  Relative property (T) has recently been used to construct a variety of new
rigidity phenomena, for example in von Neumann algebras and the study of
orbit-equivalence relations. However, until recently there were few examples of
group pairs with relative property (T) available through the literature. This
motivated the following result:
  A finitely generated group Gamma admits a real-special linear representation
with non-amenable Zariski closure if and only if it acts on an Abelian group A
(of finite nonzero Q-rank) so that the corresponding group pair (Gamma \ltimes
A, A) has relative property (T).
  The proof is constructive. The main ingredients are Furstenberg's celebrated
lemma about invariant measures on projective spaces and the spectral theorem
for the decomposition of unitary representations of Abelian groups. Methods
from algebraic group theory, such as the restriction of scalars functor, are
also employed.
