Groups, measures, and the NIP

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in saturated <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="o"> <mml:semantics> <mml:mi>o</mml:mi> <mml:annotation encoding="application/x-tex">o</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -minimal structures to compact Lie groups. We also prove some other structural results about such <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , for example the existence of a left invariant finitely additive probability measure on definable subsets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We finally introduce the new notion of “compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="o"> <mml:semantics> <mml:mi>o</mml:mi> <mml:annotation encoding="application/x-tex">o</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -minimal case.