Relations with a fixed interval exchange transformation

TL;DR

This paper investigates the relations within the group of interval exchange transformations, showing that for any given IET, there exists a dense set of related IETs sharing relations, extending prior results on the non-freeness of generic pairs.

Contribution

It extends previous work by demonstrating the existence of dense sets of IETs sharing relations with any given IET, broadening understanding of the group's structure.

Findings

For every IET, a dense open set of related IETs exists.

The group generated by a generic pair of IETs is not free under certain conditions.

The results apply to IETs with and without flips.

Abstract

We study the group of all interval exchange transformations (IETs). We show that for every IET $S$, there exists a dense open set of admissible IETs that share a relation with $S$. This is an extension of a result published by Dahmani, Fujiwara and Guirardel in 2013: the group generated by a generic pair of elements of IET([0;1[) is not free (assuming a suitable condition on the underlying permutation). Key words: interval exchange transformations, free group of rank 2, interval exchange transformations with flips.