Distortion in groups of affine interval exchange transformations

Abstract

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 $|m|\ne |n|$ , 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.