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March 2017 Symmetry gaps in Riemannian geometry and minimal orbifolds
Wouter van Limbeek
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J. Differential Geom. 105(3): 487-517 (March 2017). DOI: 10.4310/jdg/1488503005


We study the size of the isometry group $\mathrm{Isom}(M,g)$ of Riemannian manifolds $(M,g)$ as $g$ varies. For $M$ not admitting a circle action, we show that the order of $\mathrm{Isom}(M,g)$ 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 we establish a similar universal bound on the index of the deck group $\pi_1 (M)$ in the isometry group $\mathrm{Isom}(\widetilde{M},\widetilde{g})$ of the universal cover $\widetilde{M}$ in the absence of suitable actions by connected groups. We apply this to characterize locally symmetric spaces by their symmetry in covers. This proves a conjecture of Farb andWeinberger with the additional assumption of bounds on curvature, diameter, and injectivity radius. Further we generalize results of KazhdanñMargulis and Gromov on minimal orbifolds of nonpositively curved manifolds to arbitrary manifolds with only a purely topological assumption.


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Wouter van Limbeek. "Symmetry gaps in Riemannian geometry and minimal orbifolds." J. Differential Geom. 105 (3) 487 - 517, March 2017.


Received: 19 May 2014; Published: March 2017
First available in Project Euclid: 3 March 2017

zbMATH: 1364.53042
MathSciNet: MR3619310
Digital Object Identifier: 10.4310/jdg/1488503005

Rights: Copyright © 2017 Lehigh University


Vol.105 • No. 3 • March 2017
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