Let $G$ be a simply connected Lie group and consider a Lie $G$ foliation $\mathscr F$ on a closed manifold $M$ whose leaves are all dense in $M$ . Then the space of ends ${\mathscr E}(F)$ of a leaf $F$ of $\mathscr F$ is shown to be either a singleton, a two points set, or a Cantor set. Further if $G$ is solvable, or if $G$ has no cocompact discrete normal subgroup and $\mathscr F$ admits a transverse Riemannian foliation of the complementary dimension, then ${\mathscr E}(F)$ consists of one or two points. On the contrary there exists a Lie $\widetilde{SL}(2,\bm{R})$ foliation on a closed 5-manifold whose leaf is diffeomorphic to a 2-sphere minus a Cantor set.
Publié le : 2005-07-14
Classification:
foliations,
Lie G foliations,
Riemannian foliations,
leaf,
space of ends,
developing map,
holonomy homomorphism,
57R30,
57D30,
37C85
@article{1158241934,
author = {HECTOR, Gilbert and MATSUMOTO, Shigenori and MEIGNIEZ, Ga\"el},
title = {Ends of leaves of Lie foliations},
journal = {J. Math. Soc. Japan},
volume = {57},
number = {4},
year = {2005},
pages = { 753-779},
language = {en},
url = {http://dml.mathdoc.fr/item/1158241934}
}
HECTOR, Gilbert; MATSUMOTO, Shigenori; MEIGNIEZ, Gaël. Ends of leaves of Lie foliations. J. Math. Soc. Japan, Tome 57 (2005) no. 4, pp. 753-779. http://gdmltest.u-ga.fr/item/1158241934/