First of all, my knowledge on subgroups of $PSL(2,\mathbf{C})$ is not very good, so please bear with me.
Just for context, I have a hyperbolic Riemann surface and I know it's fundamental group is non-abelian. Since fundamental groups correspond to a group of deck transformations on the covering space, which is the unit disk, this group can be interpreted as a subgroup of $PSL(2,\mathbf{C})$. Apparently, from here, we can deduce that this group is non-elementary (i.e has no finite orbits), but I'm stuck on seeing how this this.
Any help would be greatly appreciated.