Lamplighter group

Example

Let $L = \bigoplus_{i \in \Z} \Z/2\Z$ be the group of finitely supported functions $f \colon \Z \to \Z/2\Z$, with pointwise addition. Let $\Z$ act on $L$ by shifts: $(n \cdot f)(i) = f(i - n)$. The lamplighter group is the semidirect product $$\Z/2\Z \wr \Z = L \rtimes \Z,$$ the restricted wreath product of $\Z/2\Z$ by $\Z$. An element is a pair $(f, n)$. Here $f$ says which lamps on the integer line are lit, finitely many of them, and $n$ is the position of the lamplighter.

It is generated by $t = (0, 1)$, which moves the lamplighter one step, and $a = (\delta_0, 0)$, which switches the lamp at $0$. So it is a finitely generated group. It is an extension of the abelian group $L$ by the abelian group $\Z$, so it is solvable of derived length $2$. Solvable groups are amenable, so it is amenable.