Invariant Random Subgroups, Soficity, and Lück's determinant conjecture
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915505799954432 |
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| author | Manzoor, Aareyan |
| author_facet | Manzoor, Aareyan |
| contents | 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 $\mathsf{MIP}^* = \mathsf{RE}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15154 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Invariant Random Subgroups, Soficity, and Lück's determinant conjecture Manzoor, Aareyan Operator Algebras Group Theory Quantum Physics 46L10 (primary), 20F69, 81P99 (secondary) 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 $\mathsf{MIP}^* = \mathsf{RE}$, 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. |
| title | Invariant Random Subgroups, Soficity, and Lück's determinant conjecture |
| topic | Operator Algebras Group Theory Quantum Physics 46L10 (primary), 20F69, 81P99 (secondary) |
| url | https://arxiv.org/abs/2508.15154 |