Boundaries of relative factor graphs and subgroup classification for automorphisms of free products

Abstract

Given a countable group $G$ splitting as a free product $G=G_1\ast \cdots \ast G_k\ast F_N$, we establish classification results for subgroups of the group $\text{Out}(G,\mathcal{ℱ})$ of all outer automorphisms of $G$ that preserve the conjugacy class of each $G_i$. We show that every finitely generated subgroup $H\subseteq \text{Out}(G,\mathcal{ℱ})$ either contains a relatively fully irreducible automorphism, or else it virtually preserves the conjugacy class of a proper free factor relative to the decomposition (the finite generation hypothesis on $H$ can be dropped for $G=F_N$, or more generally when $G$ is toral relatively hyperbolic). In the first case, either $H$ virtually preserves a nonperipheral conjugacy class in $G$, or else $H$ contains an atoroidal automorphism. The key geometric tool to obtain these classification results is a description of the Gromov boundaries of relative versions of the free factor graph $\text{FF}$ and the $\mathcal{𝒵}$–factor graph $\mathcal{𝒵}\text{F}$, as spaces of equivalence classes of arational trees and relatively free arational trees, respectively. We also identify the loxodromic isometries of $\text{FF}$ with the fully irreducible elements of $\text{Out}(G,\mathcal{ℱ})$, and loxodromic isometries of $\mathcal{𝒵}\text{F}$ with the fully irreducible atoroidal outer automorphisms.