Actions of SO(3) on 4-dimensional stable planes

Abstract

In a previous paper [13] on 4-dimensional compact projective planes admitting a group ~ ~ SO(3) = SO3 of automorphisms, we proved that either the plane is the Desarguesian plane P2 C or • has index at most 2 in the full automorphism group. Here we treat a similar question more generally for locally compact stable planes, where lines do not always meet (see [5] for a definition). We show that either the plane is one of three open subptanes of P2 C, or its full automorphism group has very small dimension (at most 4). The proof in the projective case rests on the fact that an involutory automorphism tr contained in a connected group of automorphisms has a centre and an axis. Removing, e.g., the centre and the points of the axis, one gets a stable plane with a free involution, i.e. one without fixed points. However, in section 1 we prove for SO3-actions on stable planes of dimension 4 that every point lies on the axis of some involution. This allows us to determine the possible isotropy groups on points and to some extent also on lines. According to [14], the isotropy groups determine the topological type of the action. In section 2, these general facts about SO3-actions are used to prove our main result, viz.