The Poincaré Series of the Hyperbolic Coxeter Groups with Finite Volume of Fundamental Domains

DOI

10.1142/S1402925110000842How to use a DOI?

Keywords

Hilbert-Poincaré series; Coxeter group

Abstract

The discrete group generated by reflections of the sphere, or the 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 is the generating function of the cardinalities of sets of elements of equal length. Around 1966, 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 an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. 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.

Copyright

© 2010 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Maxim Chapovalov
AU  - Dimitry Leites
AU  - Rafael Stekolshchik
PY  - 2021
DA  - 2021/01/07
TI  - The Poincaré Series of the Hyperbolic Coxeter Groups with Finite Volume of Fundamental Domains
JO  - Journal of Nonlinear Mathematical Physics
SP  - 169
EP  - 215
VL  - 17
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925110000842
DO  - 10.1142/S1402925110000842
ID  - Chapovalov2021
ER  -