CANNON'S CONJECTURE
Cannon's Conjecture (CC) gives a group-theoretic generalization of the generic case of W. P. Thurston's famous Geometrization Conjecture (GC) for 3-dimensional manifolds recently proved by G. Perelman. CC is motivated by that part of the Geometrization Conjecture that concerns the problem of promoting a metric of variable negative curvature to a metric of constant negative curvature. CC claims that, if an infinite, finitely presented group acts roughly like the fundamental group of a hyperbolic 3-manifold near infinity, then the group can be realized as a group of 2 × 2 matrices with complex entries acting conformally on the 2-dimensional sphere S2 and by rigid motion on non Euclidean hyperbolic 3-dimensional space H3. Perelman's work for manifolds and its recent generalization to orbifolds establishes the result in the case that the group is in fact the fundamental group of a 3-dimensional manifold or orbifold but leaves the general case open.
Here is a precise statement of the conjecture followed by the supporting definitions.
Cannon's Conjecture. Suppose that G is an infinite, finitely presented group whose Cayley graph is Gromov-hyperbolic and whose space at infinity is the 2-sphere S2. Then G is a Kleinian group.