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 f S ( 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 f S ( 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.