Growth series of some hyperbolic graphs and Salem numbers

Details Growth series of some hyperbolic graphs and Salem numbers Growth series of some hyperbolic graphs and Salem numbers

1999-10-13

arxiv.org/abs/math/9910067v2

Geom. Dedicata 90 (2002), 107--114

Mathematics

Group Theory

Scientific paper

10.1023/A:1014902918849

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-R0$, $\lambda>1$.

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