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

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The boundary topology is well defined and quasi-isometry invariant

Statement

For a proper geodesic hyperbolic space, the topology defined on the Gromov boundary by Gromov products is well defined. Moreover, a quasi-isometry between proper geodesic hyperbolic spaces induces a homeomorphism of their boundaries.

Facts & Assumptions

Given: Proper geodesic hyperbolic spaces $X$ and $Y$ .

[A1]

Different representatives of the same boundary point and different basepoints define equivalent neighborhood systems on the boundary.

[A2]

If $f : X \to Y$ is a quasi-isometry and $o^{'} \in Y$ stays a bounded distance from $f (o)$ , then there are constants $A \geq 1$ and $C \geq 0$ , depending only on the quasi-isometry data, such that boundary Gromov products satisfy $A^{- 1} (ξ, η)_{o} - C \leq (\partial f (ξ), \partial f (η))_{o^{'}} \leq A (ξ, η)_{o} + C .$ In particular $f$ sends Gromov sequences to Gromov sequences, preserves asymptoticity, and carries product neighborhoods to cofinal product neighborhoods. A quasi-inverse satisfies the corresponding estimates.

[L1]

Asymptoticity of Gromov sequences is an equivalence relation (Asymptoticity of Gromov sequences is an equivalence relation).

Proof

technique · direct

1.1

By [L1], the boundary is already a quotient by a genuine equivalence relation. The comparison result [A1] then shows that changing representatives or the basepoint only changes the neighborhoods $U_{o} (ξ, R)$ by bounded shifts of the parameter $R$ , so the topology is well defined.

L1A1

2.1

By [A2], a quasi-isometry induces a map on asymptoticity classes, and the two-sided product estimate makes that map continuous for the neighborhood systems from step 1.1. Applying the same argument to a quasi-inverse gives a continuous inverse. Thus the induced boundary map is a homeomorphism, and the boundary topology is quasi-isometry invariant.

A1A2step 1.1∎

Depends on

Used by

Free groups have Cantor-set boundaries Example

Cited to discharge well-definedness by The boundary topology defined by Gromov products.

Dependency tree · two levels

9 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

Brian H. Bowditch, A course on geometric group theory, Section 5.3 (standard reference, not scraped)