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

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 $g$.