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

TL;DR

This paper investigates the growth rates of Coxeter systems with 2-dimensional Davis complexes, revealing their connection to Salem, Pisot, and Perron numbers depending on the Euler characteristic of their nerve.

Contribution

It extends previous results by Floyd and Parry, classifying growth rates of Coxeter systems based on the Euler characteristic of their nerve.

Findings

Growth rates are Salem numbers when Euler characteristic is zero.

Growth rates are Pisot numbers when Euler characteristic is positive.

Infinitely many non-hyperbolic Coxeter systems have Perron number growth rates.

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.