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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.

53C20 Global Riemannian geometry, including pinching
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