Folding free-group automorphisms. (Q2922422)

Let \(F_n\) be the free group of rank \(n\). It is well known the Nilsen method of reduction which enables us to obtain finite generating sets of the automorphism group \(\Aut(F_n)\). Also the peak reduction method developed by Whitehead is well known.NEWLINENEWLINE Let \(Y\) be a finite set of elements of \(F_n\). In [J. Algebra 35, 205-213 (1975; Zbl 0325.20025)] \textit{J. McCool} uses the peak reduction method and gives an algorithm to obtain finite presentations of \(\mathrm{Fix}(Y)\) and \(\mathrm{Fix}_c(Y)\), the subgroups of \(\Aut(F_n)\) that fix \(Y\) pointwise, and fix each element of \(Y\) up to conjugacy, respectively. After the appearance of the notion of the folding [in \textit{J. R. Stallings}, Invent. Math. 71, 551-565 (1983; Zbl 0521.20013)] the study of \(\Aut(F_n)\) and its subgroups have taken a new turn. Especially an algorithm is hinted by Stallings to decompose an automorphism of a free group as a product of Whitehead automorphisms.NEWLINENEWLINE This algorithm was not widely known. In this paper the author refines this algorithm making this accessible not only to experts.NEWLINENEWLINE As applications he finds finite generating sets of \(\mathrm{Fix}(Y)\) and \(\mathrm{Fix}_c(Y)\), the subgroups of \(\Aut(F_n)\) that fix \(Y\) pointwise, and fix each element of \(Y\) up to conjugacy, respectively, where \(Y\) is a subset of a basis of \(F_n\). Moreover he finds a finite generating set for the intersection \(\mathrm{IA}_n\cap\mathrm{Fix}_c(Y)\), where \(\mathrm{IA}_n\) is the subgroup of \(\Aut(F_n)\) the elements of which induce the trivial automorphism on the Abelianization of \(F_n\).