  We study Property (T) for locally compact quantum groups, providing several
new characterisations, especially related to operator algebraic ergodic theory.
Quantum Property (T) is described in terms of the existence of various Kazhdan
type pairs, and some earlier structural results of Kyed, Chen and Ng are
strengthened and generalised. For second countable discrete unimodular quantum
groups with low duals Property (T) is shown to be equivalent to Property
(T)$^{1,1}$ of Bekka and Valette. This is used to extend to this class of
quantum groups classical 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). Finally we discuss in the Appendix
equivalent characterisations of the notion of a quantum group morphism with
dense image.
