Δ-hyperbolic space

In mathematics, a Gromov - hyperbolic space is a space with " uniformly thin triangles ". This term axiomatized and generalized spaces of negative curvature and has proved effective in many areas of mathematics as useful.

Definition

A metric space is called δ - hyperbolic for δ > 0, if all geodesic triangles are δ - thin, ie each edge of the triangle is contained in the δ - neighborhood of the union of the other two edges:

This condition is met with, for example, for geodesic triangles in trees or in the hyperbolic plane, generally for geodesic triangles in simply connected Riemannian manifolds of negative sectional curvature.

A metric space is called Gromov - hyperbolic if it is δ - hyperbolic for δ > 0.

Hyperbolic groups

A hyperbolic group is a finitely generated group whose Cayley graph is a finite set of generators δ - hyperbolic for δ > 0. ( Δ Except for the constant, this condition is independent of the choice of the finite generating system. )

• Geometry