Currents relative to a malnormal subgroup system

Yassine Guerch

doi:10.5802/afst.1779

Yassine Guerch ¹

¹ Laboratoire de mathématique d’Orsay, UMR 8628 CNRS, Université Paris-Saclay, 91405 ORSAY Cedex, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 2, pp. 493-522.

Abstract
Abstract

This paper introduces a new topological space associated with a nonabelian free group $F_{n}$ of rank $n$ and a malnormal subgroup system $𝒜$ of $F_{n}$ , called the space of currents relative to $𝒜$ , which are $F_{n}$ -invariant measures on an appropriate subspace of the double boundary of $F_{n}$ . The extension from free factor systems as considered by Gupta to malnormal subgroup systems is necessary in order to fully study the growth under iteration of outer automorphisms of $F_{n}$ , and requires the introduction of new techniques on cylinders. We in particular prove that currents associated with elements of $F_{n}$ which are not contained in a conjugate of a subgroup of $𝒜$ are dense in the space of currents relative to $𝒜$ .

Dans cet article, nous introduisons un nouvel espace topologique associé à un groupe libre non abélien $F_{n}$ de rang $n$ et à un système de sous-groupes malnormal $𝒜$ de $F_{n}$ . Appelé espace des courants relatifs à $𝒜$ , cet espace est constitué de mesures $F_{n}$ -invariantes à support dans un sous-espace approprié du double bord de $F_{n}$ . L’extension du cas des systèmes de facteurs libres considéré par Gupta au cas des systèmes de sous-groupes malnormaux est nécessaire afin d’étudier la croissance sous itération d’automorphismes extérieurs de $F_{n}$ , et requiert l’introduction de nouvelles techniques sur les cylindres. Nous démontrons en particulier que l’ensemble des courants associés aux éléments de $F_{n}$ qui ne sont contenus dans aucun conjugué de sous-groupes de $𝒜$ est dense dans l’espace des courants relatifs à $𝒜$ .

Received: 2021-12-17
Accepted: 2022-11-09
Published online: 2024-09-23

DOI: 10.5802/afst.1779

Classification: 20E05, 20E08, 20E36, 20F65
Keywords: Nonabelian free groups, outer automorphism groups, space of currents, group actions on trees

Author's affiliations:

Yassine Guerch ¹

¹ Laboratoire de mathématique d’Orsay, UMR 8628 CNRS, Université Paris-Saclay, 91405 ORSAY Cedex, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Currents relative to a malnormal subgroup system},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {493--522},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Yassine Guerch. Currents relative to a malnormal subgroup system. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 2, pp. 493-522. doi : 10.5802/afst.1779. https://afst.centre-mersenne.org/articles/10.5802/afst.1779/

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