On f-generic types in NIP groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912717485375488 |
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| author | Stonestrom, Atticus |
| author_facet | Stonestrom, Atticus |
| contents | Recall that a definable group is `definably amenable' if it admits a translation-invariant Keisler measure. We prove a combinatorial characterization of definable amenability for groups definable in NIP theories. More specifically, given a group $G$, a subset $D\subseteq G$ is said to (left) `$G$-divide' if there is some natural number $k$ and an infinite sequence of elements $g_i\in G$ such that $g_{i_1}D\cap\dots\cap g_{i_k}D=\varnothing$ for all $i_1<\dots<i_k$. Our main result is that, if $G$ is a group definable in an NIP theory, and the union of two definable $G$-dividing subsets of $G$ still $G$-divides, then $G$ is definably amenable. It follows that $G$ is definably amenable if and only if $G$ admits a global `f-generic' type. This answers a question of Chernikov and Simon and substantially generalizes a theorem of Hrushovski and Pillay. As a quick application of the main result, we show that every dp-minimal group is definably amenable, which answers a question of Chernikov, Pillay, and Simon. Finally, we show that the appropriate analogue of the main result holds also for type-definable groups, so that, in an NIP theory, a type-definable group with a global f-generic type is definable amenable; this additionally gives the first correct proof of the analogous result, claimed by Hrushovski and Pillay, for type-definable groups with a global \textit{strongly} f-generic type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_13470 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On f-generic types in NIP groups Stonestrom, Atticus Logic Recall that a definable group is `definably amenable' if it admits a translation-invariant Keisler measure. We prove a combinatorial characterization of definable amenability for groups definable in NIP theories. More specifically, given a group $G$, a subset $D\subseteq G$ is said to (left) `$G$-divide' if there is some natural number $k$ and an infinite sequence of elements $g_i\in G$ such that $g_{i_1}D\cap\dots\cap g_{i_k}D=\varnothing$ for all $i_1<\dots<i_k$. Our main result is that, if $G$ is a group definable in an NIP theory, and the union of two definable $G$-dividing subsets of $G$ still $G$-divides, then $G$ is definably amenable. It follows that $G$ is definably amenable if and only if $G$ admits a global `f-generic' type. This answers a question of Chernikov and Simon and substantially generalizes a theorem of Hrushovski and Pillay. As a quick application of the main result, we show that every dp-minimal group is definably amenable, which answers a question of Chernikov, Pillay, and Simon. Finally, we show that the appropriate analogue of the main result holds also for type-definable groups, so that, in an NIP theory, a type-definable group with a global f-generic type is definable amenable; this additionally gives the first correct proof of the analogous result, claimed by Hrushovski and Pillay, for type-definable groups with a global \textit{strongly} f-generic type. |
| title | On f-generic types in NIP groups |
| topic | Logic |
| url | https://arxiv.org/abs/2303.13470 |