A simple method to compute volumes of even-dimensional Coxeter polyhedra

Understanding and computing volumes of hyperbolic manifolds and orbifolds is a rich and fascinating subject. There are for instance deep connections with number theory, more specifically special values of arithmetic functions such as Dedekind ζ-functions, Dirichlet L-functions and polylogarithms (see [Za], [K2], Prasad’s volume formula from [P] as used in the hyperbolic case in [Be] and [BE]). In general, finding the volume of a hyperbolic manifold or orbifold is a difficult problem. However in even dimensions the Gauss-Bonnet-Chern theorem asserts that hyperbolic volume is a multiple of Euler–Poincaré characteristic (see section 3). Therefore, if one can compute the Euler characteristic of a hyperbolic manifold or orbifold (for instance if one knows a cell decomposition for it) then its volume is easily computed. Hyperbolic Coxeter groups form a large class of groups for which a cell decomposition of the quotient orbifold is known, and is in fact contained in the Coxeter diagram data (see sections 2 and 3). Indeed, by results of Vinberg, the faces of a hyperbolic Coxeter polyhedron correspond to the elliptic subdiagrams of its Coxeter diagram, with such a subdiagram giving the stabilizer of the corresponding face. Moreover, hyperbolic Coxeter polyhedra are simple (in the sense that links of faces are simplices), so that the dimension of each face is as expected (in other words, the walls of the polyhedron intersecting along that face are in general position). This means that one can compute the (orbifold) Euler characteristic of the quotient of hyperbolic space H by a hyperbolic Coxeter group Γ by the formula: