Weak notions of normality and vanishing up to rank in L2-cohomology

TL;DR

This paper investigates conditions under which L2-cohomology vanishes for certain groups, establishing new vanishing results for groups like SL(n,R) and Thompson's groups, with implications for their L2-Betti numbers.

Contribution

It introduces weak normality conditions that lead to vanishing results in L2-cohomology, providing new proofs and extending known results for specific groups.

Findings

L2-Betti numbers of SL(n,R) vanish below degree n-1

L2-Betti numbers of Thompson's groups F and T vanish

Weak normality conditions imply vanishing of L2-cohomology

Abstract

We study vanishing results for L2-cohomology of countable groups under the presence of subgroups that satisfy some weak normality condition. As a consequence we show that the L2-Betti numbers of SL(n,R) for any infinite integral domain R vanish below degree n-1. We also give a uniform proof for the vanishing of L2-Betti numbers of Thompson's groups F and T.