Coxeter systems with $2$-dimensional Davis complexes, growth rates and Perron numbers

Abstract

In this paper, we study growth rates of Coxeter systems with Davis complexes of dimension at most $2$. We show that if the Euler characteristic $\chi$ of the nerve of a Coxeter system is vanishing (resp. positive), then its growth rate is a Salem (resp. a Pisot) number. In this way, we extend results due to Floyd and Parry. Moreover, in the case where $\chi$ is negative, we provide infinitely many non-hyperbolic Coxeter systems whose growth rates are Perron numbers.

References in corpus (2)

• On the growth of a Coxeter group
• The rates of growth in a hyperbolic group