Salem numbers and growth series of some hyperbolic graphs. (English) Zbl 1032.20028
For \(l\)-regular graphs associated to regular tesselations of the hyperbolic plane by \(m\)-gons, the denominators of the growth series are shown to be essentially irreducible reciprocal Salem polynomials. Consequently the growth rates of these graphs are Salem numbers. These results extend results of J. W. Cannon and P. Wagreich [Math. Ann. 293, No. 2, 239–258 (1992; Zbl 0734.57001)]. Moreover, some regularity properties of the coefficients for the growth series are derived.
Reviewer: Brigitte Servatius (Worcester)
MSC:
20F65 | Geometric group theory |
05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
20F67 | Hyperbolic groups and nonpositively curved groups |
57M05 | Fundamental group, presentations, free differential calculus |