The volume of hyperbolic Coxeter polytopes of even dimension
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); translation from Usp. Mat. Nauk 40, No. 1(241), 29-66 (1985; Zbl 0579.51015)] on hyperbolic polytopes.
scientific article; zbMATH DE number 792824
scientific article