Cannon's conjecture: word-hyperbolic groups with 2-sphere Gromov boundary are virtually cocompact Kleinian groups

Cannon's conjecture predicts that every word-hyperbolic group whose Gromov boundary is homeomorphic to the 2-sphere is virtually a cocompact Kleinian group; in the torsion-free case, that the group is isomorphic to the fundamental group of a closed hyperbolic 3-manifold. It is the outstanding open piece of the group-level classification of closed 3-manifolds after the Thurston-Perelman geometrization theorem reduced topological classification to geometric pieces.

Background

A word-hyperbolic group, in Gromov's sense, is a finitely generated group whose Cayley graph with the word metric has uniformly thin geodesic triangles; fundamental groups of closed negatively curved manifolds are the basic examples. Such a group G has a Gromov boundary $\partial_\infty G$: the space of equivalence classes of geodesic rays in its Cayley graph, with a canonical topology, well defined up to homeomorphism (indeed up to quasi-symmetry, i.e. up to homeomorphisms that distort relative rather than absolute distances in a bounded fashion) and invariant under quasi-isometries of the group (maps between metric spaces that distort all distances by at most fixed multiplicative and additive constants). A Kleinian group is a discrete subgroup of $\mathrm{PSL}(2,\mathbb{C})$, the orientation-preserving isometry group of hyperbolic 3-space $\mathbb{H}^3$; it is cocompact when the quotient $\mathbb{H}^3/G$ is compact, and torsion-free cocompact Kleinian groups are exactly the fundamental groups of closed hyperbolic 3-manifolds. Since $\partial_\infty\mathbb{H}^3\cong S^2$, the fundamental group of any closed hyperbolic 3-manifold is word-hyperbolic with Gromov boundary $S^2$. The conjecture of J. W. Cannon, emerging from his 1984 study of the combinatorial structure of cocompact discrete hyperbolic groups and articulated in modern recognition form by Cannon-Swenson, asserts the converse.

The motivation is the group-level completion of 3-manifold classification: the geometrization theorem for 3-manifolds (treated in the monograph of Bessières-Besson-Boileau-Maillot-Porti) reduces the topological classification of closed 3-manifolds to their geometric pieces, and a purely group-theoretic recognition of the hyperbolic piece would characterize closed hyperbolic 3-manifold groups among all groups, making the classification accessible from the group alone. Bestvina-Mess showed that a torsion-free word-hyperbolic group with boundary $S^2$ is a 3-dimensional Poincaré duality group — a group whose homology obeys the Poincaré duality of a closed orientable 3-manifold, the precise group-theoretic analogue of the homology behavior of a closed aspherical 3-manifold (an aspherical space being one whose homotopy groups above dimension one vanish) — so, conjecturally combined with geometrization, such a group should be the fundamental group of a closed hyperbolic 3-manifold; the conjecture is exactly the missing step from coarse group data to hyperbolic geometry.

Several precise partial results and equivalent reductions shape current work. A quasiconvex subgroup of a hyperbolic group is a finitely generated subgroup whose inclusion is a quasi-isometric embedding; its limit set is a canonical subset of the boundary. The Ahlfors-regular conformal dimension of the boundary is the infimum of Hausdorff dimensions of Ahlfors-regular metrics (metrics in which balls have volume comparable to a fixed power of their radius) within its quasi-symmetry class. Bonk-Kleiner proved that if the boundary of a hyperbolic group $G$ is quasi-symmetrically equivalent to an Ahlfors $Q$-regular 2-sphere whose Ahlfors-regular conformal dimension equals $Q$ — that is, the conformal dimension is attained by such a metric — then $G$ acts discretely, cocompactly, and isometrically on $\mathbb{H}^3$. Marković proved a criterion: a hyperbolic group acting effectively (with trivial finite kernel) on its boundary, with boundary $S^2$, is Kleinian if and only if every two distinct boundary points are separated by the limit set of a quasiconvex surface subgroup (a subgroup isomorphic to the fundamental group of a closed surface). Cheetham-West-Nolte, building on Song's spherical Plateau problem — a variational problem minimizing the mass of metric-current cycles (cycles in the measure-theoretic theory of metric currents, where mass generalizes volume) representing a fixed group homology class in the quotient of the unit sphere of $\ell^2(\Gamma)$ (the Hilbert space of square-summable real-valued functions on $\Gamma$) by $\Gamma$ — proved that a torsion-free hyperbolic group $\Gamma$ with boundary $S^2$ is the fundamental group of a closed hyperbolic 3-manifold if and only if the solution to the associated spherical Plateau problem for a generator of $H_3(\Gamma,\mathbb{Z})\cong\mathbb{Z}$ is isometric to such a manifold. Ferry-Lück-Weinberger proved the stable version: after taking the product with any closed manifold of dimension at least 2, a simple homotopy equivalence to a closed manifold always exists (unique up to homeomorphism when that manifold is aspherical with fundamental group satisfying the Farrell-Jones conjecture, a standard conjecture in high-dimensional topology controlling a group's assembly maps).

Problem Statement

Cannon's conjecture. Let $\Gamma$ be a word-hyperbolic group whose Gromov boundary $\partial_\infty\Gamma$ is homeomorphic to the 2-sphere $S^2$. Must $\Gamma$ be virtually a cocompact Kleinian group — that is, must some finite-index subgroup of $\Gamma$ act discretely, faithfully, and cocompactly by isometries on hyperbolic 3-space $\mathbb{H}^3$? Equivalently, in the torsion-free case (hyperbolic groups are virtually torsion-free, and passing to a finite-index subgroup does not change the boundary): must a torsion-free word-hyperbolic group with $\partial_\infty\Gamma\cong S^2$ be isomorphic to the fundamental group of a closed hyperbolic 3-manifold? The problem is to prove or refute this statement at full generality, with no restriction on the group beyond word-hyperbolicity and the $S^2$ boundary (the action on the boundary is required to be effective in the general, possibly torsion, form, matching Marković's formulation).

References

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