Groups, measures, and the NIP

TL;DR

This paper explores measures, invariant measures, and genericity in definable groups within NIP structures, proving conjectures relating definably compact groups to Lie groups and introducing the concept of compact domination.

Contribution

It completes the proof of conjectures linking definably compact groups in o-minimal structures to compact Lie groups and introduces the new notion of compact domination.

Findings

Existence of a left invariant finitely additive probability measure on definable subsets of G

Proof of conjectures relating definably compact groups to compact Lie groups

Introduction of the concept of compact domination and related conjectures

Abstract

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce a new notion "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.