Optimal Linear Sofic Approximations Of Countable Groups

TL;DR

This paper investigates the optimal linear sofic approximation constants for countable groups, establishing new bounds and conditions, especially for torsion-free groups, and introduces techniques involving random walks and trigonometric sums.

Contribution

It proves that every linear sofic group is 1/2-linear sofic and that torsion-free groups are 1-linear sofic, improving previous bounds and answering a question by Arzhantseva.

Findings

Linear sofic groups are 1/2-linear sofic.

Torsion-free groups are 1-linear sofic.

Effective non-concentration estimates for random walks.

Abstract

A countable group G is called k-linear sofic (for some 0 <k \le 1) if finite subsets of G admit "approximate representations" by complex invertible matrices in the normalized rank metric, so that non-identity elements are k-away from the identity. This class of groups was systematically studied by Arzhantseva and Paunescu [AP17], where it is shown that such a group is always 1/4-linear sofic. In this paper, we will study the optimality of this result for general countable groups and show that every linear sofic group is 1/2-linear sofic, and 1/2 cannot be improved. However, if G is assumed to be torsion-free, then it is 1-linear sofic. These results answer a question posed by G. Arzhantseva in her talk in the IAS Stability and Testability lecture series. We also study the optimal linear sofic constant of finite groups over C and fields of positive characteristic. For the proof, we establish effective non-concentration estimates for random walks on finitely generated abelian groups, which may be of independent interest. Another ingredient of the proof involves bounding integrals of certain trigonometric sums.