Combinatorial structure of some hyperbolic buildings
The author presents a rather ingenious construction of certain SCABS (chamber systems that are almost buildings), as a special case of a more general construction of so-called polyhedra that are locally a given bipartite graph. This general construction provides chamber systems with arbitrary rank 2 residues. Applied to generalized polygons, one obtains chamber systems with a rank 3 Coxeter diagram. The universal cover is a hyperbolic building (where all rank 2 residues are generalized \(m\)-gons for some given \(m\geq 4\)), and the quotient defines an automorphism group which acts cocompatly. Before, there were not so many examples known of this phenomenon, but the author forgot to mention the \(G_2(3)\)-example of \textit{M. Aschbacher} and \textit{S. D. Smith} [Commun. Algebra 11, 1675-1684 (1983; Zbl 0518.51007)]. Nevertheless, the construction in the paper under review is elegant and elementary. It uses the right generalization of an idea of \textit{D. I. Cartwright}, \textit{M. A. Mantero}, \textit{T. Steger} and \textit{A. Zappa} [Geom. Dedicata 47, No. 2, 143-166 (1993; Zbl 0784.51010)] on triangle presentations of projective planes.
scientific article; zbMATH DE number 1889149