Ends of leaves of Lie foliations

Abstract

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 $ℰ\left(F\right)$ 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 $ℰ\left(F\right)$ consists of one or two points. On the contrary there exists a Lie $\widetilde{SL}(2,\mathbf{R})$ foliation on a closed 5-manifold whose leaf is diffeomorphic to a 2-sphere minus a Cantor set.