False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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FALSE: the same delta works after every finite change of generating set

Statement

False claim: once a finitely generated group is hyperbolic, one numerical slimness constant $δ$ works for the Cayley graph of every finite generating set.

Facts & Assumptions

Given: The free group $F_{2} = ⟨ a, b ⟩$ and, for each integer $n \geq 2$ , the generating set $S_{n} = {a, b, a^{n} b^{n}}$ .

[L1]

Hyperbolicity is independent of the generating set, but only up to quasi-isometry (Hyperbolicity of a finitely generated group is independent of the finite generating set).

[A1]

In the Cayley graph of $(F_{2}, S_{n})$ , the vertices $1$ , $a^{n}$ , and $a^{n} b^{n}$ form a geodesic triangle with side lengths $n$ , $n$ , and $1$ .

Refutation

technique · direct

1.1

The generating set ${a, b}$ gives a tree Cayley graph, so $F_{2}$ is $0$ -hyperbolic for that choice.

given

1.2

Let $δ \geq 0$ , and choose $n > 2 δ + 2$ . By [A1], the side from $1$ to $a^{n}$ is geodesic in the Cayley graph of $(F_{2}, S_{n})$ . Write $k = ⌊ n / 2 ⌋$ . Any word from $a^{k}$ to $a^{n} b^{n}$ has abelianization $(n - k, n)$ ; using the generator $a^{n} b^{n}$ once leaves $(- k, 0)$ and therefore still needs at least $k$ letters, while using it zero times needs at least $n$ letters. Hence $d (a^{k}, a^{n} b^{n}) \geq k + 1$ , so the distance from $a^{k}$ to the one-edge side $[1, a^{n} b^{n}]$ is at least $k > δ$ . Its distance to the side $[a^{n}, a^{n} b^{n}]$ is $n - k > δ$ . So this geodesic triangle is not $δ$ -slim.

A1choosealgebra

2.1

Since $δ$ was arbitrary, no single constant works for all finite generating sets of $F_{2}$ . The theorem [L1] preserves only existence of some constant for each generating set separately. Therefore the claim is false.

L1step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Brian H. Bowditch, A course on geometric group theory, Section 2.2 (standard reference, not scraped)