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).Comment: v2: 35 pages. Final version. To appear in the Annals of Mathematics,
Vol. 192, no. 3, November 202