Growth functions for semi-regular tilings of the hyperbolic plane
Cannon and Wagreich computed the growth function of a genus \(g\) surface group with fundamental domain a \(4g\)-gon. In the paper under review the authors study the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domain \(D\) in the hyperbolic plane an \((n; p,q)\)-gon; i.e., vertices of \(D\) have angle measures that alternate between \({{2\pi} \over p}\) and \({{2\pi} \over q}\) for some integers \(p,q\geq 3\). After classifying the values of \(n\), \(p\), \(q\) for which there is a torsion-free surface group with fundamental polygon an \((n; p,q)\)-gon, the growth functions \(f\) are computed. Finally conditions are given for the denominator of \(f\) to be a product of a Salem polynomial and cyclotomic polynomials.
scientific article; zbMATH DE number 701140