# Hyperfiniteness of boundary actions of cubulated hyperbolic groups

## TL;DR

This paper proves that hyperbolic groups acting on CAT(0) cube complexes have boundary actions that are hyperfinite, extending classical results from free groups to a broader class of groups.

## Contribution

It establishes the hyperfiniteness of boundary actions for cubulated hyperbolic groups, generalizing previous results from free groups to hyperbolic groups acting on cube complexes.

## Key findings

- Boundary actions of cubulated hyperbolic groups are hyperfinite.
- Generalizes Dougherty, Jackson, and Kechris's result from free groups.
- Connects geometric group actions with descriptive set theory.

## Abstract

We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group the natural action on its Gromov boundary is hyperfinite, which generalizes an old result of Dougherty, Jackson and Kechris for the free group case.