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 lregular graphs X associated to regular tessellations of hyperbolic plane by mgons, 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 $$K\lambda^nR<a_n<K\lambda^n+R$$ for some constants $K,R>0$, $\lambda>1$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 1999
 arXiv:
 arXiv:math/9910067
 Bibcode:
 1999math.....10067B
 Keywords:

 Mathematics  Group Theory;
 20F32;
 11R06
 EPrint:
 Geom. Dedicata 90 (2002), 107114