#### Volume 26, issue 1 (2022)

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

### Vincent Guirardel and Camille Horbez

Geometry & Topology 26 (2022) 71–126
DOI: 10.2140/gt.2022.26.71
##### 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 $\mathrm{Out}\left(G,\mathsc{ℱ}\right)$ of all outer automorphisms of $G$ that preserve the conjugacy class of each ${G}_{i}$. We show that every finitely generated subgroup $H\subseteq \mathrm{Out}\left(G,\mathsc{ℱ}\right)$ 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 $\mathsc{𝒵}$–factor graph $\mathsc{𝒵}F$, as spaces of equivalence classes of arational trees and relatively free arational trees, respectively. We also identify the loxodromic isometries of $\mathrm{FF}$ with the fully irreducible elements of $\mathrm{Out}\left(G,\mathsc{ℱ}\right)$, and loxodromic isometries of $\mathsc{𝒵}F$ with the fully irreducible atoroidal outer automorphisms.

##### Keywords
automorphism groups of free groups and free products, subgroup classification, Gromov hyperbolic spaces, Gromov boundaries, free factor graph
##### Mathematical Subject Classification 2010
Primary: 20E06, 20E07, 20E08, 20E36