Discrete and continuous aspects of local symmetry /

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Bibliographic Details
Author / Creator:van Limbeek, Wouter Herman, author.
Imprint:2015.
Ann Arbor : ProQuest Dissertations & Theses, 2015
Description:1 electronic resource (96 pages)
Language:English
Format: E-Resource Dissertations
Local Note:School code: 0330
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/10773184
Hidden Bibliographic Details
Other authors / contributors:University of Chicago. degree granting institution.
ISBN:9781321915358
Notes:Advisors: Benson Farb Committee members: Shmuel Weinberger.
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Dissertation Abstracts International, Volume: 76-12(E), Section: B.
English
Summary:In the first chapter, we give a classification of many closed Riemannian manifolds M whose universal cover possesses a continuous amount of symmetry. More precisely, we consider closed Riemannian manifolds M such that the isometry group of the universal cover has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space. This is inspired by work of Eberlein (for nonpositively curved manifolds) and Farb-Weinberger (for aspherical manifolds), and generalizes work of Frankel (for a semisimple group action). As an application, we characterize simply-connected Riemannian manifolds with both compact and finite volume noncompact quotients.
In the second chapter, we study quantitative analogues of the qualitative results obtained in Chapter 1. Namely, we study the size of the isometry group Isom(M,g) of Riemannian manifolds (M,g) as g varies. For M not admitting a circle action, we show that the order of Isom(M,g) can be universally bounded in terms of the bounds on Ricci curvature, diameter, and injectivity radius of M. This generalizes results known for negative Ricci curvature to all manifolds.
More generally, in the absence of suitable actions by connected groups, we establish a similar universal bound on the index of the deck group in the isometry group of the universal cover of M. We apply this to characterize locally symmetric spaces by their symmetry in covers. This proves a conjecture of Farb and Weinberger with the additional assumption of bounds on Ricci curvature, diameter, and injectivity radius. Further we generalize results of Kazhdan-Margulis and Gromov on minimal orbifolds of nonpositively curved manifolds to arbitrary manifolds with only a purely topological assumption.