Rigidity and regularity for almost homogeneous spaces with Ricci curvature bounds

Abstract

We say that a metric space X is (ϵ, G)-homogeneous if G ≤Iso(X) is a discrete group of isometries with diam(X/G) ≤ ϵ. A sequence of (ϵi, Gi)-homogeneous spaces Xi with ϵi → 0 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 ≥ K. We also obtain a topological rigidity theorem for (ϵ, G)-homogeneous RCD(K, N) spaces, which generalizes a recent result by Wang. Indeed, if X is an (ϵ, G)-homogeneous RCD(K, N) space and G is an almost-crystallographic group, then X/G is bi-Hölder to an infranil orbifold. Moreover, we study (ϵ, G)homogeneous spaces in the smooth setting and prove rigidity and ϵ-regularity theorems for Riemannian orbifolds with Einstein metrics and bounded Ricci curvatures respectively.peerReviewe