Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27
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Finite subgroups of a hyperbolic group have uniformly bounded order

Statement

Let G be a hyperbolic group and fix a finite generating set S. Then there exists a constant BS such that every finite subgroup F≤G satisfies ∣F∣≤BS.

Facts & Assumptions

Given: A hyperbolic group G with finite generating set S.

[A1]

Every finite subgroup of a hyperbolic group has an orbit of uniformly bounded diameter in the Cayley graph, with the bound depending only on the generating set.

[A2]

Only finitely many group elements can act faithfully on a fixed finite ball in the Cayley graph, so a uniform orbit-diameter bound yields a uniform order bound.

[L1]

Morse stability is one of the geometric tools used in the standard proof (Morse stability of quasi-geodesics).

Proof

technique · direct
1.1

Let F≤G be finite. By [A1], some F-orbit in the Cayley graph of (G,S) has diameter bounded by a constant depending only on S.

givenA1
2.1

That orbit lies in a finite ball, and the action of F on its orbit is faithful. Therefore [A2] gives a uniform bound ∣F∣≤BS. The role of [L1] in the standard proof is to supply the geometric control behind [A1].

A2L1step 1.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources