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

TL;DR

This paper investigates the stabilizer subgroup of an attractive fixed point in the boundary of a relative outer space for a free product, revealing a structure involving automorphisms of the factors and an infinite cyclic quotient.

Contribution

It establishes the structure of the stabilizer of an attractive fixed point for an IWIP automorphism relative to a free product decomposition, including a normal subgroup related to automorphisms of the factors.

Findings

The stabilizer has a normal subgroup isomorphic to a subgroup of the direct sum of automorphism groups of the factors.

The quotient of the stabilizer by this subgroup is isomorphic to the integers.

The proof uses the machinery of attractive laminations for IWIP automorphisms.

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 $\mathcal{O}$.