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

TL;DR

This paper advances the understanding of idempotent measures and convolution semigroups over definable groups, establishing generic transitivity and classifying measures, thereby extending classical harmonic analysis results to model-theoretic contexts.

Contribution

It introduces the property of generic transitivity, proves its necessity and sufficiency for stable group theory, and classifies idempotent measures in abelian groups within a model-theoretic framework.

Findings

Established generic transitivity for generically stable types in abelian and rosy theories.

Classified idempotent generically stable measures as translation-invariant on type-definable subgroups.

Constructed minimal left ideals in measure semigroups for countable NIP groups, proving the revised Ellis group 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.