What is the group $\Gamma$ such that $\mathbb{H}/\Gamma$ is a genus-n torus

Question

We know that the universal cover of genus-n torus is a unit disk ($n\ge2$), which is conformal to upper half plane $\mathbb{H}$, with automorphism group $SL(2,\mathbb{R})$. Thus the genus-n torus can be identified with $\mathbb{H}/\Gamma$. with $\Gamma$ isomorphic to the fundamental group of genus-n torus. I want to know the exact form how $\Gamma$ embedded in $SL(2,\mathbb{R})$.

Answer 1

A very explicit (and completely useless in my opinion) computation of 2-by-2 matrices for a genus $g$ surface could be found on pages 183-184 of

Dubrovin, Novikov, Fomenko, "Modern Geometry: Methods and Applications: The Geometry and Topology of Manifolds. Part 2", Springer-Verlag, Graduate Texts in Mathematics, Volume 93.

Answer 2

First, the group of orientation-preserving isometries of the upperhalf plane is not $SL(2,R)$ but rather $PSL(2,R)$. Second, this surface is not called a genus-$n$ torus but rather a genus-$n$ surface; the term torus is generally reserved for the genus-$1$ case.

So what you are asking for is an explicit Fuchsian group of a closed surface of genus $n$. These are not that easy to exhibit explicitly. One approach is to use congruence subgroups using suitable quaternion algebras, as developed here.

Sometimes it is best to "construct" these groups geometrically using fundamental domains in the upperhalf plane. This shows the "existence" of the groups without exhibiting them explicitly. I also recommend the book by S. Katok on Fuchsian groups.