Suppose S is a genus g surface with n punctures satisfying the hyperbolicity condition 2g + n - 2 > 0. If n > 0 the fundamental group of the surface is a free group on 2g + n - 1 := m generators.
If we look the universal covers of different punctured surfaces with the same m (e.g., thrice-punctured sphere and once-punctured torus for m = 2) in, say the hyperbolic plane or the Poincare disc model, how do they differ? The "only" apparent difference is in the number of punctures which should give rise to a difference in the lifts of the punctures to the boundary of the disc. The fundamental groups are isomorphic, but they must act differently to produce quotient surfaces of different genera. How?
How does the set of lifts of punctures on the boundary relate to the standard Farey set?
Thanks a lot in advance!