Asymptotics of Cheeger constants and unitarisability of groups

Abstract

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

24 pages, no figures; v2 minor update