PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + 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.

A finitely generated group is finite if and only if it is quasi-isometric to a point

Statement

A finitely generated group is finite if and only if it is quasi-isometric to a point.

Facts & Assumptions

Given: The hypotheses of the Statement.

[F1]

A finitely generated group is quasi-isometric to a metric space when its word metric for some, equivalently every, finite generating set is (The quasi-isometry type of a finitely generated group).

[L1]

Balls of a word metric are finite if and only if the generating set is finite (Balls of a word metric are finite if and only if the generating set is finite).

[L2]

The nonempty metric spaces quasi-isometric to a one-point space are exactly those of finite diameter (The nonempty metric spaces quasi-isometric to a one-point space are exactly those of finite diameter).

[L3]

Bounded subset. $A$ is bounded if $A = \emptyset$ or there are $x_{0} \in X$ and a real $r > 0$ with $A \subseteq B (x_{0}, r)$ . (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).

[L4]

A set $A$ is finite when $A \approx n$ for some $n \in N$ . (The cardinality $∣ A ∣$ of a finite set).

Proof

technique · direct

1.1

A finite group has finite diameter in any word metric, hence is quasi-isometric to a point.

F1L2L3L4

2.1

Conversely finite diameter with a finite generating set makes the whole group a ball of finite radius, and such balls are finite.

F1L1L2L3L4step 1.1∎

Depends on

Used by

Finiteness is a geometric property of finitely generated groups Proposition

Dependency tree · two levels

34 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)