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

Abstract

We study quantum isometry groups, denoted by $\mathbb{Q}(\Gamma, S)$, of spectral triples on $C^*_r(\Gamma)$ 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 $\mathbb{Q}(\Gamma, S)$ including : \begin{itemize} \item For a group $\Gamma$ with polynomial growth property, the dual of $\mathbb{Q}(\Gamma, S)$ has polynomial growth property provided the action of $\mathbb{Q}(\Gamma,S)$ on $C^*_r(\Gamma)$ has full spectrum, \item $\mathbb{Q}(\Gamma, S) \cong QISO(\hat{\Gamma}, d)$ for any abelian $\Gamma$, where $d$ is a suitable metric on the dual compact abelian group $\hat{\Gamma}$. \end{itemize} We then carry out explicit computations of $\mathbb{Q}(\Gamma,S)$ 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.