# Vanishing of $\ell^2$-Betti numbers of locally compact groups as an invariant of coarse equivalence

## TL;DR

This paper proves that the property of having zero $ ext{l}^2$-Betti numbers for unimodular locally compact second countable groups remains unchanged under coarse equivalence, establishing it as a coarse invariant.

## Contribution

It demonstrates that the vanishing of $ ext{l}^2$-Betti numbers is preserved under coarse equivalence for a broad class of groups, extending previous invariance results.

## Key findings

- Vanishing of $ ext{l}^2$-Betti numbers is a coarse invariant.
- The result applies to unimodular locally compact second countable groups.
- Provides a new tool for classifying groups via coarse geometry.

## Abstract

We provide a proof that the vanishing of $\ell^2$-Betti numbers of unimodular locally compact second countable groups is an invariant of coarse equivalence.