Automorphisms of graphs, p-subgroups of \(Out(F_ n)\) and the Euler characteristic of \(Out(F_ n)\)
type: Item
instance of: scholarly article
title: Automorphisms of graphs, p-subgroups of \(Out(F_ n)\) and the Euler characteristic of \(Out(F_ n)\)
zbMATH Open document ID: 0641.20029
DOI: 10.1016/0022-4049(87)90128-9
author: John Smillie
author: Karen Vogtmann
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: If \(F_ n\) is a free group of finite rank n, then each automorphism of \(F_ n\) induces in a natural way an automorphism on the free abelian group \(F_ n/F'_ n\cong Z\) n and therefore there exists a natural homomorphism of Aut \(F_ n\) onto GL(n,\({\mathbb{Z}})\). Since each inner automorphism is mapped onto the identity, this gives a homomorphism of Out \(F_ n\) onto GL(n,\({\mathbb{Z}})\). Here the authors, by analyzing isometry groups of graphs, study finite subgroups of \(Out(F_ n)\) and derive information about the powers of primes which divide the denominator of the rational Euler characteristic \(\chi (Out(F_ n))\) of \(Out(F_ n)\). In this way the authors get information on the kernel of the homomorphism from \(Out(F_ n)\) onto GL(n,\({\mathbb{Z}})\) for n even. For example it is proved that \(\chi (Out(F_ n))\) is non-zero whenever n is even. Details of the methods and other results are too technical to be included here.
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 4044088
zbMATH Keywords: automorphisms of free groups
zbMATH Keywords: free group
zbMATH Keywords: inner automorphism
zbMATH Keywords: isometry groups of graphs
zbMATH Keywords: finite subgroups of \(Out(F_ n)\)
zbMATH Keywords: rational Euler characteristic
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90128-9
OpenAlex ID: W2064004330
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description: scientific article; zbMATH DE number 4044088