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

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.

arXiv comment17 pages + references; edits following referee comments; to appear in JNCG

References in corpus (5)

• The Haagerup property for locally compact quantum groups
• Equivalent Notions of Normal Quantum Subgroups, Compact Quantum Groups with Properties F and FD, and Other Applications
• Around Property (T) for quantum groups
• On approximation properties of semidirect products of groups
• Kirchberg's Factorization Property for Discrete Quantum Groups