## D. Corro - J. Santos-Rodríguez - J. Núñez-Zimbrón

# Cohomogeneity one RCD-spaces

created by corro on 16 May 2024

[

BibTeX]

*preprint*

**Inserted:** 16 may 2024

**Last Updated:** 16 may 2024

**Year:** 2024

**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 each slice at a point is homeomorphic to a
non-negatively curved $\mathsf{RCD}$-space. Under the assumption that $X$ is
non-collapsed we further show that the slices are homeomorphic to metric cones
over homogeneous spaces with $\mathrm{Ric} \geq 0$. As a consequence we obtain
complete topological structural results and a principal 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 $4$.