Cannon's Conjecture for hyperbolic groups with 2-sphere boundary

Establish that every hyperbolic group whose Gromov boundary is homeomorphic to the standard 2-sphere admits a discrete, cocompact, isometric action on hyperbolic 3-space.

Background

Cannon's Conjecture is presented as a central conjecture in geometric group theory concerning when a finitely generated group is Kleinian. It predicts that the topology of a hyperbolic group's boundary determines a three-dimensional hyperbolic action: specifically, a 2-sphere boundary should force the group to be a cocompact Kleinian group.

The paper states this conjecture as background and does not resolve it. Its main results instead establish a conformal-dimension-attainment criterion related to a relative analogue for relatively hyperbolic groups.

References

One of the central conjectures in this direction is Cannon's Conjecture, which predicts that a hyperbolic group with $2$-sphere boundary is a cocompact Kleinian Group.

Let $G$ be a hyperbolic group whose Gromov boundary $\partial_\infty G$ is homeomorphic to the standard $2$-sphere $\mathbb{S}2$. Then $G$ admits a discrete, cocompact, isometric action on hyperbolic $3$-space $\mathbb{H}3$

— On 2-Sphere Bowditch Boundaries Attaining Conformal Dimension  (2608.28346 - Pal et al., 28 Aug 2026) in Section 1, Introduction