Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type

TL;DR

This paper investigates measure equivalence and rigidity properties of 2-dimensional hyperbolic Artin groups, proving boundary amenability, classifying measure equivalence classes, and establishing strong rigidity theorems with applications to orbit equivalence and von Neumann algebras.

Contribution

It establishes boundary amenability for these groups, introduces fixed set graphs as invariants for classification, and proves new rigidity theorems linking measure equivalence to automorphism groups.

Findings

Boundary amenability of 2D hyperbolic Artin groups

Classification of measure equivalent groups via fixed set graphs

Rigidity theorems relating measure equivalence to automorphism groups

Abstract

We study $2$-dimensional Artin groups of hyperbolic type from the viewpoint of measure equivalence, and establish rigidity theorems. We first prove that they are boundary amenable. So is every group acting discretely by simplicial isometries on a connected piecewise hyperbolic $\mathrm{CAT}(-1)$ simplicial complex with countably many simplices in finitely many isometry types, assuming that vertex stabilizers are boundary amenable. Consequently, they satisfy the Novikov conjecture. We then show that measure equivalent $2$-dimensional Artin groups of hyperbolic type have isomorphic fixed set graphs -- an analogue of the curve graph, introduced by Crisp. This yields classification results. We obtain strong rigidity theorems. Let $G=G_\Gamma$ be a $2$-dimensional Artin group of hyperbolic type, with $\mathrm{Out}(G)$ finite. When the automorphism groups of the fixed set graph and of the Cayley complex $\mathfrak{C}$ coincide, every countable group $H$ which is measure equivalent to $G$, is commensurable to a lattice in $\mathrm{Aut}(\mathfrak{C})$. This happens whenever $\Gamma$ is triangle-free with all labels at least $3$ -- unless $G$ is commensurable to the direct sum of $\mathbb{Z}$ and a free group. When $\Gamma$ satisfies an additional star-rigidity condition, then $\mathrm{Aut}(\mathfrak{C})$ is countable, and $H$ is almost isomorphic to $G$. This has applications to orbit equivalence rigidity, and rigidity results for von Neumann algebras associated to ergodic actions of Artin groups. We also derive a rigidity statement regarding possible lattice envelopes of certain Artin groups, and a cocycle superrigidity theorem from higher-rank lattices to $2$-dimensional Artin groups of hyperbolic type.