CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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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.

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

Statement

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

Facts & Assumptions

Given: The hypotheses of the Statement.

[L1]

The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph (The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph).

[L2]

The vertex set of a connected simple graph with its path metric is a $(1, 1)$ -quasi-geodesic space (The vertex set of a connected simple graph with its path metric is a $(1, 1)$ -quasi-geodesic space).

[F1]

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).

[L3]

A group is finitely generated when some finite subset generates it (Finitely generated groups).

Proof

technique · direct

1.1

The word metric is the path metric of the Cayley graph, which is connected because the set generates.

F1L1L3

2.1

The previous proposition then applies verbatim.

L2step 1.1∎

Depends on

Used by

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

Dependency tree · two levels

18 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)

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