Integrable measure equivalence rigidity of right-angled Artin groups via quasi-isometry
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| Format: | Preprint |
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2023
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| author | Horbez, Camille Huang, Jingyin |
| author_facet | Horbez, Camille Huang, Jingyin |
| contents | Let $G$ be a right-angled Artin group with $|\mathrm{Out}(G)|<+\infty$. We prove that if a countable group $H$ with bounded torsion 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-named author, $H$ acts properly and cocompactly on a $\mathrm{CAT}(0)$ 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-named 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 $n\ge 5$. In contrast, we also prove that if a right-angled Artin group $G$ with $|\mathrm{Out}(G)|<+\infty$ splits non-trivially 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_12147 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Integrable measure equivalence rigidity of right-angled Artin groups via quasi-isometry Horbez, Camille Huang, Jingyin Group Theory Dynamical Systems Geometric Topology Let $G$ be a right-angled Artin group with $|\mathrm{Out}(G)|<+\infty$. We prove that if a countable group $H$ with bounded torsion 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-named author, $H$ acts properly and cocompactly on a $\mathrm{CAT}(0)$ 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-named 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 $n\ge 5$. In contrast, we also prove that if a right-angled Artin group $G$ with $|\mathrm{Out}(G)|<+\infty$ splits non-trivially 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. |
| title | Integrable measure equivalence rigidity of right-angled Artin groups via quasi-isometry |
| topic | Group Theory Dynamical Systems Geometric Topology |
| url | https://arxiv.org/abs/2309.12147 |