Symmetries of planar growth functions
type: Item
instance of: scholarly article
title: Symmetries of planar growth functions
zbMATH Open document ID: 0652.20052
DOI: 10.1007/BF01410198
author: William Floyd
author: Steven P. Plotnick
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143607
review text: 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.
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 4063390
zbMATH Keywords: finitely generated group
zbMATH Keywords: finite generating set
zbMATH Keywords: growth function
zbMATH Keywords: words
zbMATH Keywords: generators
zbMATH Keywords: irreducible Euclidean Coxeter group
zbMATH Keywords: geometrically defined groups
zbMATH Keywords: Euclidean and Hyperbolic planar isometry groups
zbMATH Keywords: geometric generating sets
MaRDI profile type: Publication
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OpenAlex ID: W2165839278
description: scientific article; zbMATH DE number 4063390