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Register a new userUniform simplicity for subgroups of piecewise continuous bijections of the unit interval
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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 Unpublished Theorems of Arnoux ([Arn81b]) state that $mathcal{PC}^+(I)$ and certain groups of interval exchanges are simple, their proofs are the purpose of the Appendix.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.
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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 $overline{mathcal G}$ of all Interval Exchange Transformations with flips is simple. We establish that every element of $overline{mathcal G}$ 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 $gin mathcal G_0$ this length is at most $5$. As analogous arguments work for the involution length in $overline{mathcal G}$, we add an appendix whose purpose is to prove that every element of $overline{mathcal G}$ has an involution length not exceeding $12$.
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 PC $rightarrow$ PC splits modulo the alternating subgroup of even permutations. This is shown by constructing a nonzero group homomorphism, called signature, from PC to Z 2Z. Then we use this signature to list normal subgroups of every subgroup G of PC which contains S fin and such that G, the projection of G in PC , is simple.
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 $fin C[0,1]$ (or $fin C^infty [0,1]$) and $Dsubset E(f)$. This generalizes some ideas of Zabeti. We observe that, if $f$ is continuous, then $E(f)$ is a closed nowhere dense subset of $f^{-1}[{ 0}]$ where each $xin {0,1}cap E(f)$ is an accumulation point of $E(f)$. Our main result states that, for a closed nowhere dense set $Fsubset [0,1]$ with each $xin {0,1}cap E(f)$ being an accumulation point of $F$, there exists $fin C^infty [0,1]$ such that $F=E(f)$.
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 $delta(mu)$ of any discrete invariant random subgroup $mu$ of the locally compact group $G$ and show that $delta(mu) > frac{d}{2}$ in general and that $delta(mu) = d$ if $mu$ is of divergence type. Whenever $G$ is a rank-one simple Lie group with Kazhdans property $(T)$ it follows that an ergodic invariant random subgroup of divergence type is a lattice. One of our main tools is a maximal ergodic theorem for actions of hyperbolic groups due to Bowen and Nevo.
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