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

Guirardel, Vincent; Horbez, Camille

doi:10.2140/gt.2022.26.71

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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 $G = G_{1} * \dots * G_{k} * F_{N}$ , we establish classification results for subgroups of the group $Out (G, ℱ)$ of all outer automorphisms of $G$ that preserve the conjugacy class of each $G_{i}$ . We show that every finitely generated subgroup $H \subseteq Out (G, ℱ)$ 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 $FF$ and the $𝒵$ –factor graph $𝒵 F$ , as spaces of equivalence classes of arational trees and relatively free arational trees, respectively. We also identify the loxodromic isometries of $FF$ with the fully irreducible elements of $Out (G, ℱ)$ , and loxodromic isometries of $𝒵 F$ with the fully irreducible atoroidal outer automorphisms.

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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

References

Publication

Received: 12 March 2019

Accepted: 13 August 2020

Published: 5 April 2022

Proposed: Martin R Bridson

Seconded: David M Fisher, Mladen Bestvina

Authors

Vincent Guirardel
Institut de recherche en mathématiques de Rennes, UMR 6625 Université de Rennes 1 et CNRS Rennes France

Camille Horbez
Laboratoire de Mathématiques d’Orsay Université Paris-Sud CNRS Université Paris-Saclay Orsay France

Volume 26, issue 1 (2022)