The volume of hyperbolic Coxeter polytopes of even dimension
type: Item
instance of: scholarly article
title: The volume of hyperbolic Coxeter polytopes of even dimension
zbMATH Open document ID: 0831.51007
DOI: 10.1016/0019-3577(95)91242-N
author: Gert Heckman
published in: Indagationes Mathematicae. New Series
publication date: 1996-02-11T00:00:00Z
review text: The author considers hyperbolic Coxeter \(n\)-polytopes of finite volume. Such a \(n\)-dimensional polytope \(D\) in \(H^n\) is generated by a (finite) collection of half spaces \(H^+_s\), where the (inner) dihedral angle of two halfspaces \(H_s\) and \(H_t\) with non-empty intersection has size \(\pi/m_{st}\) for certain integers \(m_{st}\geq 2\). The set \(S\) of reflections of \(H^n\) in the corresponding hyperplanes \(H_s\) generates the discrete group \(W\) of motions of \(H^n\). Thus \(D\) is a strict fundamental domain for the action of \(W\) on \(H^n\), and \((W, S)\) is a Coxeter group. Let \(\ell(w)\) denote the length of \(w\in W\) with respect to the generating set \(S\), let \(\text{vol}_n(S^n)\) be the volume of the \(n\)-dimensional unit sphere, and let \(P_W(t)= \Sigma_W t^{\ell(w)}\) be the Poincaré series of \(W\), then for \(n\) even we have the formula \(1/P_W(1)= (- 1)^{n/2}\cdot 2\text{ vol}_n(D)/\text{vol}_n(S^n)\), which is the subject of this paper. (If \(n\) is odd, then \(1/P_W(1)\) is cero.) This result is essentially just a specialization of the Gauss-Bonnet theorem to the present situation. For \(D\) compact this formula can be derived from the work by \textit{J. P. Serre} [Prospects Math., Ann. Math. Stud. 70, 77-169 (1971; Zbl 0235.22020)] on the cohomology of discrete groups. But here, the author obtains this result as a consequence of the differential volume formula and the reduction formula of \textit{L. Schläfli} [`Theorie der vielfachen Kontinuität'. Ges. Math. Abh. 1, Birkhäuser Basel, pp. 167-387 (1949; Zbl 0035.219)], continued into the hyperbolic case. This approach is of elementary nature. He was inspired by a paper of \textit{R. Kellerhals} [Math. Z. 206, No. 2, 193-210 (1991; Zbl 0717.52011)]. Especially, he furthermore uses ideas of \textit{R. Steinberg} [Mem. Am. Math. Soc. 80, 109 p. (1968; Zbl 0164.029)] on computing \(P_W(t)\) by induction and of \textit{E. B. Vinberg} [Russ. Math. Surv. 40, No. 1, 31-75 (1985); trans…
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 51M25
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 792824
zbMATH Keywords: Coxeter groups
zbMATH Keywords: reflection groups
zbMATH Keywords: reflection geometry
zbMATH Keywords: polytopes
zbMATH Keywords: hyperbolic geometry
zbMATH Keywords: volume
reviewed by: Johannes Böhm
MaRDI profile type: Publication
cites work: The Gauss-Bonnet Theorem for Riemannian Polyhedra
cites work: Q3918670
cites work: The combinatorial structure of cocompact discrete hyperbolic groups
cites work: A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds
cites work: THE FUNCTIONS OF SCHLÄFLI AND LOBATSCHEFSKY
cites work: Growth functions on Fuchsian groups and the Euler characteristic
cites work: Q3998721
cites work: A class of hyperbolic Coxeter groups
cites work: On Schläfli's reduction formula
cites work: Volumes in hyperbolic 5-space
cites work: On the volumes of hyperbolic 5-orthoschemes and the trilogarithm
cites work: Counterexamples Involving Growth Series and Euler Characteristics
cites work: The Gauss-Bonnet theorem for \(V\)-manifolds
cites work: Q5180606
cites work: Endomorphisms of linear algebraic groups
cites work: Hyperbolic reflection groups
full work available at URL: 0019-3577(95)91242-n
OpenAlex ID: W2084063180
description: scientific article; zbMATH DE number 792824