Groups with infinitely many types of fixed subgroups.
type: Item
instance of: scholarly article
title: Groups with infinitely many types of fixed subgroups.
zbMATH Open document ID: 1098.20033
DOI: 10.1007/BF02984407
author: Tim Hsu
author: Daniel T. Wise
published in: Israel Journal of Mathematics
publication date: 2005-03-21T00:00:00Z
review text: \textit{J. A. Shor} proved in his PhD thesis [On fixed subgroups of automorphisms in hyperbolic groups, Columbia Univ. (1999)] that given any hyperbolic group, then, there are only finitely many isomorphism types of subgroups that appear as fixed subgroups of automorphisms of this group. In the paper under review, the authors show that Shor's hypothesis on the group being hyperbolic cannot be weakened to the group being CAT(0) nor biautomatic. More precisely, they prove that there exists a group \(G\) acting freely and cocompactly on the Cartesian product of two trees such that infinitely many non-isomorphic groups appear as fixed subgroups of automorphisms of \(G\). (Such a group \(G\) is known to be CAT(0) and biautomatic.)
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E08
zbMATH DE Number: 2147123
zbMATH Keywords: hyperbolic groups
zbMATH Keywords: CAT(0) groups
zbMATH Keywords: automorphism-fixed subgroups
zbMATH Keywords: centralizers of elements
zbMATH Keywords: products of trees
MaRDI profile type: Publication
cites work: Train tracks and automorphisms of free groups
cites work: Fixed points of automorphisms of free groups
cites work: Small cancellation theory and automatic groups. II
cites work: Q3856928
cites work: Q3152685
full work available at URL: https://doi.org/10.1007/bf02984407
OpenAlex ID: W2008886491
description: scientific article; zbMATH DE number 2147123