# 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 $F_k$.
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https://hal-insu.archives-ouvertes.fr/insu-02164333
Contributor : Vincent Guirardel <>
Submitted on : Tuesday, June 25, 2019 - 8:33:03 AM
Last modification on : Friday, July 10, 2020 - 4:17:38 PM

### Identifiers

• HAL Id : insu-02164333, version 1
• ARXIV : 1906.09654

### Citation

Vincent Guirardel, Gilbert Levitt. Random subgroups, automorphisms, splittings. 2019. ⟨insu-02164333⟩

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