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Math-2205 · mathematics · Topology

Kirby Problem 3.24

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Suppose a word-hyperbolic group has Gromov boundary homeomorphic to $S^2$. Must it act as a cocompact Kleinian group? More generally, if its boundary is planar, must a finite-index subgroup be a convex-cocompact Kleinian group? A Kleinian group is a discrete group of isometries of hyperbolic three-space.

Formulation references: [S1], [S2], [S3].

Short English statement

Suppose a word-hyperbolic group has Gromov boundary homeomorphic to $S^2$. Must it act as a cocompact Kleinian group? More generally, if its boundary is planar, must a finite-index subgroup be a convex-cocompact Kleinian group? A Kleinian group is a discrete group of isometries of hyperbolic three-space.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2205-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2205

The main theorem and its closed-hyperbolic-three-manifold consequences settle the S^2-boundary subcase. The record quantifies over additional boundary types, general hyperbolic groups, dimensions, or independent bundle clauses that this paper does not settle. This entry records partial support; complete coverage of the question is not established by this catalogue.

Kirby Problem 3.24

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. A hyperbolic group with two-sphere boundary is claimed to act geometrically on hyperbolic three-space with finite kernel.

Mathematical statement of the claimed result

Result 1.
Let $G$ be word-hyperbolic with Gromov boundary $\partial G\cong S^2$. The source claims a homomorphism \[ \rho:G\longrightarrow\operatorname{Isom}(\mathbb H^3) \] whose action on hyperbolic three-space is proper and cocompact, and whose kernel is finite. With the additional assumption that $G$ is torsion-free, the corollaries identify $G\cong\pi_1(M)$, where $M$ is closed and has a hyperbolic metric in dimension three, and supply a finite-index subgroup \[ H\cong\pi_1(\Sigma)\rtimes\mathbb Z, \] where $\Sigma$ is a closed connected orientable surface. They also assert residual finiteness, virtual compact specialness, and positive virtual first Betti number in that torsion-free case.

Outline of the argument

The argument applies a boundary modulus criterion. Unbounded modulus produces scalar limits analyzed by exponential records or subexponential probability flows; comparing same-depth expansions forces the required measure control and the three-dimensional hyperbolic action.

Formal verification

For family 246, the documented theorem gives a proper cocompact hyperbolic-three-space action with finite kernel from the two-sphere boundary hypothesis. The listed Comparator files are CannonGeometricAction.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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