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.
Best Answer
Yes, the magic words are "The Poincare polygon theorem". For (considerably) more detail, see Fine and Rosenberger "Algebraic generalizations of discrete groups: A path to combinatorial group theory through One-relator products" (section 4, I believe).