Abstract. In this work, it is shown that a simply-connected, rationallyelliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an application, simply-connected, rationally-elliptic manifolds admitting slice-maximal torus actions are classified up to equivariant rational homotopy. The case where the rational-ellipticity hypothesis is replaced by non-negative curvature is also discussed, and the Bott Conjecture in the presence of a slice-maximal torus action is proved.

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary we obtain the classification of closed, n-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric T n−1 action. In contrast to the one-and twodimensional cases, where it is known that an Alexandrov space is a topological manifold, in dimension 3 the classification contains, in addition to the known cohomogeneity one manifolds, the spherical suspension of RP 2 , which is not a manifold.

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 homogenous 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.

We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspension of RP 2 ; we use this to classify, up to homeomorphism, closed, positively curved Alexandrov spaces of dimension three. Second, we classify closed three-dimensional Alexandrov spaces of nonnegative curvature. Third, we study the well-known Poincaré Conjecture in dimension three, in the context of Alexandrov spaces, in the two forms it is usually formulated for manifolds. We first show that the only closed three-dimensional Alexandrov space that is also a homotopy sphere is the 3-sphere; then we give examples of closed, geometric, simply connected three-dimensional Alexandrov spaces for five of the eight Thurston geometries, proving along the way the impossibility of getting such examples for the Nil, SL2(R) and Sol geometries. We conclude the paper by proving the analogue of the geometrization conjecture for closed threedimensional Alexandrov spaces.

Abstract. We determine the structure of the fundamental group of the regular leaves of a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold. We also study closed singular Riemannian foliations whose leaves are homeomorphic to aspherical or to Bieberbach manifolds. These foliations, which we call A-foliations and B-foliations, respectively, generalize isometric torus actions on Riemannian manifolds. We apply our results to the classification problem of compact, simply connected Riemannian 4-and 5-manifolds with positive or nonnegative sectional curvature.

Abstract. We classify closed, simply connected n-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions 2 ≤ n ≤ 6. In dimensions 3k, k = 1, 2 there is only one such manifold and it is diffeomorphic to the product of k copies of the 3-sphere.

Let X be an Alexandrov space (with curvature bounded below). We determine the maximal dimension of the isometry group Isom(X) of X and show that X is isometric to a Riemannian manifold, provided the dimension of Isom(X) is maximal. We determine gaps in the possible dimensions of Isom(X). We determine the maximal dimension of Isom(X) when the boundary ∂ X is non‐empty and classify up to homeomorphism Alexandrov spaces with boundary and an isometry group of maximal dimension. We also show that a symmetric Alexandrov space is isometric to a Riemannian manifold.

ABSTRACT. We show that a closed, simply-connected, non-negatively curved 5-manifold admitting an effective, isometric T 2 action is diffeomorphic to one of S 5 , S 3 × S 2 , S 3× S 2 or the Wu manifold SU (3)/SO(3).