Hadamard spaces: rigidity and recognition theorems

• Applicant: Professorin Dr. Petra Nora Schwer
• Subject Area: Mathematics
• Term: from 2011 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 202726003

The purpose of this project is to find suitable conditions on non-positively curved metric spaces, in particular Hadamard spaces, which ideally will work in all (higher) dimensions and allow to identify a space as an affine building. Further we will study questions related to rigidity of maps between the types of spaces considered.Affine buildings were first introduced in order to understand algebraic groups defined over fields with (discrete) valuation. They do not only provide an interesting class of examples of non-positively curved spaces but they are also an important tool in other areas of pure mathematics. Their geometry is well understood and in higher rank a full classification is available. To have a space pegged as an affine building is hence quite helpful although difficult.So far not many recognition theorems are known and most of the few are only available in small dimensions. In almost all cases one assumes certain diameter bounds on the boundary or on links, that is the space of directions at a point. Rigidity of maps between affine buildings and Hadamard spaces are far less understood. One example is a generalization of Mostow's celebrated rigidity theorem to affine buildings. Our aim is to combine new techniques in order to generalize and improve existing results, to reduce assumptions on the spaces and to find higher dimensional generalizations of known theorems.

• DFG Programme: Research Grants