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



Amenable exponential growth
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


