Topological affine quadrangles (Q2493444)

In his Habilitationsschrift [Stable graphs and polygons, Würzburg (2003)] the author introduced stable graphs, which form a common generalization of topological generalized polygons and stable planes but many more types of geometries and even non-bipartite graphs are encompassed by this notion. Based on this work the paper under review shows how this tool can be applied in the investigtion of generalized quadrangles. By modifying axioms for affine quadrangles (that is, the geometries induced on generalized quadrangles minus a point star) given by \textit{H. Pralle} [Geom. Dedicata 84, 1--23 (2001; Zbl 0981.51010)] and \textit{B. Stroppel} [Point-affine quadrangles. Note Mat. 20, 21--31 (2000/01)] the author introduces a system of axioms for affine quadrangles that leads to a natural definition of topological affine quadrangles. For an incidence geometry \({\mathcal A}=(P,L,I)\) the author defines two relations \(|\) and \(||\) (strong and weak parallelism) on its line set \(L\) by \(g|h\) if and only if for all points \(a\) on \(g\) and \(b\) on \(h\) one has \(d(a,h)=d(b,g)\) and \(g||h\) if and only if the distance \(d(g,h)\) is either 0 or 6. (All incidence geometries are considered as bipartite graphs with the two parts formed by points and lines.) Then \({\mathcal A}\) is called an affine quadrangle if (A1) There are a point that is on at least 2 lines and a line that has at least 3 points. (A2) The girth of \({\mathcal A}\) is at least 6 and distance between lines is at most 6. (A3) For every \((p,h)\in P\times L\) there is a unique line \(l\) through \(p\) such that \(l|h\). (A4) For any two lines \(g,h\) such that \(g\not| h\) there is a unique line \(l\) of distance 2 from \(g\) such that \(l||h\). In this case, the weak and strong parallelisms are equivalence relations. The author shows that an incidence geometry is an affine quadrangle if and only if there is a generalized quadrangle and a point \(p\) such that the affine derivation at \(p\) is isomorphic to it. The topological case is more involved and requires the following additional crucial condition (C): For any compact set \(C_1\times C_2\subseteq\not |\) of non-parallel line pairs the line set \(D_{0,2,6}(C_1)\cap D_{0,2,6}(C_2)\) is compact where \(D_{0,2,6}(C_i)\) consists of all lines of distance 0, 2 or 6 from elements in \(C_i\). It is shown that an affine quadrangle has a completion to a topological compact generalized quadrangle if and only it is topological with locally compact and connected point and line spaces satisfying condition (C). Using pseudo-isotopic contractions on the panels of a topological locally compact connected affine quadrangle the author succeeds in verifying condition (C) so that these affine quadrangles automatically have compact completions. As an application new proofs are obtained of the facts that Tits quadrangles defined by compact connected ovoids and Lie geometries of finite-dimensional locally compact connected Laguerre planes are topological compact connected quadrangles.