PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

Right translation by a fixed element displaces every point of a word metric space by exactly the word length of that element

Statement

Right translation by a fixed element displaces every point of a word metric space by exactly the word length of that element.

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]

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).

Proof

technique · direct

1.1F1L1L2

Left invariance gives $d_{S} (h, h g) = ∣ g ∣_{S}$ for every $h$ , so right translation by $g$ displaces every point by the same amount.

2.1F1L1L2step 1.1∎

Hence right translation by $g$ is at bounded distance $∣ g ∣_{S}$ from the identity map, and it is the identity exactly when that displacement is zero.

Depends on

Used by

FALSE: every word metric is invariant under right translation False statement

Dependency tree · two levels

14 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)