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Groups, measures, and the NIP

2007; American Mathematical Society; Volume: 21; Issue: 02 Linguagem: Inglês

10.1090/s0894-0347-07-00558-9

1088-6834

Ehud Hrushovski, Ya’acov Peterzil, Anand Pillay,

Homotopy and Cohomology in Algebraic Topology

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.