On upper bounds for the first ℓ²-Betti number

Abstract

This article presents a method for proving upper bounds for the first ℓ 2 \ell ^2 -Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first ℓ 2 \ell ^2 -Betti number. Our approach extends to generalizations of ℓ 2 \ell ^2 -Betti numbers that are defined using characters. We illustrate this flexibility by generalizing results of Peterson-Thom on q-normal subgroups to this setting.