Hyperbolic metric space

In mathematics, a hyperbolic metric space is a metric space 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 and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called Gromov-hyperbolic groups.

Definitions

In this paragraph we give various definitions of a

δ

{\displaystyle \delta }

-hyperbolic space. A metric space is said to be (Gromov-) hyperbolic if it is

δ

{\displaystyle \delta }

-hyperbolic for some

δ

>

0

{\displaystyle \delta >0}

.

Definition using the Gromov product

Let

(

X

,

d

)

{\displaystyle (X,d)}

be a metric space. The Gromov product of two points

y

,

z

∈

X

{\displaystyle y,z\in X}

with respect to a third one

x

∈

X

{\displaystyle x\in X}

is defined by the formula:

(

y

,

z

)

x

=

1

2

(

d

(

x

,

y

)

+

d

(

x

,

z

)

−

d

(

y

,

z

)

)

.

{\displaystyle (y,z)_{x}={\frac {1}{2}}\left(d(x,y)+d(x,z)-d(y,z)\right).}

Gromov's definition of a hyperbolic metric space is then as follows:

X

{\displaystyle X}

is

δ

{\displaystyle \delta }

-hyperbolic if and only if all

x

,

y

,

z

,

w

∈

X

{\displaystyle x,y,z,w\in X}

satisfy the four-point condition

(

x

,

z

)

w

≥

min

(

(

x

,

y

)

w

,

(

y

,

z

)

w

)

−

δ

{\displaystyle (x,z)_{w}\geq \min \left((x,y)_{w},(y,z)_{w}\right)-\delta }

Note that if this condition is satisfied for all

x

,

y

,

z

∈

X

{\displaystyle x,y,z\in X}

and one fixed base point

w

0

{\displaystyle w_{0}}

, then it is satisfied for all

w

{\displaystyle w}

with a constant

2

δ

{\displaystyle 2\delta }

. 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

δ

{\displaystyle \delta }

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

X

{\displaystyle X}

is geodesic, i.e.