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

## TL;DR

This paper classifies subgroups of outer automorphisms of free products, revealing their structure via boundary descriptions of associated graphs and identifying key automorphism types.

## Contribution

It provides a new classification of subgroups of Out(G, F) using boundary theory and characterizes loxodromic automorphisms in terms of fully irreducible and atoroidal elements.

## Key findings

- Finitely generated subgroups either contain a fully irreducible automorphism or virtually preserve a free factor.
- Boundary of the free factor graph described via arational trees.
- Loxodromic isometries correspond to fully irreducible automorphisms.

## Abstract

Given a countable group $G$ splitting as a free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the group $Out(G,\mathcal{F})$ of all outer automorphisms of $G$ that preserve the conjugacy classes of each $G_i$. We show that every finitely generated subgroup $H\subseteq Out(G,\mathcal{F})$ 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 $\mathrm{FF}$ and the $\mathcal{Z}$-factor graph $\mathcal{Z}\mathrm{F}$, as spaces of equivalence classes of arational trees (respectively relatively free arational trees). We also identify the loxodromic isometries of $\mathrm{FF}$ with the fully irreducible elements of $Out(G,\mathcal{F})$, and loxodromic isometries of $\mathcal{Z}\mathrm{F}$ with the fully irreducible atoroidal outer automorphisms.