The Cannon conjecture asserts that every hyperbolic group whose Gromov boundary is homeomorphic to the two-dimensional sphere admits a geometric group action on three-dimensional hyperbolic space. The action is by isometries and may have a finite kernel. No assumption of torsion-freeness or preservation of orientation is part of this formulation.
Cannon Conjecture
See also
Geometric Group Action, Gromov Boundary, Hyperbolic Group, Hyperbolic SpaceExplore with Wolfram|Alpha
References
Lück, W. "Survey on Geometric Group Theory." Münster J. Math. 1, 73-108, 2008. https://arxiv.org/abs/0806.3771.Cite this as:
Weisstein, Eric W. "Cannon Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CannonConjecture.html