a compact space with any metric extending the original ones remains hyperbolic. Important properties Invariance under quasi-isometry One way to precise the meaning of "large scale" is to require invariance under quasi-isometry. This is true of hyperbolicity. If a geodesic metric space is quasi-isometric to a -hyperbolic space then there exists such that is -hyperbolic. The constant depends on and on the multiplicative and additive constants for the quasi-isometry. Approximate trees in hyperbolic spaces The definition of an hyperbolic space in terms of the Gromov product can be seen as saying that the metric relations between any four points are the same as they would be in a tree, up to the additive constant . More generally the following property shows that any finite subset of an hyperbolic space looks like a finite tree. For any there is a constant such that the following holds: if are points in a -hyperbolic space there is a finite tree and an embedding such that for all and The constant can be taken to be with and this is optimal. Exponential growth of distance and isoperimetric inequalities In an hyperbolic space we have the following property: There are such that for all with , every path joining to and staying at distance at least of has length at least . Informally this means that the circumference of a "circle" of radius grows exponentially with . This is reminiscent of the isoperimetric problem in the Euclidean plane. Here is a more specific statement to this effect. Suppose that is a cell complex of dimension 2 such that its 1-skeleton is hyperbolic, and there exists such that the boundary of any 2-cell contains at most 1-cells. Then there is a constant such that for any finite subcomplex we have Here the area of a 2-complex is the number of 2-cells and the length of a 1-complex is the number of 1-cells. The statement above is a linear isoperimetric inequality ; it turns out that having such an isoperimetric inequality characterises Gromov-hyperbolic spaces. Linear isoperimetric inequalities were inspired by the small cancellation conditions from combinatorial group theory. Quasiconvex subspaces A subspace of a geodesic metric space is said to be quasiconvex if there is a constant such that any geodesic in between two points of stays within distance of . A quasi-convex subspace of an hyperbolic space is hyperbolic. Asymptotic cones All asymptotic cones of an