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.