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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.

这个项目的主要动机是所谓的Auslander猜想，这是一个关于仿射空间上适当和不连续作用的仿射变换组的非常基本的问题。这些组大致对应于规则仿射平铺，即所有平铺具有相同形状直到仿射变换的平铺。这些群在微分几何中也被解释为平坦仿射流形。这些群保持欧几里得度规的特殊情况，在平铺语言中意味着所有瓷砖在普通意义上具有相同的形状，这已经非常著名了一个多世纪：这基本上是经典结晶学的主题。一般来说，这些群体的种类要多得多；并且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。项目的第一部分在于研究一个猜想，对于每个半单实李群G及其不可约表示V，给出在V的仿射自同构群中存在子群Gamma的一个充要条件，使得Gamma在G中具有线性部分Zariski稠密，是自由的非交换的，并且在V上适当地间断作用。项目的第二部分包括将这个抽象的代数判据转化为关于表示的最高约束权的一个简单条件，从而对这类子群的Zariski闭包进行了完全分类。提案的最后部分在于更好地理解这些群：例如，尽可能地构造一个与它们在仿射空间上的适当作用相对应的基本域；通过寻找对给定Zariski闭包的所有此类群进行分类的结果，超越证明这些群的存在或不存在的范围；并试图将它们与在其他齐次空间上适当作用的自由群联系起来。