is called QPC-space if every quasi-isometric map


171

f W X ! Z is parallel to a continuous one, i.e., if there exists a continuous map f 0 W X ! Z such that jf .x/f 0 .x/j C < 1 for all x 2 X . In that case f 0 is also quasi-isometric. Lemma 14.2.3. Suppose that X is quasi-isometric to a QPC-space Z. Then corank X D inf rank.g W Z ! T /; where the infimum is taken over all subexponential foliations of Z. Proof. If X 0 is quasi-isometric to X then X 0 is quasi-isometric to Z. Thus there is a continuous quasi-isometry X 0 ! Z. By Lemma 14.2.2, inf rank.g 0 W X 0 ! T 0 / inf rank.g W Z ! T /. Lemma 14.2.4. Every proper Hadamard space X is QPC. Proof. Assume there is an .a; b/-quasi-isometric map f W Y ! X . We take a maximal ı-separated net Yı Y with aı C b ı0 > 0 and note that every ball B2ı .˛/, ˛ 2 Yı , contains only finitely many elements of the net Yı . This is so because the set f .Yı / X is ı0 -separated, the space X is proper and hence the ball B2aıCb .f .˛// f .B2ı .˛// intersects f .Yı / over a finite set. Thus the nerve N of the covering A D fBı .˛/ W ˛ 2 Yı g of Y is a locally finite simplicial complex. Choosing a barycentric map associated with A, we obtain a continuous map g W Y ! N , see Section 9.2.2. Note that g.y/ lies in the simplex y spanned by f˛ 2 Yı W j˛yj < ıg. Next, we identify Yı with the 0-skeleton of N and extend f jYı W N0 ! X to a continuous fN W N ! X using the convexity of X and acting by induction on the dimension of the skeletons. Then fN.y / BaıCb .f .y// and thus fN B g W Y ! X is a continuous map parallel to f .

14.2.2 Properties of the subexponential corank We list some properties of the corank which easily follow from the definition. They are in parts similar to properties of the hyperbolic dimension. (1) Monotonicity: If X is quasi-isometric to QPC and X 0 is quasi-isometrically embedded in X , then corank X 0 corank X . Indeed, one can assume that X is QPC. If Z is quasi-isometric to X 0 then there is a continuous quasi-isometric map f W Z ! X . By Lemma 14.2.3, corank X D inf rank.g W X ! T /, where the infimum is taken over all subexponential foliations. By Lemma 14.2.2, every subexponential foliation g W X ! T induces a subexponential foliation g B f W Z ! T . Furthermore rank.g B f / rank.g/. Hence corank X 0 corank X . (2) The product theorem: If the metric product X1 X2 is a QPC-space, then corank.X1 X2 / corank X1 C corank X2 . Indeed, in this case both X1 , X2 are QPC, and the product of subexponential foliations gi W Xi ! Ti , i D 1; 2, is a subexponential foliation g1 g2 W X1 X2 ! T1 T2

172


with rank.g1 g2 / rank.g1 /Crank.g2 / by the product theorem for the topological dimension. (3) corank Rn D 0 for every n 0. (4) For an n-dimensional Hadamard manifold X , we have corank X n 1. Indeed, the projection to a fixed horosphere from its center is a subexponential foliation of rank n 1. Theorem 14.2.5. Assume that a metric space X is quasi-isometric to a QPC-space. Then rank h X corank X: For the proof we need the following notion. Let X be a (truncated) hyperbolic approximation (with parameter r 1=6) of a compact metric space Z (see Chapter 6, Section 6.4.1). The extended hyperbolic approximation Xy of Z is a simplicial polyhedron which contains X as the 1-skeleton and which is defined in the same way as X with the only difference that for each k k0 any collection of vertices v 2 Vk with nonempty intersection of the balls B.v/ spans a simplex of Xy . We consider a path metric on Xy for which every simplex is isometric to the standard spherical simplex with edge length 1 (in the sphere of radius 2=). For every k k0 , the combinatorial sphere Sk is a subpolyhedron of Xy spanned by Vk . The inclusion X Xy is a quasi-isometry since the distances between any vertices v; v 0 2 V in Xy and in its 1-skeleton X differ only by a universal constant. This is because any geodesic segment in Xy of length > 2 which misses the 0-skeleton contains conjugate points and cannot be a minimizer. Thus the extended hyperbolic approximation Xy of Z is hyperbolic since X is hyperbolic, and @1 Xy D @1 X D Z. The advantage of Xy over X is that for every k k0 we have a continuous barycentric map pk W Z ! Xy associated with the covering fB.v/ W v 2 Vk g of Z (the covering is finite because Z is compact). The image pk .Z/ Xy lies in the combinatorial sphere Sk , and Sk is a subset of the metric annulus fx 2 Xy W k joxj k C 1=2g, where o is the root of X . Proof of Theorem 14.2.5. We can assume that X is QPC. Let f W Y ! X be a quasiisometric embedding of a proper geodesic hyperbolic space Y into X . By Theorem 7.1.2, we can replace Y first by a hyperbolic approximation of Z D @1 Y , taken with a visual metric d , and then by the extended hyperbolic approximation according to the discussion above. Furthermore, we can also assume that f is continuous. By Lemma 14.2.2, any continuous subexponential foliation g W X ! T induces a continuous subexponential foliation g B f W Y ! T . Thus it suffices to show that dim Z rank.g W Y ! T / for any subexponential foliation g. The idea of the proof is simple. For k k0 , we consider a barycentric map pk W Z ! Sk into the combinatorial sphere Sk (of the level k). Let Kk D g.Sk / T

173


be a compact set. Its dimension is at most rank.g/ by the definition. On the other hand, any open covering of Kk is lifted by hk D g B pk W Z ! T to an open covering of Z with the same multiplicity. The main step of the proof is to show that if the mesh of a covering of Kk is sufficiently small and the level k is large, then the mesh of the induced covering Z is arbitrarily small, which implies the required inequality. The fact that the mesh of the induced covering is small easily follows from the fact that the foliation g is subexponential. Now we look at this argument in details. Given an open covering O of Kk , there is an inscribed finite closed covering C with multiplicity n C 1, n D rank.g/. We take a maximal ı-separated net Yı Y (we can assume that Yı is the subset of the vertex set V of Y ) and ı for which the subexponential growth rate condition of the foliation is fulfilled, and fix " 2 .0; 1/. In what follows, we shorten the notation sizeYı ; μ size. For every R R0 .Yı ; ; "/ we have 1 ln size.g 1 .t / \ BR .o// < "; R where o 2 Y is the root. Lemma 14.2.6. Let k R0 .Yı ; ; "/ be given. Then there is an open covering Ok D fU t W t 2 Kk g of the compact space Kk such that 1 k

ln size.g 1 .U t / \ Sk / < 2"

for every t 2 Kk . Proof. Take R D k C 1=2. The set g 1 .t / \ Sk is covered by N.k; t / D size.g 1 .t / \ Sk / open balls B .y/ with y 2 Yı . Since Sk BR .o/, we have 1 k

ln N.k; t /

2 R

ln size.g 1 .t / \ BR .o// < 2":

Let W Y be the union of the mentioned balls. We claim that there is a neighborhood U t of t 2 Kk in T such that g 1 .U t / \ Sk W . If this is not the case, then there is a sequence yi 2 Sk nW for which g.yi / ! t. One can assume that yi ! y1 2 Sk nW . By continuity, g.y1 / D lim g.yi / D t . This contradicts the fact that y1 62 g 1 .t /. Thus the covering Ok D fU t W t 2 Kk g fulfills the requirements. We take a finite closed covering Ck with multiplicity n C 1 inscribed in Ok , and consider the finite closed covering Ak D g 1 .Ck / of the combinatorial sphere Sk . Its multiplicity is n C 1. Then, by Lemma 14.2.6, size A < e 2k" for any A 2 Ak . Unfortunately, this does not mean that the diameter (along the sphere Sk ) of A is subexponential – the property which we would like to have. Thus we modify the covering Ak as follows. For every A 2 Ak , we consider the covering by those

174


open balls B .y/, y 2 Yı which intersect A. Let D Yı be the set of all y 2 Yı with B .y/ \ A ¤ ;. We say that points y; y 0 2 D are in one component if they are connected by a sequence in D for which the appropriate balls B related to consecutive members of the sequence intersect. Then D consists of finitely many components ˛ since there are S only finitely many points of the net Yı in the ball BRC .o/, R S D k C 1=2. Thus A D ˛ A˛ is the finite disjoint union of closed sets A˛ D A \ y2˛ B .y/ , which we now consider as the covering members. For every A˛ A, any x; x 0 2 A˛ can be connected in Y by a piecewise geodesic path whose vertices lie in the -neighborhood of Sk . Such a path has at most e 2k" edges all of length 2 , and all edges, except maybe the initial and the final ones, lie in the 1-skeleton of Y (these edges are geodesics in Y between the centers of balls B .y/). Finally, we consider the finite closed covering Bk D pk1 .Ak / of Z. Its multiplicity is n C 1. To complete the proof of Theorem 14.2.5, it remains to show that the mesh of this covering is arbitrarily small as k ! 1. For B 2 Bk , we take , 0 2 B with d.; 0 / D diam B. The points pk ./, pk . 0 / 2 A˛ lie in open stars stv , st v0 of some vertices v, v 0 2 Vk respectively. Let r 1=6 be the parameter of Y . Lemma 14.2.7. Assume that a piecewise geodesic path Y between v; v 0 2 Vk lies in the annulus fk joyj k C g and consists of p geodesic edges having the length 2 . Then d.v; v 0 / cp log2 r r k ; where the constant c D c.r; / is independent of k and of the path . Proof. We can assume that lies in the 1-skeleton of Y . The idea is to project down at lower levels until it collapses to a point, and estimate how many steps one needs for that. First, using Lemma 6.2.3, we can project to a horizontal path 0 in the highest level Sk 0 below , k 0 k 4 , without increasing its length L. 0 / L. / 2p . Next, let vi1 vi , vi viC1 Sk 0 be adjacent horizontal edges. As in Lemma 6.2.1, we find w 2 Vk 0 1 so that vi1 , vi , viC1 are connected with w by radial edges. Applying this construction together with Lemma 6.2.3 to 0 , we find a projection 00 Sk 0 1 of 0 , whose length is at most half of the length L. 0 /. After at most m D log2 L. 0 / steps 0 is projected to a point in Sk 0 m . Since during all projections, the end points v, v 0 are moving along radial geodesics, we have k 0 m .vjv 0 /o C c. Thus we obtain k k 0 C 4 .vjv 0 /o C log2 L. 0 / C 4 C c: 0

Since d.v; v 0 / c r .vjv /o , we obtain d.v; v 0 / c p log2 r r k :

14.3. Applications to nonembedding results

175

Applying Lemma 14.2.7, we obtain d.; 0 / d.; v/ C d.v; v 0 / C d.v 0 ; 0 / 2r k C c p log2 r r k with p e 2k" by the estimates above. Thus for sufficiently small " we have mesh Bk ! 0 as k ! 1. This completes the proof of Theorem 14.2.5.

14.3 Applications to nonembedding results 14.3.1 Subexponential corank of a product Let X D Hn1 Hnk , where n1 ; : : : ; nk 2. By Theorem 14.2.5 and property (4) of the subexponential corank, we have corank Hni D ni 1. Thus X corank X Rm .ni 1/ D dim X k i

for every m 0. In particular, for p 1 > dim X k there is no quasi-isometric embedding Hp ! X Rm . This estimate is the best possible because for p D dim X .k 1/ there is a quasi-isometric embedding Hp ! X ; see Section 14.1.

14.3.2 Subexponential corank of symmetric spaces Let X be a Riemannian symmetric space of noncompact type. Then corank X dim X rank X: Therefore, by Theorem 14.2.5, there is no quasi-isometric embedding Y ! X for any proper geodesic hyperbolic space Y with dim @1 Y > dim X rank X , in particular, there is no quasi-isometric embedding Hp ! X Rm with p 1 > dim X rank X and any m 0. Indeed, an Iwasawa decomposition G D NAK of the connected component of the identity in its isometry group allows to identify X with the solvable group NA. Fix x0 2 X and consider the orbit T D N x0 X . Then the map g W X ! T given by g.x/ D nx0 , where x D nax0 for n 2 N , a 2 A, is a subexponential foliation of rank dim T D dim N D dim X rank X . Its fibers g 1 .t / are geodesic rank X -flats.

14.4 Subexponential corank versus hyperbolic dimension The subexponential corank as well as the hyperbolic dimension may serve as obstacles to quasi-isometric embeddings. We have the following estimates of the hyperbolic

176


rank from above rank h X corank X; rank h X hypdim X 1; where the first estimate holds for all QPC-spaces, see Theorem 14.2.5, and the second one is obtained as follows: If Y ! X is a quasi-isometric embedding of a hyperbolic space Y , then hypdim Y hypdim X and, since hypdim Y dim @1 Y C 1 by Theorem 13.5.1, we get the estimate. However, the ranges of these invariants are different. For example, 2 2 corank.H H / 2 which implies by Theorem 14.2.5 that there is no quasiisometric embedding H4 ! H2 H2 Rm for any m 0. The estimate hypdim.H2 H2 / 4 prohibits quasi-isometric embeddings H5 ! H2 H2 Rm for any m 0, but it gives no information about the existence of embeddings H4 ! H2 H2 Rm (the precise value of hypdim.H2 H2 / is not known, it must be 3 or 4, and very likely it is 4). Furthermore, the hyperbolic dimension unlike the subexponential corank is hard to compute for nonhyperbolic spaces. On the other hand, in many cases the hyperbolic dimension gives better nonembedding results than the subexponential corank. For example, by Theorem 13.4.1, there is no quasi-isometric embedding H3 ! T1 T2 Rm for any metric trees T1 , T2 and any m 0 because hypdim H3 3 and hypdim Ti 1. If we take trees with exponential growth rate, e.g. binary trees, then corank Ti 1 because any continuous map of Ti into a zero dimensional space must be constant and thus there is no continuous subexponential foliation Ti ! T of rank 0. Hence, corank.T1 T2 Rm / 2, which prohibits quasi-isometric embeddings H4 ! T1 T2 Rm but gives no information about the existence of embeddings H3 ! T1 T2 Rm . In general, the situation can be described as follows. There are four quasi-isometry invariants or better two pairs of invariants relevant to the nonembedding problem: .rank h ; corank/

and

.t-rank; hypdim/:

The hyperbolic rank measures how large a space is intrinsically and the subexponential corank is complementary to it, that is, the subexponential corank serves as a tool to estimate the hyperbolic rank from above. The t-rank (t stands for trees) of a metric space X is the minimal k so that X can be quasi-isometrically embedded into the product T1 Tk Rm for some m 0. The t-rank measures how large a space is extrinsically. The hyperbolic dimension is complementary to the t-rank in the sense that it serves as a tool to estimate the t-rank from below, hypdim X t- rank X for every metric space X .

177


14.5 Supplementary results and remarks 14.5.1 Hyperbolic rank of some nonhyperbolic spaces We discuss the hyperbolic rank of Riemannian symmetric spaces and of products. Let X be a Riemannian symmetric space of noncompact type. It is proven in [Le] that rank h X dim X rank X . Thus by Section 14.3.2, rank h X D dim X rank X D corank X: The product X D X1 Xm of Hadamard manifolds with pinched negative curvature, k 2 K 1, always has rank h X

m X

rank h Xi :

iD1

This is a generalization of Proposition 14.1.1 and is proven in [Gr2], [BrFa] for Xi being real hyperbolic manifolds and in [FS1] for the general case. Using properties of the subexponential corank and Theorem 14.2.5, we obtain the product formula rank h X D

m X iD1

rank h Xi D

m X

corank Xi D corank X:

iD1

P However a product formula of the type rank h …i Xi D i rank h Xi , does not hold in general. In Chapter 10, we have described a quasi-isometric embedding of the hyperbolic plane H2 into the product Ta Tb of two metric simplicial trees. Thus we obtain rank h .Ta Tb / 1 (actually equality holds). On the other hand, dim @1 Ta D dim @1 Tb D 0 and therefore rank h Ta D rank h Tb D 0. Hence we have rank h .Ta Tb / > rank h Ta C rank h Tb : In this example, the trees Ta , Tb have infinite valence at every vertex. According to [BDS], there is a quasi-isometric embedding of Hn into the n-fold product Xn of binary trees. Thus rank h Xn n 1, while rank h T D 0 for every binary tree T . For a further discussion of the hyperbolic rank of products it is useful to review the hyperbolic product construction.

14.5.2 Hyperbolic product of hyperbolic spaces The embedding described in Proposition 14.1.1 is a particular case of a general construction which associates to any hyperbolic spaces X1 , X2 a hyperbolic subspace Y X1 X2 as follows. On X1 X2 consider the l1 product metric, i.e. j.x1 ; x2 /; .y1 ; y2 /j ´ maxfjx1 y1 j; jx2 y2 jg

for all x ; y 2 X ; D 1; 2:

178


Recall that for a, b, c 2 R and c 0 we have the notation : a Dc b

if and only if ja bj c:

Given two pointed hyperbolic metric spaces .X1 ; o1 / and .X2 ; o2 / as well as a number 0, we write o ´ .o1 ; o2 / 2 X1 X2 and define : Y ;o ´ f.x1 ; x2 / 2 X1 X2 W jo1 x1 j D jo2 x2 jg: The space Y ;o X1 X2 is endowed with the restriction of the maximum metric on X1 X2 . Theorem 14.5.1 (FS1). If X1 , X2 are hyperbolic, then Y ;o is hyperbolic.

In order to investigate the boundary of Y ;o one needs more structure: Let k 0. A k-rough geodesic is a map W I ! X from an interval I R to a metric space X with : j.s/.t /j Dk js tj for all s; t 2 I: The space X is called k-roughly geodesic if for every pair x; y 2 X there exists a k-rough geodesic W Œ0; jxyj ! X with .0/ D x and .jxyj/ D y. X is called roughly geodesic if X is k-roughly geodesic for some k 0. Theorem 14.5.2 (FS1). If X1 , X2 are ı-hyperbolic and k-roughly geodesic, then there exists 0 0 such that for all 0 the space Y ;o is roughly geodesic and the boundary @1 Y ;o is naturally homeomorphic to @1 X1 @1 X2 . There is also a version of these theorems where the basepoints o 2 X are replaced by Busemann functions based at points z 2 @1 X . The space Y ;0 is called the hyperbolic product of the pointed spaces .X ; o /. For simplicity, we denote the hyperbolic product by X1 h X2 . The operation h can be considered as a natural product construction in the class of hyperbolic spaces. Since X1 h X2 X1 X2 we clearly have rank h .X1 h X2 / rank h .X1 X2 /: By Theorem 14.5.2 we have for roughly geodesic hyperbolic spaces X that rank h .X1 h X2 / D dim.@1 X1 @1 X2 /: By the product theorem in dimension theory we obtain rank h .X1 h X2 / rank h X1 C rank h X2 : We now show that there are cases of strict inequality for both estimates of rank h .X1 h X2 /. It is proven in [Dr4] that for every prime p, there exists a hyperbolic Coxeter group p with a Pontryagin surface …p as boundary at infinity. It is


179

well known that for primes p ¤ q we have dim.…p …q / D dim …p C dim …q 1. Thus we obtain rank h .p h q / < rank h .p / C rank h .q /: We already remarked in Section 14.5.1 that rank h .T T / 1 for the binary tree T . Since rank h .T h T / rank h T C rank h T D 0, we have rank h .T h T / < rank h .T T /: We now come back to the discussion of the hyperbolic rank of a product of hyperbolic spaces. We have already seen in the last section that for Hadamard manifolds Xi with pinched negative curvature we have rank h .X1 X2 / D rank h X1 C rank h X2 ; but, for example, for the binary tree we have rank h .T T / > rank h T C rank h T: We do not know whether there are spaces X1 and X2 with rank h .X1 X2 / < rank h X1 C rank h X2 . In analogy with the hyperbolic product a possible candidate for this situation is p q . We do not know whether the strict inequality holds in that case. However using the equality obtained in [Leb] asdim. 0 / D dim.@1 @1 0 / C 2; which holds for all hyperbolic groups , 0 , we can estimate rank h .p q / hypdim.p q / 1 asdim.p q / 1 D dim.…p …q / C 1 D dim …p C dim …q D rank h p C rank h q : Here, the first inequality follows from Theorem 13.5.1. Bibliographical note. The notion of the hyperbolic rank of a metric spaces is introduced in [Gr1] in a slightly stronger form: one takes into account all hyperbolic spaces. The results of this chapter are based on [BS1] where a slightly weaker version of Theorem 14.2.5 is obtained for the hyperbolic rank, which takes into account only CAT.1/-spaces.

Appendix Models of the hyperbolic space Hn

Here we consider various models of the real hyperbolic space Hn and explain the classical result that there is one-to-one correspondence between the isometries of Hn and the Möbius transformations of the unit sphere S n1 . Furthermore, we give a proof of another classical result which characterizes the Möbius transformations as ones preserving the cross-ratio.

A.1 The pseudo-spherical model For simplicity, we assume first that n D 2. Let g W R3 ! R be the quadratic form given by g.v/ D x 2 C y 2 z 2 ; v D .x; y; z/ 2 R3 : Consider the unit pseudo-sphere (more precisely, its upper component) A D f.x; y; z/ W x 2 C y 2 z 2 D 1; z > 0g: We introduce the pseudo-spherical coordinates on A x D sinh cos ';

y D sinh sin ';

z D cosh ;

where 0 ' < 2, 0. We have x 2 C y 2 z 2 D sinh2 cosh2 1 and computing the induced quadratic form on A, we obtain dsA2 D dx 2 C dy 2 dz 2 D d2 C sinh2 d' 2 : We see that the form dsA2 is positive definite and thus is a Riemannian metric on A. Its Gaussian curvature is .sinh /00

1: KD sinh The space A with the Riemannian metric dsA2 is called the pseudo-spherical model of the hyperbolic plane H2 . The geodesics in A are the intersections of A with 2-dimensional linear subspaces E R3 .

Appendix. Models of the hyperbolic space Hn

182

A.2 The unit disc model We introduce the polar coordinates .r; '/, 0 r < 1, 0 ' < 2 in the unit disc B D f.x; y/ 2 R2 W x 2 C y 2 < 1g: x D r cos ';

y D r sin ':

A f B 1

r 1 1

Figure A.1. The map f W A ! B.

The map f W A ! B, f .; '/ D .r; '/, where sinh cosh C 1 D ; r 1 introduces the structure of the hyperbolic plane H2 on B. Let us find the corresponding metric dsB2 . We have 2 sinh 1 D cosh2 D sinh2 C 1; r

whence

2 1 1 sinh D 2 r r

and consequently sinh D On the other hand rD thus dr D

2r : 1 r2

sinh ; cosh C 1

cosh .cosh C 1/ sinh2 d d D .cosh C 1/2 cosh C 1

183

A.3. The upper half-plane model

and d D .cosh C 1/dr D

sinh dr r

D

2 dr. 1r 2

It follows that

dsB2 D d2 C sinh2 d' 2 4r 2 4 d' 2 dr 2 C 2 2 .1 r 2 /2 .1 r / 4 D .dr 2 C r 2 d' 2 /; .1 r 2 /2 D

or dsB2 D

.1

4 .dx 2 C dy 2 /: C y 2 //2

.x 2

The space B with the Riemannian metric dsB2 is called the unit disc model of H2 . An isomorphism between the models A and B is given by the map f . Exercise A.2.1. Using the isomorphism f show that any geodesic in B is either a diameter of B or the arc of a circle in R2 orthogonal to the boundary circle S 1 of B.

A.3 The upper half-plane model We denote by C D f.u; v/ 2 R2 W v > 0g the upper half-plane and consider the map g W B ! C given in complex numbers by the fractional linear transformation g.z/ D w D {

1z ; 1Cz

where z D x C {y. The map g introduces the structure of H2 on C . Let us find the corresponding metric dsC2 . Let zN D x {y be the conjugate to z. Then x 2 Cy 2 D z z, N dx 2 C dy 2 D dz d zN and thus dsC2 D Since zD

{w {Cw

4dz d zN : .1 z zN /2

and

zN D

{Cw x ; {w x

we have dz D

2{d w x : .{ w/ x 2

2{dw .{ C w/2

and d zN D

1 z zN D

2{.w w/ x .{ C w/.{ w/ x

Using that we obtain


184 and find

4 dz d zN .1 z z/ N 2 .{ C w/2 .{ w/ x 2 4dw d w x D 2 x 2 .w w/ x .{ C w/2 .{ w/ 4dw d w x 1 D D 2 dw d w: x .w w/ x 2 v

dsC2 D

The space C with the Riemannian metric dsC2 D

1 .du2 C dv 2 / v2

is called the upper half-plane model of H2 . An isomorphism between the models B and C is given by the map g. Exercise A.3.1. It is well known that any fractional linear map of C transforms a generalized circle (i.e. a circle or a line) into a generalized circle. Using this and Exercise A.2.1 show that any geodesic in C is either a vertical half-line or the arc of a circle in C orthogonal to the boundary real line R of C .

A.3.1 The angle of parallelism Consider a right-angled infinite triangle ab H2 with 2 @1 H2 and ]a .b; / D =2. Then the angle ˛ D ]b .a; / is called the angle of parallelism. There is an important formula relating the angle of parallelism with the distance d D jabj. Lemma A.3.2. Under the condition above, we have tan.˛=2/ D exp.d /:

a0 1

˛

1 0

b

11 00

1 0

˛

Figure A.2. Parallelism angle.

185

A.4. The solvable group model

Proof. It is convenient to use the upper half-plane model C . Assuming that the side ab lies on the half-circle f.u; v/ 2 C W u2 C v 2 D 1g so that a lies in fu D 0g, and D 1, we obtain that the sides a, b are vertical rays in the model C . The ray b forms the angle ˛ with the half-circle. Thus Z =2 dt ˛ jabj D D ln tan : sin t 2 ˛ Exercise A.3.3. Show using the angle parallelism formula that jxx 0 jS D 2 sinh.jxx 0 j=2/ for a horosphere S H2 and for all x; x 0 2 S , where jxx 0 jS is the distance between x, x 0 along S.

A.4 The solvable group model We fix ¤ 0 and consider on R2 D f.t; x/ W t 2 R; x 2 Rg a group structure of the semi-direct product S2 D R Ë R given by multiplication .t; x/ .t 0 ; x 0 / D .t C t 0 ; x C e t x 0 /: It is easy to see that S2 is a solvable group. It acts on itself by the left translations L.t;x/ W S2 ! S2 , L.t;x/ .t 0 ; x 0 / D .t; x/ .t 0 ; x 0 /. A metric on S2 is left invariant if the left translation L.t;x/ is an isometry for every .t; x/ 2 S2 . To find a left invariant metric on S2 , we note that the curve .0; x 0 /, x 0 2 R is shifted to the curve L.t;x/ .0; x 0 / D .t; x C e t x 0 /. The tangent vector dx 0 at .0; 0/ is shifted to the vector dL.t;x/ .dx 0 / D .0; e t dx 0 /. Furthermore, the curve .t 0 ; 0/, t 0 2 R is shifted to the curve L.t;x/ .t 0 ; 0/ D .t C t 0 ; x/, and the tangent vector dt 0 at .0; 0/ is shifted to the vector dL.t;x/ .dt 0 / D dt 0 at .t 0 ; x 0 /. Assuming that the vectors dx 0 and dt 0 are orthogonal at zero with respect to a Riemannian metric h ; i on S2 , we find that the condition to be left invariant is reduced to the equality he t dx 0 ; e t dx 0 i.t;x/ D hdx 0 ; dx 0 i.0;0/ ; or hdx 0 ; dx 0 i.t;x/ D e 2t hdx 0 ; dx 0 i.0;0/ . In other words, the metric ds 2 D dt 2 C e 2t dx 2 t 00

is left invariant on S2 . Its Gaussian curvature is constant K D .eet/ D 2 and for D 1 it equals 1. This gives one more model of the hyperbolic plane H2 , which we call the solvable group model and denote by D, 2 D dt 2 C e 2t dx 2 : dsD


186

The horosphere f.0; x/ W x 2 Rg is an abelian subgroup in S2 . The map h W C ! D, h.u; v/ D .ln v; u/ is an isomorphism between C and D: for t D ln v, x D u we have dt D dv , dx D du and v D e t . Thus v du2 C dv 2 dx 2 C e 2t dt 2 D D dt 2 C e 2t dx 2 : e 2t v2

A.5 Generalizations to an arbitrary dimension The quadratic form g.v/ D x02 .x12 C C xn2 /, v D fx0 ; x1 ; : : : ; xn g induces on An RnC1 , given by g.v/ D 1 and the condition x0 > 0, the metric with the constant sectional curvature 1 which has the form dsA2 D d2 C sinh2 d! 2 in the pseudo-spherical coordinates x0 D cosh , xi D sinh fi .!/, i 1, where d! 2 is the metric of the unit sphere in Rn . This is the pseudo-spherical model of the hyperbolic space Hn . The model of Hn in the unit disc B n D f.x1 ; : : : ; xn / W x12 C C xn2 < 1g is given by the metric 4 .dr 2 C r 2 d! 2 /; dsB2 D .1 r 2 /2 where .r; !/ are spherical coordinates in Rn . An isomorphism f W An ! B n between these models is given by the same formula as for n D 2. The boundary at infinity @1 Hn is the boundary sphere for the ball B n . The upper half-space model of Hn in the upper half-space C n D f.x1 ; : : : ; xn / W xn > 0g is given by the metric dsC2 D

1 .dx1 C C dxn2 /: xn2

Here the boundary at infinity @1 Hn is the subspace Rn1 D f.x1 ; : : : ; xn / W xn D 0g complemented by a point 1. An isomorphism g W B n ! C n can be obtained by composing the inversion '0 W Rn ! Rn with respect to the sphere of radius 2 centered at .1; 0; : : : ; 0/, and an appropriate similitude '1 W Rn ! Rn . This is slightly different from the case n D 2, where g leaves invariant the orientation. Finally, the metric 2 dsD D dt 2 C e 2t dx 2 on the solvable group Sn D R Ë Rn1 with multiplication .t; x/ .t 0 ; x 0 / D .t C t 0 ; x C e t x 0 /

187

A.6. Möbius transformations

is left invariant and has the constant sectional curvature 1, i.e., it gives another model of the hyperbolic space Hn . As an isomorphism h W C n ! D n between the models C n and D n one can take the map given by h.x1 ; : : : ; xn / D .t D ln xn ; x D .x1 ; : : : ; xn1 // :

A.6 Möbius transformations The boundary at infinity @1 HnC1 of the hyperbolic space HnC1 possesses a canonical conformal structure of the unit sphere S n RnC1 . There is an important relation between the isometries of HnC1 and the Möbius transformations of the sphere S n .

A.6.1 Inversions and isometries Let Sa .z/ Rn be the sphere of radius a centered at z. The inversion ' W Rn n z ! Rn n z with respect to Sa .z/ is defined by the condition that the point '.x/ lies in the ray .zxi with vertex z which contains x and jz'.x/j jzxj D a2 : y n D Rn [ 1, then ' is extended to the If one compactifies Rn by a point 1, R n n y ! R y , '.z/ D 1, '.1/ D z. Note that R y n can be identified involution ' W R n nC1 by the stereographic projection. We also define the with the sphere S R inversion with respect to a hyperplane E Rn as the reflection with respect to E (the hyperplane E can be viewed as the sphere S1 ./ of infinite radius centered at 2 @1 Rn ). Theorem A.6.1. Let HnC1 D f.x1 ; : : : ; xnC1 / W xnC1 > 0g RnC1 be the upper half-space model of the hyperbolic space. Then the restriction of the inversion ' with respect to the sphere Sa .z/ RnC1 to HnC1 is an isometry of HnC1 for every z 2 @1 HnC1 and every radius a. Proof. For z D 1 this is obvious since such an inversion is the reflection with respect to the vertical hyperplane. Thus we assume that z D .0; : : : ; 0/. Fix x 2 HnC1 and show that the differential d' W Tx HnC1 ! T'.x/ HnC1 is an isometry of corresponding tangent spaces. Let v 2 Tx HnC1 be a radial vector, i.e. tangent to the ray .zxi, and let w 2 Tx HnC1 be a tangential vector, i.e. tangent to the sphere centered at z and containing x. Denote by v 0 D d'.v/, w 0 D d'.w/ 2 T'.x/ HnC1 the images of that vectors. Then v 0 is radial, and w 0 tangential. It follows from similarity of corresponding triangles that jz'.x/j a2 jw 0 je D μ : D jzxj jzxj2 jwje


188

w0

z We find

jv 0 je , jvje

!

'.x/

v0

w xv

assuming that jvje D 1. The expression

1 a2 .jz'.x C t /j jz'.x/j/ D t t

1 1 jz.x C t /j jzxj

D

a2 jzxj jz.x C t /j

jv 0 je a2 y nC1 ! R y nC1 tends to jzxj 2 D as t ! 0. Thus jvje D . This implies that ' W R is a conformal map, jd'.u/je D juje for all tangent vectors at x. Let ! 2 .0; =2 be the angle between the ray .zxi and the plane xnC1 D 0. Then the hyperbolic length jujh D jzxj1sin ! juje . Thus for the image u0 of u 2 Tx HnC1 under the differential d' we have

1 jzxj ju0 je ju0 je D 2 jz'.x/j sin ! a sin ! jzxj 1 D 2 juje D juje D jujh : a sin ! jzxj sin !

ju0 jh D

This proves that the inversion ' is an isometry of HnC1 .

y n coincides with the boundary at infinity @1 HnC1 of HnC1 in Since the sphere R y n is extended to an inversion of the the upper half-space model, any inversion of R nC1 upper half-space H , which by Theorem A.6.1 is the reflection with respect to the corresponding hyperplane in the hyperbolic geometry. Recall that every isometry of HnC1 can be represented as the composition of finitely many reflections. y n is by definition the composition of finitely many A Möbius transformation of R inversions. Thus Theorem A.6.1 implies yn ! R y n is extended to an isometry Corollary A.6.2. Any Möbius transformation 'y W R nC1 nC1 nC1 'W H !H . Every isometry of H can be obtained in such a way.

A.7 Cross-ratio y n be pairwise distinct points. Their (classical) cross-ratio is defined Let a; b; c; d 2 R by jacj jbd j Œa; b; c; d D jabj jcd j

189

A.7. Cross-ratio

(distances are Euclidean). If one of the points coincides with 1, then the factors containing it cancel out. For example, Œ1; b; c; d D

jbd j : jcd j

Theorem A.7.1. The cross-ratio is invariant under any Möbius transformation. y n are pairwise distinct, and let ' W R yn ! R y n be Proof. Assume that a; b; c; d 2 R Möbius. We have to check that Œ'.a/; '.b/; '.c/; '.d / D Œa; b; c; d : One can assume that ' is the inversion with respect to the sphere Sr .o/. The triangles r2 r2 , jo'.b/j D jobj and thus oab and o'.b/'.a/ are similar, since jo'.a/j D joaj jobj jo'.a/j D : jo'.b/j joaj Therefore,

j'.a/'.b/j j'.a/oj j'.b/oj D D : jabj jobj joaj

Then j'.a/'.c/j D jacj

j'.a/oj I jocj

j'.b/'.d /j D jbd j

j'.a/'.b/j D jabj

j'.a/oj I jobj

j'.c/'.d /j D jcd j

and

j'.d /oj jobj

j'.d /oj : jocj

Thus j'.a/'.c/j j'.b/'.d /j j'.a/'.b/j j'.c/'.d /j jacj jbd j j'.a/oj jobj j'.d /oj jocj D jabj jcd j jocj j'.a/oj jobj j'.d /oj D Œa; b; c; d :

Œ'.a/; '.b/; '.c/; '.d / D

The case when one of the points coincides with 1 we leave as an exercise to the reader. yn ! R y n leaves invariant the cross-ratio. Theorem A.7.2. Assume that a map ' W R Then ' is Möbius.


190

Proof. Composing with an appropriate Möbius transformation and using Theorem A.7.1, one can assume that '.1/ D 1. Then '.Rn / D Rn . Fix c, d 2 Rn and consider the points a; b 2 Rn as variables. Then Œ1; b; c; d Œ'.1/; '.b/; '.c/; '.d / jabj D D jcd j Œa; b; 1; d Œ'.a/; '.b/; '.1/; '.d / Œ1; '.b/; '.c/; '.d / j'.a/'.b/j D D : Œ'.a/; '.b/; 1; '.d / j'.c/'.d /j Hence the ratio

j'.c/'.d /j j'.a/'.b/j DD jcd j jabj

is independent of a, b, and thus ' is a similitude.

A.7.1 Cross-ratio in hyperbolic geometry y n as the boundary at infinity @1 HnC1 . Any distinct points Consider the sphere R n y a; b 2 R are connected by the unique geodesic ab HnC1 . d B C

c a b Figure A.3. Cross-ratio.

y n let B, C 2 ad Theorem A.7.3. Given pairwise distinct points a; b; c; d 2 R nC1 be the projections of b, c respectively to the geodesic ad . Then for the hyperH bolic distance between B and C we have jBC jh D jln Œa; b; c; d j:

A.7. Cross-ratio

191

Proof. One can assume that d D 1. Then in the upper half-space model the points B and C lie in the vertical ray Œa; d /. Since the geodesics in this model are either vertical rays or vertical half-circles centered at infinity, we see that the Euclidean distances jaBj D jabj and jaC j D jacj. Thus ˇ ˇ ˇ jacj ˇ ˇ D j lnŒa; b; c; d j: jBC jh D ˇˇln jabj ˇ Historical note. The models B n , C n , D n of the hyperbolic space Hn for n 2 appeared among others in 1868 in a memoir by M. Beltrami, [Belt]. That is 14 years before the famous papers by H. Poincaré on Fuchsian functions, where the unit disk and the upper half-space models in dimensions 2 and 3 appeared, fractional linear transformations as product of inversions were represented and interpreted as motions of two and three dimensional hyperbolic geometry. For more details see [Po] and [Mi].

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Index

admissible function, 57 admissible quadruple for triples, 88 ancestor, 149 angle ı-measure, 162 angle of parallelism, 184 annulus, 130 a-path, 5 Assouad–Nagata dimension, 108 asymptotic dimension, 108 asymptotic `-dimension, 109 asymptotic rays, 20 balanced covering, 141 ball closed, 4 open, 4 barycentric map, 112 barycentric subdivision, 113 barycentric subdivision map, 113 barycentric triangulation, 115, 116 boundary at infinity, 12, 30 boundary continuous space, 32 bounded geometry, 103 Bourdon metric, 22 branch point, 72 Busemann function, 25 capacity of a covering, 137 CAT.1/-space, 7 Cayley graph, 7 central ancestor, 70 chain construction, 14 characteristic subset, 117 classical cross-difference, 36 classical cross-ratio, 36 coarse map, 36

coarsely equivalent spaces, 127 cobounded metric space, 17 coherent covering of n , 125 colored covering, 109 colored P-dimension, 120 coloring, 97 combinatorial dimension, 111 comparison point, 7 comparison triangle, 7 complete subpolyhedron, 111 cone point, 72 constant threshold, 51 control function, 36, 51, 53 convex hull, 100 coordinates horospherical, 167 corank subexponential, 169 covering, 107 balanced, 141 colored, 109 inscribed, 143 locally finite, 112 uniformly ls-doubling, 160 covering P-dimension, 120 critical level, 98, 152 cross-difference, 40 classical, 36 small, 19 cross-pair, 46, 50 cross-ratio, 49 classical, 36 curve graph, 135 cut function, 112

198 d -disjoint family, 126 degree of a vertex, 102 ı-inequality, 10 ı-triple, 10 descendant, 149 difference ordinary, 63 dimension Assouad–Nagata, 108 asymptotic, 108 asymptotic `-, 109 combinatorial, 111 hyperbolic, 161 `-, 109 linearly controlled asymptotic, 109 linearly controlled metric, 109 topological, 107 uniform asymptotic, 131 directed poset, 149 distance, 1 d -multiplicity, 126 doubling at small scales, 18 doubling metric space, 18, 97 edge horizontal, 69 radial, 69 equiradial points, 2 family d -disjoint, 126 maximal, 143 filter, 117 finite power set, 113 foliation subexponential, 169 foliation rank, 169 function admissible, 57 control, 36, 51, 53 cut, 112 level, 149

Index

geodesic, 1 quasi, 4 radial, 70 rough, 178 geodesic boundary, 20 Gromov product, 2, 26–28 group Gromov hyperbolic, 7 word hyperbolic, 7 Hadamard manifold, 7, 102 Hadamard space, 7 hereditary, 102 homothetic map, 16 horizontally close vertices, 151 horizontally distinct vertices, 152 horospherical coordinates, 167 hyperbolic approximation, 69 extended, 172 truncated, 78 hyperbolic cone, 69, 79 hyperbolic dimension, 161 hyperbolic product, 178 hyperbolic rank, 167 hyperbolic space, 11 geodesic, 3 visual, 81 hyperbolicity constant, 3 infinitely remote, 29 inscribed covering, 143 inversion, 57 -inversion, 58 large scale doubling, 159 `-dimension, 109 Lebesgue number, 108 level function, 149 levelled tree, 149 local capacity, 141 locally finite covering, 112 locally finite polyhedron, 111 locally self-similar metric space, 17

199

Index

locally similar, 17, 154 Möbius map, 56 Möbius structure, 56 Möbius transformation, 188 manifold Hadamard, 7 map barycentric, 112 barycentric subdivision, 113 barycentric triangulation, 116 bilipschitz, 4 coarse, 36 coherent with a covering, 124 homothetic, 16 Möbius, 56 power quasi-Möbius, 52 PQ-isometric, 41 PQ-Möbius, 52 quasi-homothetic, 16 quasi-isometric, 4 quasi-Möbius, 51 quasi-symmetric, 53 roughly homothetic, 81 strongly PQ-isometric, 37 uniformization, 111 maximal family, 143 metric, 1 Bourdon, 22 left invariant, 185 path, 7 product, 147 uniform, 111 visual, 15, 29 word, 7 metric space cobounded, 17 doubling, 18, 97 doubling at small scales, 18 doubling at some scale, 103 geodesic, 1 Gromov hyperbolic, 3, 11

large scale doubling, 159 locally self-similar, 17 proper, 17 uniformly perfect, 60, 84 metric tree, 3 mirror, 133 model of H2 pseudo-spherical, 181 solvable group, 185 unit disc, 183 upper half-plane, 184 multiplicity, 107 nerve, 112 net, 4 n-gon, 2 open star, 111 ordinary difference, 63 ordinary ratio, 63 parameter of a hyperbolic approximation, 69 path metric, 7 P-dimension, 117, 120, 121 polyhedral P-dimension, 121 polyhedron locally finite, 111 poset, 149 directed, 149 power quasi-Möbius map, 52 PQ-isometric map, 41 PQ-Möbius map, 52 product metric, 147 property space, 117 pseudo-spherical model, 181 Q-metric, 29 QPC-space, 170 quasi-homothetic map, 16 quasi-isometricity constants, 4 quasi-isometry, 4 quasi-Möbius map, 51

200 quasi-metric, 14, 29 quasi-metric space, 13, 29 quasi-symmetric map, 53 rank foliation, 169 hyperbolic, 167 ratio classical cross-, 36 cross-, 49 ordinary, 63 representative triple for a ball, 87 root of a tree, 149 rough geodesic, 178 rough isometry, 81 rough similarity, 81 roughly geodesic space, 178 roughly isometric, 81 saturation, 122 segment, 1 separated subset, 69 sequence(s) converging to infinity, 12, 30 equivalent, 12, 30 shadow, 162 simplex spherical, 172 simplicial metric tree, 147 size, 169 small cross-difference, 19 snow-flake transformation, 67, 97 solvable group model, 185 space CAT.1/-, 7 boundary continuous, 32 Hadamard, 7 property, 117 QPC-, 170 quasi-metric, 13, 29 roughly geodesic, 178 spherical simplex, 172

Index

standard simplex, 111 star, 111 open, 111 strongly PQ-isometric map, 37 subexponential corank, 169 subexponential foliation, 169 subpolyhedron complete, 111 threshold constant, 51 topological dimension, 107 tree levelled, 149 metric, 3 simplicial, 147 triangle, 2 triangulation barycentric, 115, 116 triple cross-difference, 19 cross-ratio, 49 ı-, 10 multiplicative K-, 50 representative for a ball, 87 tripod, 2 ultra-metric, 14 uniform asymptotic dimension, 131 uniform metric, 111 uniformization, 111 uniformization map, 111 uniformly ls-doubling covering, 160 unit disc model, 183 upper half-plane model, 184 valence, 147 visual hyperbolic space, 81 inequalities, 15 metric, 15, 29 word metric, 7

Elements of asymptotic geometry

Elements of Asymptotic Geometry (EMS Monographs in Mathematics)

Elements of noncommutative geometry

Euclid's Elements of Geometry

The elements of geometry

Elements of differential geometry

Elements of algebraic geometry

The Elements of Coordinate Geometry

The elements of solid geometry

Asymptotic Formulae in Spectral Geometry

The Elements of Non-Euclidean Geometry

Elements of algebra: Geometry, numbers, equations

The Elements of Non-Euclidean Geometry

Elements of Algebra: Geometry, Numbers, Equations

The elements of plane analytic geometry

The Elements of Non-Euclidean Geometry

Basic elements of differential geometry and topology

Techniques of asymptotic analysis

Asymptotic approximations of integrals

Asymptotic approximations of integrals

Techniques of asymptotic analysis

Derivation of Asymptotic Formulae

Asymptotic approximations of integrals

Asymptotic statistics

Asymptotic expansions

Asymptotic Statistics

Asymptotic Analysis

Asymptotic Analysis

Asymptotic expansions

Asymptotic expansions

The elements of non-Euclidean plane geometry and trigonometry

Elements of asymptotic geometry