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?
I am sort of novice in advanced group theory. All your comments are more than welcome.
Many thanks
 A: @ 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).
A: All solvable groups are amenable, but many have exponential growth. Look at J. Milnor's classic 1968 paper...
A: 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.
A: 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.

A: 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
