Real Algebraic and Analytic Geometry
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143. Ya'acov Peterzil, Anand Pillay:
Generic sets in definably compact groups.

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Submission: 2004, October 30.

Abstract:
A subset $X$ of a group $G$ is called {\em left-generic} if finitely many left-translates of $X$ cover $G$. Our main result is that if $G$ is a definably compact group in an o-minimal structure and a definable $X\sub G$ is not right-generic then its complement is left-generic.
Among our additional results are (i) a new equivalent condition to definable compactness, (ii) the existence of a finitely additive invariant measure on definable sets in a definably compact group $G$ in the case where $G$ = $^{*}H$ for some compact Lie group $H$ (generalizing results from \cite{BO2}), and (iii) in a definably compact group every definable sub-semi-group is a subgroup.
Our main result uses recent work of Alf Dolich on forking in o-minimal stuctures.

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