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

Aareyan Manzoor

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arXiv · 2508.15154

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

Aareyan Manzoor

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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.

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Aareyan Manzoor. 2025-08-21. Invariant Random Subgroups, Soficity, and L\"uck's determinant conjecture. https://arxiv.org/abs/2508.15154

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