Speaker: Atticus Stonestrom
University of Notre Dame
Will give a Model Theory Seminar entitled:
New results for NIP groups, part 2
Abstract: Let T be a first-order theory and G a definable group of T, and assume for convenience that T has constant symbols naming a model M_0. A definable subset of G is said to be `strong f-generic' if no left translate of it forks over M_0; a fundamental result in the structure theory of NIP groups, due to Hrushovski and Pillay, is that, if T is NIP, then G is definably amenable if and only if the non strong f-generic subsets of G form an ideal. (Meanwhile, by a much earlier result of Pillay, if T is simple then the non strong f-generic subsets of G always form an ideal.) Strong f-generic types play an indispensable role in the study of definably amenable NIP groups. I will present a new notion which I hope to serve as a substitute for strong f-generic sets in the study of arbitrary (ie not definably amenable) NIP groups. Namely, let us call a definable subset of G ‘piecewise strong f-generic’ if some union of finitely many left translates of it is strong f-generic. The main theorem I will present is that, if T is NIP, then the non piecewise strong f-generic subsets of G form an ideal. As a quick consequence I will give a new and improved proof of Gismatullin's result on the ‘existence’ of G000, and next week in the Part II talk I will present two other applications.
Date: 10-06-2026
Time: 10:30 am
Location: 125 Hayes Healy Center