Existence problem and deformation of compact Clifford-Klein forms

起初，我计划计算Clifford-Klein形式的变形空间。但是，我发现这样的变形空间在很多情况下是非hausdorff的，很难直观的得到变形空间。为了理解这种非hausdorff空间，我引入了一种“拓扑放大”方法。这种方法类似于代数变量的放大。将此方法应用于非Hausdorff空间，得到了一个Hausdorff空间。这个Hausdorff空间有很多关于原始非Hausdorff空间的拓扑信息。特别地，如果将其应用于Clifford-Klein形式的变形空间，我们可以从放大的空间中获得变形空间拓扑的完整信息。

At first, I had planed to calculate deformation spaces of Clifford-Klein forms. However, I found that such deformation spaces are non-Hausdorff in many cases, and it is not easy to get an intuition of the deformation space.In order to understand such non-Hausdorff spaces, I have introduced a method topological blow-up'. This method is similar to the blow-up for algebraic varieties. Applying this method for a non-Hausdorff space, we obtain a Hausdorff space. This Hausdorff space has much information about the topology of the original non-Hausdorff space. Especially, if we apply it for deformation spaces of Clifford-Klein forms, we can obtain complete information on the topology of the deformation space from the blow-upped spaces.