# Growth series.

##### November 30, 2006 general

In seminar today, Okun pointed out the following interesting observation; for any finitely generated group , you can define its growth series , where is the length of the shortest word for . The first observation is that is often a rational function, in which case makes sense. The second observation is that is “often” equal to . This is an example of weighted 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.