On f-generic types in NIP groups

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.