The geometry Anosov subgroups in Lie groups

流形上的局部齐次几何结构领域在以下意义上概括了地图学领域：绘制世界地图集就像试图使用平面几何来描述地球的形状，而局部齐次几何结构就像试图使用一些齐次几何来描述更一般世界(或流形)的形状。该主题的第一个定理之一指出，事实上不可能不失真地构建完整的世界地图集，即在二维球体上不存在欧几里得(平面)结构。在平面结构的研究中，比伯巴赫定理已经证明了几何结构和对称群之间的紧密关系。这些定理表明，直到有限个覆盖，2维上唯一的有限区域平面结构类似于Pac-man世界，即它们是通过识别平行四边形的平行边而获得的。平行四边形可以通过平移自身的多个副本来平铺平面，从而创建具有平移对称的平铺，这并不是巧合。研究对象或概念之间的对称性是纯数学和应用数学以及其他科学中的一系列主题的基础。我们说一个物体有连续对称性，如果它的对称性可以是非常小的运动，例如球体非常小的旋转就是对称性，所以它具有连续对称性。如果不是这样，我们说它具有离散对称性，例如，一个立方体具有离散对称性，因为它只有90度旋转对称性。立方体的对称性是球体对称性的子集。类似地，平面具有连续对称性，但平行四边形平铺在平面的对称性内具有离散对称性。我的专业位于几何结构研究和连续群的离散子群研究之间的交叉点，这两个研究推广了上面的对称性讨论。我将调查一系列例子，这些例子是以上的推广，并致力于解决这些情况下的分类问题。然后，这些例子可以用来表述一般的猜想和定理。我感兴趣的一些例子有爱因斯坦静态宇宙的局部几何，一个三维时空的玩具模型。一些更多的例子有局部投影几何，这是3D计算机图形中使用的几何类型。更好地理解这些更一般的几何和对称性概念，有助于深入了解两者之间的基本关系。

The field of locally homogeneous geometric structures on manifolds generalizes the field of cartography, in the following sense : drawing an atlas of the world is like trying to describe the shape of the earth using flat geometry, and a locally homogeneous geometric structure is like trying to describe the shape of a more general world (or manifold) using some homogeneous geometry. One of the first theorems in the subject states that it is in fact impossible to construct a complete atlas of the world without distortion, that is, that there exists no Euclidean (flat) structure on the two-dimensional sphere. Already in the study of flat structures, the Bieberbach theorems exemplify the tight relationship between geometric structures and groups of symmetries. These theorems imply that, up to finite covers, the only finite-area flat structures in 2 dimensions are similar to the world of Pac-man, that is, they are obtained by identifying parallel edges of a parallelogram. It is no coincidence that the parallelogram can tile the plane by translating multiple copies of itself, creating a tiling with translational symmetry. The study of symmetries between objects or concepts is foundational to a range of topics in pure and applied mathematics and in other sciences. We say that an object has continuous symmetry if its symmetries can be very small motions, for instance a very small rotation of a sphere is a symmetry, so it has continuous symmetry. We say that it has discrete symmetry if this is not the case, for instance a cube has discrete symmetry because it only has 90 degree rotational symmetry. The symmetries of the cube are a subset of the symmetries of the sphere. Similarly, the plane has continuous symmetry but a parallelogram tiling has discrete symmetry within the symmetries of the plane. My field lies at the intersection between the study of geometric structures and the study of discrete subgroups of continuous groups, which generalize the symmetry discussion above. I will investigate a range of examples which are generalizations of the above and work towards solutions to classification problems in those cases. These examples can then be used to formulate general conjectures and theorems. Some of the examples I am interested in have the local geometry of Einstein's static universe, a toy model of spacetime in 3 dimensions. Some more examples have local projective geometry, the type of geometry used in 3D computer graphics. Having a better understanding of these more general notions of geometry and symmetry provides important insight into the fundamental relationship between the two.