Hyperlinear approximations to amenable groups come from sofic approximations

TL;DR

This paper establishes a quantitative link between hyperlinear and sofic approximations for amenable groups, demonstrating that hyperlinear approximations can be derived from sofic ones, and explores related stability concepts.

Contribution

It provides a quantitative formulation of the equivalence between hyperlinearity and soficity specifically for amenable groups, connecting different types of group approximations.

Findings

Hyperlinear approximations are essentially derived from sofic approximations for amenable groups.

A quantitative relationship between Hilbert-Schmidt and permutation stability is established.

The results clarify the connection between different stability notions in group approximations.

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.