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 $N \lhd \Gamma_g$?

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