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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 Unpublished Theorems of Arnoux ([Arn81b]) state that $mathcal{PC}^+(I)$ and certain groups of interval exchanges are simple, their proofs are the purpose of the Appendix.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.
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],
Let PC be the group of bijections from [0, 1[ to itself which are continuous outside a finite set. Let PC be its quotient by the subgroup of finitely supported permutations. We show that the Kapoudjian class of PC vanishes. That is, the quotient map
We show that certain groups of piecewise linear homeomorphims of the interval are invariably generated.
For a function $fcolon [0,1]tomathbb R$, we consider the set $E(f)$ of points at which $f$ cuts the real axis. Given $fcolon [0,1]tomathbb R$ and a Cantor set $Dsubset [0,1]$ with ${0,1}subset D$, we obtain conditions equivalent to the conjunction $f
Let $X$ be a proper geodesic Gromov hyperbolic metric space and let $G$ be a cocompact group of isometries of $X$ admitting a uniform lattice. Let $d$ be the Hausdorff dimension of the Gromov boundary $partial X$. We define the critical exponent $del