Automorphisms of free groups and outer space
type: Item
instance of: scholarly article
title: Automorphisms of free groups and outer space
zbMATH Open document ID: 1017.20035
DOI: 10.1023/A:1020973910646
author: Karen Vogtmann
published in: Geometriae Dedicata
publication date: 2003-02-04T00:00:00Z
review text: This is a survey on recent results on the automorphism groups \(\Aut(F)\) and outer automorphism groups \(\text{Out}(F)\) of finitely generated free groups, concentrating mainly on results obtained by geometric methods, and in particular by studying actions of \(\Aut(F)\) and \(\text{Out}(F)\) on suitable geometric objects. Many of these recent methods and results are motivated by analogies with geometric techniques for arithmetic and mapping class groups: for example, in analogy with the actions of mapping class groups on Teichmüller spaces, there is a proper action of \(\text{Out}(F)\) on ``outer space'' which consists of equivalence classes of marked metric graphs. Also, train track techniques inspired by Thurston's theory of train tracks for surfaces are used to model a single automorphism. The first object discussed in the paper is outer space and its properties (dimension, contractibility, closure and boundary, connections with actions on \(\mathbb{R}\)-trees). The second part of the survey discusses various algebraic properties of the automorphism groups such as presentations, virtual finiteness properties, homology and Euler characteristics, fixed subgroup of an automorphism, various types of subgroups (finite, Abelian, solvable subgroups and Tits alternative), relations to arithmetic and linear groups as well as to automatic and hyperbolic groups. The paper closes with lists of open problems and of 132 references.
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 1864092
zbMATH Keywords: automorphism groups of free groups
zbMATH Keywords: outer automorphism groups
zbMATH Keywords: mapping class groups
zbMATH Keywords: train tracks
zbMATH Keywords: outer space
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: a:1020973910646
OpenAlex ID: W54401119
description: scientific article; zbMATH DE number 1864092