Hyperfiniteness and Borel asymptotic dimension of boundary actions of hyperbolic groups

TL;DR

This paper provides a new, concise proof that the boundary actions of hyperbolic groups are hyperfinite and demonstrates that these actions have finite Borel asymptotic dimension, advancing understanding of their structural properties.

Contribution

The paper introduces a simplified proof of hyperfiniteness for boundary actions of hyperbolic groups and establishes their finite Borel asymptotic dimension, extending prior results.

Findings

Boundary actions of hyperbolic groups are hyperfinite.

Such actions have finite Borel asymptotic dimension.

The new proof is shorter and more direct than previous ones.

Abstract

We give a new short proof of the theorem due to Marquis and Sabok, which states that the orbit equivalence relation induced by the action of a finitely generated hyperbolic group on its Gromov boundary is hyperfinite. Our methods permit moreover to show that every such action has finite Borel asymptotic dimension.