# On the growth of a Coxeter group

### Tommaso Terragni

Università di Milano-Bicocca, Italy

## Abstract

For a Coxeter system $(W,S)$ let $a_{n}$ 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 $(W,S)⪯(W_{′},S_{′})$ then $a_{n}≤a_{n}$ for all $n∈N_{0}$, where $⪯$ is a suitable partial order on Coxeter systems (cf. Theorem A).

It is proven that there exists a constant $τ=1.13…$ such that for any non-affine, non-spherical Coxeter system $(W,S)$ the growth rate $ω(W,S)=limsupna_{n} $ satisfies $ω(W,S)≥τ$ (cf. Theorem B). The constant $τ$ is a Perron number of degree 127 over $Q$.

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. [3]), which allows one to pass from one Coxeter generating set $S$ to another Coxeter generating set $μ(S)$. A generalisation of the diagram twisting is introduced, the *mutation*, and it is proven that Poincaré series are invariant under mutations (cf. Theorem C).

## Cite this article

Tommaso Terragni, On the growth of a Coxeter group. Groups Geom. Dyn. 10 (2016), no. 2, pp. 601–618

DOI 10.4171/GGD/358