Rigidity and regularity for almost homogeneous spaces with Ricci curvature bounds

TL;DR

This paper investigates the geometric and topological properties of almost homogeneous spaces with Ricci curvature bounds, establishing limits, rigidity, and regularity results for both smooth and metric measure spaces.

Contribution

It proves that limits of almost homogeneous RCD spaces are nilpotent Lie groups and generalizes topological rigidity theorems to RCD spaces with group actions.

Findings

GH limits are nilpotent Lie groups with Ricci bounds

Bi-Hölder equivalence to infranil orbifolds under group actions

Rigidity and ε-regularity results for Einstein orbifolds

Abstract

We say that a metric space $X$ is $(\epsilon,G)$-homogeneous if $G<Iso(X)$ is a discrete group of isometries with $diam(X/G)<\epsilon$.\ A sequence of $(\epsilon_i,G_i)$-homogeneous spaces $X_i$ with $\epsilon_i\to0$ is called a sequence of almost homogeneous spaces. In this paper we show that the Gromov-Hausdorff limit of a sequence of almost homogeneous RCD$(K,N)$ spaces must be a nilpotent Lie group with $Ric\geqslant K$. We also obtain a topological rigidity theorem for $(\epsilon,G)$-homogeneous RCD$(K,N)$ spaces, which generalizes a recent result by Wang. Indeed, if $X$ is an $(\epsilon,G)$-homogeneous RCD$(K,N)$ space and $G$ is an almost-crystallographic group, then $X/G$ is bi-H\"older to an infranil orbifold. Moreover, we study $(\epsilon,G)$-homogeneous spaces in the smooth setting and prove rigidity and $\epsilon$-regularity theorems for Riemannian orbifolds with Einstein metrics and bounded Ricci curvatures respectively.