Growth series.
2006-11-30 05:32:12 +0000 general
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
MR1408557
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.