Uniform perfectness for Interval Exchange Transformations with or without Flips

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 $\bar{\mathcal{G}}$ of all Interval Exchange Transformations with flips is simple. We establish that every element of $\bar{\mathcal{G}}$ 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 $\overline{\mathcal G}$,weaddanappendixwhosepurposeistoprovethateveryelementof$\overline{\mathcal G}$hasaninvolutionlengthnotexceeding$12$.