Asymptotics of Cheeger constants and unitarisability of groups

2018/01/29 by Gerasimova, Maria, Gruber, Dominik, Monod, Nicolas +1
#FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1801.09600

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)∈ [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 \rm Lit(Γ). Using graphical small cancellation theory, we prove that there exist groups Γ for which 1

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