Logic Seminar: Atticus Stonestrom (Notre Dame)

Historically, it has often only been possible to develop good structure theory for NIP groups either by restricting the class of theories one works in (eg looking at definable groups in o-minimal structures, the p-adics, or ACVFs), or working in arbitrary NIP theories but putting an additional hypothesis on the groups one studies. In the latter case, the most general hypothesis under which a good structure theory has been developed is that of definable amenability. In this talk I will discuss some progress on developing a structure theory for arbitrary NIP groups, without a definable amenability assumption. The first step is to find a substitute for the strong f-generic types of definably amenable NIP groups; calling a definable set piecewise strong f-generic if some union of finitely many translates of it is strong f-generic, my main theorem here is that in any NIP group the non-piecewise-strong f-generic sets form an ideal. I believe the corresponding machinery of piecewise strong f-generic types to be a robust notion for the study of arbitrary NIP groups, and I will give two applications to illustrate this, both dealing with the "Ellis group" of an NIP group. The first application is that, if T is an NIP theory, then the size of the Ellis group of a definable group in any model M of T has size at most 2^|T|, independent of the choice of M; this make major progress on the open question of whether the isomorphism type is independent of the choice of M, a natural generalization to the non-definably-amenable setting of Newelski's conjecture on definably amenable NIP groups. The second application is that, if M is a countable NIP structure that satisfies a certain natural condition of bounded VC-codensity (satisfied in o-minimal structures, the p-adics, and ACVFs), then the Ellis group of any definable group in M has "finite Archimedean rank", meaning that its connected component is profinite-by-Lie; this result is greatly influenced by a theorem of Hrushovski. In both applications a crucial tool is the recent result of Chernikov-Gannon-Krupinski and Basso-Zucker that the tau-topology on the Ellis group of an NIP group is Hausdorff.