Branched coverings of symmetric spaces and nonpositive curvature

对称空间和非正曲率的分支覆盖

• 批准号: 329206558
• 负责人: Dr. Stephan Stadler
• 金额: --
• 依托单位: Max-Planck-Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/329206558?language=en
• 关键词: Branched; coverings; symmetric; spaces; nonpositive

Branched coverings produce metric spaces of nonpositive curvature with intriguing features. For instance, I proved in my thesis that a certain 4-manifold which allows for a branched covering over a product of hyperbolic surfaces and therefore is nonpositively curved in the metric sense is not smoothable, i.e. does not carry a Riemannian metric of nonpositive curvature. This answered a question of Gromov from the year 1985. Refining and completing the techniques from my thesis makes it possible to prove rigidity theorems for total spaces of branched coverings of symmetric spaces. In particular one obtains information about the quasi-isometry groups of their universal coverings. Examples show that these groups are not rigid in the strict sense, i.e. they are not virtually isomorphic to the respective isometry groups. The aim of this project is to reveala form of semi-rigidity which virtually identifies the quasi-isometry group of a branched covering with the quasi-isometry group its ramification locus.

分支覆盖物产生具有有趣特征的非阳性曲率的度量空间。例如，我在论文中证明了一定的4个manifold允许在双曲表面的产物上进行分支覆盖，因此在度量意义上是非阳性弯曲的，即不可平整，即不会带有无阳性曲率的Riemannian公制。这回答了1985年Gromov的问题。从我的论文中提炼并完成了技术，因此可以证明对对称空间的分支覆盖物的总空间证明刚性定理。特别是，一个人获得了有关其通用覆盖的准静电群的信息。示例表明，这些组在严格的意义上并不是刚性的，即它们对各个等轴测组实际上不是同构。该项目的目的是揭示半分数的形式，该形式实际上识别了带有准时分支组的分支覆盖的准静电群，其分支基因座。