On the growth of a Coxeter group

Abstract

For a Coxeter system $(W,S)$ let $a_n^{(W,S)}$ be the cardinality of the sphere of radius $n$ in the Cayley graph of $W$ with respect to the standard generating set $S$. It is shown that, if $(W,S)\preceq(W',S')$ then $a_n^{(W,S)}\leq a_n^{(W',S')}$ for all $n\in \mathbb{N}_0$, where $\preceq$ is a suitable partial order on Coxeter systems (cf. Thm. A). It is proven that there exists a constant $\tau= 1.13\dots$ such that for any non-affine, non-spherical Coxeter system $(W,S)$ the growth rate $\omega(W,S)=\limsup \sqrt[n]{a_n}$ satisfies $\omega(W,S)\geq \tau$ (cf. Thm. B). The constant $\tau$ is a Perron number of degree $127$ over $\mathbb{Q}$. For a Coxeter group $W$ the Coxeter generating set is not unique (up to $W$-conjugacy), but there is a standard procedure, the diagram twisting (cf. [BMMN02]), which allows one to pass from one Coxeter generating set $S$ to another Coxeter generating set $\mu(S)$. A generalisation of the diagram twisting is introduced, the mutation, and it is proven that Poincar\'e series are invariant under mutations (cf. Thm. C).