Random subgroups, automorphisms, splittings

Guirardel, Vincent; Levitt, Gilbert

Annales de l'Institut Fourier, Tome 71 (2021) no. 4, pp. 1363-1391.

Nous montrons que si $H$ est un sous-groupe aléatoire d’un groupe libre de type fini ${𝔽}_k$, tout automorphisme de ${𝔽}_k$ préservant $H$ est intérieur. Nous prouvons un résultat similaire pour les sous-groupes aléatoires de groupes hyperboliques toriques, et plus généralement de groupes hyperboliques relativement à des sous-groupes sveltes. Ces résultats découlent de la non-existence de scindements au-dessus de sous-groupes sveltes qui sont relatifs à un élément aléatoire. Les sous-groupes aléatoires peuvent être définis en termes de marches aléatoires ou de boules dans le graphe de Cayley de ${𝔽}_k$.

Dans le cas du groupe libre ${𝔽}_k$, nous démontrons aussi le résultat déterministe suivant  : si un mot cycliquement réduit $h\in {𝔽}_k$ contient tous les mots réduits de longueur $L$, alors ${𝔽}_k$ n’a pas de scindement relatif à $h$ au-dessus d’un sous-groupe de rang $\le (k-1)(L-2)$.

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, and 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 ${𝔽}_k$.

In the free group ${𝔽}_k$, we also prove the following deterministic result: if a cyclically reduced word $h\in {𝔽}_k$ contains all reduced words of length $L$, then ${𝔽}_k$ has no splitting relative to $h$ over a subgroup of rank $\le (k-1)(L-2)$.

Classification : 20F28, 20E08, 20F67, 20P05
Keywords: Random subgroups, random walk, splitting, automorphisms, free group, relatively hyperbolic group
Mot clés : Sous-groupes aléatoires, marche aléatoire, scindements, automorphismes, groupes libres, groupes relativement hyperboliques

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