Efficient representatives for automorphisms of free products
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.
free product of freely indecomposable groups
scientific article; zbMATH DE number 731828
scientific article