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.

up vote 8 down vote accepted

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 $\{\pm 1\}$.

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