Many Examples of Non-cocompact Fuchsian Groups Sitting in PSL(2,Q)

We construct infinitely many noncommensurable non-cocompact Fuchsian groups $\mathrm{\Delta }$ of finite covolume sitting in PSL(2,Q) so that the set of hyperbolic fixed points of $\mathrm{\Delta }$ will contain a given finite collection of elements in the boundary of the hyperbolic plane.