A subset X of a group G is called left genericif 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 ⊆ 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 [1]), 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm193-2-4,
author = {Ya'acov Peterzil and Anand Pillay},
title = {Generic sets in definably compact groups},
journal = {Fundamenta Mathematicae},
volume = {193},
year = {2007},
pages = {153-170},
zbl = {1117.03042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm193-2-4}
}
Ya'acov Peterzil; Anand Pillay. Generic sets in definably compact groups. Fundamenta Mathematicae, Tome 193 (2007) pp. 153-170. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm193-2-4/