Myers-Steenrod theorems for metric and singular Riemannian foliations

We prove that the group of isometries preserving a metric foliation on a closed Alexandrov space X, or a singular Riemannian foliation on a manifold M is a closed subgroup of the isometry group of X in the case of a metric foliation, or of the isometry group of M for the case of a singular Riemannian foliation. We obtain a sharp upper bound for the dimension of these subgroups and show that, when equality holds, the foliations that realize this upper bound are induced by fiber bundles whose fibers are round spheres or projective spaces. Moreover, singular Riemannian foliations that realize the upper bound are induced by smooth fiber bundles whose fibers are round spheres or projective spaces.