Hyperlinear approximations to amenable groups come from sofic approximations

Peter Burton; Maksym Chaudkhari; Kate Juschenko; Kyrylo Muliarchyk

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arXiv · 2311.09202

Hyperlinear approximations to amenable groups come from sofic approximations

Peter Burton·Maksym Chaudkhari·Kate Juschenko·Kyrylo Muliarchyk

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Abstract

We provide a quantitative formulation of the equivalence between hyperlinearity and soficity for amenable groups, effectively showing how every hyperlinear approximation to such a group is simulated by a suitable sofic approximation. The proof is probabilistic, using the concentration of measure in high-dimensional spheres to control the deviation of an operator's matrix coefficients from its trace. As a corollary, we obtain a result connecting stability of sofic approximations with stability of hyperlinear approximations.

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Peter Burton, Maksym Chaudkhari, Kate Juschenko, Kyrylo Muliarchyk. 2023-11-15. Hyperlinear approximations to amenable groups come from sofic approximations. https://arxiv.org/abs/2311.09202

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