# Distortion in groups of Affine Interval Exchange transformations

## TL;DR

This paper investigates distortion phenomena in the group of Affine Interval Exchange Transformations, revealing structural properties of distorted elements and implications for group actions and subgroup classifications.

## Contribution

It characterizes distorted elements in AIET groups and shows that certain groups cannot act faithfully, providing new insights into the structure of AIET groups.

## Key findings

- Distorted elements have iterates conjugate to products of infinite order rotations.
- No faithful action of Baumslag-Solitar groups with |m| ≠ |n| on AIETs.
- Finitely generated nilpotent subgroups of AIETs are virtually abelian.

## 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 $\vert m \vert \neq \vert n \vert$, 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.