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

TL;DR

This paper investigates quantum isometry groups of spectral triples on group C*-algebras, providing general results and explicit computations for various classes of finitely generated groups, including abelian and non-abelian examples.

Contribution

It establishes new connections between quantum isometry groups and classical group properties, and computes these groups explicitly for several important classes of finitely generated groups.

Findings

Quantum isometry groups of abelian groups are isomorphic to classical isometry groups.

For groups with polynomial growth, dual quantum groups also have polynomial growth under certain conditions.

Explicit calculations of quantum isometry groups for free, direct product, Baumslag-Solitar, and Coxeter 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.