Hyperlinear approximations to amenable groups come from sofic approximations
arXiv:2110.03076
Abstract
We provide a quantitative formulation of the equivalence between hyperlinearity and soficity for amenable groups, showing that every hyperlinear approximation to such a group is essentially produced from a sofic approximation. This translates to a quantitative relationship between Hilbert-Schmidt and permutation stability for approximate homomorphisms which appropriately separate the elements of the group.
This is an early version of the article, and has been superseded by a new version with additional coauthors at arXiv:2311.09202