Stabiliser of an Attractive Fixed Point of an IWIP Automorphism of a free product

Abstract

For a group $G$ of finite Kurosh rank and for some arbiratily free product decomposition of $G$, $G=H_1\ast H_2\ast ...\ast H_r\ast F_q$, where $F_q$ is a finitely generated free group, we can associate some (relative) outer space $\mathcal{O}(G,\{H_1,...,H_r\})$. We define the relative boundary $\partial (G,\{H_1,...,H_r\})=\partial (G,\mathcal{O})$ corresponding to the free product decomposition, as the set of infinite reduced words (with respect to free product length). By denoting $Out(G,\{H_1,...,H_r\})$ the subgroup of $Out(G)$ which is consisted of the outer automorphisms which preserve the set of conjugacy classes of $H_i$'s, we prove that for the stabiliser $Stab(X)$ of an attractive fixed point in $X\in \partial (G,\{H_1,...,H_r\})$ of an irreducible with irreducible powers automorphism relative to $\mathcal{O}$, it holds that it has a (normal) subgroup $B$ isomorphic to subgroup of $\underset{i=1}{\overset{r}{⨁}}Out(H_i)$ such that $Stab(X)/B$ is isomorphic to $\mathbb{Z}$. The proof relies heavily on the machinery of the attractive lamination of an IWIP automorphism relative to $\mathcal{O}$.

30 pages