TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-09-24 (gpt-6-sol)

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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.

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Hyperbolicity is a quasi-isometry invariant of geodesic spaces

Statement

Assume the Axiom of Choice. If two geodesic metric spaces are quasi-isometric and one of them is hyperbolic, then so is the other.

Facts & Assumptions

Given: AC and a quasi-isometry between geodesic metric spaces $X$ and $Y$ .

[F1]

Under AC, a quasi-isometric embedding of geodesic spaces transports slimness from its target to its source, with an explicit bound. A quasi-isometry also has a controlled coarse inverse, so the implication works in both directions (Hyperbolicity is transported by a quasi isometry).

[A1]

AC is used by [F1] for the Morse bound and construction of the controlled inverse (The Axiom of Choice).

Proof

technique · direct

1.1givenF1A1

Let $f : X \to Y$ be the given quasi-isometry. If $Y$ is $δ$ -slim, [F1] first extracts uniform quasi-isometric embedding constants for $f$ and then gives an explicit slimness constant for $X$ . The extraction uses the supplied coarse inverse and both bounded composite errors; the transport uses the two Hausdorff inclusions of Morse stability.

2.1F1A1step 1.1∎

If instead $X$ is hyperbolic, [F1] gives a controlled quasi-isometric inverse $g : Y \to X$ . Applying the same transport assertion to $g$ makes $Y$ hyperbolic. In the empty-space case the quasi-isometry convention forces both spaces empty and the claim is vacuous. Thus hyperbolicity is invariant in both directions.

Depends on

Used by

Dependency tree · two levels

10 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

Clara Löh, Geometric Group Theory, Section 6.2.3 (standard reference, not scraped)
Brian H. Bowditch, A course on geometric group theory, Section 2.2 (standard reference, not scraped)