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Boundaries of relative
factor graphs and subgroup classification for automorphisms
of free products
Vincent Guirardel and Camille Horbez |
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Geometry & Topology 26 (2022) 71–126
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DOI: 10.2140/gt.2022.26.71
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Abstract |
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Given a countable group splitting as a free product , we establish classification results for subgroups of the group of all outer automorphisms of that preserve the conjugacy class of each . We show that every finitely generated subgroup 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 can be dropped for , or more generally when is toral relatively hyperbolic). In the first case, either virtually preserves a nonperipheral conjugacy class in , or else 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 and the –factor graph , as spaces of equivalence classes of arational trees and relatively free arational trees, respectively. We also identify the loxodromic isometries of with the fully irreducible elements of , and loxodromic isometries of 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
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Mathematical Subject Classification 2010
Primary: 20E06, 20E07, 20E08, 20E36
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References |
Publication
Received: 12 March 2019
Accepted: 13 August 2020
Published: 5 April 2022
Proposed: Martin R Bridson
Seconded: David M Fisher, Mladen Bestvina
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Authors |
| Vincent Guirardel | |
| Institut de recherche en
mathématiques de Rennes, UMR 6625 Université de Rennes 1 et CNRS Rennes France |
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| Camille Horbez | |
| Laboratoire de Mathématiques
d’Orsay Université Paris-Sud CNRS Université Paris-Saclay Orsay France |
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