Artem ChernikovVery excited about this new preprint, with Kyle Gannon and Krzysztof Krupinski!
"Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture"
https://arxiv.org/abs/2406.00912
Classical work by Wendel, Rudin, Cohen (before inventing forcing) and others classifies idempotent Borel measures on locally compact abelian groups, showing that they are precisely the Haar measures of compact subgroups.
We are interested in a counterpart of this phenomenon in the definable category. In the same way as e.g. algebraic or Lie groups are important in algebraic or differential geometry, the understanding of groups definable in a given first-order structure (or in certain classes of first-order structures) is important for model theory and its applications. The class of stable groups is at the core of model theory, and the corresponding theory was developed in the 1970s-1980s borrowing many ideas from the study of algebraic groups over algebraically closed fields. More recently, many of the ideas of stable group theory were extended to the class of NIP groups, which contains both stable groups and groups definable in o-minimal structures or over the p-adics. This led to multiple applications, e.g. a resolution of Pillay’s conjecture for compact o-minimal groups, or Hrushovski’s work on approximate subgroups. This brought to light the importance of the study of invariant measures on definable subsets of the group, as well as the methods of topological dynamics. In particular, deep connections with tame dynamical systems as studied by Glasner, Megrelishvili and others have emerged.
Definable convolution and idempotent Keisler measures III. Generic stability, generic transitivity, and revised Newelski's conjecture
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.