Although originally defined in terms of irreducible representations, property (T) can often be checked even when there is little or no explicit knowledge of the unitary dual.
Let G be a σ-compact, locally compact topological group and π : G → U(H) a unitary representation of G on a (complex) Hilbert space H. If ε > 0 and K is a compact subset of G, then a unit vector ξ in H is called an (ε, K)-invariant vector if The following conditions on G are all equivalent to G having property (T) of Kazhdan, and any of them can be used as the definition of property (T).
(2) Any sequence of continuous positive definite functions on G converging to 1 uniformly on compact subsets, converges to 1 uniformly on G. (3) Every unitary representation of G that has an (ε, K)-invariant unit vector for any ε > 0 and any compact subset K, has a non-zero invariant vector.
(5) Every continuous affine isometric action of G on a real Hilbert space has a fixed point (property (FH)).
If H is a closed subgroup of G, the pair (G,H) is said to have relative property (T) of Margulis if there exists an ε > 0 and a compact subset K of G such that whenever a unitary representation of G has an (ε, K)-invariant unit vector, then it has a non-zero vector fixed by H. Definition (4) evidently implies definition (3).