Definitions
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 π(g) ξ - ξ < ε for all g in K.
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).
(1) The trivial representation is an isolated point of the unitary dual of G with Fell topology.
(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.
(4) There exists an ε > 0 and a compact subset K of G such that every unitary representation of G that has an (ε, K)-invariant unit vector, has a nonzero 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.
Read more about this topic: Kazhdan's Property (T)
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