Following a brief review of the history of the link between Einstein’s velocity addition law of special relativity and the hyperbolic geometry of Bolyai and Lobachevski, we employ the binary operation of Einstein’s velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry. In full analogy with Euclidean geometry, we show in this article that the in...

The action of Möbius transformations with real coefficients preserves the hyperbolic metric in the upper half-plane model of the hyperbolic plane. The modular group is an interesting group of hyperbolic isometries generated by two Möbius transformations, namely, an order-two element, g2 HzL = -1 ê z, and an element of infinite order, g• HzL = z + 1. Viewing the action of the group elements on a...

In this paper we introduce a new type of Pascal’s pyramids. The new object is called hyperbolic Pascal pyramid since the mathematical background goes back to the regular cube mosaic (cubic honeycomb) in the hyperbolic space. The definition of the hyperbolic Pascal pyramid is a natural generalization of the definition of hyperbolic Pascal triangle ([2]) and Pascal’s arithmetic pyramid. We descri...

We say that a group G is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that the class of acylindrically hyperbolic groups coincides with many other classes studied in the literature, e.g., the class Cgeom introduced by Hamenstädt, the class of groups admitting a non-elementary weakly properly discontinuous action on a hyperbolic spac...

A generalized hyperbolic tetrahedra is a polyhedron (possibly noncompact) with finite volume in hyperbolic space, obtained from a tetrahedron by the polar truncation at the vertices lying outside the space. In this paper it is proved that a volume formula for ordinary hyperbolic tetrahedra devised by J. Murakami and M. Yano can be applied to such ones. There are two key tools for the proof; one...

We show that for any non–elementary hyperbolic group H and any finitely presented group Q, there exists a short exact sequence 1 → N → G → Q → 1, where G is a hyperbolic group and N is a quotient group of H . As an application we construct a representation rigid but not superrigid hyperbolic group, show that adding relations of the form xn = 1 to a presentation of a hyperbolic group may drastic...

This paper is the first in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Here we introduce Mom technology and enumerate the hyperbolic Mom-n manifolds for n ≤ 4. Our longterm goal is to show that all low-volume closed and cusped hyperbolic 3-manifolds are obtained by filling a hyperbolic Mom-n manifold, n ≤ 4 and to enumerate the lo...

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 ...

This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a 2-dimensional Artin group the Deligne complex is Gromov hyperbolic preci...

If G is a hyperbolic group, where G = H oφ Z, H is finitely presented, and φ is an automorphism of H, then H satisfies a polynomial isoperimetric inequality. Necessary and sufficient conditions of homological character are given for a finitely presented subgroup H of a hyperbolic group to be hyperbolic (resp. a quasi-convex subgroup). If Y is a connected subcomplex of the finite connected 2comp...