Gromov Hyperbolic Discrete Spaces and Their Application to Extension of Classes of Mappings

Gromov hyperbolic discrete spaces and their application to extension of classes of mappings are studied. A metric space is considered and the equality relation between sequences is defined. The extendability of a mapping requires the compatibility of its approximations in the sense that the characteristics of the corresponding standard mappings on sufficiently close balls must be close. If a mapping is approximated by similarities on each ball, the quasiconformal extendability of this mapping requires that the angle of rotation and the logarithm of the coefficient of pressure must satisfy the weak condition. For a quasiconvex metric space, function with the weak property is found to be near-Lipschitz. For quasi-convex space and a function with extendability property, a space exists with intrinsic metric.