On NIP and invariant measures

By Ehud Hrushovski, Anand Pillay. 2007

210 citations recorded by the source corpus.

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 [13]. Among key results are (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over bdd(A), (ii) analogous statements for Keisler measures and definable groups, including the fact that G 000 = G 00 for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o-minimal expansions of real closed fields.

Subfields: Artificial Intelligence, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/fef1ae34a111137870919a226f394b0361d8d65d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Computational Theory and Mathematics

Citations in the corpus

Non-Archimedean Tame Topology and Stably Dominated Types (2010, methodology): “sequence (a1,a2) based on pwill otherwise consist of distinct elements, so either a1 <a2 or a1 >a2, neither of which can be an indiscernible set. The statement on unique extensions follows from [18], Proposition 3.2(v). We shall use the following statement, Theorem 12.18 from [14]: Theorem 2.8”
Generically stable and smooth measures in NIP theories (2010, methodology): “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.”
Regularity lemma for distal structures (2015, background): “(4) Any definable, definably compact group G in an o-minimal structure or over the p-adics admits a unique G-invariant generically stable measure [21, 7], which is then smooth by distality and Fact 2.”
Introduction to theories without the independence property (2008, methodology): “So far Keisler measures have not become part of the mainstream, but this is about to change, since they were used recently in the proofs of Pillay’s o-minimal group conjecture and the o-minimal case of the compact domination conjecture [17, 18].”
Stable group theory and approximate subgroups (2009, background): “so a∈ Z−1Za i. In the local case, we say Z⊆ (X−1X)2 is right-generic if ﬁnitely many translates Zb(b∈ X) cover X−1X. Again if Z⊆ X−1Xhas positive I-measure, then Z−1Zis right-generic. Lemma 3.2. [cf. [19]] Let H be an V -deﬁnable subgroup of Ge. Then G/He is bounded iﬀ every deﬁnable set containing ”
Definably amenable NIP groups (2015, methodology): “ strongly f-generic over some small model M. Note that we change the usual terminology: our notion of strongly fgeneric corresponds to what was previously called f-generic in the literature (see e.g. [HP11]). We feel that this change is justiﬁed by the development of the theory presented here. Note ”
On central extensions and definably compact groups in o-minimal structures (2008, background): “As pointed out (with references) at the beginning of section 8 of [17], results of both Shelah and Baisalov-Poizat imply thatM∗ is weakly o-minimal (but not o-minimal).”