4-Dimensional projective planes of Lenz type III

HELMUT SALZMANN

4-DIMENSIONAL

PROJECTIVE

PLANES

OF LENZ TYPE III

A collineation group A of a projective plane P is of Lenz type III if there exists a line W and a point o ~ W such that the group Atz' aa of elations in A with center z and axis A is transitive if and only if ze W and A = owz. The compact 2-dimensional planes with a group of Lenz type III are exactly the Moulton planes; the full collineation group of a proper Moulton plane is a4-dimensional Lie group fixing o and W([2]; [6], §4; [7]; [8], 4.8;[4]; [11]). In this note, we show that there exist no complex analogues of the Moulton planes; more precisely, we prove the following THEOREM. The arguesian complex plane C = PG2 (C) is the only compact 4-dimensional projective plane admitting a collineation group A which fixes a non-incident point-line-pair and is either at least 7-dimensional or of Lenz type 11I. We begin by listing a few properties of a compact 4-dimensional topological projective plane P = (P, ~). (1) Each line L e ~ , viewed as a point set, is a 2-sphere, and P is a manifold homeomorphic to the point set of C [9]. (2) In the compact-open topology, the full collineation group Fp is a Lie group and its connected component F 1 acts regularly on the set of quadrangles, so that, in particular, dimF~< 16 ([10], 1.16 and 4.1). (3) If a connected group A acts transitively on a line L, then A induces on L one of the simple groups SO3 (~), PSL2 (C), or PGL3 (E). In the last case, A cannot act primitively on L. This can be seen by inspection of Mostow's list [3] of transitive actions on surfaces. If the full collineation group A of the real projective plane D would act primitively on a 2-sphere L, the stabilizer Ap of a point p e L would be a connected 6-dimensional maximal subgroup of A. But then Ap would also fix an element of D, and would have index 2 in the stabilizer of this element. Proof of the theorem. We show first that a group A of dimension at least 7 has Lenz type III if it fixes a pair (o, W) with oe W e ~ . From (2) we obtain dim A ~<8, and we may always assume A to be connected. If dim A = 8, it follows from (2) that A is doubly transitive and, in particular, primitive on W; therefore, A induces the M/Sbius group PSL2 (C) on W by (3), and the group E = Arwj of homologies with axis W is 2-dimensional and hence transitive. Because of the simplicity of the M6bius group, Geometriae Dedicata 1 (1972) 18-20. All Rights Reserved Copyright © 1972 by D. ReidelPublishing Company, Dordrecht-Holland

4-DIMENSIONAL PROJECTIVE PLANES

19

2: is the centre of A. Consequently, the 4-dimensional stabilizer Ap of a point p c W fixes the line oup pointwise, and A~ is a transitive group of elations. P is easily seen to be arguesian in this case. Now let dimA=7. Then A is transitive on P\W\{o}, since dimp~=3 implies by (2) that Ap has only 4-dimensional orbits outside W and oup, a contradiction. A induces again the M6bius group on W, and A has a connected one-dimensional central subgroup Z consisting of homologies with axis IV. This implies that the universal covering group of A splits into the direct product of SL2 (C) and a one-parameter group, e.g. cp. [5], Satz 94, 96. Hence A has a 6-dimensional quasi-simple normal subgroup A with A =A~. Because the M6bius group contains permutable involutions, there exists an (o, W)-reflection in the centre of A, and A =~ SL 2 (C). In the ordinary action of the M6bius group, the connected 4-dimensional subgroups are the stabilizers of points and form a full conjugaey class. Since A acts transitively on W, the stabilizer A, of a point v e W is a 2-fold covering of the linear group L2(C) = {z~--~az + b; a c e * , b e e } , in particular, each non-central normal subgroup of Ao contains the commutator group A'o. Let M=ouv\(o, v). Then the orbit space M/Z is a one-dimensional manifold, and A~ acts transitively on M/Y,. By the preceding remark, Ao has a 3-dimensional normal subgroup ~ containing A" and mapping each 2:-orbit in M onto itself. Because 2~ is in the centre of A, the stabilizer ~p of a point p e M fixes pS pointwise. An easy calculation in L 2 (C) shows that the normal subgroup consisting of those elements in 3 which induce the identity on p~ contains A~ for each peM. Acting freely on W\{v}, the group A~' is a transitive group of elations with axis owv, and A has Lenz type III. From these considerations it follows that the elations in a group of Lenz type III generate a 6-dimensional group A~SL2 (C) which acts in the same way as the MSbius group on the fixed line IV. Moreover, by the transitivity of the elation groups, A is transitive on I:, the set of flags which are not incident with the fixed elements. For (p, L)e[:, the stabilizers Ap and A L are the commutator groups of two distinct 4-dimensional connected subgroups of A. As A permutes these pairs of commutator groups transitively, by comparison with the ordinary action of SL 2 (C) the HigmanMcLaughlin construction [1] shows that P is arguesian. BIBLIOGRAPHY [1] Higman, D. G. and McLaughlin, J. E., 'Geometric ABA-Groups', Illinois d. Math. 5 (1960, 382-397.

20

HELMUT SALZMANN

[2] Lenz, H., 'Kleiner desarguesscher Satz und Dualit~t in projektiven Ebenen', Jber. dtsch. Math.-Verein. 57 (1954), 20-31. [3] Mostow, G. D., 'The Extensibility of Local Lie Groups of Transformations and Groups on Surfaces', Ann. of Math. 52 (1950), 606-636. [4] Moulton, F. R., 'A Simple Non-Desarguesian Plane Geometry', Trans. Amer. Math. Soc. 3 (1902), 192-195. [5] Pontrjagin, L. S., Topologische Gruppen, Teil 2, Teubner, Leipzig, 1958. [6] Salzmann, H., 'Zur Klassifikation topologischer Ebenen. II', Abh. Math. Sere. Hamburg 27 (1964), 145-166. [7] Salzmann, H., 'Polarit~iten yon Moulton-Ebenen', Abh. Math. Sem. Hamburg 29 (1966), 212-216. [8] Salzmann, H., 'Topological Planes', Advances in Math. 2 (1967), 1-60. [9] Salzmann, H., 'Kompakte vier-dimensionale Ebenen', Arch. Math. 20 (1969), 551-555. [10] Salzmann, H., 'Kollineationsgruppen kompakter 4-dimensionaler Ebenen', Math. Z. 117 (1970), 112-124. [11] Spencer-Yaqub, J. C. D., 'On Projective Planes of Class HI', Arch. Math. 12 (1961), 146-150. Anschr~t des Verfassers: H e l m u t Salzmann, Mathematisches Institut der Universit~it Ttibingen, 74 Tiibingen, H61dedinstrasse 19.

(Received June 4, 1971)