On f-generic types in NIP groups

[2303.13470] On f-generic types in NIP groups Skip to main content Search Submit Donate Log in Mathematics > Logic arXiv:2303.13470 (math) [Submitted on 23 Mar 2023 ( v1 ), last revised 25 Aug 2026 (this version, v5)] Title: On f-generic types in NIP groups Authors: Atticus Stonestrom View PDF HTML (experimental) Abstract: `Definable amenability' of a definable group is the model-theoretic analogue of amenability of a discrete group; precisely, a definable group is said to be definably amenable if it admits a translation-invariant finitely additive probability measure on its definable subsets. 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 non-$G$-dividing (or, equivalently, `f-generic') type. This answers a question of Chernikov and Simon and 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 definably 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. Comments: final version. changed copyright license to prohibit commercial use Subjects: Logic (math.LO) Cite as: arXiv:2303.13470 [math.LO] (or arXiv:2303.13470v5 [math.LO] for this version) https://doi.org/10.48550/arXiv.2303.13470 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Atticus Stonestrom [ view email ] [v1] Thu, 23 Mar 2023 17:42:39 UTC (9 KB) [v2] Mon, 27 Mar 2023 12:49:01 UTC (9 KB) [v3] Thu, 6 Apr 2023 17:35:39 UTC (10 KB) [v4] Wed, 19 Nov 2025 14:47:17 UTC (22 KB) [v5] Tue, 25 Aug 2026 00:00:38 UTC (23 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.LO < prev | next > new | recent | 2023-03 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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