TITLE: Equivariant geometry of Alexandrov 3-spaces

ABSTRACT: Alexandrov spaces constitute a synthetic generalization of Riemannian manifolds with a lower bound on sectional curvature to the class of metric spaces. They provide a natural setting to study many notions of global Riemannian geometry, and therefore, a fundamental problem is that of extending to Alexandrov spaces what is known for Riemannian manifolds. As in the Riemannian case, one may investigate Alexandrov spaces via their symmetries. Since the isometry group of a compact Alexandrov space is a compact Lie group, this point of view naturally leads to the study of isometric Lie group actions on Alexandrov spaces. Berestovski i showed that finite-dimensional homogeneous Alexandrov spaces actually are Riemannian manifolds. Later, Galaz-Garcia and Searle studied Alexandrov spaces of cohomogeneity one (i.e. those with an isometric action of a compact Lie group whose orbit space is one-dimensional) and classified them in dimensions at most 4.