Uniform perfectness for Interval Exchange Transformations with or without Flips

Let G be the group of all Interval Exchange Transformations. Results of Arnoux-Fathi ([Arn81b]), Sah ([Sah81]) and Vorobets ([Vor17]) state that G 0 the subgroup of G generated by its commutators is simple. In ([Arn81b]), Arnoux proved that the group G of all Interval Exchange Transformations with flips is simple. We establish that every element of G has a commutator length not exceeding 6. Moreover, we give conditions on G that guarantee that the commutator lengths of the elements of G 0 are uniformly bounded, and in this case for any g ∈ G 0 this length is at most 5. As analogous arguments work for the involution length in G, we add an appendix whose purpose is to prove that every element of G has an involution length not exceeding 12.