Entropy and affine actions for surface groups

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…

Notes on affine isometric actions of discrete groups

Consider a lattice $Γ$ in a group $G = SL_2(\R), SO(1,n), SU(1,n)$, $SL_2(\Q_p)$. We discuss actions of $Γ$ by affine isometric transformations of Hilbert spaces. We show that for irreducible affine isometric action of $G$ its restriction to $Γ$ is irreducible. We prove the existence of canonical irreducible affine isometric actions of $Γ$ associated to actions of $Γ$ on $\R$- trees. Using such actions we construct irreducible representations of semigroup of probabilistic measures on $Γ$ and construct the series of representations of the groups of diffeomorphisms of Riemann surfaces enumerated…

Path group actions induced by sigma-actions and affine Kac-Moody symmetric spaces of group type

In 1995, C.-L. Terng associated to each hyperpolar action on a compact symmetric space, a hyperpolar proper Fredholm (PF) action on a Hilbert space. This is a group action by an infinite dimensional path group and it acts on a Hilbert space via the gauge transformations. Those two hyperpolar actions are related through an equivariant Riemannian submersion called the parallel transport map and they have close relations to the infinite dimensional symmetric spaces called affine Kac-Moody symmetric spaces. In this paper we define a linear isomorphism between Hilbert spaces and show that it is equ…

球面上的$L_p$-浮动面积、曲率熵与等周不等式

The $L_p$-floating area, curvature entropy, and isoperimetric inequalities on the sphere

该研究将欧氏几何中的$L_p$-仿射表面积等概念推广到球面空间，引入球面凸体的$L_p$-浮动面积与曲率熵，证明相关等周不等式、对偶公式与单调性，并推广到非负常曲率空间形式。

Constructing proper affine actions via higher strip deformations

We introduce higher strip deformations, which give a way of constructing affine deformations of discrete free groups in the image of the irreducible representation $\operatorname{PSL}_2\mathbb{R}\to \operatorname{SO}(2n,2n-1)$. We use the Margulis invariant to find a properness condition for these deformations, showing that each convex cocompact free group admits affine deformations acting properly on $\mathbb{R}^{4n-1}$. Using our method we can also construct proper affine actions of virtually free groups, which we demonstrate by finding proper actions of $C_2 \star C_3$ on $\mathbb{R}^7.$

Affine actions with Hitchin linear part

Properly discontinuous actions of a surface group by affine automorphisms of $\mathbb R^d$ were shown to exist by Danciger-Gueritaud-Kassel. We show, however, that if the linear part of an affine surface group action is in the Hitchin component, then the action fails to be properly discontinuous. The key case is that of linear part in $\mathsf{SO}(n,n-1)$, so that the affine action is by isometries of a flat pseudo-Riemannian metric on $\mathbb R^d$ of signature $(n,n-1)$. Here, the translational part determines a deformation of the linear part into $\mathsf{PSO}(n,n)$-Hitchin representations …

Fuchsian表示的形变与真仿射作用

Deformation of Fuchsian representations and proper affine actions

本文推广Mess的工作，研究Hitchin表示中Fuchsian表示的无穷小形变，引入仿射交比、三交比与Margulis不变量，给出真仿射作用的等价判据，并证明线性部分为Hitchin表示的非阿贝尔自由群存在真仿射作用。

K-theory of affine actions

For a Lie group $G$ and a vector bundle $E$ we study those actions of the Lie group $TG$ on $E$ for which the action map $TG\times E \to E$ is a morphism of vector bundles, and call those \emph{affine actions}. We prove that the category $\mathrm{Vect}_{TG}^{\mathrm{aff}}\left(X\right)$ of such actions over a fixed $G$-manifold $X$ is equivalent to a certain slice category $\mathfrak g_X \backslash \mathrm{Vect}_G\left(X\right)$. We show that there is a monadic adjunction relating $\mathrm{Vect}_{TG}^{\mathrm{aff}}\left(X\right)$ to $\mathrm{Vect}_G\left(X\right)$, and the right adjoint of thi…

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。

The deformations of flat affine structures on the two-torus

The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group $\Aff(2)$ on $\bbR^2$. Since this action has non-compact stabiliser $\GL(2,\bbR)$, the underlying locally homogeneous geometry is highly non-Riemannian. In this article, we describe the deformation space of all flat affine structures on the two-torus. In this context interesting phenomena arise in the topology of the deformation space, which, for example, is \emph{not} a Hausdorff space. This contrasts with the case of constant curvature metri…

Affine actions on Lie groups and post-Lie algebra structures

We introduce post-Lie algebra structures on pairs of Lie algebras $(\Lg,\Ln)$ defined on a fixed vector space $V$. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtain several results on the existence of post-Lie algebra structures, in terms of the algebraic structure of the two Lie algebras $\Lg$ and $\Ln$. One result is, for example, that if there exists a post-Lie algebra structure on $(\Lg,\Ln)$, where $\Lg$ is nilpotent, then $\Ln$ must …

仿射 Anosov 表示与真作用

Affine Anosov representations and proper actions

本文定义词双曲群到特定仿射群的仿射 Anosov 表示，并证明其等价于线性部分关于极大迷向平面稳定子为 Anosov 且群像在欧氏空间上真作用。