Cannon conjecture

Cannon's conjecture predicts a converse to a basic fact about Kleinian groups. A group acting properly discontinuously, cocompactly and isometrically on $\mathbb H^3$ is Gromov-hyperbolic with boundary at infinity a $2$-sphere; the conjecture asserts that these coarse properties characterize such groups, so that any hyperbolic group $G$ with $\partial G\cong S^2$ admits an action of this kind. The 1994 date reflects Cannon's combinatorial Riemann mapping theorem [Cannon1994CombinatorialRiemann], developed to uniformize the boundary sphere from combinatorial data; the explicit group-theoretic formulation appeared around 1998 in Cannon and Swenson's work on recognizing constant-curvature discrete groups in dimension three [CannonSwenson1998RecognizingConformal].

Cannon and Swenson also gave a criterion for the existence of the desired action in terms of the combinatorics of the boundary [CannonSwenson1998RecognizingConformal], and the conjecture has been established for significant classes of groups, notably cubulated hyperbolic groups and groups with enough codimension-one subgroups [BregmanIncertiMedici2024Cannon].

A resolution would show that hyperbolic three-dimensional geometry is detected by coarse group-theoretic data alone. For a general hyperbolic group with spherical boundary, however, no construction of the required Kleinian action is known, and the conjecture remains open.