Around Property (T) for quantum groups

Kazhdan's Property (T) is a notion of fundamental importance, with numerous applications 
in various fields of mathematics such as abstract harmonic analysis, ergodic theory and 
operator algebras. By using Property (T), Connes was the first to exhibit a rigidity 
phenomenon of von Neumann algebras. Since then, the various forms of Property (T) have 
played a central role in operator algebras, and in particular in Popa's deformation/rigidity 
theory.
This talk is devoted to some recent progress in the notion of Property (T) for locally 
compact quantum groups. Most of our results are concerned with second countable discrete 
unimodular quantum groups with low duals. In this class of quantum groups, Property (T) is 
shown to be equivalent to Property (T)$^{1,1}$ of Bekka and Valette. As applications, we 
extend to this class several known results about countable groups, including theorems on 
"typical" representations (due to Kerr and Pichot) and on connections of Property (T) with 
spectral gaps (due to Li and Ng) and with strong ergodicity of weakly mixing actions on a 
particular von Neumann algebra (due to Connes and Weiss).
Joint work with Matthew Daws and Adam Skalski.
The talk will be self-contained: no prior knowledge of quantum groups or Property (T) for groups is required.