Independence Tuples and Deninger's Problem

TL;DR

This paper introduces a modified concept of independence tuples for sofic entropy, enabling the generalization of results related to Deninger's Problem and linking Fuglede-Kadison determinants to invertibility in group rings.

Contribution

It defines a new version of independence tuples that incorporate weak containment conditions, extending previous results to a broader class of groups and operators.

Findings

Fuglede-Kadison determinant equals 1 iff the operator is invertible in the group ring

Generalization of Deninger's Problem to sofic groups and operators

New independence tuple concept links entropy and operator invertibility

Abstract

Motivated by our results in "Polish Models and Sofic Entropy," we define modified version of the independence tuples for sofic entropy developed by Kerr and Li. These modified version essentially require that the independence sequences give rise to representations weakly contained in the left regular when projected onto the Koopman representation. Using this, we can generalize our previous results for Deninger's Problem. Namely, we can show that if G is a sofic group, and if f is in M_{n}(Z(G)) and is invertible as an operator on l^{2}(G)^{n}, then the Fuglede-Kadison determinant of f is 1 if and only if f is invertible in M_{n}(Z(G)).