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

## TL;DR

This paper demonstrates that the quantum isometry group of the group C*-algebra for the integers remains unchanged regardless of the symmetric generating set, but this invariance does not hold for higher-dimensional integer lattices.

## Contribution

It proves the invariance of the quantum isometry group for the integer group and shows the failure of this invariance for higher-dimensional free abelian groups.

## Key findings

- Quantum isometry group of $C_r^*(bZ)$ is independent of the generating set.
- Invariance does not extend to $bZ^n$ for $n>1$.
- Results highlight differences in quantum symmetries between one-dimensional and higher-dimensional groups.

## Abstract

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 $n>1$.