On Growth Functions of Coxeter Groups

Abstract Let ( W , S ) be a Coxeter system of rank n , and let $p_{(W, S)}(t)$ be its growth function. It is known that $p_{(W, S)}(q^{-1}) \lt \infty$ holds for all $n \leq q \in \mathbb{N}$ . 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 $p_{(W, S)}(q^{-1}) = \infty$ 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.