The size of hyperbolic Coxeter simplex
type: Item
instance of: scholarly article
title: The size of hyperbolic Coxeter simplex
zbMATH Open document ID: 0953.20041
DOI: 10.1007/BF01238563
author: Ruth Kellerhals
author: John G. Ratcliffe
author: Norman W. Johnson
author: S. Tschantz
published in: Transformation Groups
publication date: 2001-01-22T00:00:00Z
review text: Coxeter simplices of finite volume with dihedral angles submultiples of \(\pi\) or zero exist in hyperbolic \(n\)-space only up to \(n=9\) and for \(n>2\), there are finitely many. Extending and combining their earlier work, jointly and separately, the authors determine the volumes of all these hyperbolic simplices. Dissection methods using orthoschemes and Schäfli's reduction method are shown to give formulae for these volumes in the form of iterated integrals. These are used to give accurate numerical values for the volumes but also, in most cases, to give closed formulae involving the evaluation of some reasonably tractable function at a suitable value. The associated reflection groups in many cases are arithmetic so that these functions can be arithmetically defined and commensurability relationships between the groups deduced. Furthermore, since the Gauss-Bonnet formula holds in even dimensions, the main difficult cases are in odd dimensions. Complete tables are given.
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 51M25
zbMATH DE Number: 1394481
zbMATH Keywords: Coxeter simplices
zbMATH Keywords: volumes
zbMATH Keywords: hyperbolic simplices
zbMATH Keywords: reflection groups
zbMATH Keywords: Gauss-Bonnet formula
reviewed by: C. Maclachlan
MaRDI profile type: Publication
cites work: Q3927897
cites work: Q4402546
cites work: Discrete groups generated by reflections
cites work: THE FUNCTIONS OF SCHLÄFLI AND LOBATSCHEFSKY
cites work: World-structure and non-Euclidean honeycombs
cites work: Dissecting orthoschemes into orthoschemes
cites work: Quadratic Integers and Coxeter Groups
cites work: On the volume of hyperbolic polyhedra
cites work: On Schläfli's reduction formula
cites work: On the volumes of hyperbolic 5-orthoschemes and the trilogarithm
cites work: Volumes in hyperbolic 5-space
cites work: Q3916824
cites work: A lower bound for the volume of hyperbolic 3-orbifolds
cites work: Q4314293
cites work: Q4342585
cites work: Scissors congruences, I. The Gauss-Bonnet map.
cites work: Q5540809
cites work: Spiegelungsgruppen und Aufzählung halbeinfacher Liescher Ringe
full work available at URL: https://doi.org/10.1007/bf01238563
OpenAlex ID: W2079717050
description: scientific article; zbMATH DE number 1394481