Normal Fuchsian subgroup of $PSL(2,\mathbb{R})$

I've been working with Fuchsian groups and from geometrical motivations finding a cocompact normal Fuchsian subgroups of $$PSL(2,\mathbb{R})$$ would have intresting properties for my research.

It is known that $$SL(2,\mathbb{R})$$ has no connected normal subgroups other than the its centre { Id,-Id }, however it might have discrete normal subgroups. I been working with quaternion generated cocompact Fuchsian subgroups which are in addition purely hyperbolic, all its elements have Trace > 2, and I couln't find any normal subgroups of $$SL(2,\mathbb{R})$$ there, I suspect that I have to include some elliptic elements to improve my chances.

To add up it possible for $$PSL(2,\mathbb{R})$$ to have discrete normal subgroups? how about cocompact normal subgroups?

Note: Notice that I'm not talking about normal subgroups of Fuchsian subgroups.

$$PSL_2(\mathbb{R})$$ is simple, so it has no non-trivial proper normal subgroups.