Logic Seminar
Kyle Gannon
Beijing International Center for Mathematical Research
Upside down and backwards
Abstract: We investigate the semigroup of invariant types through the lens of Ellis theory; primarily focusing on definably amenable NIP groups. In the definably amenable NIP context, we observe that the collection of right strong $f$-generic types forms the unique minimal left ideal and thus, the Ellis subgroups are isomorphic to the $G/G^{00}$ via the canonical quotient map. As consequence of the Newelski-Pillay conjecture, the Ellis subgroups of the semigroup of invariant types are abstractly isomorphic to the Ellis subgroups of the semigroup of finitely satisfiable types in the definable amenable NIP setting. We are interested in the existence of natural isomorphisms from invariant Ellis subgroups to finitely satisfiable Ellis subgroups and we determine when these isomorphisms can be witnessed by (variants of) the canonical retraction map. We then provide several limiting examples. Outside of the NIP context, we provide an (dfg) abelian group in which the invariant Ellis subgroups and finitely satisfiable Ellis subgroups not isomorphic. This is joint work with Tomasz Rzepecki.
Tuesday February 24, 2026 at 3:00 PM in 636 SEO