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Symmetry gaps in Riemannian geometry and minimal orbifolds. (English) Zbl 1364.53042
Author’s abstract: We study the size of the isometry groups $$\mathrm{Isom}(M, g)$$ 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 B. Farb and S. Weinberger [Ann. Math. (2) 168, No. 3, 915–940 (2008; Zbl 1175.53055)] with the additional assumption of bounds on curvature, diameter, and injectivity radius. Further we generalize results of D. A. Kazhdan and G. A. Margulis [Math. USSR, Sb. 4, 147–152 (1969; Zbl 0241.22024); translation from Mat. Sb., n. Ser. 75(117), 163–168 (1968)] and M. Gromov [J. Differ. Geom. 13, 223–230 (1978; Zbl 0433.53028)] on minimal orbifolds of nonpositively curved manifolds to arbitrary manifolds with only a purely topological assumption.

##### MSC:
 53C20 Global Riemannian geometry, including pinching
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