Gromov-hyperbolic metric space ($\delta$)

A geodesic metric space is $\delta$-hyperbolic when each side of every geodesic triangle lies in the closed $\delta$-neighbourhood of the other two sides.

A finitely generated group is hyperbolic when one, equivalently every, Cayley graph for a finite generating set is a Gromov-hyperbolic metric space.

A quasi-tree is a geodesic metric space quasi-isometric to a tree.

A geodesic metric space has the bottleneck property if every path joining the endpoints of a geodesic passes within one uniform distance of that geodesic's midpoint. A geodesic metric space is a quasi-tree exactly when it has this property.