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

Stoll, Michael Rational and transcendental growth series for the higher Heisenberg groups. Invent. Math. 126 (1996), no. 1, 85—109.

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.