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

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Abelian subgroups of hyperbolic groups are virtually cyclic

Statement

Every abelian subgroup of a hyperbolic group contains a cyclic subgroup of finite index.

Facts & Assumptions

Given: An abelian subgroup $A$ of a hyperbolic group $G$ .

[A1]

An abelian subgroup of a hyperbolic group that is torsion is finite.

[L1]

Centralizers of infinite-order elements are virtually cyclic (The centralizer of an infinite-order element in a hyperbolic group is virtually cyclic).

Proof

technique · direct

1.1

If $A$ contains an element of infinite order, then $A \subseteq C_{G} (g)$ for that element $g$ , so [L1] shows that $A$ is virtually cyclic.

givenL1

2.1

If every element of $A$ has finite order, then [A1] says that $A$ is finite, hence virtually cyclic. Therefore every abelian subgroup of a hyperbolic group is virtually cyclic.

A1step 1.1∎

Depends on

The centralizer of an infinite-order element in a hyperbolic group is virtually cyclic

Used by

Nothing in the library uses this result yet.

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, Section 6.5.2 (standard reference, not scraped)