Symmetry gaps in Riemannian geometry and minimal orbifolds

Abstract

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 we establish a similar universal bound on the index of the deck group π1(M ) in the isometry group Isom( M , g) of the universal cover 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 and Weinberger with the additional assumption of bounds on 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.

On the work of Rodriguez Hertz on rigidity in dynamics

“…However, as in many mathematical theories, the presence of symmetries may cause further rigidity. For example in geometry, if a closed manifold has non-positive curvature or is aspherical, non-discreteness of the isometry group of the universal covering of a compact manifold forces the metric to be locally symmetric [Ebe80,Ebe82,FW08,vL13,vL14].…”