Invariant Random Subgroups, Soficity, and L\"uck's determinant conjecture

TL;DR

This paper extends L"uck's determinant conjecture to invariant random subgroups of free groups, demonstrating the existence of non-co-sofic IRS satisfying the conjecture, and explores implications for soficity and non-Connes embeddability.

Contribution

It generalizes the determinant conjecture to IRS, constructs non-co-sofic IRS satisfying it, and links non-local games to von Neumann algebra properties.

Findings

Existence of IRS satisfying the determinant conjecture but not co-sofic

Soficity may be a stronger property than satisfying the determinant conjecture

Connections between non-local games, MIP* = RE, and non-Connes embeddability

Abstract

We extend L\"uck's determinant conjecture from groups to invariant random subgroups (IRS) of free groups, a framework generalizing groups where a non-sofic object is known to exist. For every free group, we prove the existence of an IRS satisfying the determinant conjecture that is not co-hyperlinear, and hence not co-sofic. This provides evidence that satisfying the determinant conjecture might be a weaker property than soficity for groups, and consequently the conjecture possibly holds for all groups. We use techniques from non-local games and $\mathsf{MIP}^* = \mathsf{RE}$, showing more generally when the latter can be used to narrow down when a von Neumann algebra (or IRS) contains a non-Connes embeddable object.