Uniform perfectness for Interval Exchange Transformations with or without Flips

2019/10/20 by Nancy Guelman, Guelman, Nancy, Isabelle Liousse +1
Mathematics · Engineering · #Finite Group Theory Research #graph theory and CDMA systems #Mathematics and Applications

paper · doi:10.48550/arxiv.1910.08923

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 G₀ the subgroup of \mathcal G generated by its commutators is simple. In [Arn81b], Arnoux proved that the group \mathcal G of all Interval Exchange Transformations with flips is simple. We establish that every element of \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 G₀ are uniformly bounded, and in this case for any g∈ \mathcal G₀ this length is at most 5. As analogous arguments work for the involution length in \mathcal G, we add an appendix whose purpose is to prove that every element of \mathcal G has an involution length not exceeding 12.

Related

• Relations with a fixed interval exchange transformation
• Notes on the commutator group of the group of interval exchange transformations
• Subgroup of interval exchanges generated by torsion elements and rotations
• Orbit structure of interval exchange transformations with flip
• Mild mixing of certain interval exchange transformations