Permanence of the torsion-freeness property for divisible discrete quantum subgroups

TL;DR

This paper demonstrates that torsion-freeness in quantum groups remains stable under divisible discrete quantum subgroups, leading to broader stability results for quantum group constructions and the Baum-Connes conjecture.

Contribution

It establishes the preservation of torsion-freeness under divisible discrete quantum subgroups and extends stability results for the Baum-Connes conjecture in quantum groups.

Findings

Torsion-freeness is preserved under divisible discrete quantum subgroups.

Stability of the Baum-Connes conjecture is confirmed for quantum subgroups.

Improved understanding of quantum group constructions and their properties.

Abstract

We prove that torsion-freeness in the sense of Meyer-Nest is preserved under divisible discrete quantum subgroups. As a consequence, we obtain some stability results of the torsion-freeness property for relevant constructions of quantum groups (quantum (semi-)direct products, compact bicrossed products and quantum free products). We improve some stability results concerning the Baum-Connes conjecture appearing already in a previous work of the author. For instance, we show that the (resp. strong) Baum-Connes conjecture is preserved by discrete quantum subgroups (without any torsion-freeness or divisibility assumption).