# Amenable exponential growth

Dear forum members,

Does anyone have a clear example of an amenable group with exponential growth?

Is real that if G is virtually amenable (has an amenable subgroup of finite index) then it is amenable?

Many thanks

• I think the answers to both your questions ought to be found in Paterson's book "Amenability", which does what it says on the tin – Yemon Choi Feb 20 '11 at 19:35
• The answers below are good, but perhaps low on explicit examples. As Igor notes, any solvable, non-virtually nilpotent group will do. Examples include the solvable Baumslag--Solitar groups $BS(1,n)=\langle a,b\mid b^{-1}ab=a^n\rangle$. – HJRW Feb 20 '11 at 22:09

any solvable group which is not virtually nilpotent has exponential growth. For an example, take a semi-direct product of $Z$ and the direct sum $A$ of infinitely many copies of ${\mathbb Z}$ where the cyclic group acts by translations. It is not hard to see that the growth of the balls is exponential. In fact consider all sequences $(a_i)$ in $A$ with support $[1,\sqrt{n}]$ where $a_i$ is arbitrarily chosen from the set $0,1,2$. It is easy to see that the length of this element in the group is at most $$C \sum_{i=1}^{\sqrt{n}} i a_i \le 2C \sqrt{n}^2/2= C n$$ which gives you the exponential grwoth

• I think you first sentence is not correct. there are groups of intermediate growth (i.e Grigorchuk group). – Mustafa Gokhan Benli Feb 20 '11 at 19:34
• Presumably Keivan means `any solvable group which is not virtually nilpotent has exponential growth'. This is the Milnor--Wolf Theorem. – HJRW Feb 20 '11 at 22:06
• Also, for concision, it might be worth noting that Keivan's example is $\mathbb{Z}\wr\mathbb{Z}$. – HJRW Feb 20 '11 at 22:10
• Oh I see I read it as "any amenable group". – Mustafa Gokhan Benli Feb 20 '11 at 22:48
• Thank you all for the comments. Yes, as Henry mentioned I meant solvable. – Keivan Karai Feb 22 '11 at 15:02

All solvable groups are amenable, but many have exponential growth. Look at J. Milnor's classic 1968 paper...

@ HW: One could add a polycyclic example, e.g. the semi-direct product $\mathbb{Z}^2\rtimes \mathbb{Z}$, where $\mathbb{Z}$ acts by powers of some hyperbolic automorphism, e.g. the Anosov matrix (2,1,1,1).

Basilica group is amenable but not subexponentially amenable. See L.Bartholdi, B.Virág "Amenability via random walks", Duke Math. J. Volume 130, Number 1 (2005), 39-56.

The lampligher group is a nice example, see e.g. the answer to this question by Jeremy Voltz, in which he wrote:

The lamplighter group, defined as the wreath product $\mathbb Z/2\mathbb Z\wr \mathbb Z$, is amenable yet has exponential growth. It can be thought of as a bi-infinite sequence of street lamps, each of which can be turned on and off, and a lamplighter who moves along the sequence. The three generators of the group are to move the lamplighter right or left, and to switch the state of the lamp he is positioned in front of. With this picture in mind, it is easy to show the group has exponential growth.

• this is a particular case of Keivan Karai's answer. – YCor Jan 15 '17 at 12:28
• Yes, the lamplighter group is solvable and not virtually nilpotent, so it is a concrete instance of the (very general and non-concrete) first sentence in his answer. No, it is not the same as the group $\mathbb Z\wr\mathbb Z$ that the rest of the answer is devoted to describing. – pre-kidney Jan 15 '17 at 12:38