Quantum isometry groups of $0$-dimensional manifolds

Abstract

Quantum isometry groups of spectral triples associated with ap- proximately finite-dimensional C � -algebras are shown to arise as inductive limits of quantum symmetry groups of corresponding truncated Bratteli dia- grams. This is used to determine explicitly the quantum isometry group of the natural spectral triple on the algebra of continuous functions on the middle- third 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.