PropositionStatement: Literature-sourcedProof: 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.

The word metric is the largest left-invariant metric in which each generator and its inverse lie within distance one of the identity

Statement

The word metric is the largest left-invariant metric in which each generator and its inverse lie within distance one of the identity.

Facts & Assumptions

Given: The hypotheses of the Statement.

[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).

[L1]

The word length $∣ g ∣_{S}$ is the least $n$ such that $g$ is a product of $n$ elements of $S \cup S^{- 1}$ (Word length of a group element with respect to a generating set).

[L2]

Word length is defined on every element and satisfies $∣ g h ∣_{S} \leq ∣ g ∣_{S} + ∣ h ∣_{S}$ , $∣ g^{- 1} ∣_{S} = ∣ g ∣_{S}$ , and $∣ g ∣_{S} = 0$ exactly when $g$ is the identity (Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws).

[L3]

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).

[L4]

Every metric satisfies the triangle inequality (Metric space: $d (x, y) = 0$ iff $x = y$ , symmetry, and the triangle inequality; pseudometric and ultrametric).

Proof

technique · direct

1.1

For a competing left-invariant metric with the stated property, the triangle inequality along a shortest expression bounds the competing distance by the word length.

F1L1L2L4

2.1

The word metric itself gives every symmetrised generator distance one from the identity, so it satisfies the constraint and dominates every competitor.

F1L1L3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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)