Measure equivalence classification of transvection-free right-angled Artin groups

Horbez, Camille; Huang, Jingyin

doi:10.5802/jep.199

Measure equivalence classification of transvection-free right-angled Artin groups
[Classification des groupes d’Artin à angles droits sans transvections pour l’équivalence mesurée]

Horbez, Camille ¹ ; Huang, Jingyin ²

¹ Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay 91405, Orsay, France
² Department of Mathematics, The Ohio State University 100 Math Tower, 231 W 18th Ave, Columbus, OH 43210, USA

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1021-1067.

Résumé
Abstract

Nous démontrons que si deux groupes d’Artin à angles droits sans transvections sont mesurablement équivalents, alors ils ont des graphes d’extension isomorphes. En conséquence, deux groupes d’Artin à angles droits ayant des groupes d’automorphismes extérieurs finis sont mesurablement équivalents si et seulement s’ils sont isomorphes. Ceci coïncide avec la classification pour la quasi-isométrie. Par contre, contrairement au cas de la quasi-isométrie, un groupe d’Artin à angles droits ne peut jamais être super-rigide pour l’équivalence mesurée, pour deux raisons. D’abord, un groupe d’Artin à angles droits $G$ est toujours mesurablement équivalent à tout produit graphé de groupes moyennables infinis dénombrables sur le même graphe sous-jacent. Ensuite, lorsque $G$ est non abélien, le groupe d’automorphismes du revêtement universel du complexe de Salvetti de $G$ contient toujours des réseaux (non uniformes) qui ne sont pas de type fini.

We prove that if two transvection-free right-angled Artin groups are measure equivalent, then they have isomorphic extension graphs. As a consequence, two right-angled Artin groups with finite outer automorphism groups are measure equivalent if and only if they are isomorphic. This matches the quasi-isometry classification. However, in contrast with the quasi-isometry question, we observe that no right-angled Artin group is superrigid for measure equivalence in the strongest possible sense, for two reasons. First, a right-angled Artin group $G$ is always measure equivalent to any graph product of infinite countable amenable groups over the same defining graph. Second, when $G$ is nonabelian, the automorphism group of the universal cover of the Salvetti complex of $G$ always contains infinitely generated (non-uniform) lattices.

Reçu le : 2020-10-15
Accepté le : 2022-05-15
Publié le : 2022-06-09

DOI : 10.5802/jep.199

Classification : 20F36, 20F65, 37A20, 46L36
Keywords: Right-angled Artin groups, measure equivalence
Mot clés : Groupes d’Artin à angles droits, équivalence mesurée

Affiliations des auteurs :

Horbez, Camille ¹ ; Huang, Jingyin ²

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     title = {Measure equivalence classification of transvection-free right-angled {Artin} groups},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Horbez, Camille; Huang, Jingyin. Measure equivalence classification of transvection-free right-angled Artin groups. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1021-1067. doi : 10.5802/jep.199. http://www.numdam.org/articles/10.5802/jep.199/

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