Hyperlinear approximations to amenable groups come from sofic approximations
Group Theory
2024-01-12 v2
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.
Cite
@article{arxiv.2311.09202,
title = {Hyperlinear approximations to amenable groups come from sofic approximations},
author = {Peter Burton and Maksym Chaudkhari and Kate Juschenko and Kyrylo Muliarchyk},
journal= {arXiv preprint arXiv:2311.09202},
year = {2024}
}
Comments
Updated to emphasize the effective nature of the results