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

Existence and rigidity of quantum isometry groups for compact metric spaces