Many Examples of Non-cocompact Fuchsian Groups Sitting in PSL(2,Q)
  We construct infinitely many noncommensurable non-cocompact Fuchsian groups $\Delta$ of finite covolume sitting in PSL(2,Q) so that the set of hyperbolic fixed points of $\Delta$ will contain a given finite collection of elements in the boundary of the hyperbolic plane. 
Mark Norfleet
math.GT math.GR