# 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

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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

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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...

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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.

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@ 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).

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