I will answer the question of whether this also gives the classification cheaply. No. It gives the classification at the expense of proving that every surface group has a free cocompact action on the upper half plane (or on the euclidean plane, or on the 2-sphere). This in turn depends upon proving: - Every surface has a triangulation (Rado's Theorem, and now you've done 90% of the work of the classification theorem)... - Every triangulated surface has a compatible smooth structure... - Every smooth surface has a compatible conformal structure... and, finally - Every conformal structure has a compatible hyperbolic metric, euclidean metric, or spherical metric (the uniformization theorem).