Distortion in groups of affine interval exchange transformations

  • Nancy Guelman

    Universidad de la República, Montevideo, Uruguay
  • Isabelle Liousse

    Université de Lille 1, France
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Abstract

In this paper, we study distortion in the group A\mathcal A of affine interval exchange transformations (AIET). We prove that any distorted element ff of A\mathcal A has an iterate fkf^k that is conjugate by an element of A\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)(m,n) with mn\vert m \vert \neq \vert n \vert , acts faithfully by elements of A\mathcal A; every finitely generated nilpotent group of A\mathcal A is virtually abelian and there is no distortion element in AQ\mathcal A_{\mathbb Q}, the subgroup of A\mathcal A consisting of rational AIETs.

Cite this article

Nancy Guelman, Isabelle Liousse, Distortion in groups of affine interval exchange transformations. Groups Geom. Dyn. 13 (2019), no. 3, pp. 795–819

DOI 10.4171/GGD/505