Minimal orbifolds and (a)symmetry of piecewise locally symmetric manifolds

TL;DR

This paper proves that for closed piecewise locally symmetric manifolds, the universal cover's isometry group is discrete and its index over the fundamental group is uniformly bounded, regardless of the metric.

Contribution

It establishes the discreteness of the universal cover's isometry group and bounds its index relative to the fundamental group for such manifolds.

Findings

Universal cover's isometry group is discrete

Index of isometry group over fundamental group is bounded independently of the metric

Results hold for any Riemannian metric on the manifold

Abstract

We show that if $g$ is a Riemannian metric on a closed piecewise locally symmetric manifold $M$, then the lift of $g$ to the universal cover $\widetilde{M}$ has a discrete isometry group. We also show that the index $[\Isom(\widetilde{M}): \pi_1(M)]$ is bounded by a constant independent of $g$.