On upper bounds for the first ${\ell }^2$-Betti number

Abstract

This article presents a method for proving upper bounds for the first ${\ell }^2$-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 ${\ell }^2$-Betti number. Our approach extends to generalizations of ${\ell }^2$-Betti numbers, that are defined using characters. We illustrate this flexibility by generalizing results of Thom-Peterson on q-normal subgroups to this setting.

10 pages, comments welcome