Finitely generated subgroups of lattices in $\mathrm{PSL}_2\mathbb{C}$

Let $\Gamma$ be a lattice in $\mathrm{PSL}_2 (\mathbb{C})$. The pro-normal topology on $\Gamma$ is defined by taking all cosets of nontrivial normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every finitely generated subgroup $\Delta < \Gamma$ is closed in the pro-normal topology. As a corollary we deduce that if $H$ is a maximal subgroup of a lattice in $\mathrm{PSL}_2( \mathbb{C})$, then either $H$ is of finite index or $H$ is not finitely generated.