Asymptotics of Cheeger constants and unitarisability of groups

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 Lit(Gamma) is an element of [0, infinity] which we call the Littlewood exponent. Finiteness, amenability, unitarisability and the existence of free subgroups are related respectively to the thresholds 0, 1, 2 and infinity for Lit(Gamma). Using graphical small cancellation theory, we prove that there exist groups Gamma for which 1 < Lit(Gamma) < infinity. Further applications, examples and problems are discussed. (C) 2020 Elsevier Inc. All rights reserved.