A remark about the spectral radius

TL;DR

This paper demonstrates that non-amenable groups can have generating sets with arbitrarily small spectral radius, impacting percolation theory and operator space theory.

Contribution

It establishes the existence of finite symmetric generating sets with spectral radius below any positive threshold for non-amenable groups, a novel result.

Findings

Existence of generating sets with spectral radius less than any ε>0 for non-amenable groups

Applications to percolation theory and operator spaces

Advances understanding of spectral properties in group theory

Abstract

We show that for any finitely generated non-amenable group and any $\varepsilon>0$, there exists some finite symmetric generating set with spectral radius less than $\varepsilon$. We give applications to percolation theory and the theory of operator spaces.