Properly Discontinuous Actions on Homogeneous Spaces

这个项目的主要动机是所谓的Auslander猜想，这是一个非常基本的问题，关于在仿射空间上适当和不连续地作用的仿射变换群。这些群大致对应于规则的仿射平铺，也就是说，所有的平铺都具有相同的形状，直到仿射变换。这些群在微分几何中也可以解释为平面仿射流形。特殊情况下，这些群体保持欧几里得度规，在瓷砖的语言中，这意味着所有的瓷砖在普通意义上具有相同的形状，已经众所周知了一个多世纪：这基本上是经典晶体学的主题。一般来说，这些群体的种类要多得多；它们的分类问题也导致了李群及其表示理论中一些有趣的问题。项目的第一部分是研究一个猜想，对于每一个半单实李群G及其不可约表示V，给出了在具有线性部分的V的仿射自同构群中存在子群Gamma的充分必要判据，使得Gamma在G中具有线性部分Zariski-dense，项目的第二部分是将这个抽象的代数判据转化为表示的最高限制权的简单条件，从而对这些子群的zariski闭包进行完全分类。建议的最后一部分包括更好地理解这些群：例如，尽可能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，超越了证明此类群存在或不存在；并试图将它们与在其他同质空间中正常活动的自由群体联系起来。

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.