论文标题
de surface de dan lesréseauxdes de lie lie lie lie semi-simples [d'AprèsJ。Kahn，V。Marković，U。Hamenstädt，F。Labourieet S. Mozes]
Groupes de surface dans les réseaux des groupes de Lie semi-simples [d'après J. Kahn, V. Marković, U. Hamenstädt, F. Labourie et S. Mozes]
论文作者
论文摘要
半完整的Lie Group $ G $中的CoCoCompact晶格是一个离散的亚组$γ$，因此商$ g/γ$是紧凑的。这样的晶格是否总是包含表面基，即紧凑型双曲表面的基本组的亚组同构？如果是这样，它是否包含表面亚组关闭（从精确的定量意义上）到$ g $的fuchsian子组，即在$ \ operatorname {（p）sl}副本中包含的$ g $的离散子组（2，2，2，\ mathbf {r}）in $ g $？情况$ g = \ operatorname {psl}（2，\ m马理bf {c}）$ $对应于3二维双曲线歧管上的瑟斯顿的著名猜想，以及case $ g = \ operatation的定量版本\ operatatorName {psl}（2，\ mathbf {r}）$意味着在成对紧凑的双曲线表面上的Ehrenpreis的猜想；十年前，卡恩（Kahn）和马尔可（Marković）证明了这两个猜想。 Hamenstädt是出于Gromov的一个问题的动机，解决了$ G $具有实际等级的情况，除了$ G = \ operatatorName {so}（so}（2n，1）$。在最近的预印本（Arxiv：1805.10189）中，卡恩，劳里和莫兹斯对待大量半圣母谎言组的情况，包括尤其是所有复杂的简单谎言组；他们获得的表面组是劳动意义上的表示形式的图像。我们介绍了他们证明的一些想法。
A cocompact lattice in a semisimple Lie group $G$ is a discrete subgroup $Γ$ such that the quotient $G/Γ$ is compact. Does such a lattice always contain a surface group, i.e. a subgroup isomorphic to the fundamental group of a compact hyperbolic surface? If so, does it contain surface subgroups close (in a precise quantitative sense) to Fuchsian subgroups of $G$, i.e to discrete subgroups of $G$ contained in a copy of $\operatorname{(P)SL}(2,\mathbf{R})$ in $G$? The case $G=\operatorname{PSL}(2,\mathbf{C})$ corresponds to a famous conjecture of Thurston on 3-dimensional hyperbolic manifolds, and the quantitative version of the case $G=\operatorname{PSL}(2,\mathbf{R}) \times \operatorname{PSL}(2,\mathbf{R})$ implies a conjecture of Ehrenpreis on pairs of compact hyperbolic surfaces; these two conjectures were proved by Kahn and Marković around ten years ago. Motivated by a question of Gromov, Hamenstädt solved the case that $G$ has real rank one, except for $G=\operatorname{SO}(2n,1)$. In a recent preprint (arXiv:1805.10189), Kahn, Labourie, and Mozes treat the case of a large class of semisimple Lie groups, including in particular all complex simple Lie groups; the surface groups they obtain are images of representations that are Anosov in the sense of Labourie. We present some of the ideas of their proof.
