Δ-hyperbolic space

In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called (Gromov-)hyperbolic groups.

In this paragraph we give various definitions of a ${\displaystyle \delta }$-hyperbolic space. A metric space is said to be (Gromov-) hyperbolic if it is ${\displaystyle \delta }$-hyperbolic for some ${\displaystyle \delta >0}$.

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