Stable planes with large groups of automorphisms : the interplay of incidence, topology, and homogeneity

Abstract

The theory of topological planes (or stable planes, to stress the importance of the stability axiom) originates from the foundations of geometry. In fact, a simultaneous axiomatic treatment of the "classical plane geometries" - the euclidean, hyperbolic and elliptic plane - has to combine incidence properties with topological (or ordering) properties as well as some assumptions that nowadays are conveniently stated by means of a group action (distance, or angles, among others). The use of topology instead of an ordering makes it also possible to include, e.g., the complex plane geometries. Of course, the theory will be substantial only if one imposes some conditions on the topologies involved. It turns out that the assumption of locally compactness in combination with connectedness singles out a very manageable class of topological planes. This class includes the planes whose point space is a two-dimensional manifold; i.e., the (topologically) nearest relatives of the classical plane geometries.