与固定区间交换变换的关系

Distortion in groups of Affine Interval Exchange transformations

In this paper, we study distortion in the group $\mathcal A$ of Affine Interval Exchange Transformations (AIET). We prove that any distorted element $f$ of $\mathcal A$, has an iterate $f^ k$ that is conjugate by an element of $\mathcal A$ to a product of infinite order restricted rotations, with pairwise disjoint supports. As consequences we prove that no Baumslag-Solitar group, $BS(m,n)$ with $\vert m \vert \neq \vert n \vert$, acts faithfully by elements of $\mathcal A$, every finitely generated nilpotent group of $\mathcal A$ is virtually abelian and there is no distortion element in $\mat…

Free groups of interval exchange transformations are rare

We study the group IET of all interval exchange transformations. Our first main result is that the group generated by a generic pairs of elements of IET is not free (assuming a suitable irreducibility condition on the underlying permutation). Then we prove that any connected Lie group isomorphic to a subgroup of IET is abelian. Additionally, we show that IET contains no infinite Kazhdan group. We also prove residual finiteness of finitely presented subgroups of IET and give an example of a two-generated subgroup of IET of exponential growth that contains an isomorphic copy of every finite grou…

On groups of rectangle exchange transformations

We study a generalization Rec_d of the group IET=Rec_1 of interval exchange transformations in every dimension d>0, called the rectangle exchange transformations group. The subset of restricted rotations in IET is a generating subset and we prove that a natural generalization of these elements, called restricted shuffles, form a generating subset of Rec_d. We denote by T_d the subset of Rec_d made up of those transformations that permute two disjoint rectangles by translations. We prove that the derived subgroup of Rec_d is generated by T_d. We also identify the abelianization of Rec_d.

Notes on the commutator group of the group of interval exchange transformations

We study the group of interval exchange transformations and obtain several characterizations of its commutator group. In particular, it turns out that the commutator group is generated by elements of order 2.

Uniform perfectness for Interval Exchange Transformations with or without Flips

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 uni…

康托集的局部单调同胚群与广义区间交换变换群中的扭曲

Distortion in the group of locally monotone homeomorphisms of a Cantor set and in the group of generalized interval exchange transformations

研究广义区间交换变换或实直线上康托子集的局部单调同胚，证明间断点数有界、共轭于特定同胚限制、在相应群中扭曲这三个条件等价。

某些区间交换变换的初等可容许子群

Some elementary amenable subgroups of interval exchange transformations

本文研究了区间交换变换中某些初等可容许子群的性质，探讨了其非虚拟可解性和可解性条件，并展示了无限多个非线性且非虚拟可解的群以及可解群的导出长度。

区间交换变换群：几乎阿贝尔群的自由作用与动力学

Interval Exchange Transformations groups. Free actions and dynamics of virtually abelian groups

本文研究区间交换变换群的自由与非自由作用，分类包含$\mathbb Z^2$的群的自由作用，并构造非几乎幂零的IET子群例子。

Extensive amenability and an application to interval exchanges

Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group IET of interval exchange transformations that have angular components of rational rank~${\leq 2}$. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amena…

A finite interval in the subsemigroup lattice of the full transformation monoid

In this paper we describe a portion of the subsemigroup lattice of the \emph{full transformation semigroup} $Ω^Ω$, which consists of all mappings on the countable infinite set $Ω$. Gavrilov showed that there are five maximal subsemigroups of $Ω^Ω$ containing the symmetric group $\sym(Ω)$. The portion of the subsemigroup lattice of $Ω^Ω$ which we describe is that between the intersection of these five maximal subsemigroups and $Ω^Ω$. We prove that there are only 38 subsemigroups in this interval, in contrast to the $2^{2^{\aleph_0}}$ subsemigroups between $\sym(Ω)$ and $Ω^Ω$.

Abelianization of some groups of interval exchanges

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 …