The project is concerned with rigidity, compactifications and local-to-global principles in CAT(0) geometry.
One aim is to give a uniform construction of compactifications of euclidean buildings, using Gromov's embedding into spaces of continuous functions. The ultimate goal is to study the dynamics of discrete group actions on the building, using the compactification.
LG-rigidity of a metric space \(X\) means that there is some \(r>0\) such that if \(Y\) is a metric space in which every \(r\)-ball is isometric to some \(r\)-ball in \(X\), then there is a covering map \(X\to Y\) which is a local isometry on all \(r\)-balls. The project intends to investigate LG-rigidity and non-rigidity for the 1-skeletons and chamber graphs of general Bruhat-Tits buildings.
We prove an affine analog of Scharlau's reduction theorem for spherical buildings. To be a bit more precise let X be a euclidean building with spherical building ∂X at infinity. Then there exists a euclidean building X¯ such that X splits as a product of X¯ with some euclidean k-space such that ∂X¯ is the thick reduction of ∂X in the sense of Scharlau. \newline In addition we prove a converse statement saying that an embedding of a thick spherical building at infinity extends to an embedding of the euclidean building having the extended spherical building as its boundary.
20Compactifications and Local-to-Global Structure for Bruhat-Tits Buildings