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 $\bigoplus \limits_{i=1} ^{r} Out(H_i)$ 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 $\mathcal{O}$.