Salem numbers and growth series of some hyperbolic graphs

Extending the analogous result of Cannon andWagreich for the fundamental groups of surfaces, we show that, for the l-regular graphs Xl,m associated to regular tessellations of the hyperbolic plane by m-gons, the denominators of the growth series (which are rational and were computed by Floyd and Plotnick (Floyd and Plotnick, 1994)) are reciprocal Salem polynomials. As a consequence, the growth rates of these graphs are Salem numbers. We also prove that these denominators are essentially irreducible (they have a factor of X + 1 when m ≡ 2 mod 4; and when l = 3 and m ≡ 4 mod 12, for instance, they have a factor of X−X+1). We then derive some regularity properties for the coefficients fn of the growth series: they satisfy Kλ n −R < fn < Kλ n +R for some constants K,R > 0, λ > 1.