On Growth Functions of Coxeter Groups

Let $(W, S)$ be a Coxeter system of rank n and let p(W, S)(t) be its growth function. It is known that p(W, S)(q⁻¹) < ∞ holds for all n ≤ q ∈ N. In this paper we will show that this still holds for $q = n-1$, if $(W, S)$ is $2$-spherical. Moreover, we will prove that p(W, S)(q⁻¹) = ∞ holds for $q = n-2$, if the Coxeter diagram of $(W, S)$ is the complete graph. These two results provide a complete characterization of the finiteness of the growth function in the case of $2$-spherical Coxeter systems with complete Coxeter diagram.