Ends of leaves of Lie foliations

摘要

Let G be a simply connected Lie group and consider a Lie G foliation F on a closed manifold M whose leaves are all dense in M. Then the space of ends E(F) of a leaf F of 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 F admits a transverse Riemannian foliation of the complementary dimension, then E(F) consists of one or two points. On the contrary there exists a Lie SL(2, R) foliation on a closed 5-manifold whose leaf is diffeomorphic to a 2-sphere minus a Cantor set.