Generic sets in definably compact groups

Ya'acov Peterzil ¹   ; Anand Pillay ²

¹ Department of Mathematics University of Haifa Haifa, Israel
² Department of Mathematics UIUC Urbana, IL 61801, U.S.A. and School of Mathematics University of Leeds Leeds LS2 9JT, UK

Fundamenta Mathematicae, Tome 193 (2007) no. 2, pp. 153-170

Résumé

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.

Détail

DOI : 10.4064/fm193-2-4

Keywords: subset group called generic finitely many translates cover main result definably compact group o minimal structure definable subseteq right generic its complement generic among additional results condition equivalent definable compactness existence finitely additive invariant measure definable sets definably compact group where * compact lie group generalizing results nbsp cite iii definably compact group every definable subsemi group subgroup main result uses recent work alf dolich forking o minimal stuctures