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

Camille Horbez, Jingyin Huang

Abstract

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.

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

Abstract

Let be a right-angled Artin group with . We prove that if a countable group with bounded torsion is measure equivalent to , with an -integrable measure equivalence cocycle towards , then is finitely generated and quasi-isometric to . In particular, through work of Kleiner and the second-named author, acts properly and cocompactly on a cube complex which is quasi-isometric to and equivariantly projects to the right-angled building of . 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 -gon with . In contrast, we also prove that if a right-angled Artin group with splits non-trivially as a product, then there does not exist any locally compact group which contains all groups that are -measure equivalent to as lattices, even up to replacing by a finite-index subgroup and taking the quotient by a finite normal subgroup.
Paper Structure (50 sections, 69 theorems, 28 equations, 3 figures)

This paper contains 50 sections, 69 theorems, 28 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background, history and motivation
  3. Integrable measure equivalence versus quasi-isometry for RAAGs
  4. Consequences to superrigidity
  5. Lattice envelopes
  6. Discussion of the proof of the main theorem
  7. Structure of the paper
  8. Preliminaries on the geometry of right-angled Artin groups
  9. Right-angled Artin groups
  10. Extension graphs and right-angled buildings
  11. Automorphisms of the extension graph and the right-angled building
  12. From $\mathrm{Aut}(\mathbb B_\Gamma)$ to $\mathrm{Aut}(\Gamma^e)$.
  13. From $\mathrm{Aut}(\Gamma^e)$ to $\mathrm{Aut}(\mathbb B_\Gamma)$.
  14. Blow-up buildings and a quasi-isometry criterion
  15. Blow-up buildings
  16. ...and 35 more sections

Key Result

Theorem 1

Let $G$ be a right-angled Artin group with $|\operatorname{Out}(G)|<+\infty$, let $H$ be a countable group with bounded finite subgroups. If there exists an $(L^1,L^0)$-measure equivalence coupling from $H$ to $G$, then $H$ is finitely generated and quasi-isometric to $G$.

Figures (3)

  • Figure 1: Fundamental domain of the pentagon building.
  • Figure 2: A branched line and a projection map.
  • Figure 3: An axis of $w$.

Theorems & Definitions (139)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Remark 1.1
  • Theorem 4
  • Theorem 5
  • Theorem 6: see Theorem \ref{['theo:fg']}
  • Proposition 2.1: CCV
  • Definition 2.2: Extension graph KK
  • Definition 2.3: Right-angled building
  • ...and 129 more