Random subgroups, automorphisms, splittings

Vincent Guirardel (1) , Gilbert Levitt (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme

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 $F_k$.

Domains: Group Theory [math.GR]

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