Hyperbolicity

In mathematics a δ-hyperbolic space is a geodesic metric space satisfying certain metric relations (depending quantitatively on the nonnegative real number δ) between points. The definitions are inspired by 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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