Quantum isometry group of dual of finitely generated discrete groups- $\textrm{II}$

TL;DR

This paper explicitly computes quantum isometry groups for various finitely generated discrete groups, extending previous work and providing new examples and descriptions of these quantum symmetries.

Contribution

It offers new explicit calculations of quantum isometry groups for specific groups and describes certain quantum groups via free wreath products, expanding the understanding of quantum symmetries.

Findings

Computed quantum isometry groups for braid group of 3 generators and others.

Provided alternative descriptions of certain quantum groups as free wreath products.

Identified new groups with quantum isometry groups as doubling of their group C*-algebras.

Abstract

As a contribution of the programme of Goswami and Mandal (2014), we carry out explicit computations of $\mathbb{Q}(\Gamma,S)$, the quantum isometry group of the canonical spectral triple on $C_{r}^{*}(\Gamma)$ coming from the word length function corresponding to a finite generating set S, for several interesting examples of $\Gamma$ not covered by the previous work Goswami and Mandal (2014). These include the braid group of 3 generators, $\mathbb{Z}_4^{*n}$ etc. Moreover, we give an alternative description of the quantum groups $H_s^{+}(n,0)$ and $K_n^{+}$ (studied in Banica and Skalski (2012), Banica and Skalski (2013)) in terms of free wreath product. In the last section we give several new examples of groups for which $\mathbb{Q}(\Gamma)$ turn out to be doubling of $C^*(\Gamma)$.