Geometric group theory · hyperbolic geometry · low-dimensional topology

Cannon Conjecture

Collaboration beta

Does every word-hyperbolic group with a two-sphere boundary come from a geometric symmetry group of hyperbolic three-space?

\partial G ≅ S^{2} \Rightarrow G ↷ H^{3} geometrically

Known results and sources

A hyperbolic group may have a boundary homeomorphic to a sphere; Cannon asks whether that topological boundary always comes from a geometric action on hyperbolic three-space.

Research problem

Exact mathematical statement

Let $G$ be a word-hyperbolic group whose Gromov boundary is homeomorphic to the two-sphere:

\partial G \cong S^2.

The Cannon conjecture says that $G$ admits a geometric action on hyperbolic three-space $\mathbb H^3$ : an isometric action that is proper and cocompact, with a finite kernel allowed. The full conjecture remains open. The retained source studies a boundary metric-measure route and exact finite reductions; it does not supply a proof or formal verification.

Problem infographic

Problem at a glance

Problem-first Cannon conjecture infographic showing a hyperbolic group boundary homeomorphic to a two-sphere, the open Ahlfors two-regular and quasisymmetric bridge, the proper cocompact action on hyperbolic three-space, and three known criteria or special cases. — Known criteria reach hyperbolic three-space once enough quantitative conformal geometry or separating surface subgroups are available; obtaining that structure from every topological two-sphere boundary remains open.

Current mathematical picture

Where work on Cannon Conjecture stands

Open conjecture

Revision 4 makes the exact fixed-doubling ambient-net optimizer the canonical finite construction, so Revision 2's mass-selection, mesh-comparison, compactness, generic separator-equivariance, and metric-proof work orders are retained only as source-reported closed or superseded history. The exact unweighted common-annulus tail-swap statement survives, but weighted certificate preservation remains open. PUL-A common-annulus selection and PUL-B weighted uncrossing are the current load-bearing subgates, followed by total collapse or nested-separator realization; no proof of Cannon, independent proof audit, or formal verification is claimed.

Strongest supported footholdExact finite and local reductions

The current work supplies a finite maximin identity and an exact local-to-global volume-edge lemma, while separating them from the unresolved multiscale construction.

Evidence posture · Reported result

Leading routeMeasured lamination after weighted uncrossing

Start from the exact fixed-doubling ambient-net optimizer and localized escape dual; prove PUL-A common-annulus selection and PUL-B certificate-preserving weighted uncrossing, then feed the measured lamination into total collapse or nested-separator realization.

Route status · Active route

Useful failureRevision 2 finite-design route (superseded)

Historical route: choose a canonical multiscale graph mass and bridge mesh escape to exact volume-edge escape. Revision 4 eliminates this as the active design question by specifying the exact ambient-net optimizer and arbitrary-jump objective.

Route status · Eliminated route

Main reductionFixed-doubling escape mainline

Revision 4 makes uniform exact local square-root ball-volume escape on maximal vanishing-mesh nets with one fixed finite doubling constant the current main reduction; failure is narrowed to the open PUL-A/PUL-B weighted lamination gate.

Evidence posture · Reported reduction

Priority open bridgeClose the current load-bearing frontier

Prove PUL-A common-annulus selection and PUL-B weighted uncrossing with certificate-data preservation, then close either the total-collapse or nested-separator endpoint.

Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Cannon Conjecture in numbers

2.6kretained lines of mathematical investigation1,848 in the current working snapshot

Argument development: 2,131 · 83%
Explored or eliminated routes: 48 · 2%
Computational analysis: 45 · 2%
Open obligations: 132 · 5%
Definitions and setup: 204 · 8%

8selected mapped statements6routes investigated4reported milestones1open questions1contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byJames W. Cannon, Eric L. SwensonProposed1998Status checkedAug 6, 2026

Source recordedAug 7, 2026

Recommended next task

Close the current load-bearing frontier

Prove PUL-A common-annulus selection and PUL-B weighted uncrossing with certificate-data preservation, then close either the total-collapse or nested-separator endpoint.

Suggested move: Write the retained fractions and scale range for one common annulus, then formulate weighted transport or reassignment that preserves every named certificate field with universal loss.

What would count as progress

Select a nonvanishing normalized-weight family in one bounded-geometry annulus.
Transfer probability, demand, energy, shadow, and interaction data through uncrossing.
Complete a terminal total-collapse or nested-separator theorem without imported-status inflation.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeMeasured lamination after weighted uncrossing

Route status · Active route

Explored alternatives

Other routes

5 recorded

Eliminated routeRevision 2 finite-design route (superseded)

Route status · Eliminated route

Eliminated routeRevision 2 unweighted separator route (superseded)

Historical route: pass directly from a good-measure failure to a planar separator. Revision 4 eliminates that shortcut as an active route because unweighted tail swaps do not preserve the certificate; the replacement is the precise PUL-A/PUL-B weighted mainline.

Route status · Eliminated route

Eliminated routeRevision 2 compactness route (superseded)

Historical route: separately build the weak-limit compactness passage. Revision 4 replaces this work order with its fixed-doubling finite-to-continuum characterization and explicit full-support step, while retaining publication-level snapping and Portmanteau audit points.

Route status · Eliminated route

Browse 2 more explored routes

Eliminated routeRevision 2 metric-proof route (source-reported closed)

Historical proof route: fill the metric-doubling seams left in Revision 2. Revision 4 reports a complete INTERNAL–EXACT proof and retains priority audit as evidence review rather than a current mathematical construction frontier.

Route status · Eliminated route

Narrowed routeFixed-measure separator hierarchy

Structural fallback: use invariant fixed-measure zero classes and their stabilizers to organize separator and hierarchy information, without treating collapse as an obstruction to Cannon.

Route status · Narrowed route

Route statements and reductions

Statements the next route can inspect and build on

Route statementFixed-doubling ambient-net mainline

The strongest retained route characterizes the target through a uniform exact escape bound on maximal vanishing-mesh nets, with finite convex/minimax structure controlling its failure.

Source-reported route statement · dependencies incomplete

Route statementFinite-to-continuum escape criterion

The current work supplies a fixed-doubling finite-to-continuum criterion, now with the weak-limit full-support step ordered before reverse doubling and its remaining audit requirements exposed.

Source-reported route statement · dependencies incomplete

Route statementWeighted tail-swap shortcut withdrawn

Topological tail swapping does not automatically preserve the probability, demand, shadow, energy, or pole-interaction data required by the certificate; weighted preservation is a separate open theorem.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

1 featured task

Close the current load-bearing frontier

Prove PUL-A common-annulus selection and PUL-B weighted uncrossing with certificate-data preservation, then close either the total-collapse or nested-separator endpoint.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 6, 2026

Current statusOpen conjecture

The full Cannon conjecture remains open: it is not known whether every word-hyperbolic group whose Gromov boundary is a two-sphere acts properly discontinuously, cocompactly, and isometrically on hyperbolic three-space, with the finite kernel handled by passing to the effective boundary action. Strong criteria, a Coxeter-group case, a stable theorem, and recent relative reductions do not settle the unrestricted statement.

[1][6][7]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2024
PreprintGroves, Haissinsky, Manning, Osajda, Sisto, and Walsh developed drilling for hyperbolic groups and reduced the residually finite Cannon conjecture, under the paper's hypotheses, to a relative version. The…[7]
2019
Peer reviewedFerry, Lueck, and Weinberger proved a stable version for torsion-free groups: after taking a product with any closed manifold of dimension at least two, the relevant classifying-space product is…[5]
2013
Peer reviewedBourdon and Kleiner proved the Cannon conclusion for the special class of hyperbolic Coxeter groups whose boundary is a two-sphere.[4]
2013
Peer reviewedMarkovic proved that, after passing to the effective boundary action, Cannon's conjecture is equivalent to the existence of enough quasiconvex surface subgroups to separate every pair of boundary points.[3]

Mathematical neighborhood

Related results and reusable starting points

Current focusCannon conjecture

Dependency or reductionAhlfors regular conformal-dimension realization

Realizing the Ahlfors regular conformal dimension of the two-sphere boundary gives the quantitative conformal geometry needed to conjugate the boundary action to the round sphere and obtain the hyperbolic-three-space action.

[2][6]

Equivalent formulationenough separating quasiconvex surface subgroups

For the effective boundary action, having quasiconvex surface subgroups whose limit sets separate every pair of boundary points is equivalent to the Kleinian conclusion of Cannon's conjecture.

[3]

Solved special caseCannon conjecture for hyperbolic Coxeter groups

The conjecture is proved when the group is a hyperbolic Coxeter group with two-sphere boundary; this does not cover arbitrary word-hyperbolic groups.

[4]

Equivalent formulationtorsion-free closed hyperbolic three-manifold formulation

In the torsion-free case, the geometric action conclusion is equivalent to saying that the group is the fundamental group of a closed hyperbolic three-manifold.

[3][5]

Weaker or relaxed formstable Cannon theorem

The stable Cannon theorem realizes a product of the classifying space with a closed manifold as a closed manifold up to simple homotopy equivalence. The stabilization deliberately weakens the three-dimensional endpoint.

[5]

Dependency or reductionrelative Cannon conjecture

The drilling construction turns a residually finite instance, under suitable conditions, into a relative hyperbolic pair with two-sphere boundary, reducing that branch to a relative Cannon problem.

[7]

Related problemWall's conjecture for PD(3) groups

For torsion-free hyperbolic groups, a two-sphere boundary is tied to three-dimensional Poincare duality; Wall's conjecture asks more generally whether every PD(3) group is a closed aspherical three-manifold group.

[5][6]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA proof-assistant treatment of finitely generated word-hyperbolic groups, Cayley graphs, Gromov boundaries, visual metrics, and the statement that a boundary is homeomorphic to the two-sphere.
Formalization targetFormal definitions of proper and cocompact isometric group actions on hyperbolic three-space, together with the finite-kernel/effective-action reduction.
Formalization targetSubstantial formal libraries for quasisymmetric and quasi-Mobius maps, Ahlfors regularity, conformal dimension, metric-sphere uniformization, and convergence-group rigidity.
Formalization targetFor the subgroup route, formal infrastructure for quasiconvex codimension-one surface subgroups, limit sets, separation of boundary points, cubulation, and the exact hypotheses of Markovic's criterion.

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

Revision 4 supersedes the finite-design frontierRevision 4 fixes the canonical finite optimizer, retires the Revision 2 mass/mesh/compactness frontier, and makes PUL-A common-annulus selection plus PUL-B weighted uncrossing the current load-bearing gate.

Changed the research frontierLater mathematical revision

Research stage 5

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

4 mapped milestonesretained argument map

Browse all 4 mapped stages

stage 1Cannon conjecture and boundary endpoint fixed
stage 2Variable-measure formulation becomes the mainline
stage 3Exact finite and local reductions retained
stage 4First unresolved multiscale gates isolated

Cannon conjecture and boundary endpoint fixedThe source records the exact geometric-action question and its standard sphere-boundary uniformization endpoint while stating that the conjecture remains open.

Mapped research milestoneInitial research sequence

Research stage 1

Variable-measure formulation becomes the mainlineThe current work reframes the endpoint as finding one full-support doubling measure whose square-root volume chain metric has quantitative noncollapse.

Mapped research milestoneInitial research sequence

Research stage 2

Exact finite and local reductions retainedThe source supplies the finite maximin identity and a local-escape-plus-doubling lemma while preserving their exact scopes.

Mapped research milestoneInitial research sequence

Research stage 3

First unresolved multiscale gates isolatedThe revised source separates the macrocurve extremizer from the required multiscale measure and makes metric comparison, compactness, and equivariance explicit open obligations.

Mapped research milestoneInitial research sequence

Research stage 4

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements2 proposed statements4 mathematical milestones1 open questions1 narrowed routes2 conditional results

Statements by mathematical role8 selected mapped statements

theorem candidate1 of 81
lemma2 of 82
equivalence2 of 82
reduction2 of 82
negative result1 of 81

Selected mathematical clusters4 mathematical clusters

Cannon conjecture and metric-measure endpointThe exact open geometric-action question, the conditional variable-measure formulation, and Revision 4's source-reported metric-doubling equivalence, recorded as narrative-only mathematics with priority audit recommended.6 displayed rows

retained route statementCannon conjecture
retained route statementSquare-root volume chain universal propertyintermediate
retained route statementSquare-root metric-doubling equivalenceconditional
retained route statementConditional metric-measure formulation of Cannonconditional
ChallengeRevision 2 left technical seams in reverse doubling, ball-mass inversion, ball inclusions, quasisymmetry, and atom exclusion. Revision 4 supplies a complete source-internal proof and retains only independent checking recommendation; no independent verification is inferred.unsupported step · reported resolved
Research targetRevision 2 metric-equivalence proof gap (source-reported closed)completed reported

Exact ambient-net optimization and escapeThe historical finite maximin identity, the exact local-to-global lemma, Revision 4's ambient-net arbitrary-jump optimizer, and the source-reported finite-to-continuum criterion.11 displayed rows · 2 routes included

retained route statementFinite maximin identityintermediate
retained route statementLocal escape plus doubling implies global metric doublingintermediate
retained route statementRevision 2 finite dichotomy (superseded)
ChallengeRevision 2's adjacent mesh-path surrogate did not control arbitrary-jump volume-edge chains. Revision 4 withdraws the obsolete comparison challenge by replacing the surrogate mainline with the exact ambient-net arbitrary-jump model; it does not claim that the old comparison theorem was proved.unsupported step · withdrawn
ComputationSource-presented finite-dimensional maximin, KKT traffic, and homogeneous two-weight calculations.The current work reports exact algebra for the selected finite curve problem and a model two-weight collapse calculation, while keeping the multiscale and continuum conclusions open. · reported unreproduced
Research targetRevision 2 canonical-mass selection (superseded)superseded
Research targetRevision 2 mesh/volume bridge (superseded)superseded
retained route statementFixed-doubling ambient-net mainlineintermediate
retained route statementFinite-to-continuum escape criterionintermediate
Eliminated routeRevision 2 finite-design route (superseded)Historical route: choose a canonical multiscale graph mass and bridge mesh escape to exact volume-edge escape. Revision 4 eliminates this as the active design question by specifying the exact ambient-net optimizer and arbitrary-jump objective.
Eliminated routeRevision 2 unweighted separator route (superseded)Historical route: pass directly from a good-measure failure to a planar separator. Revision 4 eliminates that shortcut as an active route because unweighted tail swaps do not preserve the certificate; the replacement is the precise PUL-A/PUL-B weighted mainline.

Weighted lamination and terminal frontierThe current load-bearing frontier is PUL-A common-annulus selection plus PUL-B weighted certificate-preserving uncrossing, followed by total collapse or nested-separator realization.4 displayed rows · 1 route included

retained route statementWeighted tail-swap shortcut withdrawnintermediate
Research targetClose the current load-bearing frontieropen
Useful failureAssume extremal finite masses or their limiting separators automatically define a G-invariant decomposition.reported failure
Active routeMeasured lamination after weighted uncrossingStart from the exact fixed-doubling ambient-net optimizer and localized escape dual; prove PUL-A common-annulus selection and PUL-B certificate-preserving weighted uncrossing, then feed the measured lamination into total collapse or nested-separator realization.

Superseded and corrected shortcutsThe retained correction history rules out fixed-measure obstruction, automatic mesh/volume comparison, generic separator equivariance, and unweighted tail-swap preservation of weighted certificate data.11 displayed rows · 5 routes included

Useful failureTreat collapse of the original fixed critical metric as an intrinsic obstruction to Cannon.reported failure
Useful failureIdentify adjacent mesh-path length automatically with the square-root volume-edge chain metric.reported failure
Useful failureAssume extremal finite masses or their limiting separators automatically define a G-invariant decomposition.reported failure
ChallengeRevision 2 left technical seams in reverse doubling, ball-mass inversion, ball inclusions, quasisymmetry, and atom exclusion. Revision 4 supplies a complete source-internal proof and retains only independent checking recommendation; no independent verification is inferred.unsupported step · reported resolved
ChallengeRevision 2's adjacent mesh-path surrogate did not control arbitrary-jump volume-edge chains. Revision 4 withdraws the obsolete comparison challenge by replacing the surrogate mainline with the exact ambient-net arbitrary-jump model; it does not claim that the old comparison theorem was proved.unsupported step · withdrawn
retained route statementWeighted tail-swap shortcut withdrawnintermediate
Narrowed routeFixed-measure separator hierarchyStructural fallback: use invariant fixed-measure zero classes and their stabilizers to organize separator and hierarchy information, without treating collapse as an obstruction to Cannon.
Eliminated routeRevision 2 finite-design route (superseded)Historical route: choose a canonical multiscale graph mass and bridge mesh escape to exact volume-edge escape. Revision 4 eliminates this as the active design question by specifying the exact ambient-net optimizer and arbitrary-jump objective.
Eliminated routeRevision 2 unweighted separator route (superseded)Historical route: pass directly from a good-measure failure to a planar separator. Revision 4 eliminates that shortcut as an active route because unweighted tail swaps do not preserve the certificate; the replacement is the precise PUL-A/PUL-B weighted mainline.
Eliminated routeRevision 2 compactness route (superseded)Historical route: separately build the weak-limit compactness passage. Revision 4 replaces this work order with its fixed-doubling finite-to-continuum characterization and explicit full-support step, while retaining publication-level snapping and Portmanteau audit points.
Eliminated routeRevision 2 metric-proof route (source-reported closed)Historical proof route: fill the metric-doubling seams left in Revision 2. Revision 4 reports a complete INTERNAL–EXACT proof and retains priority audit as evidence review rather than a current mathematical construction frontier.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve PUL-A common-annulus selection and PUL-B weighted uncrossing with certificate-data preservation, then close either the total-collapse or nested-separator endpoint.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Select a nonvanishing normalized-weight family in one bounded-geometry annulus.
Transfer probability, demand, energy, shadow, and interaction data through uncrossing.
Complete a terminal total-collapse or nested-separator theorem without imported-status inflation.

Continue the mathematics

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Prepared starting pointClose the current load-bearing frontier

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Research contextPrepared context for any AI agent

Does every word-hyperbolic group with a two-sphere boundary come from a geometric symmetry group of hyperbolic three-space?

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Sources and references9 cited works · next context review by Nov 6, 2026

The mathematical context was checked on Aug 6, 2026. Status can be refreshed sooner after a material result or claim.

1
Recognizing constant curvature discrete groups in dimension 3original source · James W. Cannon, Eric L. Swenson · Transactions of the American Mathematical Society · 1998 · DOI 10.1090/S0002-9947-98-02107-2 · accessed Aug 6, 2026
2
Conformal dimension and Gromov hyperbolic groups with 2-sphere boundarypeer reviewed result · Mario Bonk, Bruce Kleiner · Geometry & Topology · 2005 · ARXIV math/0208135 · DOI 10.2140/gt.2005.9.219 · accessed Aug 6, 2026
3
Criterion for Cannon's Conjecturepeer reviewed result · Vladimir Markovic · Geometric and Functional Analysis · 2013 · ARXIV 1205.5747 · DOI 10.1007/s00039-013-0228-5 · accessed Aug 6, 2026
4
Combinatorial modulus, the combinatorial Loewner property, and Coxeter groupspeer reviewed result · Marc Bourdon, Bruce Kleiner · Groups, Geometry, and Dynamics · 2013 · ARXIV 1002.1991 · DOI 10.4171/GGD/177 · accessed Aug 6, 2026
5
On the stable Cannon Conjecturepeer reviewed result · Steve Ferry, Wolfgang Lueck, Shmuel Weinberger · Journal of Topology · 2019 · ARXIV 1804.00738 · DOI 10.1112/topo.12099 · accessed Aug 6, 2026
6
Some groups with planar boundariessurvey or monograph · Sang-hyun Kim, Genevieve S. Walsh · Surveys in Differential Geometry · 2022 · ARXIV 1907.06898 · DOI 10.4310/SDG.2020.v25.n1.a7 · accessed Aug 6, 2026
7
Drilling hyperbolic groupspreprint · Daniel Groves, Peter Haissinsky, Jason F. Manning, Damian Osajda, Alessandro Sisto, Genevieve S. Walsh · arXiv · 2024 · ARXIV 2406.14667 · accessed Aug 6, 2026
8
James W. Cannonencyclopedia · Wikipedia contributors · Wikipedia · accessed Aug 6, 2026
9
Gromov boundaryencyclopedia · Wikipedia contributors · Wikipedia · accessed Aug 6, 2026

Important qualifications

The full Cannon conjecture is kept separate from the Bonk--Kleiner and Markovic criteria, the solved Coxeter-group case, the stable theorem, and relative or residually finite reductions; none of those results settles the unrestricted conjecture.
The status search covered the original paper, peer-reviewed milestones, a current survey, the 2024 drilling preprint, and scoped 2024--2026 web and arXiv results. No later authoritative status-changing source was found, but this is not an exhaustive literature review.
The formalization search covered the current Google DeepMind Formal Conjectures repository, Lean/mathlib pages, the Archive of Formal Proofs, and scoped Lean, Coq, and Isabelle web results. Empty formalization entries mean that no Cannon-specific resource was verified in this search, not that none exists.
No external computation, certificate, software package, or dataset was identified that decides the general conjecture. The current work's finite identities and candidate arguments are internal research content, not external reproduced evidence.

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