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

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Morse stability of quasi-geodesics

Statement

Assume the Axiom of Choice. For every $δ \geq 0$ and every quasi-geodesic constants $λ \geq 1$ , $ε \geq 0$ , there exists $R = R (δ, λ, ε)$ with the following property: if $X$ is a geodesic $δ$ -hyperbolic space and $q_{1}, q_{2}$ are $(λ, ε)$ -quasi-geodesics in $X$ with the same endpoints, then the Hausdorff distance between the images of $q_{1}$ and $q_{2}$ is at most $R = 184 λ^{2} (ε + 3 δ)$ . No properness or continuity of the quasi-geodesics is required.

Facts & Assumptions

Given: AC, a geodesic $δ$ -hyperbolic space $X$ and two $(λ, ε)$ -quasi-geodesics $q_{1}, q_{2}$ with the same endpoints.

[F1]

Under AC, every such quasi-geodesic has Hausdorff distance at most $M = 92 λ^{2} (ε + 3 δ)$ from every specified endpoint geodesic, including both Hausdorff inclusions (Morse stability with explicit parameter dependence).

[A1]

AC is used through [F1] to choose its projection family (The Axiom of Choice).

Proof

technique · direct

1.1givenF1A1

Choose one geodesic $γ$ joining the common endpoints. By [F1], each image $Q_{i} = im (q_{i})$ has Hausdorff distance at most $M$ from $γ$ . This means both that every point of $Q_{i}$ is within distance $M$ of $γ$ and that every point of $γ$ is within distance $M$ of $Q_{i}$ , with infimum distances understood as in [F1].

2.1step 1.1algebra∎

Fix $x \in Q_{1}$ and $h > 0$ . Choose $z \in γ$ with $d (x, z) < M + h$ , then $y \in Q_{2}$ with $d (z, y) < M + h$ . Thus $d (x, Q_{2}) \leq 2 M + 2 h$ ; letting $h$ decrease to zero gives $d (x, Q_{2}) \leq 2 M$ . Reverse the roles of $Q_{1}, Q_{2}$ for the other inclusion. Therefore their Hausdorff distance is at most $2 M = 184 λ^{2} (ε + 3 δ)$ , as claimed.

Depends on

Used by

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Dependency tree · two levels

19 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 2.1 (standard reference, not scraped)