Partially ordered groups which act on oriented order trees
  It is well known that a countable group admits a left-invariant total order if and only if it acts faithfully on R by orientation preserving homeomorphisms. Such group actions are special cases of group actions on simply connected 1-manifolds, or equivalently, actions on oriented order trees. We characterize a class of left-invariant partial orders on groups which yield such actions, and show conversely that groups acting on oriented order trees by order preserving homeomorphism admit such partial orders as long as there is an action with a point whose stabilizer is left-orderable. 
Matthew E. Horak (Trinity College), Melanie I. Stein (Trinity College)