Event № 245
In the late 60s, Ottmar Loos gave a surprising and beautiful characterization of affine symmetric spaces as smooth reflection spaces with a weak isolation property for fixed points. The first half of this talk is intended as a survey on the structure of Riemannian and affine symmetric spaces from this reflection space point of view. In particular, we explain how geometric representations of finite reflection group arise from the local geometry of flats in such spaces. The second half of this talk is then devoted to exotic examples of topological reflection spaces, which satisfy all of Loos' axioms except for smoothness. This part is based on ongoing joined work with W. Freyn, M. Horn and R. Köhl. We show that for any 2-spherical Coxeter group W there exists an infinite-dimensional such reflection space of finite rank whose local geometry is governed by the geometric representation of W. Our examples are based on split-real Kac-Moody groups and have a number of geometric properties not observed in this context before. For example, any two points in the reflection space can be joined by a piecewise geodesic curve, but the reflection space is not midpoint convex. Time permitting we will discuss further properties of the construction, such as the classification of automorphisms and its relation to the natural boundary action of elliptic subgroups of the automorphism group.