A notion of a quantum automorphism group of a finite quantum group, generalising that of a classical automorphism group of a finite group, is proposed and a corresponding existence result proved.
For $mu in (0,1), c> 0,$ we identify the quantum group $SO_mu(3)$ as the universal object in the category of compact quantum groups acting by `orientation and volume preserving isometries in the sense of cite{goswami2} on the natural spectral triple
on the Podles sphere $S^2_{mu, c}$ constructed by Dabrowski, DAndrea, Landi and Wagner in cite{{Dabrowski_et_al}}.
Quantum isometry groups of spectral triples associated with approximately finite-dimensional C*-algebras are shown to arise as inductive limits of quantum symmetry groups of corresponding truncated Bratteli diagrams. This is used to determine explici
tly the quantum isometry group of the natural spectral triple on the algebra of continuous functions on the middlethird Cantor set. It is also shown that the quantum symmetry groups of finite graphs or metric spaces coincide with the quantum isometry groups of the corresponding classical objects equipped with natural Laplacians.
We formulate a quantum group analogue of the group of orinetation-preserving Riemannian isometries of a compact Riemannian spin manifold, more generally, of a (possibly $R$-twisted in the sense of a paper of one of the authors, and of compact type) s
pectral triple. The main advantage of this formulation, which is directly in terms of the Dirac operator, is that it does not need the existence of any `good Laplacian as in our previous works on quantum isometry groups. Several interesting examples, including those coming from Rieffel-type deformation as well as the equivariant spectral triples on $SU_mu(2)$ and $S^2_{mu 0}$ are dicussed.
