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Math-2205-A1 · revision r1 · on Math-2205

OpenAI claim summary for Math-2205

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Kirby Problem 3.24
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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