TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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Hyperbolicity of a finitely generated group is independent of the finite generating set

Statement

Assume the Axiom of Choice. Let $G$ be a finitely generated group. If the Cayley graph of $G$ is hyperbolic for one finite generating set, then it is hyperbolic for every finite generating set.

Facts & Assumptions

Given: AC, a finitely generated group $G$ and two finite generating sets $S, T$ .

[L1]

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

[L2]

Hyperbolicity is a quasi-isometry invariant of geodesic spaces (Hyperbolicity is a quasi-isometry invariant of geodesic spaces).

[A1]

AC is used in [L2] through its Morse and controlled-inverse suppliers (The Axiom of Choice).

Proof

technique · direct

1.1givenL1

By [L1], the identity map on the vertex sets is bilipschitz for the two word metrics. Extend it over each edge of $Γ (G, S)$ by a chosen shortest $T$ -path for that edge label, and conversely for $T$ -edges. Because the generating sets are finite, these paths can be fixed by finitely many choices. The resulting maps are quasi-isometries of the geometric Cayley graphs: every point is within $1 / 2$ of a vertex, and the vertex metrics have the bilipschitz bounds from [L1].

2.1L2A1step 1.1∎

By [L2], under [A1] hyperbolicity transfers from one geometric Cayley graph to the other. Since $S, T$ were arbitrary finite generating sets, the definition does not depend on the set.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

Clara Löh, Geometric Group Theory, Section 6.3 (standard reference, not scraped)