Random subgroups, automorphisms, splittings

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...

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Bibliographic Details
Published in:arXiv.org 2019-06
Main Authors: Guirardel, Vincent, Levitt, Gilbert
Format: Article
Language:English
Subjects:
Automorphisms
Random walk
Subgroups
Online Access:Get full text
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Summary: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 \(F_k\).
ISSN:2331-8422