TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pendingverified 2026-10-22 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Mostow–Prasad rigidity and knot volume

Statement

Assume the Axiom of Choice. Let $M$ and $N$ be interiors of compact connected oriented knot exteriors carrying complete finite-volume hyperbolic metrics. Every isomorphism $π_{1} (M) \to π_{1} (N)$ , considered up to the conjugacy induced by basepoint changes, is realized by a unique isometry in the corresponding proper homotopy class. If the isomorphism is induced by an orientation-preserving proper homotopy equivalence, the isometry preserves orientation. In particular, the two metrics have equal volume, where volume is the integral of their Riemannian volume densities.

Statement references

Gopal Prasad, Strong Rigidity of Q-Rank 1 Lattices, Theorem C

Danny Calegari, 3-Manifolds, §3.1.5

William P. Thurston, The Geometry and Topology of Three-Manifolds, Chapter 6

Facts & Assumptions

Given: AC, the two connected oriented knot-exterior interiors with their complete finite-volume hyperbolic metrics, and the stated outer fundamental-group isomorphism.

[A1]

Full choice is The Axiom of Choice and supplies the countable-choice hypothesis in The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

Mostow–Prasad rigidity for complete finite-volume hyperbolic three-manifolds proves that every isomorphism between the fundamental groups of connected oriented complete finite-volume curvature- $- 1$ three-manifolds is induced by a unique isometry, properly homotopic to every proper homotopy equivalence inducing the same outer isomorphism. It proves orientation agreement and equality of volume integrals. Its quotient presentation and finite torus-cusp decomposition are proved consequences of its metric hypotheses, not extra assumptions.

Proof

1.1A1F1givenalgebra

The two given knot-exterior interiors are nonempty connected oriented boundaryless three-manifolds. Their metrics are complete, have sectional curvature $- 1$ and have finite volume by hypothesis. Thus every geometric hypothesis of [F1] holds. No generic hyperbolization, fibering, triangulation or deformation theorem is needed, since the metrics are already supplied.

2.1F1step 1.1algebra∎

Apply [F1] to the supplied group isomorphism. It gives its unique realizing isometry and the asserted proper homotopy classification. In the stated orientation-preserving proper homotopy class that isometry preserves orientation, and its Riemannian volume integral agrees on the two manifolds. These are exactly the claimed conclusions.

Depends on

Used by

Worked case: the figure-eight complement Example

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gopal Prasad, Strong Rigidity of Q-Rank 1 Lattices (standard reference, not scraped)
Danny Calegari, 3-Manifolds (standard reference, not scraped)
William P. Thurston, The Geometry and Topology of Three-Manifolds (standard reference, not scraped)