We study the geometry of nonrelatively 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 nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. ...

In this paper, we prove that every metric line in the Poincare ball model of hyperbolic geometry is exactly a classical line of itself. We also proved nonexistence of periodic lines in the Poincare ball model of hyperbolic geometry.

in this paper, we prove that every metric line in the poincare ball model of hyperbolic geometry is exactly a classical line of itself. we also proved nonexistence of periodic lines in the poincare ball model of hyperbolic geometry.

There are several ways to generalise the hyperbolic plane and its isometry group to objects in higher dimensions. Perhaps the most familiar is (real) hyperbolic three space, popularised by the work of Thurston [14]. The Poincaré disc and half plane models of the hyperbolic plane naturally come with a complex structure and it is natural to generalise them to complex hyperbolic space in higher co...

A closed, and say orientable, Riemannian 3–manifold (M, ρ) is hyperbolic if the metric ρ has constant sectional curvature κρ = −1. Equivalently, there is a discrete and torsion free group Γ of isometries of hyperbolic 3–space H3 such that the manifolds (M, ρ) and H3/Γ are isometric. It is well-known that the fundamental group π1(M) of every closed 3–manifold which admits a hyperbolic metric is ...

Let Mg,n be the moduli space of Riemann surfaces of genus g with n punctures. From a complex perspective, moduli space is hyperbolic. For example, Mg,n is abundantly populated by immersed holomorphic disks of constant curvature −1 in the Teichmüller (=Kobayashi) metric. When r = dimC Mg,n is greater than one, however, Mg,n carries no complete metric of bounded negative curvature. Instead, Dehn ...

We introduce the functor ◦∗ which assigns to every metric space X its symmetric join ◦∗X . As a set, ◦∗X is a union of intervals connecting ordered pairs of points in X . Topologically, ◦∗X is a natural quotient of the usual join of X with itself. We define an Isom(X)–invariant metric d∗ on ◦∗X . Classical concepts known for H and negatively curved manifolds are defined in a precise way for any...

We give a new proof of the complete integrability of the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose non-parameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperbolic metric inside the ellipsoid (Klein’s model). Mathematics Subject Classi...

The space of shapes of a polyhedron with given total angles less than 2π at each of its n vertices has a Kähler metric, locally isometric to complex hyperbolic space CH. The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point. The metric completion is a complex hyperbolic conemanifold. In some interesting special cases, the metric completion...