Cannon's conjecture

The explanation

Draw a square and cut it into a 10 by 10 grid of small cells. A path from the left edge to the right edge must cross every column. So it passes through at least 10 cells. Give every cell a weight of 0.1. Then every such path collects a total weight of at least 1. Now add up the squares of all the weights. That is 100 cells times 0.01, which is 1. Cut finer, into 1,000 by 1,000 cells, each weighing 0.001. The sum of squares is still 1. The cost of blocking every crossing does not grow as the cells shrink. A ceiling of exactly this kind is the main thing this paper has to prove, in a much stranger setting.

That setting is a group, a collection of moves that can be combined and undone. Pick a finite set of basic moves. Draw a dot for each element and a line for each basic move. The result is a network called a Cayley graph. A group is hyperbolic when every triangle of shortest paths in this network is uniformly thin. Thin means each side stays within a fixed distance of the other two. Follow shortest paths outward forever, and you reach the group's boundary. The boundary is the group's view at infinity.

In 1998, Cannon and Swenson stated the conjecture now named after Cannon. Suppose a hyperbolic group has a boundary that is a sphere, up to stretching without tearing. Then the group should act as symmetries of hyperbolic three-space, a standard curved three-dimensional space. The action should also be geometric. Proper means each bounded region overlaps its own copies under only finitely many elements. Cocompact means copies of one bounded region cover the whole space. The question links the shape of a group at infinity to three-dimensional geometry.

Much of the groundwork was in place. Cannon's 1994 combinatorial Riemann mapping theorem recovered coordinates on a sphere from discrete counts like the grid example. In 1999, Cannon, Floyd and Parry reduced the problem to bounds on such counts at small scales. Bonk and Kleiner proved the conjecture in 2005 under an extra assumption. That assumption is that a best possible metric on the boundary exists. Markovic gave a criterion in 2013 using enough surfaces inside the group. A 2026 preprint by Groves and five coauthors showed that a related conjecture would imply a weaker version. Each of these routes needed an extra hypothesis.

The paper follows the analytic routeSource 1. Cover the boundary with small balls at a scale indexed by n. Give each ball a weight, so that every path of a fixed size collects at least 1. Call the cheapest total of squared weights M(n)Source 2. This cheapest total is a kind of modulus. A known criterion of Bourdon and Kleiner makes it enough to bound M(n)Source 3. If M(n) stays below one ceiling for every n, the boundary matches the round sphereSource 4. The obstacle is getting that ceiling. The paper must get it from two facts alone: thin triangles, and the sphere shape of the boundary.

The argument runs by contradiction. Suppose the costs M(n) grow without limit. From the cheapest weights, the paper builds a continuous function on part of the boundary that is not constantSource 5. The group's own symmetries then restrict how this function can vary. The proof splits by how fast the costs growSource 6. If they grow exponentially, magnifying the function at finer and finer scales leads to a contradictionSource 7. If they grow more slowly, the paper spreads the function's variation over many small balls. Comparing these in pairs demands more weight on small balls than their overlap allowsSource 8. Both cases fail, so the costs stay boundedSource 9.

With the bound in hand, known methods finish the job. The boundary gets coordinates matching the round sphere. Methods of Sullivan and Tukia then straighten the group's action into symmetries of hyperbolic three-spaceSource 10. The theorem allows a finite kernel, a finite set of elements that move nothingSource 11. The symmetries may also include mirror flips, which reverse orientation.

A group is torsion-free when no element except the identity returns to the start after finitely many repeats. For such groups, the paper claims more. It claims the group is the fundamental group of a closed hyperbolic three-manifoldSource 12. A three-manifold looks like ordinary space near each point, and its fundamental group records its loops. That manifold may be one-sided in a mirror sense, called nonorientable. Using work of Agol and of Bergeron and Wise, the paper also claims such groups are residually finiteSource 13. That means each element except the identity still acts as a real move in some finite version of the group.

How sure is this? A computer proof checker called Lean has checked a formal version of the main theorem. It assumes thin triangles in a Cayley graph and a boundary matching the sphere. It concludes a proper, cocompact action on hyperbolic three-space with finite kernel. The Lean scope note does not mention the three-manifold consequences, so those lie outside the stated check. No specialist review of the paper has been published.

What it doesn't show. This does not give a way to tell whether a group's boundary is a sphere. It starts from groups already known to have one.