344 12. Approximation Properties for Groups
1
= ~ L v(s)ll~ - 7r(s)~ll^2
sES
::; max 117r(s)~ - ~112
sESfor every~ EH. This implies that (S, .J2E) is a Kazhdan pair. D
The following lemma will be needed in the next chapter. Its proof is
similar to the proof of Lemma 12.1.8 and hence will be omitted.
Lemma 12.1.9. Let I' be a group and S c r be a finite generating subset
which contains the unit. Then, r has property (T) if and only if the following
is true: For any unitary representation ( 7r, 1i) without a nonzero invariant
vector, one has
II L 7r(s)ll < ISi.
sESProperty (T) for SL(3, Z). Our next goal is to prove that SL(3, Z) has
property (T), following ideas of Y. Shalom.^4 The proof takes several pages
and consists of two parts. We first prove relative property (T) for certain
subgroups of SL(3, Z) and then deduce property (T) from a bounded gener-
ation trick.
For 1 :S i # j :S n, we denote by Eij E SL( n, Z) the matrix with 1 's on
the diagonal, 1 in the ( i, j)-th entry, and O's elsewhere. It is not hard to see
that the set S = {Eij : i # j} generates SL(3, Z). Let G, H c SL(3, Z) be
the subgroups given by
a~[~~~] andH~[~ ~ ~]We note that G ~ SL(2, Z) normalizes H ~ Z^2 and the subgroup GH
is canonically isomorphic to the semidirect product Z^2 ><1 SL(2, Z), where
SL(2, Z) acts on Z^2 by linear transformations.
Theorem 12.1.10. The inclusion (Z^2 c Z^2 ><1 SL(2, Z)) has relative prop-
erty (T). More precisely, if So= {E12,E21,E13,E23}, then (So,10-^1 ) is a
Kazhdan pair for (Z^2 c Z^2 ><1 SL(2, Z)).
Proof. For notational simplicity, we set 91 = E12, g2 = E21, hl = E13 and
h2 = E23. Let (7r, 1i) be a unitary representation of Z^2 ><1 SL(2, Z) which
does not have a nonzero invariant vector. Suppose by contradiction that
there exists a unit vector ~ E 1i such that
5=max117r(s)~ - ~II< 10-^1.
sESo(^4) The argument works in greater generality, but we'll be content with this case since it's all
we need for applications.