DefinitionDefinition: Literature-sourcedProof: Not applicableverified 2026-09-24 (gpt-6-sol)

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The boundary topology defined by Gromov products

Definition

Let $(X, d)$ be a proper geodesic hyperbolic space, let $o \in X$ , and let $\partial X$ be the boundary defined by Gromov sequences.

For boundary classes $ξ = [(x_{n})]$ and $η = [(y_{n})]$ , define

$(ξ, η)_{o} : = \sup {lim inf}_{m, n \to \infty} (x_{m}, y_{n})_{o},$

where the supremum runs over all representatives of the two classes.

For $ξ \in \partial X$ and $R > 0$ , let

$U_{o} (ξ, R) : = {η \in \partial X : (ξ, η)_{o} > R} .$

The boundary topology consists of the sets $O \subseteq \partial X$ such that, for every $ξ \in O$ , some $R > 0$ satisfies $U_{o} (ξ, R) \subseteq O$ . Each $U_{o} (ξ, R)$ is a neighbourhood of $ξ$ ; it need not itself be open. The next theorem proves that this criterion defines a topology and is independent of the chosen basepoint.

Depends on

Used by

Free groups have Cantor-set boundaries Example
FALSE: a proposed Gromov boundary quotient needs no equivalence check False statement
The boundary topology is well defined and quasi-isometry invariant Theorem

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

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