Profinite rigidity, Kleinian groups, and the cofinite Hopf property

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