1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.1. Kazhdan's property (T) 347

Proof. Let

s ~ [: ; :J E SL(3,Z)

be given. By the Chinese Remainder Theorem, there exists m E Z such that

x+mz = 1 (mod p) for all prime divisors p of y which do not divide z. Since

s E SL(3, Z), we have gcd(x, y, z) = 1 and thus no prime divisor of gcd(y, z)

divides x. It follows that gcd(x + mz, y) = 1. Thus,

[x y z : : :] [ m ~ ~ 0 ~] 1 [ x' : : y z :]

with gcd( x', y) = 1. Hence, there exists a, b E Z such that ax'+ by+ z = 1,

or equivalently,

[x' : : y z :] [~ 0 ~ 0 1 ~] [: x' : y ~] 1

for some u, v E Z. Finally, we have

[~ ~ 0 0 =~] 1 [ : : x' y ~] 1 [ -x' ~ -y ~ ~] 1

Therefore, s E HiGH2H1H2 = HiH2H1H2G. D
Definition 12.1.13. Let r be a group and G, Hi, H2 c r be subgroups.
We say the subgroups satisfy the Shalom property if

(1) r is generated by Hi and H2;

(2) G normalizes both Hi;

(3) r is boundedly generated by G, Hi and H2, i.e., r = Hi(l) · · · Hi(l)G;

(4) both (Hi c r), i = 1, 2, have relative property (T).

We have seen that there is a system of subgroups G, Hi, H2 C SL(3, Z)
which satisfy the Shalom property. (That Hi and H2 generate r is a trivial
exercise.)
Theorem 12.1.14. Let r be a group which admits a system of subgroups
with the Shalom property. Then r has property (T). In particular, SL(3, Z)
has property (T) and (S, 10-^3 ) is a Kazhdan pair, where S = {Eij: i =/: j}.

Proof. By Theorem 12.1. 7, there exist a finite subset E c r and K, > 0

such that

II~ - pi~ll ~ K,-l sup 117r(s)~ -~II

sEE