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The boundary topology is well defined and quasi-isometry invariant
Statement
Assume the Axiom of Choice. 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: AC and proper geodesic hyperbolic spaces and .
The supremal boundary product and any supplied representative product differ by at most , changing basepoints shifts products by at most their distance, and the threshold-neighbourhood criterion gives a Hausdorff topology (Boundary products have controlled representative and basepoint dependence).
Under AC a quasi-isometry of geodesic hyperbolic spaces induces a continuous boundary map, bounded-distance maps induce the same map, and a controlled quasi-inverse supplies a continuous inverse (Quasi isometries extend to boundary homeomorphisms).
AC is used in [F2] for the Morse projection families and coarse-inverse selection (The Axiom of Choice).
Proof
The boundary product in The boundary topology defined by Gromov products is the supremal product of [F1]. Its comparison with every representative product proves independence of representatives; the basepoint inequality gives cofinal threshold neighbourhoods at any two basepoints.
The definition's open-set criterion is exactly the one proved in [F1], including its treatment of threshold sets as neighbourhoods that need not be open. Hence it is a topology and is Hausdorff.
By [F2] under [A1], the quasi-isometry induces a continuous map of these boundary topologies. Its controlled quasi-inverse induces a continuous inverse because the bounded-distance composites induce identity maps. This proves the claimed homeomorphism; properness is included in the statement but not required by [F1] or [F2].
Depends on
Used by
Cited to discharge well-definedness by The boundary topology defined by Gromov products.
Dependency tree · two levels
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Sources
- Brian H. Bowditch, A course on geometric group theory, Section 5.3 (standard reference, not scraped)