Centralisers of linear growth automorphisms of free groups
The structure of a group \(G\) is related in various ways to that of the centralizers \(C_{G}(g)\) of its elements. In the paper under review, the authors consider elements \(\Phi\) of \(\mathrm{Out}(F_{n})\), the outer automorphism group of a free group \(F_{n}\) of rank \(n\). \textit{J. McCool}, in [J. Algebra 35, 205--213 (1975; Zbl 0325.20025)], proved that the stabilizer \(Mc(\mathcal{C})\) of a finite set of conjugacy classes \(\mathcal{C} \subseteq F_{n}\) is finitely presented.\N\NIn this paper the authors prove that if \(\Phi\) is a linearly growing element of \(\mathrm{Out}(F_{n})\), then \(C(\Phi)\) has a finite index subgroup mapping onto a direct product of certain \textit{equivariant McCool groups} with kernel a finitely generated free abelian group. In particular, \(C(\Phi)\) is of Type \(\mathsf{VF}\) and hence finitely presented.\N\NThe reviewer points out that a group \(G\) is of Type \(\mathsf{F}\) if it has a finite Eilenberg-Maclane space. A group is said to be of Type \(\mathsf{VF}\) if it has a finite index subgroup of Type \(\mathsf{F}\). It is known, from a work by \textit{K. Rafi} et al. [Arnold Math. J. 6, No. 2, 271--290 (2020; Zbl 1473.57048)], that groups of Type \(\mathsf{VF}\) are finitely generated and finitely presented.
scientific article; zbMATH DE number 7957633
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