Invariant Random Subgroups, Soficity, and Lück's determinant conjecture

Abstract

We extend Lück'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 ${\mathsf{MIP}}^{\ast }=\mathsf{RE}$, 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.

15 pages, v2: the proof of theorem 3.4 needed a computability condition on lemma 3.5, which was added. Made minor expositional changes