Distortion in groups of affine interval exchange transformations

In this paper, we study distortion in the group $\mathcal A$ of affine interval exchange transformations (AIET). We prove that any distorted element $f$ of $\mathcal A$ has an iterate $f^k$ that is conjugate by an element of $\mathcal A$ to a product of infinite order restricted rotations, with pairwise disjoint supports. As consequences, we prove that no Baumslag–Solitar group, BS$(m,n)$ with $\vert m \vert \neq \vert n \vert $, acts faithfully by elements of $\mathcal A$; every finitely generated nilpotent group of $\mathcal A$ is virtually abelian and there is no distortion element in $\mathcal A_{\mathbb Q}$, the subgroup of $\mathcal A$ consisting of rational AIETs.