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 PSL(2, Z[ω]) with ω2 + ω + 1 = 0 is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in PSL(2, C) and the fundamental group of the Weeks manifold (the closed hyperbolic 3–manifold of minimal volume)