Subgroup of interval exchanges generated by torsion elements and rotations

TL;DR

This paper characterizes a subgroup of interval exchange transformations generated by torsion elements and rotations, using the SAF invariant, and explores conditions for membership based on interval length dependencies.

Contribution

It provides a constructive characterization of the subgroup generated by torsion elements and rotations in the group of IETs, using the SAF invariant and recent results on the commutator subgroup.

Findings

Elements of the subgroup are characterized by their SAF invariant.

A non-rotation 3-IET is in the subgroup if interval lengths are rationally dependent.

The subgroup is strictly contained within the full group of IETs.

Abstract

Denote by $G$ the group of interval exchange transformations (IETs) on the unit interval. Let $G_{per}\subset G$ be the subgroup generated by torsion elements in $G$ (periodic IETs), and let $G_{rot}\subset G$ be the subset of 2-IETs (rotations). The elements of the subgroup $G_1=< G_{per},G_{rot}>\subset G$ (generated by the sets $G_{per}$ and $G_{rot}$) are characterized constructively in terms of their Sah-Arnoux-Fathi (SAF) invariant. The characterization implies that a non-rotation type 3-IET lies in $G_1$ if and only if the lengths of its exchanged intervals are linearly dependent over $\Q$. In particular, $G_1\subsetneq G$. The main tools used in the paper are the SAF invariant and a recent result by Y. Vorobets that $G_{per}$ coincides with the commutator subgroup of $G$.