Properly Discontinuous Actions on Homogeneous Spaces

这个项目的主要动机是所谓的Auslander猜想，这是一个关于仿射变换群的非常基本的问题，这些仿射变换在仿射空间上正确且不连续地起作用。 这些组大致对应于规则仿射平铺，即，其中所有瓦片具有直到仿射变换的相同形状的瓦片。这些群体也承认在微分几何解释为平坦的仿射流形。这些群保持欧几里得度量的特殊情况，在平铺的语言中，这意味着所有的瓦片在普通意义上具有相同的形状，这在世纪以来一直非常有名：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;而它们的分类问题也导致了李群及其表示理论中的一些迷人问题。该项目的第一部分包括研究一个猜想，对于每个半单真实的李群G及其不可约表示V，给出了V的仿射自同构群中存在子群Gamma的充要条件，使得Gamma在G中具有线性部分Zakki-稠密，是自由nonabelian和行为适当不连续的V。该项目的第二部分包括在翻译这个抽象的代数标准成一个简单的条件上的最高限制重量的代表，从而完全地分类了这样的子群的Zurkiki闭包。建议的最后一部分是更好地理解这些群：例如，尽可能地构造一个基本域，对应于它们在仿射空间上的适当作用;超越证明这些群的存在或不存在，寻找对给定Zebriki闭包分类所有这些群的结果;并试图将它们与在其他齐性空间上适当作用的自由群联系起来。

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.