Efficient representatives for automorphisms of free products
type: Item
instance of: scholarly article
title: Efficient representatives for automorphisms of free products
zbMATH Open document ID: 0820.20035
author: Donald J. Collins
author: Edward C. Turner
published in: Michigan Mathematical Journal
publication date: 1995-09-14T00:00:00Z
review text: The authors prove the Scott conjecture in the general case of the free product of a finite number of freely indecomposable groups. The Scott conjecture says that if \(\varphi\) is an automorphism of the free group \(F\) of finite rank \(n\), then the subgroup \(\text{Fix}(\varphi)\), of elements fixed by \(\varphi\), is free of rank at most \(n\). This conjecture was proved in the affirmative by \textit{M. Bestvina} and \textit{M. Handel} [Ann. Math., II. Ser. 135, 1-51 (1992; Zbl 0757.57004)]. Now if \(G = *^ m_{i = 1} G_ i\) is the free product of freely indecomposable groups \(G_ i\), \(i = 1,2,\dots,m\), then \(m\) is the Kurosh rank of \(G\), denoted by \(\text{K-rank}(G)\). A subgroup \(H\) of \(G\) is by the Kurosh subgroup theorem a free product of the form \(H = F_ s *^ t_{j = 1} H_ j\), where \(F_ s\) is a free group of rank \(s\) and the \(H_ j\) are intersections of the factors \(G_ i\). The sum \(s+t\) (possibly infinite) is termed the Kurosh subgroup rank of \(H\) in \(G\), denoted \(\text{K-rank}(G,H)\). Now the authors prove an analogue of the above mentioned theorem of Bestvina and Handel (l.c.). More precisely they prove the nice theorem that if \(\varphi\) is an automorphism of \(G\), then the Kurosh subgroup rank of \(\text{Fix}(\varphi)\) in \(G\) is at most the Kurosh rank of \(G\). The proof is topological and uses a modification and adaption of the Bestvina-Handel method. Details are too difficult to be given here.
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 731828
zbMATH Keywords: subgroup of fixed elements
zbMATH Keywords: Scott conjecture
zbMATH Keywords: freely indecomposable groups
zbMATH Keywords: automorphisms
zbMATH Keywords: free product of freely indecomposable groups
zbMATH Keywords: Kurosh rank
zbMATH Keywords: Kurosh subgroup theorem
zbMATH Keywords: Kurosh subgroup rank
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005072
OpenAlex ID: W1963944594
DOI: 10.1307/MMJ/1029005072
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description: scientific article; zbMATH DE number 731828