On groups of rectangle exchange transformations

TL;DR

This paper introduces the rectangle exchange transformations group Rec_d, generalizing interval exchange transformations, and analyzes its structure, including generators, derived subgroup, and abelianization.

Contribution

It defines Rec_d, shows restricted shuffles generate Rec_d, and characterizes the derived subgroup and abelianization of Rec_d.

Findings

Restricted shuffles generate Rec_d

Derived subgroup is generated by transformations permuting rectangles

Identified the abelianization of Rec_d

Abstract

We study a generalization Rec_d of the group IET=Rec_1 of interval exchange transformations in every dimension d>0, called the rectangle exchange transformations group. The subset of restricted rotations in IET is a generating subset and we prove that a natural generalization of these elements, called restricted shuffles, form a generating subset of Rec_d. We denote by T_d the subset of Rec_d made up of those transformations that permute two disjoint rectangles by translations. We prove that the derived subgroup of Rec_d is generated by T_d. We also identify the abelianization of Rec_d.