Solvable Groups of Automorphisms of Stable Planes

An interesting problem in the foundations of geometry is the following question: What is the impact of the interplay of topological assumptions and homogeneity? One possibility to make this (rather philosophical) question treatable for a mathematician is the classiication project for stable planes. We shall brieey outline the necessary deenitions and basic (though occasionally deep) results. In section 2, we treat solvable groups of automorphisms of stable planes. This may serve as an example how the project works. Note, however, that the case of (semi-)simple groups of automorphisms is where Lie structure theory shows its full strength, cf. the nal Remark. 1. Stable planes Stable planes are topological incidence structures that share fundamental properties of the \classical compact connected projective planes" (namely, the pro-jective planes over the real division algebras R, C , H , O) and their aane and hyperbolic 1 subplanes. To be precise: Deenition 1.1. A stable plane M = (M; M) is a topological incidence geometry (with point space M and line space M) such that the following holds: Any two points x; y 2 M are incident with exactly one line x _ y 2 M, but two lines K; L need not meet. If they do, we denote the (unique) common point by K ^ L. The spaces M and M are locally compact Hausdorr spaces of positive and nite covering dimension.