Hyperbolic Coxeter groups of large dimension.
type: Item
instance of: scholarly article
title: Hyperbolic Coxeter groups of large dimension.
zbMATH Open document ID: 1068.20043
published in: Commentarii Mathematici Helvetici
publication date: 2004-01-05T00:00:00Z
review text: For any \(n\) the authors construct a Gromov hyperbolic Coxeter group \(W\) with virtual cohomological dimension \(\text{vcd}(W)=n\). The question of existence such groups with large vcd was raised by \textit{G. Moussong} [Hyperbolic Coxeter groups, Ph.D. thesis, Ohio State Univ. (1987)], \textit{M. Gromov} [Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. (1993; Zbl 0841.20039)] and Bestvina (on his problem list). The result is surprising in light of the conjecture of Moussong that there should be a universal bound on the vcd of any Gromov hyperbolic Coxeter group and the belief expressed by Gromov that any construction of high dimensional negatively curved spaces would involve, in an essential way, nontrivial number theory in the guise of arithmetic groups. The authors give an explicit construction of these groups using complexes of groups techniques from \textit{M. R. Bridson} and \textit{A. Haefliger} [Metric spaces of non-positive curvature. (Grundlehren der Math. Wiss. 319, Springer) (1999; Zbl 0988.53001)]. The authors contrast their groups of large dimension with fundamental groups of closed aspherical manifolds by extending a result of \textit{Eh. B. Vinberg} [Usp. Mat. Nauk 40, No. 1(241), 29-66 (1985; Zbl 0579.51015)]. They show that if \((W,S)\) is a virtual Poincaré duality Coxeter group of dimension \(n\) that is Gromov hyperbolic, then \(n\leq 61\). Moreover, in the case that \((W,S)\) is right angled, \(n\leq 4\) and this upper bound is sharp. As a result they conclude that if \(M\) is a manifold with dimension greater than \(4\), then \(M\) admits no negatively curved piecewise hyperbolic cubical metric.
reviewed by: Gregory C. Bell
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 57P10
zbMATH DE Number: 2021085
zbMATH Keywords: Coxeter groups
zbMATH Keywords: Gromov hyperbolic groups
zbMATH Keywords: Poincaré duality groups
zbMATH Keywords: virtual cohomological dimension
author: Tadeusz Januszkiewicz
author: Jacek Świątkowski
MaRDI profile type: Publication
full work available at URL: s00014-003-0763-z
OpenAlex ID: W2120504215
DOI: 10.1007/S00014-003-0763-Z
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description: scientific article; zbMATH DE number 2021085