Wreath product of group of integers with group of integers

Statement
The wreath product of group of integers with group of integers is defined as the restricted external wreath product:

$$\mathbb{Z} \wr \mathbb{Z}$$,

where $$\mathbb{Z}$$ is the additive group of integers, and the permutation action is the regular group action. It can also be viewed as the semidirect product of the additive group of the Laurent polynomial ring over $$\mathbb{Z}$$ with the multiplicative cyclic group generated by $$x$$.

Facts

 * This group gives an example of a finitely generated group that is not finitely presented.
 * This group occurs as a subgroup in the general affine group $$GA(1,\R)$$, as follows: let $$\alpha$$ be any transcendental real number. Consider the subgroup comprising maps of the form $$x \mapsto \alpha^k x + p(\alpha)$$, where $$k$$ is any integer (possibly negative) and $$p$$ is any Laurent polynomial in $$\alpha$$. Here, the base of the wreath product is realized as translations $$x \mapsto x + n$$, where $$n$$ is an integer, while the acting elements are of the form $$x \mapsto \alpha^k x$$.