Random subgroups, automorphisms, splittings
[Sous-groupes aléatoires, automorphismes, scindements]
Annales de l'Institut Fourier, Online first, 29 p.

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 h𝔽 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 (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 h𝔽 k contains all reduced words of length L, then 𝔽 k has no splitting relative to h over a subgroup of rank (k-1)(L-2).

Reçu le :
Révisé le :
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DOI : https://doi.org/10.5802/aif.3426
Classification : 20F28,  20E08,  20F67,  20P05
Mots clés : Sous-groupes aléatoires, marche aléatoire, scindements, automorphismes, groupes libres, groupes relativement hyperboliques
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Guirardel, Vincent; Levitt, Gilbert. Random subgroups, automorphisms, splittings. Annales de l'Institut Fourier, Online first, 29 p.

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