Cannon's conjecture

Every word-hyperbolic group whose Gromov boundary is homeomorphic to the 2-sphere acts geometrically (properly, cocompactly, isometrically) on hyperbolic 3-space.

Cannon's combinatorial Riemann mapping theorem (1994) rebuilt conformal geometry from finite subdivision rules; the conjecture asks whether the passage from boundary combinatorics to interior geometry always works. Bonk–Kleiner proved it under an Ahlfors-regularity hypothesis; Markovic settled the case of surface-by-free groups. The general case — does every hyperbolic group with 2-sphere boundary act geometrically on hyperbolic 3-space? — remains open, a meeting point of group theory, dynamics, and 3-manifold topology.