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 ~