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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 orderof Isom(M, g) can be universally bounded in terms of the bounds on Riccicurvature, diameter, and injectivity radius of M. This generalizes resultsknown for negative Ricci curvature to all manifolds. More generally weestablish a similar universal bound on the index of the deck group pi_1(M) inthe isometry group Isom(X, g) of the universal cover X in the absence ofsuitable actions by connected groups. We apply this to characterize locallysymmetric spaces by their symmetry in covers. This proves a conjecture of Farband Weinberger with the additional assumption of bounds on curvature, diameter,and injectivity radius. Further we generalize results of Kazhdan-Margulis andGromov on minimal orbifolds of nonpositively curved manifolds to arbitrarymanifolds with only a purely topological assumption.

Symmetry gaps in Riemannian geometry and minimal orbifolds