TITLE: The fundamental group of a closed surface without classification of surfaces?
QUESTION [16 upvotes]: The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation 
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know is in two steps: 1. draw your favorite presentation of the surface onto a sheet of paper and compute 
the fundamental group using Seifert-van Kampen. 2. Appeal to the classification of surfaces to prove that any surface 
is diffeomorphic to what you drew.
Here is my question: 
''Can one avoid using the classification of surfaces? More specifically, can one prove that a discrete subgroup of $PSL_2 (R) $ that acts freely and cocompactly on the upper half plane must be of the above form - using a group-theoretic argument and without refering to the classification of surfaces or something that comes close to it?''
By ''something that comes close to it'', I mean an argument using Morse theory or another device that decomposes a 
surface into simpler parts.
Background: while contemplating again about the well-known paper by Earle and Eells ('A fibre bundle description of 
Teichmueller theory'), I realized that their main arguments can be upgraded slightly to give at once

closed surfaces are determined up to diffeomorphism by their fundamental groups
each isomorphism of fundamental groups is realized by a diffeomorphism which is unique up to isotopy (Dehn-Nielsen-Baer-Epstein)
the group of diffeomorphisms homotopic to the identity is contractible (the original Earle-Eells result)

and I would like to know whether this also gives the classification cheaply. According to the above two step argument, 
I would be happy with an argument that proves that two surfaces with the same genus (defined by the relation $\chi = 2 -2g$) 
must have isomorphic fundamental groups.

REPLY [8 votes]: 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).