Uniform simplicity for subgroups of piecewise continuous bijections of the unit interval

Let $I=[0,1)$ and $mathcal{PC}(I)$ [resp. $mathcal{PC}^+(I)$] be the quotient group of the group of all piecewise continuous [resp. piecewise continuous and orientation preserving] bijections of $I$ by its normal subgroup consisting in elements with finite support (i.e. that are trivial except at possibly finitely many points). Unpublished Theorems of Arnoux ([Arn81b]) state that $mathcal{PC}^+(I)$ and certain groups of interval exchanges are simple, their proofs are the purpose of the Appendix. Dealing with piecewise direct affine maps, we prove the simplicity of the group $mathcal A^+(I)$ (see Definition 1.6). These results can be improved. Indeed, a group $G$ is uniformly simple if there exists a positive integer $N$ such that for any $f,phi in Gsetminus{Id}$, the element $phi$ can be written as a product of at most $N$ conjugates of $f$ or $f^{-1}$. We provide conditions which guarantee that a subgroup $G$ of $mathcal{PC}(I)$ is uniformly simple. As Corollaries, we obtain that $mathcal{PC}(I)$, $mathcal{PC}^+(I)$, $PL^+ (mathbb S^1)$, $mathcal A(I)$, $mathcal A^+(I)$ and some Thompson like groups included the Thompson group $T$ are uniformly simple.