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Let G be a finitely generated discrete group. The standard spectral triple onthe group C*-algebra C*(G) is shown to admit the quantum group of orientationpreserving isometries. This leads to new examples of compact quantum groups. Inparticular the quantum isometry group of the C*-algebra of the free group onn-generators is computed and turns out to be a quantum group extension of thequantum permutation group A_{2n} of Wang. The quantum groups of orientation andreal structure preserving isometries are also considered and construction ofthe Laplacian for the standard spectral triple on C*(G) discussed.

Quantum isometry groups of noncommutative manifolds associated to group C*-algebras