Cohomogeneity one RCD-spaces

We study RCD-spaces (X, d, m) with group actions by isometries preserving the reference measure m and whose orbit space has dimension one, i. e. cohomogeneity one actions. To this end we prove a Slice Theorem asserting that each slice at a point is homeomorphic to a non-negatively curved RCD-space. Under the assumption that X is non-collapsed we further show that the slices are homeomorphic to metric cones over homogeneous spaces with Ric 0. As a consequence we obtain complete topological structural results and a principal orbit representation theorem. Conversely, we show how to construct new RCD-spaces from a cohomogeneity one group diagram, giving a complete description of RCD-spaces of cohomogeneity one. As an application of these results we obtain the classification of cohomogeneity one, non-collapsed RCD-spaces of essential dimension at most 4.