Growth functions and Euler series
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.
scientific article; zbMATH DE number 3922950