On f-generic types in NIP groups

TL;DR

The paper characterizes definable amenability for groups definable in NIP theories through a combinatorial notion of G-dividing and f-genericity, proving that the existence of a global f-generic type is equivalent to definable amenability. The central result shows that in an NIP setting, a group with a global f-generic type is definably amenable, with the equivalence extending to the existence of a G^{00}-invariant global type and the non-f-generic sets forming an ideal. This framework yields immediate corollaries, notably that every dp-minimal group is definably amenable, and the main theorems extend to type-definable groups, offering the first robust proof of the analogous claims for strongly f-generic types in that context. Collectively, the work advances the structural theory of definable groups in NIP theories, resolving open questions and broadening the applicability to dp-minimal and type-definable groups, with the central role played by invariant Keisler measures and G^{00}-invariance.

Abstract

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.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.5
• Proposition 2.8
• proof
• Corollary 2.9
• proof
• Proposition 2.10