Existence and Rigidity of Quantum Isometry Groups for Compact Metric Spaces

We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed probability measure. In the former case it also follows from recent results of the second author that the quantum isometry group is classical, i.e. the commutative C^∗-algebra of continuous functions on the Riemannian isometry group.