Absolute profinite rigidity and hyperbolic geometry

We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{Z}[\omega ])$ with ${\omega }^2+\omega +1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).