主 题: Bruhat – Tits Geometry and Nonegative Curvature
报告人: Professor Karsten Grove (University of Notre Dame, USA)
时 间: 2014-12-11 14:00-15:00
地 点: Room 77201 at #78 courtyard, Beijing International Center for Mathematical Research（主持人：刘小博）
(Joint work with Fuquan Fang and Gudlaugur Thorbergsson.) There is a well-known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three.
We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but one that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a topological building in the sense of Burns and Spatzier. Using this structure we classify all polar actions on (simply connected) positively curved manifolds of cohomegeneity at least two.
For the much broader class of polar actions in nonnegative curvature, the structure of chamber systems plays a significant role as well for two of its three basic building blocks referred to as “affine”-, “spherical”- and “book”- polar. The above mentioned type of links for the
“spherical” case have not been yet been established for the “affine case”, but a conjectural picture is emerging, indication that symmetric spaces are the source of all example of the “affine” kind as well. The “book” types are completely different and structurally completely understood.