Growth series of some hyperbolic graphs and Salem numbers

Abstract

Extending the analogous result of Cannon and Wagreich for the fundamental groups of surfaces, we show that, for the l-regular graphs X associated to regular tessellations of hyperbolic plane by m-gons, the denominators of the growth series (which are rational and were computed by Floyd and Plotnick) are reciprocal Salem polynomials. As a consequence, the growth rates of these graphs are Salem numbers. We then derive some regularity properties for the coefficients $a_n$ of the growth series: they satisfy $$K\lambda^n-R<a_n<K\lambda^n+R$$ for some constants $K,R>0$, $\lambda>1$.