On the growth of cocompact hyperbolic Coxeter groups

For an arbitrary cocompact hyperbolic Coxeter group G with finite generator set S and 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 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 4space, by considering the three combinatorially different families discovered and classified by Lannér, Kaplinskaya and Esselmann, respectively. 1. Overview and results Let G be a discrete group generated by finitely many reflections in hyperplanes (mirrors) of hyperbolic space H such that the orbifold H/G is compact. We call G a cocompact hyperbolic Coxeter group and denote by S the (natural) set of generating reflections. For each generator s ∈ S , one has s = 1 while two distinct elements s, s ∈ S satisfy either no relation if the corresponding mirrors admit a common perpendicular or provide the relation (ss) = 1 for an integer m = m(s, s) > 1 if the mirrors intersect. The images of the mirrors decompose H into connected components each of whose closures gives rise to a compact convex fundamental polytope P ⊂ H for G with dihedral angles of type π/p where p ≥ 2 is an integer. Hence, P is a simple polytope so that each k-face is contained in exactly n − k facets. We call P a Coxeter polytope * Partially supported by Schweizerischer Nationalfonds 200020-121506/1, 200020-113199/1. 2000 Mathematics Subject Classification. Primary 20F55, 22E40, 51F15. 2 Ruth Kellerhals and Geneviève Perren and use the standard notation by means of the associated Coxeter graph simultaneously for G and P (cf. [D, Chapter 3] and [V, Chapter 5]). In particular, two nodes in the Coxeter graph Γ of G corresponding to mirrors intersecting under the angle of π/3 (respectively π/p) are connected by a simple edge (respectively by an edge with label p). If two mirrors are perpendicular (or admit a common perpendicular), their nodes are not joined at all (are joined by a dotted line). The focus of this work is the growth series of G defined by