Fields interpretable in superrosy groups with NIP (the non-solvable case)

Abstract

Let G be a group definable in a monster model 𝔠 of a rosy theory satisfying NIP. Assume that G has hereditarily finitely satisfiable generics and 1 < U^þ(G) < ∞. We prove that if G acts definably on a definable set of U^þ-rank 1, then, under some general assumption about this action, there is an infinite field interpretable in 𝔠. We conclude that if G is not solvable-by-finite and it acts faithfully and definably on a definable set of U^þ-rank 1, then there is an infinite field interpretable in 𝔠. As an immediate consequence, we get that if G has a definable subgroup H such that U^þ(G)=U^þ(H)+1 and G/⋂_{g ∈ G}H^g is not solvable-by-finite, then an infinite field interpretable in 𝔠 also exists.