Cohomogeneity one RCD-spaces

TL;DR

This paper investigates cohomogeneity one RCD-spaces, establishing structural theorems, constructing new examples, and classifying non-collapsed cases up to dimension four, thus advancing understanding of these geometric spaces.

Contribution

It provides a Slice Theorem for non-collapsed RCD-spaces, characterizes their topological structure, and classifies low-dimensional non-collapsed cohomogeneity one RCD-spaces.

Findings

Slices are homeomorphic to metric cones over homogeneous spaces with Ric ≥ 0

Complete topological structural results for RCD-spaces with group actions

Classification of non-collapsed cohomogeneity one RCD-spaces up to dimension four

Abstract

We study $\mathsf{RCD}$-spaces $(X,d,\mathfrak{m})$ with group actions by isometries preserving the reference measure $\mathfrak{m}$ and whose orbit space has dimension one, i.e. cohomogeneity one actions. To this end we prove a Slice Theorem asserting that when $X$ is non-collapsed the slices are homeomorphic to metric cones over homogeneous spaces with $\mathrm{Ric} \geq 0$. As a consequence we obtain complete topological structural results (also in the collapsed case) and a regular orbit representation theorem. Conversely, we show how to construct new $\mathsf{RCD}$-spaces from a cohomogeneity one group diagram, giving a complete description of $\mathsf{RCD}$-spaces of cohomogeneity one. As an application of these results we obtain the classification of cohomogeneity one, non-collapsed $\mathsf{RCD}$-spaces of essential dimension at most $4$.