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Let ( M , g ) be a smooth Riemannian manifold and G a compact Lie group acting on M effectively and by isometries. It is well known that a lower bound of the sectional curvature of ( M , g ) is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreover, the analogous stability property holds for metric foliations and submersions. The goal of the paper is to prove the corresponding stability properties for synthetic Ricci curvature lower bounds. Specifically, we show that such stability holds for quotients of RCD ⁎ ( K , N ) -spaces, under isomorphic compact group actions and more generally under metric-measure foliations and submetries. An RCD ⁎ ( K , N ) -space is a metric measure space with an upper dimension bound N and weighted Ricci curvature bounded below by K in a generalized sense. In particular, this shows that if ( M , g ) has Ricci curvature bounded below by K ∈ R and dimension N , then the quotient space is an RCD ⁎ ( K , N ) -space. Additionally, we tackle the same problem for the CD / CD ⁎ and MCP curvature-dimension conditions. We provide as well geometric applications which include: A generalization of Kobayashi's Classification Theorem of homogeneous manifolds to RCD ⁎ ( K , N ) -spaces with essential minimal dimension n ≤ N ; a structure theorem for RCD ⁎ ( K , N ) -spaces admitting actions by large (compact) groups ; and geometric rigidity results for orbifolds such as Cheng's Maximal Diameter and Maximal Volume Rigidity Theorems. Finally, in two appendices we apply the methods of the paper to study quotients by isometric group actions of discrete spaces and of (super-)Ricci flows. [ABSTRACT FROM AUTHOR]

In this paper we investigate the relationship between a general existence of transport maps of optimal couplings with absolutely continuous first marginal and the property of the background measure called essentially non-branching introduced by Rajala–Sturm (2014) [27] . In particular, it is shown that the qualitative non-degeneracy condition introduced by Cavalletti–Huesmann (2015) [6] implies that any essentially non-branching metric measure space has a unique transport maps whenever the initial measure is absolutely continuous. This generalizes a recently obtained result by Cavalletti–Mondino (2017) [8] on essentially non-branching spaces with the measure contraction condition MCP ( K , N ) . In the end we prove a measure rigidity result showing that any two essentially non-branching, qualitatively non-degenerate measures on a fixed metric spaces must be mutually absolutely continuous. This result was obtained under stronger conditions by Cavalletti–Mondino (2016) [7] . It applies, in particular, to metric measure spaces with generalized finite dimensional Ricci curvature bounded from below. [ABSTRACT FROM AUTHOR]