The growth function \(\gamma(n)\) of a finitely generated group (with respect to a given finite generating set) counts the number of elements that can be written as a product of up to \(n\) generators (or their inverses). R. I. Grigorchuk was the first to construct a group of so-called intermediate growth (see for example his survey [in Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 325-338 (1991; Zbl 0749.20016)]). He proved that the growth is between \(e^{\sqrt n}\) and \(e^{n^\beta}\) with \(\beta=\log 31/\log 32\approx 0.991\).
In this note, the upper bound is improved to \(e^{n^\alpha}\) with \(\alpha=\log 2/\log(2/\eta)\approx 0.767\) where \(\eta\) is the real root of \(X^3+X^2+X-2\). The proof relies on a careful analysis of the recursive structure of Grigorchuk’s group.
Grigorchuk’s lower bound was recently improved by Yu. G. Leonov [Preprint (1998); Mat. Zametki 64, No. 4, 573-583 (1998); translation in Math. Notes 64, No. 4, 496-505 (1998; see the review Zbl 0942.20011 above)].