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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?

I am sort of novice in advanced group theory. All your comments are more than welcome.

Many thanks

flag	asked Feb 20 2011 at 18:47 AlGoRiS 127●4

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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 2011 at 19:35

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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 $BS(1,n)=\langle a,b\mid b^{-1}ab=a^n\rangle$. – HW Feb 20 2011 at 22:09

Keivan Karai · Accepted Answer · 2011-02-22 15:02:36Z UTC

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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

edited Feb 22 2011 at 15:02

answered Feb 20 2011 at 19:17

Keivan Karai
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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 2011 at 19:34

Presumably Keivan means `any *solv*able group which is not virtually nilpotent has exponential growth'. This is the Milnor--Wolf Theorem. – HW Feb 20 2011 at 22:06

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Also, for concision, it might be worth noting that Keivan's example is $\mathbb{Z}\wr\mathbb{Z}$. – HW Feb 20 2011 at 22:10

Oh I see I read it as "any amenable group". – Mustafa Gokhan Benli Feb 20 2011 at 22:48

Thank you all for the comments. Yes, as Henry mentioned I meant solvable. – Keivan Karai Feb 22 2011 at 15:02

Igor Rivin · Answer 2 · 2011-02-20 18:59:46Z UTC

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

answered Feb 20 2011 at 18:59

Igor Rivin
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Ievgen Bondarenko · Answer 3 · 2011-02-20 19:05:13Z UTC

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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.

answered Feb 20 2011 at 19:05

Ievgen Bondarenko
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Alain Valette · Answer 4 · 2011-04-19 05:25:16Z UTC

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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).

answered Apr 19 2011 at 5:25

Alain Valette
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Amenable exponential growth

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