# Characters, $L^2$-Betti numbers and an equivariant approximation theorem

## TL;DR

This paper introduces $L^2$-multiplicities, generalizing $L^2$-Betti numbers, and proves approximation theorems for these invariants, with applications to the cohomology of arithmetic groups.

## Contribution

It defines $L^2$-multiplicities for group actions and establishes new approximation theorems extending previous results for $L^2$-Betti numbers.

## Key findings

- $L^2$-multiplicities generalize $L^2$-Betti numbers.
- Approximation theorems are extended to $L^2$-multiplicities.
- Applications to the cohomology of arithmetic groups.

## Abstract

Let $G$ be a group with a finite subgroup $H$. We define the $L^2$-multiplicity of an irreducible representation of $H$ in the $L^2$-homology of a proper $G$-CW-complex. These invariants generalize the $L^2$-Betti numbers. Our main results are approximation theorems for $L^2$-multiplicities which extend the approximation theorems for $L^2$-Betti numbers of L\"uck, Farber and Elek-Szab\'o respectively. The main ingredient is the theory of characters of infinite groups and a method to induce characters from finite subgroups. We discuss applications to the cohomology of (arithmetic) groups.