Properly Discontinuous Actions on Homogeneous Spaces

均匀空间上的适当不连续动作

基本信息

批准号：
1709952
负责人：
Yair Minsky
金额：
$ 9.38万
依托单位：
Yale University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-15 至 2019-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1709952&HistoricalAwards=false
关键词：
Properly Discontinuous Actions Homogeneous Spaces

项目摘要

The main motivation for this project is the so-called Auslander's conjecture, which is a very fundamental question about groups of affine transformations that act properly and discontinuously on the affine space. These groups roughly correspond to regular affine tilings, i.e., tilings where all the tiles have the same shape up to an affine transformation. These groups also admit an interpretation in differential geometry as flat affine manifolds. The special case where these groups preserve a Euclidean metric, which in the language of tilings means that all the tiles have the same shape in an ordinary sense, has been very well-known for over a century: this is basically the subject of classical crystallography. In the general case, there is a much larger variety of these groups; and the question of their classification also leads to some fascinating questions in the theory of Lie groups and their representations.The first part of the project consists in working on a conjecture that would give, for every semisimple real Lie group G and irreducible representation V thereof, a necessary and sufficient criterion for the existence of a subgroup Gamma in the group of affine automorphisms of V with linear part in G, such that Gamma has linear part Zariski-dense in G, is free nonabelian and acts properly discontinuously on V. The second part of the project consists in translating this abstract algebraic criterion into a simple condition on the highest restricted weight of the representation, and thus completely classifying Zariski-closures of such subgroups. The last part of the proposal consists in gaining a better understanding of these groups: for instance, constructing whenever possible a fundamental domain corresponding to their proper action on the affine space; going beyond proving existence or non-existence of such groups, by looking for results that classify all such groups for a given Zariski closure; and trying to link them with free groups acting properly on other homogeneous spaces.

该项目的主要动机是所谓的奥斯兰德猜想，这是一个关于在仿射空间上正确且不连续地作用的仿射变换群的非常基本的问题。这些组大致对应于常规仿射平铺，即所有平铺在仿射变换之前都具有相同形状的平铺。这些群也承认微分几何中的解释为平面仿射流形。这些组保留欧几里得度量的特殊情况，用平铺语言来说意味着所有平铺在普通意义上都具有相同的形状，一个多世纪以来一直是众所周知的：这基本上是经典晶体学的主题。在一般情况下，这些群体的种类要多得多。它们的分类问题也引出了李群及其表示理论中的一些有趣的问题。该项目的第一部分包括研究一个猜想，对于每个半单实李群 G 及其不可约表示 V ，在 G 中具有线性部分的 V 的仿射自同构群中存在子群 Gamma 的充要条件，使得 Gamma 在 G 中具有扎里斯基稠密的线性部分，是自由非交换且行为该项目的第二部分包括将这个抽象代数准则转化为表示的最高限制权重的简单条件，从而对此类子群的 Zariski 闭包进行完全分类。该提案的最后一部分在于更好地理解这些群体：例如，尽可能构建一个与它们在仿射空间上的适当行为相对应的基本域；通过寻找针对给定 Zariski 闭包对所有此类群体进行分类的结果，超越证明此类群体的存在或不存在；并试图将它们与在其他同质空间上正确行动的自由团体联系起来。