Growth functions and Euler series
type: Item
instance of: scholarly article
title: Growth functions and Euler series
zbMATH Open document ID: 0577.20020
DOI: 10.1007/BF01388836
published in: Inventiones Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143158
review text: Let G denote a finitely generated group and \(g_ 1,...,g_ d\) a finite generating set for G. Let \(a_ n\) denote the number of elements of the group G which can be written as a word of length n but no shorter in the given generators and their inverses. The associated growth function is the power series \(f(x)=a_ nx^ n\). This series f(x), which depends on the generating set, may be considered a generalized order for a group which applies in particular for infinite groups. The symmetries, zeros, and poles of this function seem to reflect interesting properties of the group, though most properties of f(x) remain mysterious. The author explores the observed connection between f(x) and Euler series for the group. He establishes a clear connection via homological algebra, though he develops an error term whose significance is unknown. More recent work of Walter Parry, William Floyd, and Steve Plotnick clarifies the connections even more.
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 3922950
zbMATH Keywords: finitely generated group
zbMATH Keywords: finite generating set
zbMATH Keywords: growth function
zbMATH Keywords: power series
zbMATH Keywords: symmetries
zbMATH Keywords: zeros
zbMATH Keywords: poles
zbMATH Keywords: Euler series
author: N. F. Smythe
MaRDI profile type: Publication
cites work: Non-commutative graded algebras and their Hilbert series
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description: scientific article; zbMATH DE number 3922950