Optimal Linear Sofic Approximations Of Countable Groups
Keivan Mallahi-Karai, Maryam Mohammadi Yekta
TL;DR
This work sharpens the understanding of linear sofic approximations for countable groups, proving that every linear sofic group over a characteristic-zero field is 1/2-linear sofic and that the 1/2 constant is optimal in general, while torsion-free groups achieve the stronger 1-linear sofic bound. The authors adapt and extend the amplification framework of Arzhantseva–Paunescu by incorporating two coupled quantities and an eigenvalue-mass tracking mechanism, together with effective non-concentration bounds for random walks on abelian groups and a Jordan-structure analysis under tensor powers. They also determine κ(G) for finite groups, establish stability results in the linear sofic setting, and exhibit genuine differences between characteristic-zero and positive-characteristic cases, including explicit κ-values for Z_p^n and related finite groups. The methods combine rank-metric matrix analysis, probabilistic non-concentration estimates, and representation-theoretic arguments to yield quantitative amplification and optimality results with potential implications for Gottschalk-type conjectures and stability questions in metric group approximations.
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.