  We define a complete Riemannian manifold X to be large-scale conformally
rigid if all groups that are quasi-isometric to some complete Riemannian
manifold of bounded geometry conformal to X are quasi-isometric to X. We prove
that many 3-manifolds, including Euclidean 3-space, hyperbolic 3-space and the
product of the hyperbolic plane with the real line are large-scale conformally
rigid. This implies new characterizations of groups that can act properly,
cocompactly by isometries on those manifolds.
