Generic sets in definably compact groups
Volume 193 / 2007
Fundamenta Mathematicae 193 (2007), 153-170
MSC: 03C64, 22E15.
DOI: 10.4064/fm193-2-4
Abstract
A subset $X$ of a group $G$ is called 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 $X\subseteq G$ is not right generic then its complement is left generic.
Among our additional results are (i) a new condition equivalent 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 subsemi-group is a subgroup.
Our main result uses recent work of Alf Dolich on forking in o-minimal stuctures.
