Integrable measure equivalence rigidity of right-angled Artin groups via quasi-isometry

Horbez, Camille; Huang, Jingyin

doi:10.2140/gt.2026.30.1451

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ISSN (electronic): 1364-0380

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Integrable measure equivalence rigidity of right-angled Artin groups via quasi-isometry

Camille Horbez and Jingyin Huang

Geometry & Topology 30 (2026) 1451–1514

DOI: 10.2140/gt.2026.30.1451

Abstract

Let $G$ be a right-angled Artin group with $| Out (G) | < + \infty$ . We prove that if a countable group $H$ with bounded finite subgroups is measure equivalent to $G$ , with an $L^{1}$ -integrable measure equivalence cocycle towards $G$ , then $H$ is finitely generated and quasi-isometric to $G$ . In particular, through work of Kleiner and the second author, $H$ acts properly and cocompactly on a $CAT (0)$ cube complex which is quasi-isometric to $G$ and equivariantly projects to the right-angled building of $G$ .

As a consequence of work of the second author, we derive a superrigidity theorem in integrable measure equivalence for an infinite class of right-angled Artin groups, including those whose defining graph is an $n$ -gon with $n \geq 5$ . In contrast, we also prove that if a right-angled Artin group $G$ with $| Out (G) | < + \infty$ splits nontrivially as a product, then there does not exist any locally compact group which contains all groups $H$ that are $L^{1}$ -measure equivalent to $G$ as lattices, even up to replacing $H$ by a finite-index subgroup and taking the quotient by a finite normal subgroup.

Keywords

measure equivalence, quasi-isometry, right-angled Artin groups, rigidity, integrability of cocycles, CAT(0) cube complexes, right-angled buildings, lattice embeddings

Mathematical Subject Classification

Primary: 20F36, 20F65, 22D50, 37A20

Secondary: 20F67, 20F69, 22D40, 22F10

References

Publication

Received: 22 April 2024

Revised: 17 March 2025

Accepted: 24 September 2025

Published: 9 July 2026

Proposed: Roman Sauer

Seconded: David Fisher, Mladen Bestvina

Authors

Camille Horbez
Laboratoire de Mathématiques d’Orsay CNRS - Université Paris Saclay Orsay France

Jingyin Huang
Department of Mathematics The Ohio State University Columbus, OH United States

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Volume 30, issue 4 (2026)