Hyperbolic metric space

The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees.

[1] 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.

by a constant multiple, there is an equivalent geometric definition involving triangles when the metric space

[2][3] [4] Note that the definition via Gromov products does not require the space to be geodesic.

This definition is generally credited to Eliyahu Rips.

-approximate center of a geodesic triangle: this is a point which is at distance at most

[5] Thus the notion of a hyperbolic space is independent of the chosen definition.

[6] Note that in this case the Gromov product also has a simple interpretation in terms of the incircle of a geodesic triangle.

In fact the quantity (A,B)C is just the hyperbolic distance p from C to either of the points of contact of the incircle with the adjacent sides: for from the diagram c = (a – p) + (b – p), so that p = (a + b – c)/2 = (A,B)C.[7] The Euclidean plane is not hyperbolic, for example because of the existence of homotheties.

The two-dimensional grid is not hyperbolic (it is quasi-isometric to the Euclidean plane).

It is the Cayley graph of the fundamental group of the torus; the Cayley graphs of the fundamental groups of a surface of higher genus is hyperbolic (it is in fact quasi-isometric to the hyperbolic plane).

The hyperbolic plane (and more generally any Hadamard manifolds of sectional curvature

we see that in this example the more (negatively) curved the space is, the lower the hyperbolicity constant.

Similar examples are CAT spaces of negative curvature.

While curvature is a property that is essentially local, hyperbolicity is a large-scale property which does not see local (i.e. happening in a bounded region) metric phenomena.

[citation needed] One way to make precise the meaning of "large scale" is to require invariance under quasi-isometry.

[8] 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

we have the following property:[10] Informally this means that the circumference of a "circle" of radius

[12] Linear isoperimetric inequalities were inspired by the small cancellation conditions from combinatorial group theory.

All asymptotic cones of an hyperbolic space are real trees.

[13] Generalising the construction of the ends of a simplicial tree there is a natural notion of boundary at infinity for hyperbolic spaces, which has proven very useful for analysing group actions.

is the set of equivalence classes of sequences which converge to infinity,[14] which is denoted

is metrisable and there is a distinguished family of metrics defined using the Gromov product.

is proper then the set of all such embeddings modulo equivalence with its natural topology is homeomorphic to

[17] A similar realisation is to fix a basepoint and consider only quasi-geodesic rays originating from this point.

is a simplicial regular tree the boundary is just the space of ends, which is a Cantor set.

is proper then its boundary is homeomorphic to the space of Busemann functions on

This action can be used[19] to classify isometries according to their dynamical behaviour on the boundary, generalising that for trees and classical hyperbolic spaces.

, then one of the following cases occur: Subsets of the theory of hyperbolic groups can be used to give more examples of hyperbolic spaces, for instance the Cayley graph of a small cancellation group.

For example, the following hyperbolicity results have led to new phenomena being discovered for the groups acting on them.