Definable convolution and idempotent Keisler measures

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) translationinvariant 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.

An invitation to extension domination

“…Proof. Again, the proofs are similar and so we only proof (1). Consider the definable function f : By transitivity, we conclude that µ ≥ E,Z rν + sη.…”

“…By (3), the space M x (A) can be identified with a subset of V * . The induced topology on M x (A) from V * is the same as the topology described in (1). Moreover, M x (A) is a convex subset of V * .…”

“…Proof. The proofs are similar and so we only prove (1). It is clear that for each p ∈ supp(ν), p is Z-invariant (even Z-definable).…”