Topological affine quadrangles

Abstract

The affine derivation of a generalized quadrangle is the geometry induced on the vertices at distance $3$ or $4$ of a given point. We characterize these geometries by a system of axioms which can be described as a modified axiom system for affine planes with an additional parallel relation and parallel axiom. A second equivalent description which makes it very easy to verify that, for example, ovoids and Laguerre planes yield generalized quadrangles is given. We introduce topological affine quadrangles by requiring the natural geometric operations to be continuous and characterize when these geometries have a completion to a compact generalized quadrangle. In the connected case it suffices to assume that the topological affine quadrangle is locally compact. Again this yields natural and easy proofs for the fact that many concrete generalized quadrangles such as those arising from compact Tits ovoids are compact topological quadrangles. In an appendix we give an outline of the theory of stable graphs which is fundamental to this work.