On automorphism-fixed subgroups of a free group
type: Item
instance of: scholarly article
title: On automorphism-fixed subgroups of a free group
zbMATH Open document ID: 0965.20013
published in: Journal of Algebra
publication date: 2001-05-07T00:00:00Z
full work available at URL: http://hdl.handle.net/2117/24777
review text: Bestvina and Handel proved that each 1-auto-fixed subgroup, i.e. the subgroup of fixed elements of one automorphism of the free group \(F\) of rank \(n\) has rank at most \(n\). This result has been generalized to endo-fixed subgroups of \(F\) (the elements fixed by each element of a set of endomorphisms). The present paper raises the question whether each auto-fixed subgroup of \(F\) is a 1-auto-fixed subgroup. Its main result implies that each auto-fixed subgroup of \(F_n\) is a free factor of some 1-auto-fixed subgroup (and consequently has rank at most \(n\)). On the other hand, it is shown that there exist free factors of 1-auto-fixed subgroups of \(F\) which are not auto-fixed (or endo-fixed). It is also proved that, for a maximum-rank 1-auto-fixed subgroup \(H\) of \(F\) (i.e. of rank \(n\)), the group of automorphisms of \(F\) which fix every element of \(H\) is free Abelian of rank at most \(n-1\). The results of the paper apply also to endomorphisms.
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1512981
zbMATH Keywords: automorphisms of free groups
zbMATH Keywords: endomorphisms
zbMATH Keywords: fixed subgroups
author: Armando Martino
author: Enric Ventura Capell
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
OpenAlex ID: W2049440645
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DOI: 10.1006/JABR.2000.8329
description: scientific article; zbMATH DE number 1512981