340 12. Approximation Properties for GroupsNote that a unitary representation 7r has almost I'-invariant vectors if
and only if there exists a nonzero (E, c:)-invariant vector for any finite subset
E c r and c: > 0.^1
Definition 12.1.2. Let A c r be a subgroup. We say the inclusion A c r
has relative property (T) if any unitary representation ( 7r, H) of r which has
almost I'-invariant vectors has a nonzero A-invariant vector. We say r has
Kazhdan's property (T) if the identity inclusion r c r has relative property
(T). A pair (E, "'), where E c r and"' > 0, is called a Kazhdan pair for
the inclusion A c r (or for r, when A = r) if any unitary representation
of r which has a nonzero (E, "')-invariant vector has a nonzero A-invariant
vector.Kazhdan's motivation for considering property (T) was the following two
facts: A group with property (T) is finitely generated (Corollary 6.4.7) and
has finite abelianization (as explained below).
Example 12.1.3. Finite groups have property (T). It is a simple matter to
prove that an amenable group with property (T) is finite (amenability pro-
vides almost invariant vectors in the left regular representation, so property
(T) yields an honest invariant vector - which is impossible for an infinite
group). Thus an amenable group has property (T) if and only if it is finite.
Since property (T) clearly passes to quotients, it follows that the abelian-
ization r /[r, r] of a property (T) group is always finite.The mere existence of an infinite group with property (T) is surprising,
but it turns out that examples are abundant.
Example 12.1.4. Lattices in higher rank semisimple Lie groups, as well as
lattices in Sp(l, n), have property (T). (See [15] -we will prove the case of
SL(3, Z) shortly.)
Lemma 12.1.5. For any group I', the pair (r, J2) is Kazhdan.Proof. Let a unitary representation (7r, H) and a nonzero (r, v'2)-invariant
vector e be given. Then, the unique circumcenter (of the subset 7r(r)e (see
Exercise D.l) is I'-invariant since 7r(r)e is globally I'-invariant. Letting ~
denote the real part of a complex number, we have~(e, () ::::: inf ~(e, 7r(s)e) = 1-~sup 11e - 7r(s)e11^2 > 0
sEI' 2 sEI'
and hence ( i-0. 0(^1) In terms of the Fell topology ([15]), 7r has a nonzero invariant vector if and only if the
trivial representation is contained in 7r, and 7r has almost invariant vectors if and only if the trivial
representation is weakly contained in 7r.