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 ${\displaystyle \delta >0}$.

Let ${\displaystyle (X,d)}$ be a metric space. The Gromov product of two points ${\displaystyle y,z\in X}$ with respect to a third one ${\displaystyle x\in X}$ is defined by the formula:

$${\displaystyle (y,z)_{x}={\frac {1}{2}}\left(d(x,y)+d(x,z)-d(y,z)\right).}$$

Gromov's definition of a hyperbolic metric space is then as follows: ${\displaystyle X}$ is ${\displaystyle \delta }$-hyperbolic if and only if all ${\displaystyle x,y,z,w\in X}$ satisfy the four-point condition

$${\displaystyle (x,z)_{w}\geq \min \left((x,y)_{w},(y,z)_{w}\right)-\delta }$$

Note that if this condition is satisfied for all ${\displaystyle x,y,z\in X}$ and one fixed base point ${\displaystyle w_{0}}$, then it is satisfied for all ${\displaystyle w}$ with a constant ${\displaystyle 2\delta }$. Thus the hyperbolicity condition only needs to be verified for one fixed base point; for this reason, the subscript for the base point is often dropped from the Gromov product.

Up to changing ${\displaystyle \delta }$ by a constant multiple, there is an equivalent geometric definition involving triangles when the metric space ${\displaystyle X}$ is geodesic, i.e. any two points ${\displaystyle x,y\in X}$ are end points of a geodesic segment ${\displaystyle [x,y]}$ (an isometric image of a compact subinterval ${\displaystyle [a,b]}$ of the reals). Note that the definition via Gromov products does not require the space to be geodesic.