  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 ${\rm{PSL}}(2,\mathbb{C})$.

Group Theory;Geometric Topology