Cannon’s conjecture

Imagine the directions in which an infinite network extends. They can form a spherical surface, much as directions around a point in ordinary three-dimensional space do. Does that spherical shape force the network’s symmetry rules to come from motions of hyperbolic three-space, a space with constant negative curvature? Here the network records a group: its vertices are group elements and an edge applies one of a finite list of generators. The group is hyperbolic when each shortest-path triangle stays within one fixed distance of its other two sides. Its ideal boundary records directions of endless shortest paths, rather than vertices at a finite distance.[1] [2]

An infinite group can have a network of shortest paths whose directions at infinity form a sphere. Does that topological condition alone force the group to come from a compact hyperbolic three-dimensional space? A finite generating set gives a Cayley graph; uniform thinness of its geodesic triangles makes $G$ word-hyperbolic. The hypothesis is $\partial_\infty G\cong S^2$, a boundary homeomorphism, not a specified round metric. For example, $(1,0,0)$ and $(0,1,0)$ are points on the coordinate sphere $x^2+y^2+z^2=1$; this sphere pictures the topology required of the boundary, not a construction of the group’s action. The conclusion is a proper cocompact isometric action on $\mathbb H^3$. Equivalently, $1\to F\to G\to\Gamma\to1$, where $F$ is finite and $\Gamma$ is a cocompact lattice in $\operatorname{Isom}(\mathbb H^3)$. The finite kernel consists of group elements acting trivially.[1] [2]

The conclusion is established for groups with an additional cubical geometry: they act properly with compact quotient on a nonpositively curved cube complex. Such groups are called cubulated. The missing step is showing that every hyperbolic group with a spherical ideal boundary has this extra structure. The cited higher-dimensional and relative constructions retain additional assumptions; they do not supply this step for every group in the stated two-sphere case.[1] [2]

For the two-sphere boundary, the published Introduction explains that the conjecture is equivalent to cubulation. Established work obtains sufficiently many surface subgroups from cubulation and conversely cubulates cocompact hyperbolic three-space lattices. The general sphere-boundary hypothesis has not been shown to produce the required cubical action. Results for boundaries of other dimensions may instead produce aspherical manifolds under separation and local-flatness hypotheses; they are not the same hyperbolic three-space conclusion. The branched-cover preprint also retains drilling, filling and relative hypotheses.[1] [2]

A group can be given by generators and rules without specifying any three-dimensional space. Its boundary describes only the arrangement of directions far away. Recovering a compact hyperbolic three-dimensional quotient from that information would connect a description by algebraic rules with an actual geometric space. It would also let questions about such abstract groups be studied using the geometry of hyperbolic manifolds, rather than just their presentations.[1] [2]

The boundary gives a topological description of how the group extends to infinity, whereas a lattice action gives a rigid geometric realization and a compact quotient. Relating them makes the algebra of a group available to three-manifold geometry. For torsion-free groups the quotient is a closed hyperbolic manifold, so loops and their composition acquire that geometric interpretation. The finite-kernel allowance also matters: algebraic information invisible in the action should not be confused with a failure of geometric realization. Higher-dimensional examples in the cited published paper show why the dimension and the exact geometric conclusion cannot be discarded.[1] [2]