The Asymptotic Upper Curvature Of Hyperbolic Products

The Gromov hyperbolic space introduced by the mathematician M.Gromov is a metric space in which the points in the space satisfy some metric relation.As a generalization of the classical hyperbolic space,the Gromov hyperbolic space plays an important role in the study of geometric group theory.In general,the Cartesian product of two Gromov hyperbolic metric spaces is no longer a Gromov hyperbolic space.Recently,T.Foertsch and V.Schroeder introduced the notion of hyperbolic products for two Gromov hyperbolic spaces.They proved that the hyperbolic product of two Gromov hyperbolic spaces is still a Gromov hyperbolic space.M.Bonk and T.Foertsch defined the asymptotic upper curvature of the Gromov hyperbolic space,and suggested to study the asymptotic upper curvature of hyperbolic products.The problem is studied in this paper.Firstly,we investigate the asymptotic upper curvature of the Gromov hyperbolic space in depth.We show that a metric space X is a Gromov hyperbolic space if and only if it is an ACu???-space for some ?<0 and prove that rough-isometric Gromov hyperbolic spaces have the same asymptotic upper curvature.Moreover,another sufficient and necessary condition for X is an ACu???-space is found.Secondly,we research the asymptotic upper curvature of first class hyperbolic product,and prove the following claim.Assume that Y_?,o is the hyperbolic product of Gromov hyperbolic spaces?X₁,o₁?and?X₂,o₂?,thenK_u(Y_?,o)?max{K_u?X₁?,K_u?X₂?},Where K_u?X?denote the asymptotic upper curvature of Gromov hyperbolic metric spaces X.Furthermore,if there are some special relations between Gromov hyperbolic spaces X₁ and X₂,thenK_u{Y_?,o)?K_u?X₁?.Especially under some conditions,we haveK_u?X₁??K_u(Y_?,o)?K_u?X₂?.Lastly,we research the asymptotic upper curvature of second class hyperbolic product,and prove the following claim.Assume that Y_?,?,o is the hyperbolic product of Gromov hyperbolic spaces?X₁,?₁,o₁?and?X₂,?₂,o₂?,thenK_u(Y_?,?,o)?max{K_u?X₁?,K_u?X₂?}.Furthermore,if there are some special relations between Gromov hyperbolic spaces X₁ and X₂,thenKv(Y_?,?,o)?K_u?X₁?.Especially under some conditions,we haveK_u?X₁??K_u(Y_?,?,o)?K_u?X₂?.