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

Normal complements to quasiconvex subgroups.

(With Juan Souto and Peter Storm)

Abstract.

Let $\Gamma$ be a hyperbolic group. The normal topology on $\Gamma$ is defined by taking all cosets of infinite normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every quasiconvex subgroup $\Delta < \Gamma$ is closed in the normal topology. For a uniform lattice $\Gamma < \PSL_2(\C)$ we prove, using the tameness theorem of Agol and Calegary-Gabai, that every finitely generated subgroup of $\Gamma$ is closed in the normal topology.