On growth functions of Coxeter groups

Let (W, S) be a Coxeter system of rank n, and let (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 a complete Coxeter diagram.