On the other hand "P" is a covering map, so by the monodromy theorem "P" is one-one and the translates of Δ tessellate the upper half plane.
On the other hand "P" is a local homeomorphism, so a covering map. Since the upper half plane is simply connected, it follows that "P" is one-one and hence the translates of Δ tessellate the upper half plane.