A complex hyperbolic quasi-Fuchsian group is a discrete, faithful, type preserving and geometrically finite representation of a surface group as a subgroup of the group of holomorphic isometries of complex hyperbolic space. Such groups are direct complex hyperbolic generalisations of quasi-Fuchsian groups in three dimensional (real) hyperbolic geometry. In this article we present the current st...

In this article we introduce a new geometric object called hyperbolic Pascal simplex. This new object is presented by the regular hypercube mosaic in the 4-dimensional hyperbolic space. The definition of the hyperbolic Pascal simplex, whose hyperfaces are hyperbolic Pascal pyramids and faces are hyperbolic Pascals triangles, is a natural generalization of the definition of the hyperbolic Pascal...

It is a well-known fact that every Riemann surface with negative Euler characteristic admits a hyperbolic metric. But this metric is by no means unique – indeed, there are uncountably many such metrics. In this paper, we study the space of all such hyperbolic structures on a Riemann surface, called the Teichmüller space of the surface. We will show that it is a complete metric space, and that i...

In this paper we construct a natural geometric structure for the space of shapes of a star-shaped polygon. Roughly speaking we find: The set of similarity classes of marked stars forms a hyperbolic right angle pentagon bundle over the space of external angle sets of inscribed pentagons. The assignment of the shape of its fiber to each angle set forms a hyperbolic plane bundle over the Teichmiil...

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...

We prove an “Earthquake Theorem” for closed hyperbolic surfaces with cone singularities where the total angle is less than π: the action of the space of measured laminations on Teichmüller space by left earthquakes is simply transitive. This is strongly related to another result: the space of “globally hyperbolic” AdS manifolds with cone singularities (of given angle) along time-like geodesics ...

The papers [Gálvez et al. 2000, Kokubu et al. 2003, Kokubu et al. 2005] gave a method of constructing flat surfaces in the three-dimesnional hyperbolic space. Such surfaces have generically singularities, since any closed nonsigular flat surface is isometric to a horosphere or a hyperbolic cylinder. In the paper [Sasaki et al. 2006], we defined a map, called the hyperbolic Schwarz map, from the...

Some gyrocommutative gyrogroups, also known as Bruck loops or K-loops, admit scalar multiplication, turning themselves into gyrovector spaces. The latter, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry. In classical mechanics the centroid of a triangle in velocity space is the velocity of the center of momentum of three massive ob...

Brownian motion in the three dimensional Lobachevsky space or hyperbolic space is considered in the paper written by F.I.Karpelevich, V.N.Tutubalin and M.G.Shur. A solution for radial symmetric diffusion equation in the three dimensional hyperbolic space is given in that paper. However, derivation of it is not explicitly shown. Therefore, the diffusion equation is solved analytically with an in...