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arXiv · 1705.00144

Distortion in groups of Affine Interval Exchange transformations

Nancy Guelman·Isabelle Liousse
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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.

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Nancy Guelman, Isabelle Liousse. 2017-04-29. Distortion in groups of Affine Interval Exchange transformations. https://arxiv.org/abs/1705.00144

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