The Poincare series of the hyperbolic Coxeter groups with finite volume of fundamental domains

Abstract

The discrete group generated by reflections of the sphere, or Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series (a.k.a. growth function) is the generating function of the cardinalities of sets of elements of equal length. Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave a recurrence formula. The explicit expression of the Poincaré series was known for the spherical and Euclidean Coxeter groups, and 3-generated Coxeter groups, and (with mistakes) Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, and denominators for each (quasi-)Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the Poincaré series. The non-real poles behave as in Eneström's theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem's requirements.