Related papers: Volumes in Hyperbolic Space
Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the…
After classifying 3-dimensional hyperbolic Coxeter pyramids by means of elementary plane geometry, we calculate growth functions of corresponding Coxeter groups by using Steinberg formula and conclude that growth rates of them are always…
We study relatively hyperbolic Coxeter groups of type $HM$ with maximal Euclidean Coxeter subgroups of codimension 1. Our main result in this paper is that the dimension of these groups is bounded above.
We construct an explicit lower bound for the volume of a complex hyperbolic orbifold that depends only on dimension.
In this paper, we establish that the non-zero dihedral angles of hyperbolic Coxeter polyhedra of large dimensions are not arbitrarily small. Namely, for dimensions $n\geq 32$, they are of the form $\frac{\pi}{m}$ with $m\leq 6$. Moreover,…
We classify Coxeter decompositions of hyperbolic tetrahedra, i.e. simplices in the hyperbolic space H^3. The paper completes the classification of Coxeter decompositions of hyperbolic simplices.
It is known that the volume function for hyperbolic manifolds of dimension $\geq 3$ is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by…
A polyhedron in a three-dimensional hyperbolic space is said to be generalized if finite, ideal and truncated vertices are admitted. In virtue of Belletti's theorem (2021) the exact upper bound for volumes of generalized hyperbolic…
We define the injectivity radius of a Coxeter polyhedron in H^3 to be half the shortest translation length among hyperbolic/loxodromic elements in the orientation-preserving reflection group. We show that, for finite-volume polyhedra, this…
We overview the volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in $H^3$ and $S^3$. We also present some results, which provide a…
A three-dimensional orthoscheme is defined as a tetrahedron whose base is a right-angled triangle and an edge joining the apex and a non-right-angled vertex is perpendicular to the base. A generalization, called complete orthoschemes, of…
In a variety of settings we provide a method for decomposing a 3-manifold $M$ into pieces. When the pieces have the appropriate type of hyperbolicity, then the manifold $M$ is hyperbolic and its volume is bounded below by the sum of the…
In this paper an explicit formula for a lower bound on the volume of a hyperbolic orbifold, dependent on dimension and the maximal order of torsion in the orbifolds' fundamental group, is constructed.
The volume of a Meissner polyhedron is computed in terms of the lengths of its dual edges. This allows to reformulate the Meissner conjecture regarding constant width bodies with minimal volume as a series of explicit finite dimensional…
We compare the volume of a hyperbolic 3-manifold $M$ of finite volume and the complexity of its fundamental group.
Atkinson [2] found a sequence of three-dimensional hyperbolic polyhedra whose dihedral angles are $\pi /3$. In this paper, we construct another sequence of such polyhedra. We also determine the volumes of some of these polyhedra.
In this paper we provide a geometric condition satisfied by certain closed subsets of the Riemann sphere which implies that their hyperbolic convex hulls in $\mathbb{H}^3$ have infinite volume. As a corollary, we characterize continua in…
By gluing together copies of an all-right angled Coxeter polytope a number of open hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of hyperbolic 6-manifolds having the smallest possible…
In this paper, we find lower bounds for volumes of hyperbolic 3-manifolds with various topological conditions. Let V_3 = 1.01494 denote the volume of a regular ideal simplex in hyperbolic 3-space. As a special case of the main theorem, if a…
We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact…