Rigidity and regularity for almost homogeneous spaces with Ricci curvature bounds

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.