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Upside down and backwards

Kyle Gannon, Tomasz Rzepecki

TL;DR

This work analyzes the dynamics of invariant types in definably amenable NIP groups via Ellis theory, proving that strong right $f$-generics form the unique minimal left ideal and that Ellis subgroups are abstractly isomorphic to $G(\\mathcal{U})/G^{00}(\\mathcal{U})$. Leveraging the Newelski-Pillay result, invariant and finitely satisfiable Ellis subgroups are shown to be abstractly isomorphic, while the paper investigates when natural retraction maps $F_{M}$ and $K_{M}$ witness concrete isomorphisms or anti-isomorphisms in various cases: abelian, fsg, and dfg groups. It presents a sequence of explicit theorems and limiting examples, including semidirect products, to delineate when these maps preserve Ellis subgroup structure and when they fail, thereby clarifying the interplay between invariant and finitely satisfiable dynamics. The results illuminate the delicate structure of definably amenable NIP groups and contribute to broader questions about decomposing such groups into fsg and dfg components. Overall, the paper advances understanding of model-theoretic dynamics through concrete descriptions of Ellis semigroups and natural morphisms between them.

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.

Upside down and backwards

TL;DR

This work analyzes the dynamics of invariant types in definably amenable NIP groups via Ellis theory, proving that strong right -generics form the unique minimal left ideal and that Ellis subgroups are abstractly isomorphic to . Leveraging the Newelski-Pillay result, invariant and finitely satisfiable Ellis subgroups are shown to be abstractly isomorphic, while the paper investigates when natural retraction maps and witness concrete isomorphisms or anti-isomorphisms in various cases: abelian, fsg, and dfg groups. It presents a sequence of explicit theorems and limiting examples, including semidirect products, to delineate when these maps preserve Ellis subgroup structure and when they fail, thereby clarifying the interplay between invariant and finitely satisfiable dynamics. The results illuminate the delicate structure of definably amenable NIP groups and contribute to broader questions about decomposing such groups into fsg and dfg components. Overall, the paper advances understanding of model-theoretic dynamics through concrete descriptions of Ellis semigroups and natural morphisms between them.

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 -generic types forms the unique minimal left ideal and thus, the Ellis subgroups are isomorphic to 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 -definable (dfg) type in which the invariant Ellis subgroups and finitely satisfiable Ellis subgroups not isomorphic.

Paper Structure

This paper contains 18 sections, 44 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ellis theory
  4. Model theoretic dynamics
  5. Ellis analysis of definably amenable groups in NIP theories
  6. Left, right and bi f-generics
  7. Retraction and inversion
  8. The retraction map: the basics
  9. When is the retraction an isomorphism?
  10. Abelian NIP groups
  11. NIP fsg groups
  12. NIP dfg groups
  13. Abelian NIP dfg groups
  14. Keisler measures
  15. Limiting Examples
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose that $T$ is NIP. Let $\mathcal{U}$ be a monster model of $T$ and $M$ a small elementary substructure of $\mathcal{U}$. Let $G(x)$ be an $\emptyset$-definable group which is definably amenable. We let $\mathcal{F}_{r}$ denote the collection of types which are strong right $f$-generic over $M$ See Theorem theorem:main for details.

Theorems & Definitions (121)

  • Theorem 1.1
  • Theorem 1.3
  • proof
  • Definition 2.3
  • Definition 2.5
  • Definition 2.6
  • Definition 2.9
  • Definition 2.10
  • Definition 2.12
  • Corollary 2.14
  • ...and 111 more