# Finitely generated subgroups of lattices in $̊m PSL_2\Bbb C$ Academic Article

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### abstract

• Abstract: 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.

### publication date

• January 1, 2010