Uniform perfectness for Interval Exchange Transformations with or without Flips

TL;DR

This paper investigates the algebraic structure of groups of interval exchange transformations, establishing bounds on commutator and involution lengths, and providing conditions for uniform boundedness of these lengths.

Contribution

It proves that all elements of the group with flips have a commutator length at most 6 and identifies conditions for bounded commutator lengths in the subgroup, also bounding involution length.

Findings

Elements of the group with flips have commutator length ≤ 6.

Conditions are given for uniform boundedness of commutator lengths.

Involution length of all elements with flips is at most 12.

Abstract

Let $\mathcal G$ be the group of all Interval Exchange Transformations. Results of Arnoux-Fathi ([Arn81b]), Sah ([Sah81]) and Vorobets ([Vor17]) state that $\mathcal G_0$ the subgroup of $\mathcal G$ generated by its commutators is simple. In [Arn81b], Arnoux proved that the group $\overline{\mathcal G}$ of all Interval Exchange Transformations with flips is simple. We establish that every element of $\overline{\mathcal G}$ has a commutator length not exceeding $6$. Moreover, we give conditions on $\mathcal G$ that guarantee that the commutator lengths of the elements of $\mathcal G_0$ are uniformly bounded, and in this case for any $g\in \mathcal G_0$ this length is at most $5$. As analogous arguments work for the involution length in $\overline{\mathcal G}$, we add an appendix whose purpose is to prove that every element of $\overline{\mathcal G}$ has an involution length not exceeding $12$.