Pseudofinite groups and VC-dimension

TL;DR

This paper extends local NIP group theory to pseudofinite groups, establishing measure and definability properties for certain formulas, advancing understanding of their structure and measure-theoretic behavior.

Contribution

It introduces the concept of local fsg in pseudofinite groups with NIP formulas, proving measure uniqueness and definable connected components.

Findings

Existence of a $oldsymbol{ ext{delta}^r}$-type-definable connected component.

Uniqueness of the pseudofinite counting measure as a left-invariant measure.

Establishment of generic compact domination for $oldsymbol{ ext{delta}^r}$-definable sets.

Abstract

We develop local NIP group theory in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure $G$ expanding a group, and left invariant NIP formula $\delta(x;\bar{y})$, we prove various aspects of "local fsg" for the right-stratified formula $\delta^r(x;\bar{y},u):=\delta(x\cdot u;\bar{y})$. This includes a $\delta^r$-type-definable connected component, uniqueness of the pseudofinite counting measure as a left-invariant measure on $\delta^r$-formulas, and generic compact domination for $\delta^r$-definable sets.