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

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Alex -- what is known about the subgroup for which you would like to compute a system of generators? – algori May 15 '11 at 21:48
algori: It is subgroups, described by geometrical construction with polygon on Lobachevsky plane I mentioned. I.e. there is some visual construction, but how to write it as matrices... – Alex 'qubeat' May 16 '11 at 11:45
Alex -- I still don't get it. What exactly is "some visual information"? There may be several distinct Fuchsian grous with the same fundamental domain: one has to specify how edges are identified. On the other hand, once one has done that, then there are your generators. – algori May 16 '11 at 11:58
@algori: Yes, of course. I mean, there is geometric construction with domain and way to identify edges described eg in book below. I want to know if there is some clear way to construct some example of matrices for that. – Alex 'qubeat' May 16 '11 at 12:38
Alex -- once one has two oriented segments of the same length in the hyperbolic plane (in this example, edges of the fundamental polygon), there is a unique hyperbolic isometry taking one to the other; one can write a matrix explicitly starting from the coordinates of the points. – algori May 16 '11 at 13:13
up vote 4 down vote accepted

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 a link to the copy at Google books.

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As an element of $SL_2\mathbb{R}$ is determined by where it sends three points, via the fact that it preserves cross ratios, this is an elementary, but tedious exercise. The hard part is going to be coordinatizing your $4g$-gon. There is a great passage in Jakob Nielsen's long paper on transformations of surfaces where he gives a ruler and compass construction of the symmetric $4g$-gon. The actual transformations are easy from there. I did it when I was graduate student in about 1980. There were lots of square roots involved. – Charlie Frohman May 16 '11 at 2:44
A truly great book on $SL(2,\mathbb{R})$ from an elementary perspective is Lester Ford's book "Automorphic Functions". Also Wilhelm Magnus book "Discrete Groups" is very elementary and constructive. – Charlie Frohman May 16 '11 at 2:47
Finally, there is a theorem of Poincare that says if you have a convex hyperbolic polyhedron and identifications of the edges so that the angles add up to rational multiples of $\pi$at the identified edges then the group generated by the congruence transformations will be discrete. If I am remembering it correctly. I am not sure if Poincare proved it, but John Millson was really interested in it at some point, and might have given a careful statement and proof. – Charlie Frohman May 16 '11 at 2:51
Thanks for answer and comments. I got the "Indra's pearls" and Ford's book. – Alex 'qubeat' May 16 '11 at 17:38

A fairly detailed construction of arithmetic triangle groups can be found in Takeuchi's paper, Arithmetic triangle groups, J. Math. Soc. Japan Volume 29, Number 1 (1977), 91-106.

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I took the liberty of replacing your link by a permanent one (the direct link copied from the browser's window can cause trouble and is not persistent - usually journals provide a permanent link somewhere on the article's page) and adding some bibliographical information. – Theo Buehler May 16 '11 at 22:11
Despite of finite number of such groups it is really instructive. – Alex 'qubeat' May 23 '11 at 12:26

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