Stabilizers of collineation groups of smooth stable planes

Abstract

Let S be a smooth stable plane of dimension n (see Definition 1.2) and let Δ be a closed subgroup of the collineation group of S which fixes some point p. We derive some results on the group-theoretical structure of Δ, e.g. that Δ is a linear Lie group (Theorem 3.7). As a by-product this shows that no (affine or projective) Moulton plane can be turned into a smooth plane. If Δ fixes some flag, then any Levi subgroup Ψ of Δ is a compact group and Δ is contained in the flag stabilizer of the classical Moufang plane of dimension n (Corollary 3.1 and Theorem 3.7). Let Δ fix three concurrent lines through the point p. If S is one of the classical projective planes over the reals, the complex numbers, the quaternions, or the Cayley numbers, then the dimension of Δ is d class = 3, 6, 15, or 38, respectively. We show that for a smooth stable (projective) plane S of dimension 2l either S is an almost projective translation plane (classical projective plane) or that dim Δ ≤ d class − l holds (Theorems 4.1 and 4.2).