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

TL;DR

This paper analyzes the Poincaré series of hyperbolic Coxeter groups with finite volume fundamental domains, providing explicit formulas for their numerators and denominators, and exploring the properties of their poles.

Contribution

It offers a clear description of the numerator of the Poincaré series for any Coxeter group and explicit denominators for (quasi-)Lannér groups, enhancing understanding of their growth functions.

Findings

Poincaré series are rational functions with zeros on the unit circle.

Explicit formulas for numerators and denominators of Poincaré series are provided.

Non-real poles lie in a narrow annulus, similar to Eneström's theorem.

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-)Lann\'er 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 Poincar\'e 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 Poincar\'e 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 Poincar\'e series was known for the spherical and Euclidean Coxeter groups, and 3-generated Coxeter groups, and (with mistakes) Lann\'er groups. Here we give a lucid description of the numerator of the Poincar\'e series of any Coxeter group, and denominators for each (quasi-)Lann\'er group, and review the scene. We give an interpretation of some coefficients of the denominator of the Poincar\'e series. The non-real poles behave as in Enestr\"om's theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem's requirements.