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 ⊆ G is said to (left) ‘ G -divide’ if there is some natural number k and an infinite sequence of elements g i ∈ G such that g i 1 D ∩ … ∩ g i k D = ∅ for all i 1 < … < 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 strongly f-generic type.