$L^2$-Betti numbers of totally disconnected groups and their approximation by Betti numbers of lattices

TL;DR

This paper establishes an approximation theorem for normalized Betti numbers of Farber sequences in totally disconnected groups and provides new computations related to their $L^2$-Betti numbers.

Contribution

It introduces a general approximation theorem for Betti numbers in totally disconnected groups and offers new computational insights into their $L^2$-Betti numbers.

Findings

Proves an approximation theorem for Betti numbers of lattices in totally disconnected groups.

Provides new computations and theoretical complements for $L^2$-Betti numbers in this setting.

Abstract

The main result is a general approximation theorem for normalised Betti numbers for Farber sequences of lattices in totally disconnected groups. Further, we contribute some computations and complements to the general theory of $L^2$-Betti numbers of totally disconnected groups.