I know about geometrical method of construction of discrete subgroups of $SL(2,\mathbb{R})$ using Lobachevsky plane (e.g. B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry --- Methods and Applications, Springer) via fundamental polygon. Such construction has many applications and some relevant themes were already discussed in MO.

I do not know, if my question is appropriate here, but I would like to know rather opposite thing: how to construct explicitly the matrices itself. In book mentioned above it was demonstrated only for simple case of $4g$-polygon with sum of angles $2\pi$. I mean, if there is some analytical equations or an algorithm of calculation of parameters of matrices for discrete subgroups of $SL(2,\mathbb{R})$.