4
$\begingroup$

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?

  • 1
    I [posted the question](http://mathover$f$low.net/questions/95207/does-every-$g$roup-grow-either-polynomially-or-superpolynomially) on Math Overflow.2012-04-25

1 Answers 1

2

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

  • 1
    You might as well accept your answer as well to show that this question has now been dealt with.2012-04-26