TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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Finite C'(1/6) presentations define hyperbolic groups

Statement

Let $G = ⟨ X ∣ R ⟩$ be a finite presentation satisfying the metric small-cancellation condition $C^{'} (1 / 6)$ . Then $G$ is hyperbolic.

Facts & Assumptions

Given: A finite presentation $⟨ X ∣ R ⟩$ satisfying $C^{'} (1 / 6)$ .

[A1]

Finite $C^{'} (1 / 6)$ presentations satisfy a linear isoperimetric inequality.

[L1]

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

Proof

technique · direct

1.1

By [A1], the given presentation satisfies a linear isoperimetric inequality.

givenA1

2.1

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

L1step 1.1∎

Depends on

Linear isoperimetric characterisation of hyperbolic groups

Used by

A small-cancellation presentation gives a hyperbolic group Example

Dependency tree · two levels

5 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

Nicholas Touikan, An introduction to combinatorial and geometric group theory, Section 3.5 (standard reference, not scraped)
Clara Löh, Geometric Group Theory, Section 6.4 (standard reference, not scraped)