The automorphism group of a free product of finite groups
type: Item
instance of: scholarly article
title: The automorphism group of a free product of finite groups
zbMATH Open document ID: 0654.20036
DOI: 10.1007/BF01196497
author: Donald J. Collins
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: The author studies the virtual cohomological dimension vcd(Out G) of the outer automorphism group Out G of a free product \(G=*G_ i\) of n finite groups. This is motivated by the papers by \textit{S. M. Gersten} [Preprint, Univ. Utah] and \textit{M. Culler} and \textit{K. Vogtmann} [Invent. Math. 84, 91-119 (1986; Zbl 0589.20022)] who proved that the outer automorphism group of a free group \(F_ n\) of rank n has vcd(Out \(F_ n)=2n-3\). In this paper the author proves that the group Out G has finite virtual cohomological dimension, and conjectures that vcd(Out G)\(=n-2\), where G is a free product of n finite groups, and proves this for \(n=3\). The proof is achieved by embedding Out G in the outer automorphism group of a free group of finite rank.
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E05
zbMATH DE Number: 4067204
zbMATH Keywords: free product of finite groups
zbMATH Keywords: virtual cohomological dimension
zbMATH Keywords: outer automorphism group
zbMATH Keywords: free group of finite rank
MaRDI profile type: Publication
cites work: Cohomological dimension and symmetric automorphisms of a free group
cites work: Moduli of graphs and automorphisms of free groups
cites work: The Automorphism Group of a Free Group Is Not a Cat(0) Group
cites work: Presentations of the Automorphism Group of a Free Product
cites work: Residual Properties of Infinite Soluble Groups
full work available at URL: https://doi.org/10.1007/bf01196497
OpenAlex ID: W2051306671
description: scientific article; zbMATH DE number 4067204