On the growth of cocompact hyperbolic Coxeter groups

For an arbitrary cocompact hyperbolic Coxeter group G with a finite generator set S and a complete growth function fS(x)=P(x)/Q(x), we provide a recursion formula for the coefficients of the denominator polynomial Q(x). It allows us to determine recursively the Taylor coefficients and to study the arithmetic nature of the poles of the growth function fS(x) in terms of its subgroups and exponent variety. We illustrate this in the 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 Lannér, Kaplinskaya and Esselmann, respectively.