assume by contradiction that for every K and ε there is a unitary representation that has a (K, ε)-invariant unit vector and does not have an invariant vector. Look at the direct sum of all such representation and that will negate (4). The equivalence of (4) and (5) (Property (FH)) is the Delorme-Guichardet theorem. The fact that (5) implies (4) requires the assumption that G is σ-compact (and locally compact) (Bekka et al., Theorem 2.12.4). General properties Property (T) is preserved under quotients: if G has property (T) and H is a quotient group of G then H has property (T). Equivalently, if a homomorphic image of a group G does not have property (T) then G itself does not have property (T). If G has property (T) then G/[G, G] is compact. Any countable discrete group with property (T) is finitely generated. An amenable group which has property (T) is necessarily compact. Amenability and property (T) are in a rough sense opposite: they make almost invariant vectors easy or hard to find. Kazhdan's theorem: If Γ is a lattice in a Lie group G then Γ has property (T) if and only if G has property (T). Thus for n ≥ 3, the special linear group SL(n, Z) has property (T). Examples Compact topological groups have property (T). In particular, the circle group, the additive group Zp of p-adic integers, compact special unitary groups SU(n) and all finite groups have property (T). Simple real Lie groups of real rank at least two have property (T). This family of groups includes the special linear groups SL(n, R) for n ≥ 3 and the special orthogonal groups SO(p,q) for p > q ≥ 2 and SO(p,p) for p ≥ 3. More generally, this holds for simple algebraic groups of rank at least two over a local field. The pairs (Rn ⋊ SL(n, R), Rn) and (Zn ⋊ SL(n, Z), Zn) have relative property (T) for n ≥ 2. For n ≥ 2, the noncompact Lie group Sp(n, 1) of isometries of a quaternionic hermitian form of signature (n,1) is a simple Lie group of real rank 1 that has property (T). By Kazhdan's theorem, lattices in this group have property (T). This construction is significant because these lattices are hyperbolic groups; thus, there are groups that are hyperbolic and have property (T). Explicit examples of groups in this category are provided