  We give a partial solution to a long-standing open problem in the theory of
quantum groups, namely we prove that all finite-dimensional representations of
a wide class of locally compact quantum groups factor through matrix quantum
groups (Admissibility Conjecture for quantum group representations). We use
this to study Kazhdan's Property (T) for quantum groups with non-trivial
scaling group, strengthening and generalising some of the earlier results
obtained by Fima, Kyed and So{\l}tan, Chen and Ng, Daws, Skalski and Viselter,
and Brannan and Kerr. Our main results are:
  (i) All finite-dimensional unitary representations of locally compact quantum
groups which are either unimodular or arise through a special bicrossed product
construction are admissible.
  (ii) A generalisation of a theorem of Wang which characterises Property (T)
in terms of isolation of finite-dimensional irreducible representations in the
spectrum.
  (iii) A very short proof of the fact that quantum groups with Property (T)
are unimodular.
  (iv) A generalisation of a quantum version of a theorem of Bekka--Valette
proven earlier for quantum groups with trivial scaling group, which
characterises Property (T) in terms of non-existence of almost invariant
vectors for weakly mixing representations.
  (v) A generalisation of a quantum version of Kerr-Pichot theorem, proven
earlier for quantum groups with trivial scaling group, which characterises
Property (T) in terms of denseness properties of weakly mixing representations.
