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

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