The group of affine similarities \(\operatorname{SAff}(\mathbb{R}^d)\) consists of transformations of the form
\[
X \mapsto aGX+b,
\]
where \(G\) is a rotation, \(a \in \mathbb{R}\setminus\{0\}\), and \(b \in \mathbb{R}^d\).

Let \(\mu\) be a probability measure on \(\operatorname{SAff}(\mathbb{R}^d)\), and let \((L_n)\) denote the associated left random walk. In this talk, I will sketch the proof of a generalisation of a result of Brofferio concerning the way in which \(L_n\) diverges to infinity. I will also discuss some consequences for proving the genericity of free semigroups in \(\operatorname{SAff}(\mathbb{R}^d)\). This is based on ongoing joint work with Richard Aoun.