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。然而，我发现这种变形空间在很多情况下是非Hausdorff的，而且很难直观地了解这种非Hausdorff空间。为了理解这种非Hausdorff空间，我引入了一种拓扑爆破的方法。这种方法类似于代数簇的爆破。将该方法应用于非Hausdorff空间，得到了Hausdorff空间。这个Hausdorff空间有很多关于原始非Hausdorff空间的拓扑的信息。特别是，如果我们将它应用于Clifford-Klein形式的形变空间，我们可以从爆破空间中得到关于形变空间的拓扑的完整信息。

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.