Abstract
Extending the analogous result of Cannon and Wagreich for the fundamental groups of surfaces, we show that, for the ℓ-regular graphs \(\mathcal{X}_{\ell ,m} \) 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) 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 2 − X + 1). We then derive some regularity properties for the coefficients f n of the growth series: they satisfy Kλn − R < f n < Kλn + R for some constants K, R < 0, λ < 1.
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References
Bertin, M.-J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M. and Schreiber, J.-P.: Pisot and Salem Numbers. Birkhäuser, Basel, 1992.
Cannon, J. W.: The growth of the closed surface groups and the compact hyperbolic Coxeter groups, unpublished.
Cannon, J. W. and Wagreich, P.: Growth functions of surface groups, Math. Ann. 293(2) (1992), 239–257.
Coornaert, M.: Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math. 159(2) (1993), 241–270.
Coxeter, H. S. M.: Regular honeycombs in hyperbolic apace, In: ICM Amsterdam, Vol. 3., 1954, pp. 155–169.
Floyd, W. J.: Growth of planar coxeter groups, P. V. numbers, and Salem numbers, Math. Ann. 293 (1992), 475–483.
Floyd, W. J. and Plotnick, S. P.: Growth functions on Fuchsian groups and the Euler characteristic, Invent. Math. 88(1) (1987), 1–29.
Floyd, W. J. and Plotnick, S. P.: Growth functions for semi-regular tilings of the hyperbolic plane, Geom. Dedicata 53 (1994), 1–23.
Graham, R. L., Knuth, D. E. and Patashnik, O.: Concrete Mathematics, 2nd edn, Addison-Wesley: Reading, Mass, 1994.
de la Harpe, P.: Topics in Geometric Group Theory, Univ. Chicago Press, Chicago, IL, 2000.
Kronecker, L.: Werke, Vol. I-V. Teubner, Leipzig, 1899.
Parry, W.: Growth series of Coxeter groups and Salem numbers, J. Algebra 154 (1993), 406–415.
Rudin, W.: Functional Analysis, 2nd edn, McGraw-Hill, New York, 1991.
Salem, R.: Power series with integral coefficients, Duke Math. J. 12 (1945), 153–172.
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Barthold, L., Ceccherini-Silberstein, T.G. Salem Numbers and Growth Series of Some Hyperbolic Graphs. Geometriae Dedicata 90, 107–114 (2002). https://doi.org/10.1023/A:1014902918849
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DOI: https://doi.org/10.1023/A:1014902918849