The volume of hyperbolic Coxeter polytopes of even dimension

Abstract

Let H denote hyperbolic space of dimension n, and let S be an index set for a finite collection of open half spaces H s in H n bounded by codimension one hyperplanes Hs. We assume that for all distinct s, t ∈ S either Hs ∩ Ht is not empty and the (interior) dihedral angle of H s ∩ H + t along Hs ∩ Ht has size π mst for certain integers mst = mts ≥ 2, or Hs ∩Ht is empty while H + s ∩H + t is not empty. In the latter case we put mst = mts = ∞ and we also put mss = 1. Under these assumptions the intersection C = ⋂ s H + s is not empty, and its closure D is called a hyperbolic Coxeter polytope. By abuse of notation let s ∈ S also denote the reflection of H in the hyperplane Hs. Now the group W of motions of H n generated by the reflections s ∈ S is discrete, and D is a strict fundamental domain for the action of W on H. Moreover (W,S) is a Coxeter group with Coxeter matrix M = (mst), i.e. W has a presentation with generators s ∈ S and relations (st)s,t = 1 for s, t ∈ S. Let l(w) denote the length of w ∈ W with respect to the generating set S, and let PW (t) ∈ Z[[t]] be the Poincare series of W defined by PW (t) = ∑ w t .