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Measure equivalence classification of right-angled Artin groups: the finite $\mathrm{Out}$ classes

Camille Horbez, Jingyin Huang

TL;DR

This paper advances the measure and orbit equivalence classification for right-angled Artin groups (RAAGs) by focusing on those with finite outer automorphism groups. It develops a groupoid-based framework using action-like cocycles and extension graphs to identify invariant structures, notably untransvectable cyclic parabolic subgroups, and proves rigidity results that force strong correspondence between the subgroup structure of equivalent RAAGs. The main theorem shows that, for $G$ with $|\,\operatorname{Out}(G)\,|<\infty$, a RAAG $H$ is measure equivalent to $G$ iff $H$ is a graph product of infinite finitely generated abelian groups over the defining graph of a finite-index RAAG subgroup $G^0\subseteq G$; orbit equivalence has a parallel, graph-product description. Consequently, ME and OE classifications diverge from quasi-isometric classifications in this setting, and the authors establish a robust combinatorial rigidity via the untransvectable extension graphs to realize these classifications.

Abstract

Given a right-angled Artin group $G$ with finite outer automorphism group, we determine which right-angled Artin groups are measure equivalent (or orbit equivalent) to $G$.

Measure equivalence classification of right-angled Artin groups: the finite $\mathrm{Out}$ classes

TL;DR

This paper advances the measure and orbit equivalence classification for right-angled Artin groups (RAAGs) by focusing on those with finite outer automorphism groups. It develops a groupoid-based framework using action-like cocycles and extension graphs to identify invariant structures, notably untransvectable cyclic parabolic subgroups, and proves rigidity results that force strong correspondence between the subgroup structure of equivalent RAAGs. The main theorem shows that, for with , a RAAG is measure equivalent to iff is a graph product of infinite finitely generated abelian groups over the defining graph of a finite-index RAAG subgroup ; orbit equivalence has a parallel, graph-product description. Consequently, ME and OE classifications diverge from quasi-isometric classifications in this setting, and the authors establish a robust combinatorial rigidity via the untransvectable extension graphs to realize these classifications.

Abstract

Given a right-angled Artin group with finite outer automorphism group, we determine which right-angled Artin groups are measure equivalent (or orbit equivalent) to .

Paper Structure

This paper contains 12 sections, 14 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Background
  4. Background on right-angled Artin groups
  5. Right-angled Artin groups and graph products.
  6. Background on measured groupoids
  7. Generalities on measured groupoids.
  8. Cocycles.
  9. Non-abelian equivalence classes and measure equivalence
  10. Strongly untransvectable cyclic parabolic subgroups
  11. A combinatorial rigidity statement
  12. Measure equivalence and orbit equivalence classification

Key Result

Theorem 1

Let $G$ and $H$ be two right-angled Artin groups. Assume that $|\operatorname{Out}(G)|<+\infty$. Then $G$ and $H$ are measure equivalent if and only if there exists a finite-index right-angled Artin subgroup $G^0\subseteq G$ such that, denoting by $\Lambda$ the defining graph of $G^0$, the group $H$

Theorems & Definitions (40)

  • Theorem 1: Measure equivalence classification, see Theorem \ref{['theo:me-classification']}
  • Theorem 2: Orbit equivalence classification, see Theorem \ref{['theo:oe-classification']}
  • Definition 2.1: Collapsible, clique-reduced
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Example 2.5
  • Proposition 3.1
  • Lemma 3.2
  • ...and 30 more