On the First-Order Free Group Factor Alternative
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917881612075008 |
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| author | Goldbring, Isaac Pi, Jennifer |
| author_facet | Goldbring, Isaac Pi, Jennifer |
| contents | We investigate the problem of elementary equivalence of the free group factors, that is, do all free group factors $L(\mathbb{F}_n)$ share a common first-order theory? We establish a trichotomy of possibilities for their common first-order fundamental group, as well as several possible avenues for establishing a dichotomy in direct analog to the free group factor alternative of Dykema and Radulescu. We also show that the $\forall \exists$-theories of the interpolated free group factors are increasing, and use this to establish that the dichotomy holds on the level of $\forall \exists$-theories. We conclude with some observations on related problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_08168 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the First-Order Free Group Factor Alternative Goldbring, Isaac Pi, Jennifer Logic Operator Algebras We investigate the problem of elementary equivalence of the free group factors, that is, do all free group factors $L(\mathbb{F}_n)$ share a common first-order theory? We establish a trichotomy of possibilities for their common first-order fundamental group, as well as several possible avenues for establishing a dichotomy in direct analog to the free group factor alternative of Dykema and Radulescu. We also show that the $\forall \exists$-theories of the interpolated free group factors are increasing, and use this to establish that the dichotomy holds on the level of $\forall \exists$-theories. We conclude with some observations on related problems. |
| title | On the First-Order Free Group Factor Alternative |
| topic | Logic Operator Algebras |
| url | https://arxiv.org/abs/2305.08168 |