CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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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.
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A map is a quasi-isometry exactly when it is a quasi-isometric embedding with coarsely dense image

Statement

Assume the Axiom of Choice (The Axiom of Choice).

A map is a quasi-isometry exactly when it is a quasi-isometric embedding with coarsely dense image.

Facts & Assumptions

Given: The hypotheses of the Statement, including the Axiom of Choice.

[F1]

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

[L1]

Under the Axiom of Choice, a quasi-isometric embedding with coarsely dense image admits a quasi-inverse quasi-isometric embedding (A quasi-isometric embedding with coarsely dense image has a quasi-inverse quasi-isometric embedding).

Proof

technique · direct

1.1

If a map is a quasi-isometric embedding with coarsely dense image, the previous theorem supplies a quasi-inverse quasi-isometric embedding, so the map is a quasi-isometry.

F1L1

2.1

Conversely, if $g$ is a quasi-inverse of $f$ and $d_{Y} (f (g (y)), y) \leq R$ for every $y \in Y$ , then every target point lies within distance $R$ of $f [X]$ , so the image of $f$ is coarsely dense.

F1step 1.1∎

Depends on

Used by

Two metric spaces are quasi-isometric if and only if each contains a separated net and the two nets are bilipschitz equivalent Theorem

Dependency tree · two levels

7 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

C. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)