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 $(W,S)⪯(W^{\prime },S^{\prime })$ then $a_n^{(W,S)}\le a_n^{(W^{\prime },S^{\prime })}$ for all $n\in {\mathbb{N}}_0$, where $⪯$ 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 $\omega (W,S)=\mathrm{limsup}\sqrt[n]{a_n}$ satisfies $\omega (W,S)\ge \tau$ (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 Poincaré series are invariant under mutations (cf. Thm. C).

Major revision since v2. To appear in "Groups, Geometry, and Dynamics" (modulo minor edits). 12 pages

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