Quasiconvexity of the fixed-point set of an automorphism of a hyperbolic group
type: Item
instance of: scholarly article
title: Quasiconvexity of the fixed-point set of an automorphism of a hyperbolic group
zbMATH Open document ID: 0941.20046
DOI: 10.1007/BF02674300
published in: Siberian Mathematical Journal
publication date: 2000-04-16T00:00:00Z
review text: Let \(G\) be a word hyperbolic group, i.e., \(G\) is finitely generated and the geodesic triangles in the Cayley graph of \(G\) are \(\delta\)-thin. Let \(\phi\) be an automorphism of \(G\). It was shown by \textit{W.~D.~Neumann} [Invent. Math. 110, No. 1, 147-150 (1992; Zbl 0793.20033)] that the fixed group \(G^\phi\), consisting of the fixed points of \(\phi\), is quasiconvex, i.e., there exists a constant \(c\) such that every geodesic in the Cayley graph of \(G\), joining points of \(G^\phi\), lies in the \(c\)-neighborhood of \(G^\phi\). In the article under review a new proof of this fact is given. The proof uses the Gromov boundary of a hyperbolic group.
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1432184
zbMATH Keywords: word hyperbolic groups
zbMATH Keywords: Gromov boundaries
zbMATH Keywords: fixed subgroups
zbMATH Keywords: quasiconvexity
zbMATH Keywords: Cayley graphs
zbMATH Keywords: automorphisms
zbMATH Keywords: geodesics
author: Gennady A. Noskov
MaRDI profile type: Publication
cites work: The fixed group of an automorphism of a word hyperbolic group is rational
cites work: Points fixes des automorphismes de groupe hyperbolique. (Fixed points of automorphisms of the hyperbolic group)
full work available at URL: https://doi.org/10.1007/bf02674300
OpenAlex ID: W2333706926
description: scientific article; zbMATH DE number 1432184