Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question

2 Answers 2

up vote 9 down vote accepted

Yes. This is a result of Grigorchuk, see his Mittag-Leffler notes.link text (Milnor's problem on the growth of groups and its consequences, available on line for free; see page 28).

share|improve this answer
    
seems to be p. 19, Cor. 8.6 –  Ian Agol Apr 26 '12 at 2:59
    
@Agol, yes, true, but the same result is mentioned in a broader "Gap" context on pp 28-29. –  Igor Rivin Apr 26 '12 at 3:57
    
Thank you very much! It is indeed mentioned in cor. 8.6 and on page 28-29. On page 18 there is mention of work by van der Dries and Wilkie also proving it. –  Daan Michiels Apr 26 '12 at 8:43

It is interesting that another asymptotic invariant of a group, the Dehn function, can be arbitrary large (even non-recursive) but still bounded by a polynomial on an infinite set. In fact the polynomial can be made quadratic. See this paper.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.