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

本建议涉及欧几里得建筑和非紧凑型对称空间之间的相似之处。文献中包含了著名的定理和性质的例子，这些定理和性质首先是在一种情况下得到的，然后被证明在另一种情况下也是成立的，例如无限大的球形建筑的存在。在这种情况下，人们可能还会问，在这两种情况下是否可以得到统一的陈述和证明，以及可以得到哪些统一的证明。这个项目的主要目标是展示从一个世界到另一个世界的已知性质，并对这两类空间的陈述给出统一的证明。在前一种情况下，我们想要研究视觉边界的自同构可扩性，以及S算术群的填充性质。关于统一的表述和结果，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，一致地建立无穷远处边界的建筑结构，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。整个研究过程中将使用的数学方法包括CAT(0)几何，群论，组合和同调技术，以及来自粗几何和折叠画廊的组合的方法。

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.