Suppose we have a Riemann Surface $S$ of constant negative curvature $-1$. What is the canonical map from the fundamental group $\pi_1(S)$ to the discrete subgroup $\Delta \subset PSL_2(\mathbb{R})$ where $\mathbb{H}^2/\Delta \cong S$?
I'm struggling to see a connection between the two groups.
I think I'm probably missing some important facts here. Any help would be greatly appreciated.
The important fact you're missing is called the uniformization theorem, which says in this case that the hyperbolic plane is the universal cover of $S$.