Groups, Geometry, and Dynamics

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Volume 10, Issue 2, 2016, pp. 601–618
DOI: 10.4171/GGD/358

Published online: 2016-06-09

On the growth of a Coxeter group

Tommaso Terragni[1]

(1) Università di Milano-Bicocca, Italy

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. Theorem 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 $\omega (W, S)= \mathrm {lim sup} \sqrt [n]{a_n}$ satisfies $\omega (W, S) \geq \tau$ (cf. Theorem 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. [3]), 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. Theorem C).

Keywords: Coxeter groups, growth of groups

Terragni Tommaso: On the growth of a Coxeter group. Groups Geom. Dyn. 10 (2016), 601-618. doi: 10.4171/GGD/358