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arXiv · 1801.09600

Asymptotics of Cheeger constants and unitarisability of groups

Maria Gerasimova·Dominik Gruber·Nicolas Monod·Andreas Thom
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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 ${\rm Lit}(\Gamma)\in [0, \infty]$ 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 ${\rm Lit}(\Gamma)$. Using graphical small cancellation theory, we prove that there exist groups $\Gamma$ for which $1<{\rm Lit}(\Gamma)<\infty$. Further applications, examples and problems are discussed.

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Maria Gerasimova, Dominik Gruber, Nicolas Monod, Andreas Thom. 2018-01-29. Asymptotics of Cheeger constants and unitarisability of groups. https://arxiv.org/abs/1801.09600

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