TITLE: Amenable exponential growth
QUESTION [7 upvotes]: 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

REPLY [3 votes]: 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.