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

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Slim triangles, the Gromov product, and the four-point condition are equivalent up to constants

Statement

Let $(X, d)$ be a nonempty geodesic metric space. The following are equivalent up to changing the constant:

$X$ is hyperbolic, that is, all geodesic triangles are $δ$ -slim for some $δ \geq 0$ .
For every basepoint $o \in X$ there exists $δ_{o}^{'} \geq 0$ such that one has

(x, z)_{o} \geq \min {(x, y)_{o}, (y, z)_{o}} - δ_{o}^{'}

for all $x, y, z \in X$ . 3. For some $δ^{''} \geq 0$ , one has

d (x, z) + d (y, w) \leq \max {d (x, y) + d (z, w), d (x, w) + d (y, z)} + δ^{''}

for all $x, y, z, w \in X$ .

Facts & Assumptions

Given: A nonempty geodesic metric space $(X, d)$ .

[A1]

In a nonempty geodesic metric space, $δ$ -slim triangles imply the displayed Gromov-product inequality at every chosen basepoint, with a possibly larger constant depending on that basepoint.

[A2]

The displayed Gromov-product inequality implies the four-point condition, again with a controlled change of constant.

[A3]

The four-point condition implies slim geodesic triangles, with another controlled change of constant.

Proof

technique · direct

1.1

The implication $(1) \Rightarrow (2)$ is exactly [A1]: slim triangles give a lower bound for the branch length measured by the Gromov product.

2.1

The implication $(2) \Rightarrow (3)$ is exactly [A2]: rewriting the Gromov-product inequality in terms of distances yields the four-point form.

A2step 1.1

3.1

The implication $(3) \Rightarrow (1)$ is exactly [A3]: in a geodesic space the four-point inequality forces each side of a geodesic triangle to stay within a uniform neighborhood of the other two. Hence the three formulations are equivalent up to changed constants.

A3step 2.1∎

Depends on

Used by

FALSE: the same delta works after every finite change of generating set False statement
Asymptoticity of Gromov sequences is an equivalence relation Lemma
Morse stability of quasi-geodesics Theorem

Dependency tree · two levels

4 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, Sections 6.2.1-6.2.2 (standard reference, not scraped)
Brian H. Bowditch, A course on geometric group theory, Sections 2.1-2.2 (standard reference, not scraped)