1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.1. Kazhdan's property (T) 341

Proposition 12.1.6. Let r be a group, A <1 r be a normal subgroup and

( E, 11,) be a K azhdan pair for the inclusion A c r. Let ( n, H) be a unitary

representation of r and denote by P the orthogonal projection from H onto

the subspace of all A-invariant vectors. Then, for every f, EH, one has

llf. -Pf.II ::; K-^1 sup lln(s)f, - f.11-

sEE

Proof. Let 6 = f,-Pf,. Since A is normal in r, the subspace PH is globally
r-invariant, and hence its orthogonal complement is too. Thus 7r restricts
to a unitary representation n1 on Hl =He PH. Since n1 has no nonzero
A-invariant vector, the assumption on (E, 11,) implies that

sup lln(s)f.-f.11 =sup lln1(s)6 - 611 2: Kll611,
sEE sEE
as desired. D

Here are a few characterizations of relative property (T). (See Appen-
dix D for more on 1-cocycles and isometric actions.)

Theorem 12.1.7. Let r be a countable group and A c r be a subgroup.
The following are equivalent:
(1) the inclusion Ac r has relative property (T);
(2) there exist a finite subset E C r and 11, > 0 with the following
property:^2 If (n, H) is a unitary representation of r and P is the
orthogonal projection from H onto the subspace of all A-invariant
vectors, then, for every f, E H, one has
llf.-Pf.II::; K-^1 sup lln(s)f,-f.ll;
sEE
(3) any sequence of positive definite functions on r that converges point-
wise to the constant function 1 converges uniformly on A;
(4) every 1-cocycle b: r ----t H of r is bounded on A;
(5) every action of r by affine isometries on a (real) Hilbert space has
a A-fixed point.

Moreover, if A = r, then the above conditions are equivalent to

(6) the group r is finitely generated and for any generating subsets c

r, there exists 11, = 11,(r, S) > 0 such that (S, 11,) is a Kazhdan pair.

Proof. (1) =} (4): We prove the contrapositive. Suppose that there exists
a 1-cocycle b on r which is unbounded on A. By Theorem D.11 and the
lemma following it, there exists a unitary representation 7r = ffin nt;n which

contains almost invariant vectors (f.t;n)n but no nonzero A-invariant vectors.

2rn particular, (E, K,) is a Kazhdan pair.