Example of a group whose quantum isometry group does not depend on the generating set

In this article we have shown that the quantum isometry group of $C_r^*(\mathbb{Z})$, denoted by $\mathbb{Q}(\mathbb{Z},S)$ as in \cite{gos_man}, with respect to a symmetric generating set $S$ does not depend on the generating set $S$. Moreover, we have proved that the result is no longer true if the group $\mathbb{Z}$ is replaced by $\underbrace{\mathbb{Z} \times \mathbb{Z} \times\cdots \times \mathbb{Z}}_{n \ copies}$ for $n>1$.