Profinite Rigidity, Kleinian Groups, and the Cofinite Hopf Property

Let Γ be a nonelementary Kleinian group and H<Γ finitely generated, proper subgroup. We prove that if has finite covolume, then the profinite completions of H are not isomorphic. If index in Γ, there is onto which maps but does not. These results streamline existing proofs exist full-sized groups profinitely rigid absolute sense. They build on circle ideas can used to distinguish among subgroups other contexts, for example, limit groups. construct new examples groups, including fundamental hyperbolic 3-manifold Vol(3) 4-fold cyclic branched cover figure-eight knot. also lattice PSL(2,C) rigid, so its normalizer PSL(2,C).