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. any two points x , y ∈ X {\displaystyle x,y\in X} are end points of a geodesic segment [ x , y ] {\displaystyle [x,y]} (an isometric image of a compact subinterval [ a , b ] {\displaystyle [a,b]} of the reals). Note that the definition via Gromov products does not require the space to be geodesic. Let x , y , z ∈ X {\displaystyle x,y,z\in X} . A geodesic triangle with vertices x , y , z {\displaystyle x,y,z} is the union of three geodesic segments [ x , y ] , [ y , z ] , [ z , x ] {\displaystyle [x,y],[y,z],[z,x]} (where [ p , q ] {\displaystyle [p,q]} denotes a segment with endpoints p {\displaystyle p} and q {\displaystyle q} ).
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