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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

Given a countable group G splitting as a free product G = G1 Gk FN, we establish classification results for subgroups of the group Out (G,) of all outer automorphisms of G that preserve the conjugacy class of each Gi. We show that every finitely generated subgroup H Out (G,) 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 = FN, 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 𝒵F , 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,), and loxodromic isometries of 𝒵F with the fully irreducible atoroidal outer automorphisms.

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
Received: 12 March 2019
Accepted: 13 August 2020
Published: 5 April 2022
Proposed: Martin R Bridson
Seconded: David M Fisher, Mladen Bestvina
Vincent Guirardel
Institut de recherche en mathématiques de Rennes, UMR 6625
Université de Rennes 1 et CNRS
Camille Horbez
Laboratoire de Mathématiques d’Orsay
Université Paris-Sud
Université Paris-Saclay