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We study quantum isometry groups, denoted by $\mathbb{Q}(\Gamma, S)$, ofspectral triples on $C^*_r(\Gamma)$ for a finitely generated discrete groupcoming from the word-length metric with respect to a symmetric generating set$S$. We first prove a few general results about $\mathbb{Q}(\Gamma, S)$including : \begin{itemize} \item For a group $\Gamma$ with polynomial growthproperty, the dual of $\mathbb{Q}(\Gamma, S)$ has polynomial growth propertyprovided the action of $\mathbb{Q}(\Gamma,S)$ on $C^*_r(\Gamma)$ has fullspectrum, \item $\mathbb{Q}(\Gamma, S) \cong QISO(\hat{\Gamma}, d)$ for anyabelian $\Gamma$, where $d$ is a suitable metric on the dual compact abeliangroup $\hat{\Gamma}$. \end{itemize}  We then carry out explicit computations of $\mathbb{Q}(\Gamma,S)$ for severalclasses of examples including free and direct product of cyclic groups,Baumslag-Solitar group, Coxeter groups etc. In particular, we have computedquantum isometry groups of all finitely generated abelian groups which do nothave factors of the form $\mathbb{Z}_2^k$ or $\mathbb{Z}_4^l$ for some $k,l$ inthe direct product decomposition into cyclic subgroups.

Quantum Isometry group of dual of finitely generated discrete groups and quantum groups