Criterion for Cannon's conjecture. (Q2376323)

The Hyperbolization Theorem (proved by Perelman) states that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group is hyperbolic. Cannon devised a two part program towards proving this theorem. The first part was to prove the Weak Hyperbolization Conjecture that states that the fundamental group of a closed, irreducible and atoroidal 3-manifold with infinite fundamental group is hyperbolic and thus by Bestvina and Mess is homeomorphic to the 2-sphere. The second part is to prove Cannon's Conjecture. Combining these two parts would imply that the fundamental group of a closed, irreducible and atoroidal 3-manifold with infinite fundamental group is isomorphic to a torsion free, co-compact lattice in \(\text{PSL}(2,\mathbb C)\), that is, it is isomorphic to the fundamental group of a hyperbolic manifold. The Hyperbolization Theorem would then follow from the result of Gabai et al. which states that 3-manifolds that are homotopy equivalent to hyperbolic 3-manifolds are themselves hyperbolic. Thus proving Cannon's Conjecture would offer a new proof of a major chunk of the Hyperbolization Theorem. The authors in this paper prove the following Criterion for Cannon's Conjecture: a hyperbolic group \(G\) whose boundary is homeomorphic to the 2-sphere is isomorphic to a Kleinian group if and only if every two points in the boundary of \(G\) are separated by a quasi-convex surface subgroup. Thus, Cannon's conjecture is reduced to showing that such a group contains ``enough'' quasi-convex surface subgroups.