  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.
