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

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 χ 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 [4,17].Moreover, in the case where χ is negative, we provide infinitely many non-hyperbolic Coxeter systems whose growth rates are Perron numbers.