Let $\Gamma\_g$ be a surface group of genus $g \geq 2$. That is, there is a presentation $$\Gamma\_g = \langle x\_1, y\_1, \dots, x\_g, y\_g \vert \prod\_{i = 1}^{g}[x\_i,y\_i] = 1\rangle$$


Is there a nontrivial, finitely generated $N \lhd \Gamma\_g$ of infinite index?


More generally, Can there be a finitely generated $K \leq \Gamma\_g$ of infinite index, containing a nontrivial $N \lhd \Gamma\_g$?



A Karrass-Solitar theorem for surface groups
The answer to both questions is 'no'. This was proved by Greenberg for Fuchsian groups. One outline of the proof is as follows.


1. Any finitely generated subgroup $H$ of a surface group $\Gamma$ is quasiconvex. So the map $H\to\Gamma$ induces an injection of Gromov boundaries $\partial H\to\partial\Gamma\cong S^1$.
2. If $K\lhd \Gamma$ is non-trivial then the only closed, $K$-invariant subset of $\partial\Gamma$ is the whole of $\partial\Gamma$.
3. Therefore $\partial H=\partial\Gamma$.
4. Therefore $H$ is of finite index in $\Gamma$.


As you can see, this works for any quasiconvex subgroup of any hyperbolic group. This was proved by Gersten--Short.


