Ruth Kellerhals

On the volume of hyperbolic polyhedra

where V1k is the volume Vol~_ 2 (Sj~) of the apex Sjk:= Sj c~ S~ to wjk. Schl~ifli proved this formula for spherical simplices. In 1936, H. Kneser gave a second, very skilful proof of(l) (see I l l ] and [4, Sect. 5.1]) for both the spherical and hyperbolic cases (up to a change of sign in the latter case). But even for a three-dimensional non-euclidean simplex, the integration of this Schl~li differential is practically impossible. In fact, the most basic objects in polyhedral geometry are orthoschemes (or orthogonal-simplices) first introduced by Schl~fli: An n-orthoscheme R is an n-simplex with vertices Po . . . . . Pn such that

The size of a hyperbolic Coxeter simplex

We determine the covolumes of all hyperbolic Coxeter simplex reflection groups. These groups exist up to dimension 9. the volume computations involve several different methods according to the parity of dimension, subgroup relations and arithmeticity properties.

Quaternions and Some Global Properties of Hyperbolic 5-Manifolds

We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds M of dimension 5. The result implies improved universal lower bounds for the volume vol5(M) and, for M compact, new estimates relating the injectivity radius and the diameter of M with vol5(M). The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group PSL(2, H) of quaternionic 2 × 2-matrices with Dieudonndeterminant � equal to 1 and isolation properties of PSL(2, H).

Commensurability classes of hyperbolic Coxeter groups

In this paper, we classify all the hyperbolic Coxeter n-simplex reflection groups up to widecommensurability for all n 3. We also determine all the subgroup relationships among the groups.

On the growth of cocompact hyperbolic Coxeter groups

For an arbitrary cocompact hyperbolic Coxeter group G with a finite generator set S and a complete growth function f S ( x ) = P ( x ) / Q ( x ) , we provide a recursion formula for the coefficients of the denominator polynomial Q ( x ) . It allows us to determine recursively the Taylor coefficients and to study the arithmetic nature of the poles of the growth function f S ( x ) in terms of its subgroups and exponent variety. We illustrate this in the case of compact right-angled hyperbolic n -polytopes. Finally, we provide detailed insight into the case of Coxeter groups with at most 6 generators, acting cocompactly on hyperbolic 4-space, by considering the three combinatorially different families discovered and classified by Lanner, Kaplinskaya and Esselmann, respectively.

Volumes of cusped hyperbolic manifolds

Abstract For n-dimensional hyperbolic manifolds of finite volume with m ⩾ 1 cusps a new lower volume bound is presented which is sharp for n = 2,3. The estimate depends upon m and the ideal regular simplex volume. The proof makes essential use of a density argument for ball packings in Euclidean and hyperbolic spaces and explicit formulae for the simplicial density function. Examples, consequences for the Gromov invariant, and-for n even-the maximal number of cusps are discussed.

VOLUMES IN HYPERBOLIC 5-SPACE

In this work, we study the problem of determining volumes of five-dimensional hyperbolic polytopes. Generally, the non-Euclidean volume problem has not progressed very much since its origin last century with the works of N.I. Lobachevsky [Lo] in hyperbolic 3-space H and of L. Schlafli [Sc] on the n-sphere S. While concrete results are available only for small dimensions n, which, for n = 3, were reinterpreted and unified by H.S.M. Coxeter [Co1], their methods however are of timeless value. A first observation is that each simplex in an n-dimensional space X of constant curvature is equidissectable into orthoschemes (see 1.2). An orthoscheme R ⊂ X is a simplex bounded by hyperplanes H0, . . . ,Hn such that Hi ⊥ Hj for |i−j| > 1. It is, up to congruence, uniquely determined by its (at most n) non-right dihedral angles, and, as the nomenclature indicates, it is described very conveniently by means of schemes or weighted graphs (see 1.1). For hyperbolic orthoschemes R with vertices pi opposite to Hi, i = 0, . . . , n, at most the endpoints p0 and pn of the orthogonal edge path p0p1, . . . , pn−1pn may be points at infinity. In such cases, R is called simply or doubly asymptotic. Secondly, by Schlafli’s differential formula (cf. [Sc, No. 22]; see 2.1, Theorem 1), there is the following very simple, but fundamental expression for the volume differential of an orthoscheme R ⊂ H in terms of infinitesimal angle variations:

Regular simplices and lower volume bounds for hyperbolicn-manifolds

For an-dimensional compact hyperbolic manifoldMn a new lower volume bound is presented. The estimate depends on the volume of a hyperbolic regularn-simplex of edge length equal to twice the in-radius ofMn. Its proof relies upon local density bounds for hyperbolic sphere packings.

The minimal growth rate of cocompact Coxeter groups in hyperbolic 3-space

Due to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on

The three smallest compact arithmetic hyperbolic 5–orbifolds

H^3