Fixed subgroups of automorphisms of relatively hyperbolic groups. (Q2907043)

Let \(G\) be a finitely generated group and let \(\text{Fix}(\varphi)\) denote the subgroup of \(G\) that consists of the elements fixed by an automorphism \(\varphi\). \textit{S. M. Gersten} proved that if \(G\) is a free group then \(\text{Fix}(\varphi)\) is finitely generated for every \(\varphi\in\Aut(G)\) [Adv. Math. 64, 51-85 (1987; Zbl 0616.20014)]. Among several known generalizations of Gersten's result we can mention the work of \textit{W. D. Neumann} who showed that for every word hyperbolic group \(G\) and every \(\varphi\in\Aut(G)\), \(\text{Fix}(\varphi)\) is quasiconvex in \(G\) [Invent. Math. 110, No. 1, 147-150 (1992; Zbl 0793.20033)].NEWLINENEWLINE The main theorem of the present article generalizes these results to a much bigger class of groups:NEWLINENEWLINE Theorem 1. Let \(G\) be a relatively hyperbolic group such that no peripheral subgroup of \(G\) is hyperbolic relative to a collection of proper subgroups. Then for every \(\varphi\in\Aut(G)\), the fixed subgroup \(\text{Fix}(\varphi)\) is relatively quasiconvex in \(G\).NEWLINENEWLINE The theorem has several interesting corollaries discussed in the introduction to the paper. First, the authors point out that \(\text{Fix}(\varphi)\) is itself relatively hyperbolic with respect to a natural family of peripheral subgroups. Then they show that if all peripheral subgroups of \(G\) are slender (resp., slender and coherent) then \(\text{Fix}(\varphi)\) is finitely generated (resp., finitely presented). In particular, this happens when \(G\) is a limit group and implies in this case that \(\text{Fix}(\varphi)\) is a limit subgroup of \(G\) for any \(\varphi\in\Aut(G)\).