Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups

We construct for every connected locally finite graph $\mathrm{\Pi }$ the quantum automorphism group $\text{QAut }\mathrm{\Pi }$ as a locally compact quantum group. When $\mathrm{\Pi }$ is vertex transitive, we associate to $\mathrm{\Pi }$ a new unitary tensor category $\mathcal{C}(\mathrm{\Pi })$ and this is our main tool to construct the Haar functionals on $\text{QAut }\mathrm{\Pi }$. When $\mathrm{\Pi }$ is the Cayley graph of a finitely generated group, this unitary tensor category is the representation category of a compact quantum group whose discrete dual can be viewed as a canonical quantization of the underlying discrete group. We introduce several equivalent definitions of quantum isomorphism of connected locally finite graphs $\mathrm{\Pi }$, ${\mathrm{\Pi }}^{\mathrm{\prime }}$ and prove that this implies monoidal equivalence of $\text{QAut }\mathrm{\Pi }$ and $\text{QAut }{\mathrm{\Pi }}^{\mathrm{\prime }}$.