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

Let ΓΓ be a finitely generated discrete group. The standard spectral triple on the group C∗C∗-algebra C∗(Γ)C∗(Γ) is shown to admit the quantum group of orientation preserving isometries. This leads to new examples of compact quantum groups. In particular, the quantum isometry group of the C∗C∗-algebra of the free group on nn generators is computed and turns out to be a quantum group extension of the quantum permutation group A2nA2n of Wang. The quantum groups of orientation and real structure preserving isometries are also considered and the construction of the Laplacian for the standard spectral triple on C∗(Γ)C∗(Γ) is discussed.