On the growth of a Coxeter group

  • Tommaso Terragni

    Università di Milano-Bicocca, Italy


For a Coxeter system let be the cardinality of the sphere of radius in the Cayley graph of with respect to the standard generating set . It is shown that, if then for all , where is a suitable partial order on Coxeter systems (cf. Theorem A).

It is proven that there exists a constant such that for any non-affine, non-spherical Coxeter system the growth rate satisfies (cf. Theorem B). The constant is a Perron number of degree 127 over .

For a Coxeter group the Coxeter generating set is not unique (up to -conjugacy), but there is a standard procedure, the diagram twisting (cf. [3]), which allows one to pass from one Coxeter generating set to another Coxeter generating set . 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