Examples of relatively hyperbolic groups.
type: Item
instance of: scholarly article
title: Examples of relatively hyperbolic groups.
zbMATH Open document ID: 1046.20031
DOI: 10.1023/A:1020380603501
published in: Geometriae Dedicata
publication date: 2002-12-15T00:00:00Z
review text: The author uses ideas on weak hyperbolicity and relative hyperbolicity which originate in work of Gromov. He uses the following definition of weak hyperbolicity that has been formulated by \textit{B. Farb} [in Geom. Funct. Anal. 8, No. 5, 810-840 (1998; Zbl 0985.20027)]. One starts with the Cayley graph \(\Gamma\) of a group \(G\) equipped with a finite generating system, and defines a new graph \(\widehat\Gamma=\widehat\Gamma(H)\) from \(\Gamma\) by adding a vertex \(v(gH)\) for each coset \(gH\) of \(H\) and an edge of length \(1/2\) from each element of \(gH\) to the vertex \(v(gH)\). The graph \(\widehat\Gamma\) is equipped with its natural path-metric. The group \(\Gamma\) is said to be `weakly hyperbolic with respect to \(H\)' if \(\widehat\Gamma\) is a negatively curved space in the sense of Gromov. The result of this paper is the following Theorem: Let \(H\) be a finitely presented group. Then \(H\) is a subgroup of a finitely presented group \(G\) which retracts onto \(H\) so that \(G\) is weakly hyperbolic with respect to \(H\). The proof uses ideas of relative hyperbolization of polyhedra originating in \textit{M. Gromov}'s paper [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)], as well as developments of these ideas by \textit{M. W. Davis} and \textit{T. Januszkiewicz} [in J. Differ. Geom. 34, No. 2, 347-386 (1991; Zbl 0723.57017)]. The paper also contains open problems.
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1845455
zbMATH Keywords: hyperbolic spaces
zbMATH Keywords: relatively hyperbolic groups
zbMATH Keywords: weakly hyperbolic groups
zbMATH Keywords: relative hyperbolizations
zbMATH Keywords: finitely presented groups
zbMATH Keywords: Cayley graphs
author: Andrzej Szczepański
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: a:1020380603501
OpenAlex ID: W134711009
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description: scientific article; zbMATH DE number 1845455