标题： 不连续域的Anosov子群在丛上的作用 显示英文标题

标题： Domain of Discontinuity of Anosov Subgroups Acting on Bundles

摘要： 设$\rho: \Gamma \to G$为一个Anosov表示，其中$\Gamma$为一个word双曲群，$G$为一个半单李群。 先前的研究（由Guichard--Wienhard、Kapovich--Leeb--Porti以及最近的Carvajales--Stecker）构造了一个不连续域$\Omega_\rho \subset G/H$，其中$H$为一个抛物型或对称子群。 在本文中，我们将适当的不连续的$\Gamma$-作用（通过$\rho$）扩展到在$\Omega_\rho$上的切丛的拉回空间的连接空间。当$\Omega_\rho$是一个复曲面时，我们证明了$\Gamma$-作用在这些拉回丛的复化结构的$(1,0)$部分的全纯丛结构的并集上是适当的不连续的。 我们进一步构造了一个由这些全纯线丛生成的拓扑自由群$F$，并证明$\rho(\Gamma)$在$F \setminus \{\mathrm{id}\}$上作用真正不连续。 这个自由群被证明在Zariski稠密Anosov表示的特征簇上至多同构意义下是定义良好的。 最后，我们将Anosov表示的空间赋予一个范畴结构，并构造了一个到自由群范畴的自然函子。 显示英文摘要

摘要： Let $\rho: \Gamma \to G$ be an Anosov representation, with $\Gamma$ a word hyperbolic group and $G$ a semisimple Lie group. Previous works (by Guichard--Wienhard, Kapovich--Leeb--Porti, and recently Carvajales--Stecker) constructed an open domain of discontinuity $\Omega_\rho \subset G/H$, where $H$ is a parabolic or symmetric subgroup. In this paper, we extend the properly discontinuous $\Gamma$-action (via $\rho$) to the space of connections on the pullback of the tangent bundle over $\Omega_\rho$. When $\Omega_\rho$ is a complex surface, we show that the $\Gamma$-action is properly discontinuous on the union of the Higgs bundle structures of the $(1,0)$ part of the complexification of these pullback bundles. We further construct a topological free group $F$ generated by these holomorphic line bundles and show that $\rho(\Gamma)$ acts properly discontinuously on $F \setminus \{\mathrm{id}\}$. This free group is shown to be well-defined up to isomorphism over the character variety of Zariski dense Anosov representations. Finally, we endow the space of Anosov representations with a categorical structure and construct a natural functor to the category of free groups.