Cannon's Conjecture

Theorem 1.1: for every hyperbolic group $G$ with $\partial G\cong S^2$ there is $\rho:G\to\mathrm{Isom}(\mathbb H^3)$ whose action is proper and cocompact with finite kernel; the image may contain orientation-reversing isometries. For torsion-free $G$ the quotient is a closed, possibly nonorientable, hyperbolic 3-manifold, and the paper records virtual-structure consequences. The route is the Bonk-Kleiner uniformization program: the paper proves a uniform upper bound on the combinatorial 2-modulus of curve families at a fixed small scale on the visual boundary, then applies the Bourdon-Kleiner criterion (Corollary 3.5) to get a quasi-Mobius parametrization by the round sphere.

Theorem 1.1: for every hyperbolic group $G$ with $\partial G\cong S^2$ there is $\rho:G\to\mathrm{Isom}(\mathbb H^3)$ whose action is proper and cocompact with finite kernel; the image may contain orientation-reversing isometries. For torsion-free $G$ the quotient is a closed, possibly nonorientable, hyperbolic 3-manifold, and the paper records virtual-structure consequences. The route is the Bonk-Kleiner uniformization program: the paper proves a uniform upper bound on the combinatorial 2-modulus of curve families at a fixed small scale on the visual boundary, then applies the Bourdon-Kleiner criterion (Corollary 3.5) to get a quasi-Mobius parametrization by the round sphere.