The fixed group of an automorphism of a word hyperbolic group is rational
type: Item
instance of: scholarly article
title: The fixed group of an automorphism of a word hyperbolic group is rational
zbMATH Open document ID: 0793.20033
DOI: 10.1007/BF01231328
author: Walter D. Neumann
published in: Inventiones Mathematicae
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/144046
review text: The author affirmatively answers a question of \textit{S. Gersten} and \textit{H. Short} [Ann. Math., II. Ser. 134, 125-158 (1991; Zbl 0744.20035)] that, for a word hyperbolic group \(G\) and its automorphism \(\varphi\), the fixed subgroup \(G^ \varphi\) is in fact rational. For instance, for a discrete co-compact \(G\subset \text{Isom}(H^ n)\), it is equivalent to geometric finiteness of \(G^ \varphi\). As shown by Gersten and Short (see above) rationality of a subgroup \(H\) (for any automatic structure \(L\) on \(G\)) is equivalent to the condition that it is quasiconvex, i.e. there is a bound \(N\) such that the \(N\)-neighborhood of \(H\) contains any path defined by an element of \(L\) from 1 to a point of \(H\). Moreover, due to \textit{W. D. Neumann} and \textit{M. Shapiro} [Int. J. Algebra Comput. 2, No. 4, 443-469 (1992; Zbl 0767.20013)], for the case of word hyperbolic groups, both these notions only depend on the equivalence class of the automatic structure. The author's proof of quasiconvexity of \(G^ \varphi\) is based on three geometric Lemmas. It is also shown that, for any quasi-isometric homomorphisms \(\varphi,\psi: G \to G_ 1\) of word hyperbolic groups, \(\{g \in G'\mid \varphi(g) = \psi(g)\}\) is a rational subgroup.
reviewed by: Boris N. Apanasov
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 167390
zbMATH Keywords: automorphisms
zbMATH Keywords: fixed subgroups
zbMATH Keywords: geometric finiteness
zbMATH Keywords: rationality of subgroups
zbMATH Keywords: automatic structures
zbMATH Keywords: word hyperbolic groups
zbMATH Keywords: quasiconvexity
zbMATH Keywords: quasi-isometric homomorphisms
MaRDI profile type: Publication
cites work: Q4003861
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cites work: Points fixes des automorphismes de groupe hyperbolique. (Fixed points of automorphisms of the hyperbolic group)
cites work: Q3755639
description: scientific article; zbMATH DE number 167390