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Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture

Artem Chernikov, Kyle Gannon, Krzysztof Krupiński

Abstract

We study idempotent measures and the structure of the convolution semigroups of measures over definable groups. We isolate the property of generic transitivity and demonstrate that it is sufficient (and necessary) to develop stable group theory localizing on a generically stable type, including invariant stratified ranks and connected components. We establish generic transitivity of generically stable idempotent types in important new cases, including abelian groups in arbitrary theories and arbitrary groups in rosy theories, and characterize them as generics of connected type-definable subgroups. Using tools from Keisler's randomization theory, we generalize some of these results from types to generically stable Keisler measures, and classify idempotent generically stable measures in abelian groups as (unique) translation-invariant measures on type-definable fsg subgroups. This provides a partial definable counterpart to the classical work of Rudin, Cohen and Pym for locally compact topological groups. Finally, we provide an explicit construction of a minimal left ideal in the convolution semigroup of measures for an arbitrary countable NIP group, from a minimal left ideal in the corresponding semigroup on types and a canonical measure constructed on its ideal subgroup. In order to achieve it, we in particular prove the revised Ellis group conjecture of Newelski for countable NIP groups.

Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture

Abstract

We study idempotent measures and the structure of the convolution semigroups of measures over definable groups. We isolate the property of generic transitivity and demonstrate that it is sufficient (and necessary) to develop stable group theory localizing on a generically stable type, including invariant stratified ranks and connected components. We establish generic transitivity of generically stable idempotent types in important new cases, including abelian groups in arbitrary theories and arbitrary groups in rosy theories, and characterize them as generics of connected type-definable subgroups. Using tools from Keisler's randomization theory, we generalize some of these results from types to generically stable Keisler measures, and classify idempotent generically stable measures in abelian groups as (unique) translation-invariant measures on type-definable fsg subgroups. This provides a partial definable counterpart to the classical work of Rudin, Cohen and Pym for locally compact topological groups. Finally, we provide an explicit construction of a minimal left ideal in the convolution semigroup of measures for an arbitrary countable NIP group, from a minimal left ideal in the corresponding semigroup on types and a canonical measure constructed on its ideal subgroup. In order to achieve it, we in particular prove the revised Ellis group conjecture of Newelski for countable NIP groups.
Paper Structure (31 sections, 52 theorems, 81 equations)

This paper contains 31 sections, 52 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Idempotent generically stable types
  3. Generically stable types
  4. Setting
  5. Generically stable groups
  6. Idempotent generically stable types and generic transitivity: main conjecture
  7. Abelian groups
  8. Inp-minimal groups
  9. Stable theories
  10. Stable types in arbitrary theories
  11. Simple theories
  12. Rosy theories
  13. Generic transitivity of $p$ and stratified rank localized on $p$
  14. Stabilizer of a generically transitive type is an intersection of definable groups
  15. Chain conditions for type-definable groups on a generically stable type
  16. ...and 16 more sections

Key Result

Theorem 1.3

Assume $p \in S_{G}(\mathcal{U})$ is generically stable and idempotent, and one of the following holds: Then $p$ is the unique left-/right-invariant type on a type-definable subgroup of $G(\mathcal{U})$ (namely, the left-/right-stabilizer of $p$).

Theorems & Definitions (169)

  • Theorem 1.3
  • Theorem 1.4: Proposition \ref{['prop: type-def grp is intersec of def grps']}
  • Theorem 1.5
  • Theorem 1.6: Theorem \ref{['thm: unif gen stab meas']}
  • Theorem 1.7: Corollary \ref{['cor: min ideal ctbl group']}
  • Theorem 1.8
  • Conjecture 1.9
  • Theorem 1.10: Theorem \ref{['thm: revised New conj']}
  • Remark 2.2
  • Definition 2.4
  • ...and 159 more