Property (T), property (F) and residual finiteness for discrete quantum groups

TL;DR

This paper explores the relationships between residual finiteness, property (T), and the Kirchberg factorization property in discrete quantum groups, establishing implications and equivalences among these properties.

Contribution

It introduces a notion of residual finiteness for discrete quantum groups and proves its equivalence with the factorization property when combined with property (T).

Findings

Residual finiteness implies the Kirchberg factorization property.

Property (T) and the factorization property together imply residual finiteness.

Results apply to quantum groups from bicrossed product constructions and their extensions.

Abstract

We investigate connections between various rigidity and softness properties for discrete quantum groups. After introducing a notion of residual finiteness, we show that it implies the Kirchberg factorization property for the discrete quantum group in question. We also prove the analogue of Kirchberg's theorem, to the effect that conversely, the factorization property and property (T) jointly imply residual finiteness. We also apply these results to certain classes of discrete quantum groups obtained by means of bicrossed product constructions and study the preservation of the properties (factorization, residual finiteness, property (T)) under extensions of discrete quantum groups.