Combinatorial structure of some hyperbolic buildings
type: Item
instance of: scholarly article
title: Combinatorial structure of some hyperbolic buildings
zbMATH Open document ID: 1020.51009
author: Alina Vdovina
published in: Mathematische Zeitschrift
publication date: 2003-03-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 51E24
zbMATH DE Number: 1889149
zbMATH Keywords: generalized polygon
zbMATH Keywords: SCABS
zbMATH Keywords: bipartite graph
zbMATH Keywords: hyperbolic building
reviewed by: Hendrik Van Maldeghem
MaRDI profile type: Publication
full work available at URL: s002090200423
OpenAlex ID: W2094083425
DOI: 10.1007/S002090200423
description: scientific article; zbMATH DE number 1889149