Hausdorff convergence and universal covers

Sormani, Christina; Wei, Guofang

doi:10.1090/s0002-9947-01-02802-1

Cited by 51 publications

(110 citation statements)

References 17 publications

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“…The next theorem shows that in the case when X D Y the boundedness of the Kruglov operator is a necessary and sufficient condition for the inequalities of the type (13). In the Banach setting, an analogous result was earlier proved in [Astashkin 2008].…”

Section: Khintchine Inequality In Quasi-banach Spacessupporting

confidence: 56%

“…We will point out the difference only when it is not clear from the context and notation. If r is given as in ( 10) for some R-matrix R, the Poisson structure in G N given by the formula (13) fᏲ;…”

Section: Background and Definitionsmentioning

confidence: 99%

“…where Ᏼ m is the Hausdorff measure of dimension m. Of course, Cheeger and Colding studied more than just manifolds with nonnegative Ricci curvature and more than just noncollapsing sequences in their work, but these theorems are the only ones needed in this paper. See also work of the second author with Wei for an adaption of their volume convergence theorem which deals with Hausdorff measures defined using restricted versus intrinsic distances [13].…”

Section: An Estimation Of Simplicial Volume Of Alexandrov Spacesmentioning

confidence: 99%

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2014

Pacific J. Math.

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We demonstrate the relevance of the Prokhorov inequality to the study of Khintchine-type inequalities in symmetric function spaces. Our main result shows that the latter inequalities hold for a pair of quasi-Banach symmetric function spaces X and Y if and only if the Kruglov operator K acts from X to Y . We also obtain an extension of von Bahr-Esseen and Esseen-Janson L p -estimates for sums of independent mean zero random variables to the class of quasi-Banach symmetric spaces. In particular, in contrast to the well-known Esseen-Janson theorem, we do not assume that the summands are equidistributed.

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Section: Khintchine Inequality In Quasi-banach Spacessupporting

confidence: 56%

Section: Background and Definitionsmentioning

confidence: 99%

Section: An Estimation Of Simplicial Volume Of Alexandrov Spacesmentioning

confidence: 99%

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Untitled

2014

Pacific J. Math.

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“…Definition 8.1. If (X, d) is a compact length space that admits a universal covering ( X, d X ) → (X, d), we define the systole of (X, d) as sys(X, d) := inf{d X (x, γ • x) : x ∈ X, γ ∈ π1 (X) − {Id}}, where π1 (X) is the group of deck tranformations of X, which is referred to as the revised fundamental group of X in [42]. Proof of Theorem 1.4.…”

Section: Is Measure Preserving Homeomorphic To a Warped Productmentioning

confidence: 99%

Maximal volume entropy rigidity for RCD∗(−(N−1),N) spaces

Connell

Dai

Núñez-Zimbrón

et al. 2021

J. London Math. Soc.

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For n-dimensional Riemannian manifolds M with Ricci curvature bounded below by −(n − 1), the volume entropy is bounded above by n − 1. If M is compact, it is known that the equality holds if and only if M is hyperbolic. We extend this result to RCD * (−(N − 1), N) spaces. While the upper bound is straightforward, the rigidity case is quite involved due to the lack of a smooth structure in RCD * spaces. As an application, we obtain an almost rigidity result which partially recovers a result by Chen-Rong-Xu for Riemannian manifolds.

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“…with uniform lower bounds on Ricci curvature and upper bounds on dimension and diameter. In some cases, Sormani and Wei, and later Mondino and Wei have shown existence of a traditional covering map with a universal property ( [24], [18]).…”

Section: Introductionmentioning

confidence: 99%

Every Continuum has a Compact Universal Cover

Plaut¹

2021

Preprint

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We define the compact universal cover of a compact, metrizable connected space (i.e. a continuum) X to be the inverse limit of all continua that regularly cover X. We show that such covers do indeed form an inverse system with bonding maps that are regular covering maps, and the projection φ : X → X = X/πP (X) of the inverse limit is a generalized regular covering map, where X is a continuum that is "compactly simply connected" in the sense that πP ( X) = 1. We call πP (X) the "profinite fundamental group" of X. We prove a Galois Correspondence for closed normal subgroups of πP (X), uniqueness, universal and lifting properties. As an application we prove that every non-compact manifold that regularly covers a compact manifold has a unique "profinite compactification", i.e. an imbedding as a dense subset in a compactly simply connected continuum. As part of the proof of metrizability of X we show that every continuum has at most n! non-equivalent n-fold covers by continua.

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Hausdorff convergence and universal covers

Cited by 51 publications

References 17 publications

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Maximal volume entropy rigidity for RCD∗(−(N−1),N) spaces

Every Continuum has a Compact Universal Cover

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