Groups, measures, and the NIP

By Ehud Hrushovski, Ya'acov Peterzil, Anand Pillay. 2006

238 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/40506ca4050672243d5828484f631acd0d77b8ca

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Citations in the corpus

Stable group theory and approximate subgroups (2009, methodology): “The methods here are very close to [22]; however we do not assume NIP.”
Generically stable and smooth measures in NIP theories (2010, background): “The reader is referred to say [17], [1], [18], [20] as well as [8], [9], for extensive and detailed material around stable theories, NIP theories as well as the adaptation/interpretation of forking to types and measures in NIP theories.”
ONE-DIMENSIONAL ASYMPTOTIC CLASSES OF FINITE STRUCTURES (2008, background): “A proof by induction on k is included in [14], but it also follows, for example, from Lemma 2.8 of [21].”
On NIP and invariant measures (2007, methodology): “We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16].”
On central extensions and definably compact groups in o-minimal structures (2008, background): “This paper is motivated partly by questions coming out of our paper [13], especially whether, for a definably compact group G in an o-minimal structure (say expanding a real closed field), G and G/G00 are elementarily equivalent as groups.”
On theories of random variables (2009, unknown): “…in the metric randomisation are very natural objects, namely regular Borel probability measures on the space of types of the original theory, also referred to nowadays as Keisler measures [Kei87], and which turn out to be particularly useful for the study of dependent theories, e.g., in [HPP08].”
Introduction to theories without the independence property (2008, methodology): “In particular, a superrosy NIP group with finitely satisfiable generics must be abelian-by-finite if it is of U-rank 1, and solvable-by-finite if it is of U-rank 2 [17, 10].”