PropositionStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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Free abelian groups of rank at least two are not hyperbolic

Statement

If $A$ is a free abelian group of rank at least $2$ , then $A$ is not hyperbolic.

Facts & Assumptions

Given: A free abelian group $A$ of rank $n \geq 2$ .

[L1]

Hyperbolicity of a finitely generated group is independent of the chosen finite generating set (Hyperbolicity of a finitely generated group is independent of the finite generating set).

[L2]

Hyperbolic spaces have uniformly thin geodesic quadrilaterals (Hyperbolic spaces have thin geodesic quadrilaterals).

[A1]

With the standard basis of $Z^{n}$ , the Cayley graph contains geodesic rectangles of arbitrarily large width inside the first two coordinate directions.

Proof

technique · direct

1.1

By the rank hypothesis, $A ≅ Z^{n}$ with $n \geq 2$ . In the standard Cayley graph, the points $(0, 0)$ , $(m, 0)$ , $(m, m)$ , and $(0, m)$ in the first two coordinates form a geodesic square of side length $m$ for every $m \geq 1$ .

givenA1algebra

2.1

If the standard Cayley graph were hyperbolic, [L2] would give a uniform thinness constant for all geodesic quadrilaterals. But the midpoint of one side of the square from step 1.1 has distance $m / 2$ from the union of the opposite sides, and $m$ is arbitrary. So the standard Cayley graph is not hyperbolic, and [L1] shows that $A$ itself is not hyperbolic.

A1L1L2step 1.1∎

Depends on

Used by

A product of two infinite groups need not be hyperbolic Counterexample
Z² is not hyperbolic Counterexample
FALSE: every abelian group is hyperbolic False statement

Dependency tree · two levels

12 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 6.5.4 (standard reference, not scraped)