Transformation groups of certain flat affine manifolds

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…

Rigidity for group actions on homogeneous spaces by affine transformations

We give a criterion for the rigidity of actions on homogeneous spaces. Let $G$ be a real Lie group, $Λ$ a lattice in $G$, and $Γ$ a subgroup of the affine group Aff$(G)$ stabilizing $Λ$. Then the action of $Γ$ on $G/Λ$ has the rigidity property in the sense of S. Popa, if and only if the induced action of $Γ$ on $\mathbb{P}(\frak{g})$ admits no $Γ$-invariant probability measure, where $\frak{g}$ is the Lie algebra of $G$. This generalizes results of M. Burger, and A. Ioana and Y. Shalom. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nil…

Diophantine property in the group of affine transformations of the line

We investigate the Diophantine property of a pair of elements in the group of affine transformations of the line. We say that a pair of elements g_1,g_2 in this group is Diophantine if there is a number A such that a product of length l of elements of the set {g_1,g_2,g_1^{-1},g_2^{-1}} is either the unit element or of distance at least A^{-l} from the unit element. We prove that the set of non-Diophantine pairs in a certain one parameter family is of Hausdorff dimension 0.

仿射变换的可逆性

Reversibility of Affine Transformations

本研究围绕仿射变换的可逆性展开，明确群中可逆元与强可逆元的定义，针对仿射群$\mathrm{GL}(n, \mathbb{D}) \ltimes \mathbb{D}^n $中的可逆元与强可逆元进行分类研究，为仿射变换相关群论研究提供分类依据。

仿射变换离散群的真作用

Proper actions of discrete groups of affine transformations

本文围绕Milnor1977年提出的问题，介绍Margulis关于自由群三维真仿射作用存在性的突破，梳理相关历史背景、领域最新进展及该发现启发的未来研究方向。

洛伦兹仿射群作用的不连续域

Domains of discontinuity of Lorentzian affine group actions

本文研究仿射洛伦兹变换 Anosov 群的作用，证明其真不连续域具有非空性。

关于共逆元仿射变换的乘积

On the Product of Coninvolutory Affine Transformations

本文研究了共逆元仿射变换的乘积性质，证明了仿射变换可表示为两个或更多共逆元的乘积，且满足特定条件。

Spectral gap in the group of affine transformations over prime fields

We study random walks on the semi-direct product of F_p^d and SL_d(F_p). We estimate the spectral gap in terms of the spectral gap of the projection to the linear part SL_d(F_p). This problem is motivated by an analogue in the isometry group of Euclidean space, which have application to smoothness of self-similar measures.