Asymptotics of Cheeger constants and unitarisability of groups

TL;DR

This paper explores the relationship between the unitarisability of group representations and the asymptotic behavior of isoperimetric constants of Cayley graphs, introducing the Littlewood exponent as a key invariant.

Contribution

It establishes a novel connection between unitarisability and Cheeger constants via the Littlewood exponent, and constructs groups with specific exponent values using graphical small cancellation theory.

Findings

Existence of groups with Littlewood exponent between 1 and infinity.

Finiteness, amenability, unitarisability, and free subgroups linked to thresholds of the Littlewood exponent.

New applications and open problems discussed.

Abstract

Given a group $\Gamma$, we establish a connection between the unitarisability of its uniformly bounded representations and the asymptotic behaviour of the isoperimetric constants of Cayley graphs of $\Gamma$ for increasingly large generating sets. The connection hinges on an analytic invariant ${\rm Lit}(\Gamma)\in [0, \infty]$ which we call the \emph{Littlewood exponent}. Finiteness, amenability, unitarisability and the existence of free subgroups are related respectively to the thresholds $0, 1, 2$ and $\infty$ for ${\rm Lit}(\Gamma)$. Using graphical small cancellation theory, we prove that there exist groups $\Gamma$ for which $1<{\rm Lit}(\Gamma)<\infty$. Further applications, examples and problems are discussed.