A unified approach to Euclidean Buildings and symmetric spaces of non-compact type

欧几里得建筑和非紧凑型对称空间的统一方法

• 批准号: 441721480
• 负责人: Professor Dr. Linus Kramer
• 金额: --
• 依托单位: Mathematisches Institut
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/441721480?language=en
• 关键词: unified; approach; Euclidean; Buildings; symmetric

The present proposal concerns similarities between Euclidean buildings and symmetric spaces of non-compact type. The literature contains famous examples of theorems and properties which were first obtained in one case and then shown to hold true also in the other setting, for instance the existence of the spherical building at infinity. In this set-up one may also ask whether and which uniform statements and proofs can be obtained in both worlds. The main goal of this project is to show properties known from one world also in the other world, and to give uniform proofs of statements for both classes of spaces. In the former case, we would like to investigate the extendability of automorphisms of the visual boundary, and also filling properties of S-arithmetic groups. Regarding uniform Statements and results we aim to develop metric methods allowing for a simultaneous treatment of trees and rank-1 symmetric spaces, to uniformly establish the building-structure of the boundary at infinity, to investigate strongly transitive actions from a unified perspective, and to obtain a uniform proof of Kostant's convexity theorem.Mathematical methods to be used throughout the proposed research include CAT(0) geometry, group-theoretical, combinatorial and homological techniques as well as methods from coarse geometry and the combinatorics of folded galleries.

本建议关注欧氏建筑物和非紧型对称空间之间的相似性。文献包含著名的例子定理和性质，首先获得在一种情况下，然后证明也成立，在其他设置，例如存在的球形建设在无穷大。在这种设置中，人们也可能会问，是否以及哪些统一的陈述和证明可以在两个世界中获得。这个项目的主要目标是显示从一个世界也在另一个世界已知的属性，并给出两类空间的声明的统一证明。在前一种情况下，我们想研究的可拓性的自同构的视觉边界，也填充性质的S-算术群。关于统一的陈述和结果，我们的目标是发展度量方法，允许同时处理树和秩-1对称空间，统一地建立边界在无穷远处的建筑结构，从统一的角度研究强传递作用，并获得Kostant凸性定理的统一证明。整个研究中使用的数学方法包括CAT（0）几何，群论，组合和同调技术，以及从粗几何和组合学的折叠画廊的方法。