Growth series.

 November 30, 2006 general

\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}}

\newenvironment{question}[1][]{\par\textbf{Question (#1).}}{} \newenvironment{theorem}[1][]{\par\textbf{Theorem (#1).}}{} \newenvironment{lemma}[1][]{\par\textbf{Lemma (#1).}}{} \newenvironment{proof}{\textit{Proof.}}{}

In seminar today, Okun pointed out the following interesting observation; for any finitely generated group G , you can define its growth series G(t) = \sum_{g \in G} t^{\ell(g)} , where \ell(g) is the length of the shortest word for g . The first observation is that G(t) is often a rational function, in which case G(1) makes sense. The second observation is that G(1) is “often” equal to \chi(G) . This is an example of weighted L^2 cohomology.

Grigorchuk’s group (and generally any group with intermediate (i.e., subexponential but not polynomial) growth) does not have a rational growth function; the coefficients in a power series for a rational function grow either polynomially or exponentially. This observation appears in [1]. More significantly, this paper constructs groups which, being nilpotent, have polynomial growth, but nonetheless have generating sets for which that the corresponding growth series is not rational.

[1] M. Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1996) 85–109.