Generic sets in definably compact groups

Tom 193 / 2007

Ya'acov Peterzil, Anand Pillay Fundamenta Mathematicae 193 (2007), 153-170 MSC: 03C64, 22E15. DOI: 10.4064/fm193-2-4


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.


  • Ya'acov PeterzilDepartment of Mathematics
    University of Haifa
    Haifa, Israel
  • Anand PillayDepartment of Mathematics
    Urbana, IL 61801, U.S.A.
    School of Mathematics
    University of Leeds
    Leeds LS2 9JT, UK

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