Symmetry gaps in Riemannian geometry and minimal orbifolds

TL;DR

This paper establishes universal bounds on the size of isometry groups of Riemannian manifolds based on curvature, diameter, and injectivity radius, extending previous results and characterizing symmetric spaces.

Contribution

It generalizes bounds on isometry groups to all manifolds without negative Ricci curvature and characterizes symmetric spaces via their symmetry in covers.

Findings

Bound on isometry group size in terms of geometric bounds

Characterization of locally symmetric spaces by their symmetry

Extension of minimal orbifold results to arbitrary manifolds

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 pi_1(M) in the isometry group Isom(X, g) of the universal cover X 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.