GROMOV HYPERBOLIC SPACES - UC Davis Mathematics.

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In this paper, we generalize the classical definition of Gromov hyperbolicity to the context of directed graphs and we extend one of the main results of the theory: the equivalence of the Gromov hyperbolicity and the geodesic stability. This theorem has potential applications to the development of solutions for secure data transfer on the internet.The theory of Gromov hyperbolic spaces, introduced by M. Gromov in the eighties, has been considered in the books (CDP), (GdH), (Sh), (Bow), (BH), (BBI), (Ro) and in several papers, but it is often assumed that the spaces are geodesic and usually also proper (closed bounded sets are compact).

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Hyperbolic groups are sometimes called word hyperbolic groups or negatively curved groups. A finite group is a trivial example of a hyperbolic group. Free groups of finite rank (cf. Free group ) and fundamental groups of compact Riemannian manifolds of negative sectional curvature (cf. Fundamental group; Riemannian manifold ) are hyperbolic.

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M. Gromov, Hyperbolic groups, in Essays in Group Theory, S. Gersten, editor, 75-263, Mathematical Sciences Research Institute Publications 8, Springer-Verlag, 1987. Lipschitz functions on spaces.

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