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

TL;DR

This paper introduces a geometric method to establish upper bounds for the first 2-2-Betti number of groups, demonstrating its effectiveness on Burnside groups and extending to character-based generalizations.

Contribution

It provides a novel geometric approach to bound 2-2-Betti numbers and extends results to character-based 2-2-Betti numbers, including applications to Burnside groups.

Findings

Burnside groups of large prime exponent have zero first 2-2-Betti number.

The method applies to generalized 2-2-Betti numbers defined via characters.

Extension of Thom-Peterson results to the setting of character-based 2-2-Betti numbers.

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.