On the growth of cocompact hyperbolic Coxeter groups

TL;DR

This paper develops a recursive method to analyze the growth functions of cocompact hyperbolic Coxeter groups, linking their algebraic properties to geometric structures and subgroup classifications.

Contribution

It introduces a recursion formula for the denominator polynomial of the growth function, enabling detailed analysis of these groups' growth behavior.

Findings

Recursion formula for growth function denominator polynomial

Analysis of Coxeter groups with up to 6 generators in hyperbolic 4-space

Application to classes classified by Lanner, Kaplinskaya, and Esselmann

Abstract

For an arbitrary cocompact hyperbolic Coxeter group G with finite generator set S and complete growth function P(x)/Q(x), we provide a recursion formula for the coefficients of the denominator polynomial Q(x) which allows to determine recursively the Taylor coefficients and the pole behavior of the growth function of G in terms of its Coxeter subgroup structure. We illustrate this in the easy case of compact right-angled hyperbolic n-polytopes. Finally, we provide detailed insight into the case of Coxeter groups with at most 6 generators, acting cocompactly on hyperbolic 4-space, by considering the three combinatorially different families discovered and classified by Lanner, Kaplinskaya and Esselmann, respectively.