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 $SL(2,\mathbb{R})$.



A construction of generators of discrete subgroups of SL(2,R)
For a pleasant introduction that includes many beautiful pictures, I suggest the book "Indra's Pearls" by Mumford, Series, and Wright. They also give examples of explicit computation of the relevant matrices. [Here's](http://books.google.co.uk/books?id=kC5kdUirHHoC&printsec=frontcover&dq=Indra%2527s+pearls&hl=en&ei=JUrQTbS4L5Dbsgbln7iwCw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDoQ6AEwAA#v=onepage&q&f=false) a link to the copy at Google books.


