Finitely generated subgroups of lattices in 𝑃𝑆𝐿₂ℂ

Glasner, Yair; Souto, Juan; Storm, Peter

doi:10.1090/S0002-9939-10-10310-4

Finitely generated subgroups of lattices in $\mathrm {PSL}_2\mathbb {C}$
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by Yair Glasner, Juan Souto and Peter Storm PDF

Proc. Amer. Math. Soc. 138 (2010), 2667-2676 Request permission

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.

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