Definable convolution and idempotent Keisler measures II

TL;DR

This paper investigates convolution semigroups of Keisler measures in NIP groups, revealing structural properties, explicit constructions, and classifications of invariant measures, including cases beyond definable amenability.

Contribution

It provides new insights into the structure of convolution semigroups of Keisler measures, including explicit constructions and classifications in non-definably amenable groups.

Findings

Ellis subgroups are trivial in this context

Minimal left ideals form a Bauer simplex in definably amenable groups

Explicit construction of minimal left ideals in non-amenable cases like SL(2,R)

Abstract

We study convolution semigroups of invariant/finitely satisfiable Keisler measures in NIP groups. We show that the ideal (Ellis) subgroups are always trivial and describe minimal left ideals in the definably amenable case, demonstrating that they always form a Bauer simplex. Under some assumptions, we give an explicit construction of a minimal left ideal in the semigroup of measures from a minimal left ideal in the corresponding semigroup of types (this includes the case of SL$_{2}(\mathbb{R})$, which is not definably amenable). We also show that the canonical push-forward map is a homomorphism from definable convolution on $\mathcal{G}$ to classical convolution on the compact group $\mathcal{G}/\mathcal{G}^{00}$, and use it to classify $\mathcal{G}^{00}$-invariant idempotent measures.