In mathematics, a hyperbolic metric space is a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by
Mikhael Gromov, generalizes the metric properties of classical
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P' ...
and of
trees
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite
groups called
Gromov-hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abst ...
s.
Definitions
In this paragraph we give various definitions of a
-hyperbolic space. A metric space is said to be (Gromov-) hyperbolic if it is
-hyperbolic for some
.
Definition using the Gromov product
Let
be a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
. The
Gromov product
In mathematics, the Gromov product is a concept in the theory of metric spaces named after the mathematician Mikhail Gromov. The Gromov product can also be used to define ''δ''-hyperbolic metric spaces in the sense of Gromov.
Definition
...
of two points
with respect to a third one
is defined by the formula:
:
Gromov's definition of a hyperbolic metric space is then as follows:
is
-hyperbolic if and only if all
satisfy the ''four-point condition''
:
Note that if this condition is satisfied for all
and one fixed base point
, then it is satisfied for all
with a constant
. 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.
Definitions using triangles
Up to changing
by a constant multiple, there is an equivalent geometric definition involving triangles when the metric space
is ''geodesic'', i.e. any two points
are end points of a geodesic segment