L^2-Betti numbers and Plancherel measure

TL;DR

This paper links L^2-Betti numbers of certain groups to their representation theory and Plancherel measure, enabling explicit calculations for various classes of groups including Lie groups and automorphism groups of trees.

Contribution

It introduces a method to compute L^2-Betti numbers using reduced cohomology and Plancherel measure, extending previous results to broader classes of groups.

Findings

Computed L^2-Betti numbers for semi-simple Lie groups with finite center.

Derived explicit formulas for automorphism groups of locally finite trees.

Connected L^2-Betti numbers with representation theory and harmonic analysis.

Abstract

We compute $L^2$-Betti numbers of postliminal, locally compact, unimodular groups in terms of ordinary dimensions of reduced cohomology with coefficients in irreducible unitary representations and the Plancherel measure. This allows us to compute the $L^2$-Betti numbers for semi-simple Lie groups with finite center, simple algebraic groups over local fields, and automorphism groups of locally finite trees acting transitively on the boundary.