Finitely generated subgroups of lattices in PSL(2,C)

TL;DR

This paper proves that finitely generated subgroups of lattices in PSL(2,C) are closed in the pro-normal topology, leading to a classification of maximal subgroups as either finite index or infinitely generated.

Contribution

It establishes the closedness of finitely generated subgroups in the pro-normal topology for lattices in PSL(2,C), a new topological property with implications for subgroup classification.

Findings

Finitely generated subgroups are closed in the pro-normal topology.

Maximal subgroups are either finite index or not finitely generated.

The pro-normal topology is finer than the pro-finite topology but not discrete.

Abstract

Let G be a lattice in PSL(2,C). The pro-normal topology on G is defined by taking all cosets of non-trivial 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 H<G is closed in the pro-normal topology. As a corollary we deduce that if M is a maximal subgroup of a lattice in PSL(2,C) then either M is finite index or M is not finitely generated.