Distortion in groups of Affine Interval Exchange transformations

2017/04/29 by Guelman, Nancy, Liousse, Isabelle
#37E05 37C15 37C85 20E07 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1705.00144

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 ≠ \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.

Related

• Interval Exchange Transformations groups. Free actions and dynamics of virtually abelian groups
• Interval exchanges that do not embed in free groups
• Subgroup of interval exchanges generated by torsion elements and rotations
• On groups of rectangle exchange transformations
• Reversibility of Affine Transformations