论文标题
某些间隔交换组的ABELIANIAD
Abelianization of some groups of interval exchanges
论文作者
论文摘要
让IET成为$ \ Mathopen {[} 0,1 \ Mathclose {[} $的二类二十架射击，它们都在有限的集合外，直接和分段翻译之外连续。 Abelianization同构$ f：\ text {iet} \ to to to a $，称为Saf-homormormorphism，由Arnoux-Fathi和Sah描述。 Abelian Group $ a $是皇家对理由的第二个外部力量。 对于$ \ mathbb {r/z} $的每个子组$γ$，我们将$ \ text {iet}（γ）$定义为$ \ text {iet} $的子组，由所有元素$ f $组成，以便$ f $ you $ f $是连续的$γ$。令$ \tildeγ$为$ \ MATHBB {r} $中的$γ$的预先映射。我们建立了$ \ text {iet}（γ）$的阿elianization与$ \tildeγ$的第二个偏斜 - 对称能力，$ \ mathbb {z} $由$ {}。该小组通常具有非平凡的$ 2 $ torsion，这是SAF骑行的检测到的。 然后，我们定义$ \ text {iet}^{\ bowtie} $与翻转所有间隔交换转换的组。 Arnoux证明了这个群体很简单，因此很完美。 However for every subgroup $\text{IET}^{\bowtie}(Γ)$ we establish an isomorphism between its abelianization and $\langle \lbrace a \otimes a ~ [\text{mod}~2] \mid a \in \tildeΓ \rbrace \rangle \times \langle \lbrace \ell \ wedge \ ell〜 [\ text {mod} 〜2] \ mid \ ell \ in \ in \tildeγ\ rbrace \ rangle $，是$ 2 $ -2 $ - elementary abelian abelian子$ \ bigotimes^2 _ { （2 \ bigotimes^2 _ {\ mathbb {z}} \tildeγ）\ times {}^\ circleddash \！\！\！ {}^\ circleddash \！\！\ bigWedge^2 _ {\ Mathbb {z}}} \tildeγ）$。
Let IET be the group of bijections from $\mathopen{[}0,1 \mathclose{[}$ 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 $Γ$ of $\mathbb{R/Z}$ we define $\text{IET}(Γ)$ as the subgroup of $\text{IET}$ consisting of all elements $f$ such that $f$ is continuous outside $Γ$. Let $\tildeΓ$ be the preimage of $Γ$ in $\mathbb{R}$. We establish an isomorphism between the abelianization of $\text{IET}(Γ)$ and the second skew-symmetric power of $\tildeΓ$ over $\mathbb{Z}$ denoted by ${}^\circleddash\!\!\bigwedge^2_{\mathbb{Z}} \tildeΓ$. 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 $\text{IET}^{\bowtie}(Γ)$ we establish an isomorphism between its abelianization and $\langle \lbrace a \otimes a ~ [\text{mod}~2] \mid a \in \tildeΓ \rbrace \rangle \times \langle \lbrace \ell \wedge \ell ~ [\text{mod}~2] \mid \ell \in \tildeΓ \rbrace \rangle$ which is a $2$-elementary abelian subgroup of $\bigotimes^2_{\mathbb{Z}} \tildeΓ / (2\bigotimes^2_{\mathbb{Z}} \tildeΓ) \times {}^\circleddash\!\!\bigwedge^2_{\mathbb{Z}} \tildeΓ / (2 {}^\circleddash\!\!\bigwedge^2_{\mathbb{Z}} \tildeΓ)$.
