Isometry groups of Alexandrov spaces

Galaz-García, Fernando; Guijarro, Luis

doi:10.1112/blms/bds101

Cited by 23 publications

(30 citation statements)

References 26 publications

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“…The theorem generalizes a previous result of Guijarro and the first author in the framework of finite dimensional Alexandrov spaces [31]. Recall that for some suitable parameters such spaces are RCD * (K, N ), thus they satisfy (GTB).…”

Section: Introductionsupporting

confidence: 82%

“…The theorem states that the orbits of m-almost all points in M are of a unique maximal type. For this result, we are inspired by a paper of Guijarro and the first author [31] in the framework of Alexandrov spaces. In order to compensate the lack of information about branching of geodesics in the present context, we introduce the notion of good optimal transport behavior requiring, roughly speaking, that each optimal transport from an absolutely continuous measure is induced by an optimal map (see Definition 4.1).…”

Section: Principal Orbit Theorem and Cohomogeneity One Actionsmentioning

confidence: 99%

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On quotients of spaces with Ricci curvature bounded below

Galaz-García

Kell

Mondino

et al. 2018

Journal of Functional Analysis

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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 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.

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Section: Introductionsupporting

confidence: 82%

Section: Principal Orbit Theorem and Cohomogeneity One Actionsmentioning

confidence: 99%

On quotients of spaces with Ricci curvature bounded below

Galaz-García

Kell

Mondino

et al. 2018

Journal of Functional Analysis

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“…Alexandrov spaces (of curvature bounded below) appear naturally as generalizations of Riemannian manifolds of sectional curvature bounded below. Many results for Riemannian manifolds admit suitable generalizations to the Alexandrov setting and this class of metric spaces has been studied from several angles, including recently the use of transformation groups [8,9,13].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Closed three-dimensional Alexandrov spaces with isometric circle actions

Núñez-Zimbrón¹

2018

Tohoku Math. J. (2)

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We obtain a topological and weakly equivariant classification of closed threedimensional Alexandrov spaces with an effective, isometric circle action. This generalizes the topological and equivariant classifications of Raymond [26] and Orlik and Raymond [23] of closed three-dimensional manifolds admitting an effective circle action. As an application, we prove a version of the Borel conjecture for closed three-dimensional Alexandrov spaces with circle symmetry.2010 Mathematics Subject Classification. Primary 53C23; Secondary 57M60, 57S25.

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“…In this final section we look at symmetric spaces (compare with [6] and [16]). We will say that an RCD * (K, N ) space (X, d, m) is locally (uniformly) symmetric if for every p ∈ X there exists a neighbourhood p ∈ U (p) and r > 0 such that for all q ∈ U (p) the ball B q (r, X) admits an isometric involution that only fixes q.…”

Section: Symmetric Spacesmentioning

confidence: 99%

“…In [6] Berestovskiȋ studied locally symmetric spaces as well and he showed that in simply connected G−spaces the two notions coincide. However, examples 8.1, 8.2 and 8.3 constructed in [16] exhibit that this is no longer true for Alexandrov spaces.…”

Section: Symmetric Spacesmentioning

confidence: 99%

Invariant Measures and Lower Ricci Curvature Bounds

Santos-Rodríguez

2019

Potential Anal

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Given a metric measure space (X, d, m) that satisfies the Riemannian Curvature Dimension condition, RCD * (K, N ), and a compact subgroup of isometries G ≤ Iso(X) we prove that there exists a G−invariant measure, mG, equivalent to m such that (X, d, mG) is still a RCD * (K, N ) space. We also obtain some applications to Lie group actions on RCD * (K, N ) spaces. We look at homogeneous spaces, symmetric spaces and obtain dimensional gaps for closed subgroups of isometries.

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Isometry groups of Alexandrov spaces

Cited by 23 publications

References 26 publications

On quotients of spaces with Ricci curvature bounded below

On quotients of spaces with Ricci curvature bounded below

Closed three-dimensional Alexandrov spaces with isometric circle actions

Invariant Measures and Lower Ricci Curvature Bounds

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