4-Dimensional compact projective planes with small nilradical

Geometriae Dedicata 58: 53-62, 1995. © 1995 KluwerAcademic Publishers. Printed in the Netherlands.

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4-Dimensional Compact Projective Planes with Small Nilradical Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday H A U K E KLEIN Mathematisches Seminar, Universitiit Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany (Received: 7 October 1994) Abstract. We consider 4-dimensional compact projective planes with a solvable 6-dimensional

collineation group E and with orbit type _> (2, 1), i.e. E fixes a flag v E W, acts transitively on /~v\ {W} and fixesno point in the set W\ {v }. We prove a series of lemmas concerning the action of invariant subgroups of E. These lemmas are applied to prove that the maximal connected nilpotent invariant subgroup of E has dimension at least 4. Mathematics Subject Classifications (1991): 51H 10, 51H20.

1. Introduction In the fundamental papers ([16], [17]), Salzmann studies 4-dimensional compact projective planes, i.e. topologicalprojective planes with point space homeomorphic to the classical plane over the complex numbers, We denote by E the connected component of the group of all continuous collineations of a 4-dimensional compact projective plane 7r. The plane is called flexible if E has an open orbit in the space of flags. Since this space has dimension 6 we have dim E _> 6 for such a plane. The group E is known to be a Lie group with dim E < 16. All planes with dim ~ _> 7 are completely classified ([4], [13]). All flexible translation planes (and hence also the dual translation planes) are classified ([2]) and also all flexible shift planes ([1], [11], [12], [3]). By [11] the plane 7r is a shift plane, a translation plane or a dual translation plane if and only if the group E contains a subgroup isomorphic to the vector group R 4. As a general reference for 4-dimensional compact projective planes see Chapter 7 of [18]. In this paper we will consider the remaining case that there is no 4-dimensional abelian subalgebra in ~(E) and dim ~ = 6. If Z is not solvable then ~ --- G L + R ~ (2, 1) in the sense of [5], i.e. ~ does not fix a line Y E L v \ { W } and dually ~ fixes no point u E W \ { v } and acts transitively on Z~v\{W} or on W \ { v } . Up to duality we may assume that Z acts transitively on £ v \ { W } , Next we denote by N the nilradical of E, i.e. the maximal connected nilpotent invafiant subgroup of E. By [14] we know dim N >_ 3. All planes with dim N >_ 5 are known ([6], [7]). By [14], a 6-dimensional solvable Lie algebra with a 3-dimensional nilradical is isomorphic to either 123or to Ic × 12, where 12 denotes the 2-dimensional, nonabelian Lie algebra and Ic is the analogous complex Lie algebra considered as a 4-dimensional real algebra. In order to exclude the possibility g(E) = 13 we consider the base ideal of £(E), i.e. the ideal generated by all 1-dimensional ideals of £(E) ([10, II.5]). We denote the corresponding invariant subgroup of E by B ( ~ ) . The Lie algebra 13 has a big base ideal and in Section 2 we will see that this leads to a contradiction. The most frequently used lemma on the structure of the set of fixed points of collineations q~ E ~ is the lemma on quadrangles: lfq~ E ~fixes a quadrangle, i.e. four points, no three of which are collinear, then q5 = 1. The organization of this paper is as follows. In Section 2 we prove a series of lemmas under the assumption of orbit type >_ (2, 1). These results hold independently of the structure of the nilradical N and they seem to be useful for the general classification in orbit type >_ (2, 1). In Section 3 we describe the subalgebra structure of the Lie algebra Ic × 12. This algebra occurs in Section 4 as the Lie algebra of ~ in the hypothetic case dim N = 3. In Section 4 we will exclude the group of type l c × 12, and thus prove that the nilradical N of ~ has dimension at least 4.

2. General Facts in Orbit Type _> (2, 1) Let 7r = (7~, £ ) be a 4-dimensional compact projective plane and denote by ~ the connected component of the group of all continuous collineations of 7r. We assume the following conditions: • ~r is neither a translation plane nor a dual translation plane, nor a shift plane, i.e. ~ contains no subgroup isomorphic to the vector group R 4. • The group ~ is 6-dimensional and solvable. By [5], ~ fixes a flag v E W. • ~ acts transitively on £ v \ { W ) and fixes no point of the set W \ { v } . In particular, ~, fixes no point in 7:'\{v) and no line in £ \ { W ) . We prove a series of lemmas in this situation. LEMMA 1. Let 1 # N 4. Assume that we are in the situation of the theorem but dim N < 4. We will show that these assumptions lead to a contradiction. LEMMA 1. We have g(S) TM 1c2 × 12. Proof Since d i m U _< 3 we have g(~) ~- 12c × 12 or g(~) ~ l 3. The latter possibility is excluded by Lemma 8 of Section 2. Lemma I implies that S = L 2,~ c × L2 for some n E N tO {c~}. LEMMA 2. E acts transitively on P \ W, and L~2 C_ S[~,w]. The invariant subgroup C i (L2,n) has a trivial centralizer in So for each affine point o E P \ W. Proof By Lemma 7 of Section 2 we know that L~ C S[~,w]. Let o E 79\W be an affine point. If C~o(L~n )' ¢ 1, then (L2,n) c , C ~[v,v] by Lemma 5 of Section 2. But then Lemma 3 of Section 2 implies that (Lz,n) c t = E[v,w], contradicting the fact that U2 C_ Sly,W]. If S is not transitive on P \ W , then there exists an affine point o E 79\W with dim So _> 3. By an inspection of the list of subalgebras of 12 c × 12 given in Section 3, we see that (Lz,n) c I = exp((B, C)) has a non-trivial centralizer in So; a contradiction.

LEMMA 3. Sly,W] = L2 and n = 1, i.e. S = Le2 × L2. Proof Choose an affine point o E P \ W . By Lemma 2 the stabilizer Eo is a 2-dimensional connected subgroup of E which contains no invariant subgroup of E, and the centralizer of (Lz,,~) c in Eo is trivial. By the list of 2-dimensional subalgebras of 12 c × 12 of Section 3 there are four possible cases for the subalgebra

g(Zo) (up to an automorphism of 1c × 12). These cases are:

1. (bs + f , R + b'S)(b, b' e R), 2. ( a R + b S , R + e ) ( ( a , b ) E R2\{0}), 3. ( S , R + f),

4 (a, s>. In any case, CEoL 2 • 1 and Lemma 5 and Lemma 3 of Section 2 imply L2 = ~[~,w]. By L e m m a 6 of Section 2, we know that Z(LC2,,~) C_ Z ( S ) C_ E[v,w] = L2, hence Z ( L ~ ) = 1, i.e. n = 1. Choose o E 79\W and let Y := o V v E £ ~ \ { W } . We choose the coordinates in g(S) in such a way that e(So) is one of the four subalgebras in the proof of Lemma 3. If we let C := C~oL2, then we have seen that d i m C > 1, and obviously C C_ L2c f3 S[r-]. The group M := exp(R, S) is 2-dimensional abelian with C M M # 1, hence M C S y . This implies S y = M × L2 i.e. g ( S v ) = (R, S, e, f ) . We describe L2c in the coordinates of Section 3, i.e. L c = { (z, b) : z E C\{0}, b E C}. In these coordinates we have M = {(z, 0)" z E C\{0}}. The

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HAUI~ KLEIN

group L2 is described similarly as L2 = {(s, t) : s, t E R} (with multiplication

= (,1 + , 2 , e %

+ t2)).

L E M M A 4. E acts transitively on Wk {v} and £\£v. Proof Assume to the contrary that there exists a 1-dimensional orbit B of E on W \ { v ) . Since C C E[y], there exists a b E B with C ~ Eb, hence L c g Eb. By the list of 5-dimensional subalgebras, a 5-dimerlsional subalgebra of 1c × 12 which does not contain the subalgebra Ic contains the commutator of 1c x 12. This implies Eb = E[B] __DE'. Since L2 = E[v,w] is a regular normal subgroup of the action of Eb on L b \ { W ) , we get E[b] # 1. Since each element of Eb fixes B pointwise, we know E[b] = E[b,W] C_ E[w,w]. Therefore the group E[w,w] must be abelian ([15, 8.1 ]), but it contains L2; a contradiction. A dual application of Lemma 2 yields the last assertion. Let K := (LC)' = {(1,b): b E C} and A := K x L2 _