Group actions on simply connected 1-manifolds and nonexistence of foliations

I will discuss work with Rachel Roberts and Melanie Stein.  If a 3-manifold M admits a Reebless foliation (a nice decomposition into surfaces) then the fundamental group of M acts without a global fixed point on a (not necessarily Hausdorff) simply connected 1-manifold, which is a tree-like object.  We showed that infinitely many hyperbolic 3-manifolds have fundamental groups that admit no such action, hence these manifolds do not admit a Reebless foliation.  I will aim to explain all of this to mathematicians not familiar with 3-manifold theory, concentrating on group actions.