Abelianization of some groups of interval exchanges

Abstract

Let IET be the group of bijections from $[0,1[$ to itself that are continuous outside a finite set, right-continuous and piecewise translations. The abelianization homomorphism $f:\text{IET}\to A$, called SAF-homomorphism, was described by Arnoux-Fathi and Sah. The abelian group $A$ is the second exterior power of the reals over the rationals. For every subgroup $\Gamma$ of $\mathbb{R}/\mathbb{Z}$ we define $\text{IET}(\Gamma )$ as the subgroup of $\text{IET}$ consisting of all elements $f$ such that $f$ is continuous outside $\Gamma$. Let $\tilde{\Gamma }$ be the preimage of $\Gamma$ in $\mathbb{R}$. We establish an isomorphism between the abelianization of $\text{IET}(\Gamma )$ and the second skew-symmetric power of $\tilde{\Gamma }$ over $\mathbb{Z}$ denoted by ${}^{⊝}{\bigwedge }_{\mathbb{Z}}^2\tilde{\Gamma }$. This group often has non-trivial $2$-torsion, which is not detected by the SAF-homomorphism. We then define ${\text{IET}}^{\bowtie }$ the group of all interval exchange transformations with flips. Arnoux proved that this group is simple thus perfect. However for every subgroup ${\text{IET}}^{\bowtie }(\Gamma )$ we establish an isomorphism between its abelianization and $⟨\{a\otimes a[\text{mod}2]\mid a\in \tilde{\Gamma }\}⟩\times ⟨\{\ell \wedge \ell [\text{mod}2]\mid \ell \in \tilde{\Gamma }\}⟩$ which is a $2$-elementary abelian subgroup of ${⨂}_{\mathbb{Z}}^2\tilde{\Gamma }/(2{⨂}_{\mathbb{Z}}^2\tilde{\Gamma })\times {}^{⊝}{\bigwedge }_{\mathbb{Z}}^2\tilde{\Gamma }/(2{}^{⊝}{\bigwedge }_{\mathbb{Z}}^2\tilde{\Gamma })$.

42 pages, 9 figures