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

We study quantum isometry groups, denoted by $\mathbb{Q}(\mathrm{\Gamma },S)$, of spectral triples on $C_r^{\ast }(\mathrm{\Gamma })$ for a finitely generated discrete group coming from the word-length metric with respect to a symmetric generating set $S$. We first prove a few general results about $\mathbb{Q}(\mathrm{\Gamma },S)$ including : \begin{itemize} \item For a group $\mathrm{\Gamma }$ with polynomial growth property, the dual of $\mathbb{Q}(\mathrm{\Gamma },S)$ has polynomial growth property provided the action of $\mathbb{Q}(\mathrm{\Gamma },S)$ on $C_r^{\ast }(\mathrm{\Gamma })$ has full spectrum, \item $\mathbb{Q}(\mathrm{\Gamma },S)\cong QISO(\widehat{\mathrm{\Gamma }},d)$ for any abelian $\mathrm{\Gamma }$, where $d$ is a suitable metric on the dual compact abelian group $\widehat{\mathrm{\Gamma }}$. \end{itemize} We then carry out explicit computations of $\mathbb{Q}(\mathrm{\Gamma },S)$ for several classes of examples including free and direct product of cyclic groups, Baumslag-Solitar group, Coxeter groups etc. In particular, we have computed quantum isometry groups of all finitely generated abelian groups which do not have factors of the form ${\mathbb{Z}}_2^k$ or ${\mathbb{Z}}_4^l$ for some $k,l$ in the direct product decomposition into cyclic subgroups.