# Random subgroups, automorphisms, splittings

Abstract : We show that, if $H$ is a random subgroup of a finitely generated free group $F_k$, only inner automorphisms of $F_k$ may leave $H$ invariant. A similar result holds for random subgroups of toral relatively hyperbolic groups, more generally of groups which are hyperbolic relative to slender subgroups. These results follow from non-existence of splittings over slender groups which are relative to a random group element. Random subgroups are defined using random walks or balls in a Cayley tree of $F_k$.
Document type :
Journal articles
Domain :

https://hal-insu.archives-ouvertes.fr/insu-02164333
Contributor : Vincent Guirardel Connect in order to contact the contributor
Submitted on : Tuesday, June 25, 2019 - 8:33:03 AM
Last modification on : Friday, May 20, 2022 - 9:04:50 AM

### Citation

Vincent Guirardel, Gilbert Levitt. Random subgroups, automorphisms, splittings. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2021, 71 (3), pp.1363-1391. ⟨10.5802/aif.3426⟩. ⟨insu-02164333⟩

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