Equivariant outer space and automorphisms of free-by-finite groups
type: Item
instance of: scholarly article
title: Equivariant outer space and automorphisms of free-by-finite groups
zbMATH Open document ID: 0805.20030
DOI: 10.1007/BF02565817
author: Sava Krstić
author: Karen Vogtmann
published in: Commentarii Mathematici Helvetici
publication date: 1995-02-05T00:00:00Z
full work available at URL: https://eudml.org/doc/140300
review text: The authors study the automorphism group of a finitely generated free-by- finite group. By results of \textit{J. McCool} [Bull. Lond. Math. Soc. 20, No. 2, 131-135 (1988; Zbl 0641.20028)] these groups can be studied by considering centralizers of finite subgroups of the group \(\text{Out}(F_ n)\) of the outer automorphisms of a finitely generated free group \(F_ n\). Using results of McCool and independently \textit{S. Kalajdžievski} [J. Algebra 150, No. 2, 435-502 (1992; Zbl 0780.20015)] and \textit{S. Krstić} [Proc. Lond. Math. Soc., III. Ser. 64, No. 1, 49-69 (1992; Zbl 0773.20008)] proved that these groups are finitely presented. In the present paper the authors, for a finite subgroup \(G\) of \(\text{Out}(F_ n)\), construct a simplicial complex \(L_ G\) on which the centralizer \(C(G)\) acts with finite stabilizers and finite quotient. This complex is an equivariant deformation retract of the fixed point subcomplex of outer space \(X_ n\). They prove that the complex \(L_ G\) is contractible, they compute the dimension of \(L_ G\) and thus they give an upper bound on the virtual cohomological dimension (vcd) of \(C(G)\). This implies that \(C(G)\) has finitely generated homology in all dimensions. These homological finiteness properties translate directly into similar properties for automorphism groups of free-by- finite groups. In particular they prove that the vcd of the outer automorphism group of a free product of \(n\) finite groups is equal to \(n - 2\), proving thus in the affirmative a conjecture of \textit{D. J. Collins} [Arch. Math. 50, No. 5, 385-390 (1988; Zbl 0654.20036)].
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 57M60
zbMATH DE Number: 484346
zbMATH Keywords: automorphism group
zbMATH Keywords: finitely generated free-by-finite group
zbMATH Keywords: centralizers of finite subgroups
zbMATH Keywords: outer automorphisms
zbMATH Keywords: finitely presented
zbMATH Keywords: simplicial complex
zbMATH Keywords: equivariant deformation retract
zbMATH Keywords: fixed point subcomplex
zbMATH Keywords: outer space
zbMATH Keywords: virtual cohomological dimension
zbMATH Keywords: finitely generated homology
zbMATH Keywords: homological finiteness
zbMATH Keywords: free product
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 484346