Gromov hyperbolic spaces, also known as $\delta$-hyperbolic spaces, are geodesic spaces in which every triangle is thin. Hyperbolic groups are fundamental examples of Gromov hyperbolic spaces in geometric group theory.

Let $(X,d)$ be a metric space. For all $x,y,z \in X$, the Gromov product of $y$ and $z$ with respect to $x$ is defined by $$(y,z)_x= \frac{1}{2}(d(y,x)+d(x,z)-d(y,z)).$$

Then $(X,d)$ is said $\delta$-hyperbolic if for all $p,x,y,z \in X$, $$(x,z)_p \geq \min ((x,y)_p, (y,z)_p ) - \delta.$$

When $(X,d)$ is a geodesic metric space, there are several equivalent definitions of $\delta$-hyperbolicity.

For example, a geodesic metric space is $\delta$-hyperbolic if and only if every geodesic triangle $\Delta= [x,y] \cup [y,z] \cup [z,x]$ is $\delta$-thin, that is the distance between a point of $\Delta$ and the union of the opposite sides of $\Delta$ is at most $\delta$.

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