Given two pointed Gromov hyperbolic metric spaces (Xi, di, zi), i = 1, 2, and ∆ ∈ R+0 , we present a construction method, which yields another Gromov hyperbolic metric space Y∆ = Y∆((X1, d1, z1), (X2, d2, z2)). Moreover, it is shown that once (Xi, di) is roughly geodesic, i = 1, 2, then there exists a ∆′ ≥ 0 such that Y∆ also is roughly geodesic for all ∆ ≥ ∆ ′.

Let (X, d) a metric space and BX = X × R denote the partially ordered set of (generalized) formal balls in X. We investigate the topological structures of BX, in particular the relations between the Lawson topology and the product topology. We show that the Lawson topology coincides with the product topology if (X, d) is a totally bounded metric space, and show examples of spaces for which the ...

the aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]. in [1], ungar and chen showed that the algebra of the group sl(2,c) naturally leads to the notion of gyrogroups and gyrovector spaces for dealing with the lorentz group and its underlying hyperbolic geometry. they defined the chen addition and then chen model of hyperbolic geometry. in this paper,...

We study the geometry of non-relatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with non-relatively hyperbolic peripheral subgroups is a quasi-isometry invariant...

In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes what has until now been an ad hoc collection of results by many authors. In addition to providing much greater generality than any prior work, our results also give new insight into the nature of the connection between Dio...

Providing each simplex in C(S) with the standard euclidean metric of side-length 1 equips the complex of curves with the structure of a geodesic metric space whose isometry group is just M̃g,m (except for the twice punctured torus). However, this metric space is not locally compact. Masur and Minsky [MM1] showed that nevertheless the geometry of C(S) can be understood quite explicitly. Namely, C...

Suppose G is a hyperbolic group whose boundary ∂∞G has topological dimension k. If ∂∞G is quasi-symmetrically homeomorphic to an Ahlfors kregular metric space, then, modulo a finite normal subgroup, G is isomorphic to a uniform lattice in the isometry group Isom(Hk+1) of hyperbolic (k+1)-space.

Any Kähler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dimension.

In this paper we prove that Urysohn univeral space is hyperconvex. We also examine the Gromov hyperbolicity and hyperconvexity of metric spaces. Using fourpoint property, we give a proof of the fact that hyperconvex hull of a δ-Gromov hyperbolic space is also δ-Gromov hyperbolic.