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2022 Boundaries of relative factor graphs and subgroup classification for automorphisms of free products
Vincent Guirardel, Camille Horbez
Geom. Topol. 26(1): 71-126 (2022). DOI: 10.2140/gt.2022.26.71


Given a countable group G splitting as a free product G=G1GkFN, 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 HOut(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.


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Vincent Guirardel. Camille Horbez. "Boundaries of relative factor graphs and subgroup classification for automorphisms of free products." Geom. Topol. 26 (1) 71 - 126, 2022.


Received: 12 March 2019; Accepted: 13 August 2020; Published: 2022
First available in Project Euclid: 9 May 2022

Digital Object Identifier: 10.2140/gt.2022.26.71

Primary: 20E06 , 20E07 , 20E08 , 20E36

Keywords: automorphism groups of free groups and free products , free factor graph , Gromov boundaries , Gromov hyperbolic spaces , subgroup classification

Rights: Copyright © 2022 Mathematical Sciences Publishers


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Vol.26 • No. 1 • 2022
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