Torsion and K-theory for some free wreath products

TL;DR

This paper classifies torsion actions of free wreath products of quantum groups and demonstrates how certain properties and K-theory computations extend from quantum groups to their wreath products, advancing understanding in quantum algebra.

Contribution

It provides a classification of torsion actions for free wreath products and extends the strong Baum-Connes property and K-theory calculations to these structures.

Findings

Torsion actions of free wreath products are classified.

Strong Baum-Connes property is preserved under certain conditions.

K-theory of free wreath products with quantum groups is computed.

Abstract

We classify torsion actions of free wreath products of arbitrary compact quantum groups and use this to prove that if $\mathbb{G}$ is a torsion-free compact quantum group satisfying the strong Baum-Connes property, then $\mathbb{G}\wr_{\ast}S_{N}^{+}$ also satisfies the strong Baum-Connes property. We then compute the K-theory of free wreath products of classical and quantum free groups by $SO_{q}(3)$.