DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-27

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The Gromov boundary via asymptotic sequences

Definition

Let $(X, d)$ be a proper geodesic hyperbolic space and fix a basepoint $o \in X$ .

A sequence $(x_{n})$ in $X$ is a Gromov sequence if

(x_{m}, x_{n})_{o} \to \infty as m, n \to \infty .

Two Gromov sequences $(x_{n})$ and $(y_{n})$ are asymptotic if

(x_{m}, y_{n})_{o} \to \infty as m, n \to \infty .

After Asymptoticity of Gromov sequences is an equivalence relation ↗ shows that this is an equivalence relation, the Gromov boundary $\partial X$ is defined to be the set of asymptoticity classes of Gromov sequences.

Depends on

Used by

The boundary topology defined by Gromov products Definition
FALSE: a proposed Gromov boundary quotient needs no equivalence check False statement
Asymptoticity of Gromov sequences is an equivalence relation Lemma

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)