Quantum isometry groups of log-Laplacians on Cuntz--Krieger algebras

We compute the quantum isometry groups of Cuntz-Krieger algebras endowed with the spectral triples coming from the Ahlfors regular structure of the underlying topological Markov chain.

This allows us to exhibit a new family of compact quantum groups, mixing features from quantum automorphism groups of graphs and easy quantum groups.

Contrary to the classical isometry groups, whose actions on the Cuntz-Krieger algebras are never ergodic, the quantum isometry group acts ergodically in the case of the Cuntz algebra.

This also leads to the construction of a (genuinely quantum) ergodic action of a compact matrix quantum group on the Cantor space.