TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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Morse stability of quasi-geodesics

Statement

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 $R$ .

Facts & Assumptions

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

[A1]

In a hyperbolic space, every $(λ, ε)$ -quasi-geodesic and every geodesic segment with the same endpoints have bounded Hausdorff distance.

[A2]

If two subsets each lie in the $R$ -neighborhood of the same geodesic segment, then their Hausdorff distance is at most $2 R$ .

[L1]

Thin quadrilaterals provide the local geometric mechanism behind that bound (Hyperbolic spaces have thin geodesic quadrilaterals).

Proof

technique · direct

1.1

Let $γ$ be a geodesic segment joining the common endpoints of $q_{1}$ and $q_{2}$ . By [A1], there is a constant $R_{1} = R_{1} (δ, λ, ε)$ such that each of $im (q_{1})$ and $im (q_{2})$ has Hausdorff distance at most $R_{1}$ from $γ$ .

givenA1

2.1

The comparison fact [A2] then gives Hausdorff distance at most $2 R_{1}$ between $im (q_{1})$ and $im (q_{2})$ . The lemma [L1] is the geometric input used in the standard proof of [A1]. Therefore the stated Morse-stability bound holds.

A2L1step 1.1∎

Depends on

Used by

Dependency tree · two levels

9 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)