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.
We prove that every flock of a finite-dimensional locally compact connected circle plane is homeomorphic to R or S 1 and that every flock of a real Miquelian circle plane defines a compact 4-dimensional translation plane. Furthermore we investigate (topological) properties of regulizations. These properties are used to relate the automorphism group of a flock to the automorphism group of the corresponding translation plane.
We prove that the Lie geometry of a locally compact connected Laguerre space forms a compact connected generalized quadrangle. Moreover, if the geometric dimension of such a Laguerre space is at least three, this generalized quadrangle is isomorphic to a generalized quadrangle of Tits type; all compact connected generalized quadrangles of Tits type arise from Laguerre spaces.
We introduce stable graphs as a common generalization of compact generalized polygons with closed adjacency, stable planes and other types of graphs with continuous geometric operations; non-bipartite structures like Moore graphs are also included. Topological and graph-theoretical properties of stable graphs are established, and generalized polygons are characterized among all stable graphs by means of topological properties. Some results about Moore graphs, which might help to find infinite examples, are included.
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