JournalsjemsVol. 13, No. 4pp. 1005–1061

On NIP and invariant measures

  • Anand Pillay

    University of Leeds, UK
  • Ehud Hrushovski

    Hebrew University, Jerusalem, Israel
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Abstract

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIPNIP (not the independence property), continuing aspects of the paper \cite{NIP}. Among key results are (i) if p=tp(b/A)p = tp(b/A) does not fork over AA then the Lascar strong type of bb over AA coincides with the compact strong type of bb over AA and any global nonforking extension of pp is Borel definable over bdd(A)bdd(A), (ii) analogous statements for Keisler measures and definable groups, including the fact that G000=G00G^{000} = G^{00} for GG 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 oo-minimal expansions of real closed fields.

Cite this article

Anand Pillay, Ehud Hrushovski, On NIP and invariant measures. J. Eur. Math. Soc. 13 (2011), no. 4, pp. 1005–1061

DOI 10.4171/JEMS/274