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 is always measure equivalent to any graph product of infinite countable amenable groups over the same defining graph. Second, when is nonabelian, the automorphism group of the universal cover of the Salvetti complex of 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 est toujours mesurablement équivalent à tout produit graphé de groupes moyennables infinis dénombrables sur le même graphe sous-jacent. Ensuite, lorsque est non abélien, le groupe d’automorphismes du revêtement universel du complexe de Salvetti de contient toujours des réseaux (non uniformes) qui ne sont pas de type fini.
Accepted:
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Keywords: Right-angled Artin groups, measure equivalence
Mot clés : Groupes d’Artin à angles droits, équivalence mesurée
Camille Horbez 1; Jingyin Huang 2
@article{JEP_2022__9__1021_0, author = {Camille Horbez and Jingyin Huang}, title = {Measure equivalence classification of transvection-free right-angled {Artin} groups}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {1021--1067}, publisher = {\'Ecole polytechnique}, volume = {9}, year = {2022}, doi = {10.5802/jep.199}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.199/} }
TY - JOUR AU - Camille Horbez AU - Jingyin Huang TI - Measure equivalence classification of transvection-free right-angled Artin groups JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 1021 EP - 1067 VL - 9 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.199/ DO - 10.5802/jep.199 LA - en ID - JEP_2022__9__1021_0 ER -
%0 Journal Article %A Camille Horbez %A Jingyin Huang %T Measure equivalence classification of transvection-free right-angled Artin groups %J Journal de l’École polytechnique — Mathématiques %D 2022 %P 1021-1067 %V 9 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.199/ %R 10.5802/jep.199 %G en %F JEP_2022__9__1021_0
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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