Kazhdhan property (T) for quantum groups: summary and new results.

Property (T) was introduced in the mid 60’s by D. Kazhdan, as a tool to demonstrate that a large class of lattices are finitely generated. The discovery of Property (T) was a cornerstone in group theory and the last decade saw its importance in many different subjects like ergodic theory, abstract harmonic analysis, operator algebra and some of the very recent topics like C*-tensor categories. In the late 1980’s the subject of operator algebraic quantum groups gained prominence starting with the seminal work of Woronowicz, followed by works of Baaj, Skandalis, Woronowicz, Kustermans, Vaes and others. Quantum groups can be looked upon as noncommutative analogues of locally compact groups and in this sense it was quite natural to explore the possibility of extending the notion of Property (T) to the realm of quantum groups. This was done in the following sequence: Property (T) was first studied within the framework of Kac algebras (a precursor to the theory of locally compact quantum groups), then for algebraic quantum groups and discrete quantum groups, and finally for locally compact quantum groups by Joita, Petrescu, Fima, Soltan, Kyed, Skalski, Viselter, Daws, Brannan and Kerr. Thus far, Property (T) was studied only for quantum groups with trivial scaling automorphism group. Quite recently we found a way of extending all the above studies to quantum groups with non-trivial scaling automorphisms. This talk will be a summary of the results that we have. We will start from the situation with groups abd then move over gradually to the quantum groups.