Generic differentiability and $P$-minimal groups
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915857988321280 |
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| author | Johnson, Will |
| author_facet | Johnson, Will |
| contents | We prove generic differentiability in $P$-minimal theories, strengthening an earlier result of Kuijpers and Leenknegt. Using this, we prove Onshuus and Pillay's $P$-minimal analogue of Pillay's conjectures on o-minimal groups. Specifically, let $G$ be an $n$-dimensional definable group in a highly saturated model $M$ of a $P$-minimal theory. Then there is an open definable subgroup $H \subseteq G$ such that $H$ is compactly dominated by $H/H^{00}$, and $H/H^{00}$ is a $p$-adic Lie group of the expected dimension. Additionally, the generic differentiability theorem immediately implies a classification of interpretable fields in $P$-minimal theories, by work of Halevi, Hasson, and Peterzil. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17234 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generic differentiability and $P$-minimal groups Johnson, Will Logic 03C45, 03C60, 12L12 We prove generic differentiability in $P$-minimal theories, strengthening an earlier result of Kuijpers and Leenknegt. Using this, we prove Onshuus and Pillay's $P$-minimal analogue of Pillay's conjectures on o-minimal groups. Specifically, let $G$ be an $n$-dimensional definable group in a highly saturated model $M$ of a $P$-minimal theory. Then there is an open definable subgroup $H \subseteq G$ such that $H$ is compactly dominated by $H/H^{00}$, and $H/H^{00}$ is a $p$-adic Lie group of the expected dimension. Additionally, the generic differentiability theorem immediately implies a classification of interpretable fields in $P$-minimal theories, by work of Halevi, Hasson, and Peterzil. |
| title | Generic differentiability and $P$-minimal groups |
| topic | Logic 03C45, 03C60, 12L12 |
| url | https://arxiv.org/abs/2404.17234 |