DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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.

Geometric actions on a metric space

Definition

An action of a group on a metric space is geometric if it is isometric, proper, and cobounded in the sense of Isometric, proper, and cobounded actions on metric spaces.

Depends on

Isometric, proper, and cobounded actions on metric spaces

Used by

Free groups act geometrically on regular trees Example
The integers act geometrically on the real line Example
Zⁿ acts geometrically on Euclidean n-space Example
Cobounded proper geodesic actions produce finite generating sets Lemma
The Svarc-Milnor lemma Theorem

Dependency tree · two levels

3 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. Löh, Geometric Group Theory, Sections 4.4 and 5.1 (standard reference, not scraped)
C. Drutu and M. Kapovich, Lectures on Geometric Group Theory, Chapter 5 (standard reference, not scraped)