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Cannon's conjecture
Every word-hyperbolic group with boundary homeomorphic to S admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold.
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A Modulus Proof of Cannon’s Conjecture
Sep 23, 2026·paper.pdf
We prove that every hyperbolic group whose boundary is homeomorphic to the two-sphere admits a proper cocompact isometric action on hyperbolic three-space with finite kernel. This resolves Cannon's conjecture positively.