发表机构
Notre Dame（圣母大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在NIP群中发展分段f-generic理论，证明非分段f-generic可定义集构成理想，还得到NIP群Ellis群大小上界、有限阿基米德秩等相关结果，为解决同构类型问题提供进展。
AI 中文摘要
本文包含多个部分：首先，我们在NIP群中发展了“分段（强）f- generic”理论，其中若一个可定义集的有限多个平移的并是（强）f-generic，则称该可定义集为分段（强）f-generic。我们证明，在NIP群中，非分段（强）f-generic的可定义集构成一个理想。我们希望相应的分段（强）f-generic类型能为任意NIP群中的可定义亲和NIP群的（强）f-generic类型提供替代，在论文其余部分我们给出了若干应用。其中两项应用涉及NIP群的Ellis群：设T为NIP理论，G为可定义群，M为模型，我们的第一项结果证明，G(M)的Ellis群的大小上界为2^{|T|}，与M的选择无关，为解决其同构类型是否与M无关的问题迈出了重要一步；第二项结果受Hrushovski定理启发，证明若T和M可数且T的公式具有一致有界VC-密度，则G(M)的Ellis群具有“有限阿基米德秩”，即其连通分支是profinite-by-Lie。两项结果的关键工具是Chernikov-Gannon-Krupiński与Basso-Zucker的最新结论：Ellis群上的τ-拓扑是豪斯多夫的。最后，我们运用自身技术在无一致有界VC-密度假设的任意NIP理论中得到“局部”结果：对任意“双不变”公式φ(x,y)，群G/G^{00}_φ具有有限阿基米德秩；更精确地，若φ(x,y)的VC-密度至多为δ，则G/G^{00}_φ是维数至多为(4δ)^2的紧李群的逆极限，该结果与Hrushovski的一个问题相关但不同。
英文摘要
This paper has several parts. We begin by developing a theory of `piecewise (strong) f-genericity' in NIP groups, where we call a definable set piecewise (strong) f-generic if some union of finitely many translates of it is (strong) f-generic. We show that, in an NIP group, the definable sets that are not piecewise (strong) f-generic form an ideal. Our hope is that the corresponding piecewise (strong) f-generic types can provide a substitute in arbitrary NIP groups for the (strong) f-generic types of definably amenable NIP groups, and in the rest of the paper we give several applications. Two of the applications deal with the Ellis group of an NIP group. Let $T$ be an NIP theory, $G$ a definable group, and $M$ a model. In our first result we show that the size of the Ellis group of $G(M)$ is bounded above by $2^{|T|}$, independent of the choice of $M$, giving a substantial step towards the question of whether the isomorphism type is independent of $M$. In our second result, inspired by a theorem of Hrushovski, we show that, if $T$ and $M$ are countable and the formulas of $T$ have uniformly bounded VC-codensity, then the Ellis group of $G(M)$ has `finite Archimedean rank', ie its connected component is profinite-by-Lie. A crucial tool for us in both results is the recent result of Chernikov-Gannon-Krupiński and Basso-Zucker that the $τ$-topology on the Ellis group is Hausdorff. Finally, we use our techniques to obtain a `local' result valid in arbitrary NIP theories, without the assumption of uniformly bounded VC-codensity: for any `bi-invariant' formula $ϕ(x,y)$, the group $G/G^{00}_ϕ$ has finite Archimedean rank. More precisely, if the VC-codensity of $ϕ(x,y)$ is at most $δ$, then $G/G^{00}_ϕ$ is an inverse limit of compact Lie groups of dimension at most $(4δ)^2$. This connects to, though is different than, a question of Hrushovski.
Commentssimplified proof of Lemma 3.15. removed section 3.6 and changed NTP2 question. updated discussion at the end of Section 7 // AI declaration: I strongly request that the central new concept I develop here, piecewise strong f-genericity, not be investigated in any capacity with AI, nor used in any paper that includes proofs produced by AI or otherwise involves substantial AI contributions
