Measure equivalence classification of transvection-free right-angled Artin groups
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 1021-1067.

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.

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.

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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
Camille Horbez 1; Jingyin Huang 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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Camille Horbez; Jingyin Huang. Measure equivalence classification of transvection-free right-angled Artin groups. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 1021-1067. doi : 10.5802/jep.199. https://jep.centre-mersenne.org/articles/10.5802/jep.199/

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