  The Cannon Conjecture from the geometric group theory asserts that a word
hyperbolic group that acts effectively on its boundary, and whose boundary is
homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the
following Criterion for Cannon's Conjecture: A hyperbolic group $G$ (that acts
effectively on its boundary) 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, the Cannon's
conjecture is reduced to showing that such a group contains "enough"
quasi-convex surface subgroups.
