Ends of leaves of Lie foliations

Abstract

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 Fadmits 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.