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I am reading an introduction to growth of groups. 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), but I do not know whether these occur as the growth function of some group.

The thesis of a Nick Scott 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?

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I would suggest posting this question on Math Overflow. – Jim Belk Apr 25 '12 at 21:43
I posted the question on Math Overflow. – Daan Michiels Apr 25 '12 at 22:19
up vote 2 down vote accepted

The question has been answered on Math Overflow. The answer is yes, by the way.

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You might as well accept your answer as well to show that this question has now been dealt with. – Tara B Apr 26 '12 at 9:21
Yes, I tried this. It says "you can accept your own answer tomorrow". – Daan Michiels Apr 26 '12 at 13:39

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