Pseudofinite groups and VC-dimension

We develop “local NIP group theory” in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure [Formula: see text] expanding a group, and left invariant NIP formula [Formula: see text], we prove various aspects of “local fsg” for the right-stratified formula [Formula: see text]. This includes a [Formula: see text]-type-definable connected component, uniqueness of the pseudofinite counting measure as a left-invariant measure on [Formula: see text]-formulas and generic compact domination for [Formula: see text]-definable sets.