仿射变换离散群的真作用

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…

Affine actions on Nilpotent Lie groups

To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on the Lie algebra level. As an example of the possible use of this translation, we then consider the case where dim(G)=dim(N) less than 6. Finally, we specialize to the case of abelian simply transitive affi…

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.

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…

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.

Notes on Actions of Sets and Groups and Generalized Affine Spaces

Some well-known and less well-known or new notions related to group actions are surveyed. Some of these notions are used to generalize affine spaces. Actions are seen as functions with values in transformation monoids

Affine Sieve

We establish the main saturation conjecture in [BGS10] connected with executing a Brun sieve in the setting of an orbit of a group of affine linear transformations. This is carried out under the condition that the Zariski closure of the group is Levi-semisimple. It is likely that this condition is also necessary for such saturation to hold.