The following question may be trivial or inappropriate; I am not sure though.

It is known that a cocompact oriented Fuchsian group $\Gamma$ admits a presentation: for given $m,g,d_i \geq 0$ $$ \Gamma := \langle x_1, \ldots, x_m , y_1, \ldots, y_g, z_1, \ldots, z_g | x_1^{d_1}, \ldots ,x_m^{d_m}, x_1 \ldots x_m [y_1,z_1] \ldots [y_g,z_g] \rangle $$ In the case where $m =0$ and $g > 1$ one obtains surface group of genus $g$. The question is whether there are homomorphisms with interesting properties between Fuchsian groups and surface groups, with the same genus $g$?