Finitely generated subgroups of lattices in PSL₂ℂ
Yair Glasner, Juan Souto, Peter A. Storm · Proceedings of the American Mathematical Society · 2010
Let Γ \Gamma be a lattice in P S L 2 ( C ) \mathrm {PSL}_2 (\mathbb {C}) . The pro-normal topology on Γ \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 Δ > Γ \Delta > \Gamma is closed in the pro-normal topology. As a corollary we deduce that if H H is a maximal subgroup of a lattice in P S L 2 ( C ) \mathrm {PSL}_2( \mathbb {C}) , then either H H is of finite index or H H is not finitely generated.