I am reading [an introduction to growth of groups](http://arxiv.org/abs/math/0607384). The notions of polynomial and superpolynomial growth are introduced, as are exponential and subexponential growth.


I can prove that the growth of a group is always either exponential or subexponential (it is exercise 1.6). However, there seems to be no mention of an analogous result for (super)polynomial growth (i.e. the growth of a group is always either polynomial or superpolynomial).


There exist strictly increasing functions which grow faster than polynomially but are not superpolynomial (this is pretty clear; a more detailed explanation can be found in the second section of [this document](http://arxiv.org/abs/1101.0599v1)), but I do not know whether these occur as the growth function of some group.


[The thesis of a Nick Scott](http://www.ms.unimelb.edu.au/publications/ScottNick.pdf) claims to prove that every group grows either polynomially or superpolynomially, but I don't see it (it is in subsection 1.4.1, on p.12; it seems to me the proof assumes that the limit $\log(\beta(k))/\log(k)$ exists, but I don't know why).


So my question is: **does every group grow either polynomially or superpolynomially?**



Does every group grow either polynomially or superpolynomially?
Yes. This is a result of Grigorchuk, see his Mittag-Leffler notes.[link text](http://www.mittag-leffler.se/preprints/files/IML-1112s-06.pdf) (Milnor's problem on the growth of groups and its consequences, available on line for free; see page 28).


