# Normal Fuchsian subgroups

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 bigger 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 admit discrete normal subgroups? how about cocompact normal subgroups?

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

Let $$\Gamma$$ be a discrete subgroup of a connected Lie group $$G$$. Suppose that $$\Gamma$$ is normal. For given $$\gamma\in\Gamma$$ the conjugacy class is the image of $$G$$ under the continuous map $$x\mapsto x\gamma x^{-1}$$, therefore it is connected. Since it lies in $$\Gamma$$, it consists of one point only, and as $$x=1$$ occurs, this point is $$\gamma$$. This means that $$\Gamma$$ lies in the center of $$G$$. In the case $$G=SL_2({\mathbb R})$$ the center is $$\{ \pm 1\}$$. So $$\Gamma$$ is either trivial or $$\{\pm 1\}$$.