In mathematics a δ-hyperbolic space is a geodesic metric space satisfying certain metric relations (depending quantitatively on the nonnegative real number δ) between points. The definitions are inspired by 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.
Contents
- Definitions
- Definitions using triangles
- Definition using the Gromov product
- Examples
- Hyperbolicity and curvature
- Invariance under quasi isometry
- Approximate trees in hyperbolic spaces
- Exponential growth of distance and isoperimetric inequalities
- Quasiconvex subspaces
- Asymptotic cones
- The boundary of an hyperbolic space
- Definition for proper spaces using rays
- Busemann functions
- The action of isometries on the boundary and their classification
- More examples
- References
Definitions
In this paragraph we give various definitions of a
Definitions using triangles
Let
If for any point
A definition of a
Another definition can be given using the notion of a
These two definitions of a
Definition using the Gromov product
Let
Gromov's definition of an hyperbolic metric space is then as follows:
Up to a multiplicative constant on
Examples
Two "degenerate" examples of hyperbolic spaces are spaces with bounded diameter (for example finite or compact spaces) and the real line.
Metric trees and more generally real trees are the simplest interesting examples of hyperbolic spaces as they are 0-hyperbolic (i.e. all triangles are tripods).
The hyperbolic plane is hyperbolic (every triangle is contained in an ideal triangle and those have a circumcircle of fixed radius). The Euclidean plane is not hyperbolic, for example because of the existence of homotheties.
The 1-skeleton of the triangulation by Euclidean equilateral triangles is not hyperbolic (it is in fact quasi-isometric to the Euclidean plane). A triangulation of the plane
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).
Hyperbolicity and curvature
The hyperbolic plane (and more generally any Hadamard manifolds of sectional curvature
Similar examples are CAT spaces of negative curvature. With respect to curvature and hyperbolicity it should be noted however that 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. For example, the union of an hyperbolic space with a compact space with any metric extending the original ones remains hyperbolic.
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 metric spaceThe constant
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
The constant
Exponential growth of distance and isoperimetric inequalities
In an hyperbolic space
Informally this means that the circumference of a "circle" of radius
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
Asymptotic cones
All asymptotic cones of an hyperbolic space are real trees. This property characterises hyperbolic spaces.
The boundary of an hyperbolic space
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.
In this paragraph
Definition using the Gromov product
A sequence
If
which is finite and does not depend on
Definition for proper spaces using rays
Let
A similar realisation is to fix a basepoint and consider only quasi-geodesic rays originating from this point. In case
Examples
When
When
which is related to the Poisson kernel for the unit disk. The boundary of
Busemann functions
If
The action of isometries on the boundary and their classification
A quasi-isometry between two hyperbolic spaces
In particular the group of isometries of
More examples
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. It is also known that the Cayley graphs of certain models of random groups (which is in effect a randomly-generated infinite regular graph) tend to be hyperbolic very often.
It can be difficult and interesting to prove that certain spaces are hyperbolic. For example, the following hyperbolicity results have led to new phenomena being discovered for the groups acting on them.