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.