Uniform perfectness for Interval Exchange Transformations with or without Flips

• Published in: arXiv.org 2021-09
• Main Authors: Guelman, Nancy, Liousse, Isabelle
• Format: Article
• Language: English
• Subjects: Commutators; Exchanging; Subgroups; Transformations
• ISSN: 2331-8422

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 \(12\).