Upside down and backwards

TL;DR

This paper explores the structure of invariant types in definably amenable NIP groups using Ellis theory, establishing isomorphisms between different Ellis subgroups and examining conditions for their naturality, with examples outside NIP.

Contribution

It demonstrates the isomorphism between invariant and finitely satisfiable Ellis subgroups in NIP groups and analyzes when these can be connected via canonical maps, extending understanding of Ellis semigroup structure.

Findings

Invariant Ellis subgroups form the unique minimal left ideal.

Ellis subgroups are isomorphic to $G/G^{00}$ via the canonical quotient.

Outside NIP, invariant and finitely satisfiable Ellis subgroups may not be isomorphic.

Abstract

We investigate the semigroup of invariant types through the lens of Ellis theory; primarily focusing on definably amenable NIP groups. In this context, we observe that the collection of strong right $f$-generic types forms the unique minimal left ideal and thus, the Ellis subgroups are isomorphic to $G/G^{00}$ 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 NIP retraction map. Several limiting examples are provided. Outside of the NIP context, we provide an abelian group (and thus definably amenable) with an $\emptyset$-definable (dfg) type in which the invariant Ellis subgroups and finitely satisfiable Ellis subgroups not isomorphic.