# Explicitly realizing Riemann surfaces as a quotient of the upper-half plane

Let $\Sigma_g$ be a Riemann surface of genus $g \ge 2$. Then it is known that $\Sigma_g$ is (holomorphically) a quotient of the upper-half-plane (or unit disk) by a group $\Gamma$ of hyperbolic isometries. Where can one find a nice explicit description of $\Gamma$?

EDIT: Sorry for being vague. Basically I just want to work out certain computations on Riemann surfaces and so am looking for nice, explicit groups of isometries of the upper half plane (so that I can compute equivariantly on the universal cover instead of on the surface). I don't care about finding the group corresponding to a given holomorphic structure, but am just looking for nice ones to work with.

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Please, clarify what you mean by "explicit description". If you want, say, matrix coefficients if generators in terms of, say equations defining the surface as a projective curve, then it is utterly impossible (unless your surface admits a very large group of symmetries). Comparing to this problem, Shimura-Taniyama conjecture is just a triviality. –  studiosus Oct 11 '13 at 17:57
Here is a reference: "Matrices For Fenchel-Nielsen Coordinates" -- Bernard Maskit (2001) ; [link] www.emis.de/journals/AASF/Vol26/maskit.pdf‎ –  Alan Oct 11 '13 at 19:51
Draw a regular polygon in the upper-half-plane, with the appropriate area, and then take the subgroup generated by the isometries which identify the appropriate sides. That's pretty explicit, but I don't know what exactly you're looking for! –  user641 Oct 11 '13 at 23:33
@studiosus thanks for the comment, i've updated my question –  Eric O. Korman Oct 12 '13 at 1:38
@Alan thanks for the reference-- it looks promising though maybe more advanced then what I'm looking for. –  Eric O. Korman Oct 12 '13 at 1:38