Asymptotics of Cheeger constants and unitarisability of groups

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 $\mathrm{Lit}(\Gamma )\in [0,\infty ]$ 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 $\mathrm{Lit}(\Gamma )$. Using graphical small cancellation theory, we prove that there exist groups $\Gamma$ for which $1<\mathrm{Lit}(\Gamma )<\infty$. Further applications, examples and problems are discussed.

24 pages, no figures; v2 minor update