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TheoremStatement: Literature-sourcedProof: AI-generatedverified 2026-09-24 (gpt-6-sol)
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  • 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.
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Finite C'(1/6) presentations define hyperbolic groups

Statement

Assume the Axiom of Choice. Let G=⟨X∣R⟩ be a finite presentation satisfying the metric small-cancellation condition C′(1/6). Then G is hyperbolic.

Facts & Assumptions

Given: AC and a finite presentation ⟨X∣R⟩ satisfying C′(1/6).

[L0]

Finite C′(1/6) presentations satisfy a linear isoperimetric inequality for van Kampen area (Finite C prime(1/6) presentations satisfy a linear isoperimetric inequality).

[L1]

A finite presentation with linear isoperimetric inequality defines a hyperbolic group (Linear isoperimetric characterisation of hyperbolic groups).

[A1]

AC is used through [L1]'s linear-area-to-slimness supplier (The Axiom of Choice).

Proof

technique · direct
1.1givenL0

By [L0], the given presentation satisfies a linear isoperimetric inequality.

2.1L1A1step 1.1∎

Therefore [L1] applies under [A1], and the presented group is hyperbolic.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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