This is joint work with Pete Storm and Juan Souto.
Let G be the fundamental group of a 3 dimensional hyperbolic manifold of finite volume.
We define an invariant topology on G, by taking the normal subgroups and
their cosets as a basis for the topology.
This is a refinement of the profinite topology.
We prove that every finitely generated subgroup of G is closed in this topology.
As a corollary we deduce that a maximal subgroup of infinite index in G cannot
be finitely generated.