Parameters of translation generalized quadrangles (Q2501394)

The author considers triples \((E,{\mathcal O}, T)\) where \(E\) is a group, \({\mathcal O}\) a set of subgroups, and \(T\) maps any \(A\in {\mathcal O}\) to a subgroup \(T_A\) containing \(A\). Such a triple is called a fourgonal family, if certain additional axioms hold. These axioms are chosen such that fourgonal families are in one-to-one correspondence with elation generalized quadrangles. Given natural numbers \(s, t\) with \(s\leq t\), an infinite skew field \(F\), and a \(2s+t\)-dimensional \(F\)-vector space \(E\), the author constructs using transfinite recursion a fourgonal family \((E,{\mathcal O}, T)\) where \(\dim A = s\) and \(\dim T_A = s+t\) for any \(A\in {\mathcal O}\), thus an elation generalized quadrangle (Thm. 2.1). His construction even yields a linear translation generalized quadrangle with algebraic parameters \((s,t)\) and kernel \(F\) (Cor. 2.3). Point-rows and line-pencils of a compact generalized quadrangle are homology spheres whose dimensions are called topological parameters of the quadrangle. In a compact connected translation generalized quadrangle only the topological parameters \((1,t)\), \((2,2)\), \((3,4t)\) and \((7,8t)\) \((t\in \mathbb N)\) are possible (Thm. 4.2). The proof relies on work of \textit{T. Buchanan, H. Hähl} and \textit{R.~Löwen} [Geom. Dedicata 9, 401--424 (1980; Zbl 0453.51007)], \textit{M.~Joswig} [Results Math. 38, 72--87 (2000; Zbl 0956.51005]), \textit{E.~Markert} [Isoparametric hypersurfaces and generalized quadrangles, Diplomarbeit, Würzburg (1999)], and \textit{A.~E.~Schroth} [Topological circle planes and topological quadrangles, Pitman Research Notes in Mathematics Series 337, Longman Group, Essex (1995; Zbl 0839.51013)]. Furthermore, it makes use of a characterization of continuous spreads of \(\mathbb R^n\) which are planar at an element of dimension \(l\) (Thm. 3.2). As a corollary of this characterization the author obtains that compact planar spreads of \(\mathbb R^n\) exist only if \(n=2, 4, 8, 16\) [cf. \textit{P.~Breuning}, Translations\-ebenen und Vektorraumbündel. Mitt. Math. Sem. (Gießen) 86 (1970; Zbl 0201.53203)].