CounterexampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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Taking $Z$ itself as a generating set gives a word metric of diameter one, not bilipschitz equivalent to the standard one

Statement refuted

The comparison theorem for word metrics remains true for arbitrary generating sets.

Facts & Assumptions

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

[F1]

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).

[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]

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).

[L3]

The identity map between the word metrics of two finite generating sets of a group is a bilipschitz equivalence (The identity map between the word metrics of two finite generating sets is a bilipschitz equivalence).

[L4]

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).

Counterexample

technique · direct

1.1

Taking the whole group of integers as generating set gives every nonzero integer word length one.

F1L1

2.1

So that word metric has diameter one, while the metric for the generating set ${1}$ is unbounded: the distance from $0$ to $n$ is $∣ n ∣$ . Its individual balls are finite, as [L2] requires, but their radii are not bounded uniformly.

L1L2L4step 1.1

3.1

The two are therefore not bilipschitz equivalent; the failing hypothesis of the comparison theorem is the finiteness used to take a maximum over the symmetrised set.

L3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 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)

Taking ZZ itself as a generating set gives a word metric of diameter one, not bilipschitz equivalent to the standard one