False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

FALSE: a nontrivial finitely generated group with a word metric is a geodesic metric space

Statement refuted

a nontrivial finitely generated group with a word metric is a geodesic metric space.

Facts & Assumptions

Given: The proposed claim together with the witness named in the Statement refuted.

[F1]

A geodesic of length $L$ in a metric space is an isometric embedding of the interval $[0, L]$ , and the space is geodesic when every two points are the endpoints of one (Geodesics and geodesic metric spaces).

[L1]

The word metric of $G$ with respect to $S$ is $d_{S} (g, h) = ∣ g^{- 1} h ∣_{S}$ (The word metric of a group with respect to a generating set).

[L2]

A group with the word metric of any generating set is a $(1, 1)$ -quasi-geodesic space (A group with the word metric of any generating set is a $(1, 1)$ -quasi-geodesic space).

Refutation

technique · contradiction

1.1

The claim asserts that every two elements are the endpoints of an isometric embedding of a real interval.

F1assume-contra

2.1

A geodesic between two elements at distance one supplies points at every intermediate real distance, while a word metric takes only integer values.

F1L1step 1.1

3.1

So no nontrivial group with a word metric is geodesic; the correct statement is that it is $(1, 1)$ -quasi-geodesic.

L2step 2.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

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

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)