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Groups, measures, and the NIP

Authors: Ehud Hrushovski, Ya’acov Peterzil and Anand Pillay
Journal: J. Amer. Math. Soc. 21 (2008), 563-596
MSC (2000): Primary 03C68, 03C45, 22C05, 28E05
Published electronically: February 2, 2007
MathSciNet review: 2373360
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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 $G$ 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 the new notion of “compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.

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Additional Information

Ehud Hrushovski
Affiliation: Hebrew University of Jerusalem, Department of Mathematics, Jerusalem, Israel

Ya’acov Peterzil
Affiliation: University of Haifa, Department of Mathematics and Computer Science, Haifa, Israel

Anand Pillay
Affiliation: University of Illinois, Department of Mathematics, Altgeld Hall, 1409 W Green Street Urbana, IL 61801, and University of Leeds, School of Mathematics, Leeds, LS2 9JT England
MR Author ID: 139610

Keywords: $o$-minimal, independence property, compact Lie group, Keisler measure.
Received by editor(s): July 16, 2006
Published electronically: February 2, 2007
Additional Notes: The first author was supported by the Israel Science Foundation grant no. 244/03
The last author was supported by NSF grants DMS-0300639 and FRG DMS-0100979, as well as a Marie Curie chair
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.