Abstract

A guiding principle in CAT(0) geometry is that in the presence of enough symmetry failure of hyperbolic behaviour can be attributed to flat pieces -- isometric embeddings of unbounded higher dimensional  Euclidean regions. For instance, an axial isometry acts with north-south dynamics unless one (and then every) axis  bounds a flat half-plane. In the smooth setting, the structure theory of Ballmann et al. implies the following striking result. If a discrete group G acts on a Hadamard manifold H with finite volume quotient, and every G-axis bounds a flat half-plane,  then H has to be isometric to a symmetric space or split as a metric product. Motivated by this and the general tendency in non-positive curvature that much of the smooth theory generalizes to the synthetic setting, Ballmann and Buyalo formulated two conjectures which predict a dichotomy in the geometric behaviour of periodic CAT(0) spaces according the appearing amount of flatness. In the talk I will discuss these conjectures, I will survey what is
known and report on recent progress.