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.