explicit linear representations of fundamental groups of surfaces I am looking for an explicit representation of the fundamental group of a closed orientable surface of genus >1. I guess they should be abundant in degree 2. Did anyone see the explicit matrix construction of such a representation? Are there any integral ones? Maybe in higher degrees?
 A: Yes, this has been done.
MR1292919 (96f:30045) 
Maskit, Bernard(1-SUNYS)
Explicit matrices for Fuchsian groups. (English summary) The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), 451–466, 
Contemp. Math., 169, Amer. Math. Soc., Providence, RI, 1994. 
You can find the paper with Google Books, if you don't have easy access to the library.
A: Here are the examples from Narasimhan and Seshadri that I mentioned in my comment above.  (I could have the reference wrong; I'm actually taking this out of notes from a talk I gave many years ago.)
These examples are really not that exciting, but at least they give irreducible unitary representations of each degree.  Writing the generators of $\pi_1 M^g$ as $a_i$ and $b_i$ ($i=1, \ldots, g$) we define a representation $\rho$ by sending $a_1$ to the diagonal matrix
$$A_1 = \left[\begin{array}{rrr} z_1 \\ & \ddots\\ && z_n \end{array}\right],$$ 
where the $z_i$ are distinct, and sending $a_2$ to the permutation matrix 
$$A_2 = \left[\begin{array}{rrrrrr} 0 & 0 &  \cdots & 0 &1 \\ 1 & 0  & \cdots && 0 \\
0 & \ddots & 0 & \cdots & 0\\
 0 & \cdots &   1&      0 & 0\\
  0 & \cdots &  0 &      1 & 0\end{array}\right].$$  All the other generators are sent to the identity matrix.  Since the $b_i$ all go to the identity, all the commutators $[\rho(a_i), \rho(b_i)]$ are trivial and we have a representation.  It's irreducible because the invariant subspaces of $A_1$ are just sums of eigenspaces, and these eigenspaces are permuted transitively by $A_2$.  
One can vary this example in a number of ways to get other interesting examples.
