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 Lit(Γ) ∈ [0, ∞] 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 ∞ for Lit(Γ). Using graphical small cancellation theory, we prove that there exist groups Γ for which 1 < Lit(Γ) < ∞. Further applications, examples and problems are discussed. Contents