Describing of thin geometries using Hecke groups (Q1180795)

Let \(\Gamma\) be a strongly connected geometry of rank \(n\) which is thin, cf. \textit{J. Tits} [in: The Geometric Vein, the Coxeter Festschr., 519-547 (1982; Zbl 0496.51001)]. Then \(i\)-adjacency is an involutory permutation \(\sigma_ i\) of the set of all chambers of \(\Gamma\). The involution \(\sigma_ i\) with \(1\leq i\leq n\) generate the Hecke group \(H(\Gamma)\), cf. \textit{U. Ott} [in: Finite Geometries and Designs, Lond. Math. Soc. Lect. Note Ser. 49, 260-271 (1981; Zbl 0471.05017)] for the related notion of Hecke algebras. The geometry \(\Gamma\) can be realized by double cosets in \(H(\Gamma)\), and the special automorphism group of \(\Gamma\) is a section of \(H(\Gamma)\). Coverings \(\Gamma\to\Pi\) of these geometries (and the corresponding deck transformations) are described by special group epimorphisms \(H(\Gamma)\to H(\Pi)\). As a corollary, \(\Gamma\) is always 1-covered by a Coxeter geometry \(\Sigma\), and if \(\Gamma\) has type \(M\), then \(\Sigma\) is the universal 2-cover and has type \(M\).