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Let $G$ be a finitely generated group, I'm interested in whether there is any relationship between amenability of $G$ (as a discrete group) and its growth rate. To make the question more precise let's divide the class of finitely generated groups into groups of polynomial, intermediate and exponential growth and into groups which are not amenable, elementary amenable and amenable but not elementary so. This gives $9$ possible combinations of growth rate and amenability, can they all occur? Thanks to Ycor's comments the table is now complete.

$$\begin{array}{c|ccc} \text{amenable/growth} & \text{polynomial} & \text{intermediate} &\text{exponential} \\ \hline \text{no} & \varnothing & \varnothing & F_2 \\ \text{elementary} & \Bbb Z & \varnothing & BS(1,n)\\ \text{yes but not elementary} & \varnothing & \Gamma & \Bbb Z/(2)\wr\Gamma\end{array}$$

where $F_2$ is the free group on two generators, $BS(1,n)=\langle a,b\mid b^{-1}ab=a^n\rangle$ is a Baumslag-Solitair group, and $\Gamma$ is Grigorchuk's group.

I'm looking for examples to fill in the remaining cells or proofs that some of them are empty

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    $\begingroup$ Finitely generated groups of subexponential growth are amenable. $\endgroup$
    – mathworker21
    Jan 17, 2019 at 12:59
  • $\begingroup$ Thanks @mathworker21, that settles two cases! Do you have a reference I can read for a proof? $\endgroup$
    – Alessandro Codenotti
    Jan 17, 2019 at 13:33
  • $\begingroup$ I copied and pasted that from wikipedia. I'm sure wikipedia has references. If not, there's a book on amenability by paterson, and there are course notes on amenability by Kate Juschenko $\endgroup$
    – mathworker21
    Jan 17, 2019 at 14:32
  • $\begingroup$ The box (elementary amenable group of intermediate growth) is $\emptyset$. Indeed for such group Chou showed that unless virtually nilpotent, they have a free subsemigroup on 2 generators, and hence exponential growth. $\endgroup$
    – YCor
    Jul 17, 2020 at 22:13
  • $\begingroup$ The box (amenable but not elementary amenable)—(polynomial growth) is empty by Gromov's theorem. $\endgroup$
    – YCor
    Jul 17, 2020 at 22:15

1 Answer 1

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Here the short proof of the fact that every f.g. group of subexponential growth is amenable.

Let $S$ be a symmetric generating subset with 1. If $G$ has subexponential growth, then clearly $\liminf |S^{n+1}|/|S^{n}|=1$. So we can extract from $(S^n)$ a Følner sequence.

The converse fails: many f.g. solvable groups have exponential growth.

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