Symmetries of planar growth functions
Let G be a finitely generated group, with \(\Sigma\) a finite generating set. The growth function of G,\(\Sigma\) is the generating function, \(f(z)=f_{G,\Sigma}=\sum^{\infty}_{n=0}a_ nz\) n which has as its coefficients \(a_ n\) the number of elements of G which can be expressed as words of length n in the generators \(\Sigma\) but not by words of shorter length. The growth functions of a group exhibit in their analytic properties many of the combinatorial properties of the group. \textit{J.-P. Serre} [Cohomologie des groupes discrets, in Prospects Math., Ann. Math. Stud. 70, 77-169 (1971; Zbl 0235.22020)] showed that \(f(z)=\pm f(1/z)\) in the case that G is a compact hyperbolic or irreducible Euclidean Coxeter group with respect to the standard generating sets. It has been long conjectured that this symmetry or reciprocity condition should hold widely for geometrically defined groups. The authors show that symmetry or reciprocity holds for Euclidean and Hyperbolic planar isometry groups with respect to arbitrary geometric generating sets in almost all cases. They explicitly calculate the recursive combinatorial structure of these groups in all cases and thereby determine their growth functions.
Euclidean and Hyperbolic planar isometry groups
scientific article; zbMATH DE number 4063390