1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

350 12. Approximation Properties for Groups

s ES, and we set lvl = 2:: v(s) = IEI. Let L^2 (S, v) be the weighted L^2 -space

on S with inner product given by

(!, g)v = ~^1 """" 6 v(s) f(s)g(s), -

sES.

for functions f and g on S.

Recall from Appendix E that the combinatorial Laplacian is defined as

.6. = ~d*d E JIB(L^2 (S, v)),

where d: L^2 (S, v)--+ L^2 (E) is given by d(f)((s, t)) = f(t) - f(s). If the link

L(I', S) is connected, then 0 is a simple eigenvalue of .6. and the first nonzero

eigenvalue of .!.l is denoted by .\1.

Theorem 12.1.15. Let r be a discrete group which is generated by a finite

symmetric set S withe f/. S. Suppose that the link L(r, S) is connected and

.\1 > 1/2. Then r has property (T). More precisely, (S, V2(2 - .\1^1 )) is a

K azhdan pair.

Proof. Let ( 1l', 1-l) be a universal unitary representation and let

1

h = ~ L v(s)1T'(s).

sES

Since L(I',S) is connected, we have v(s) = v(s-^1 ) > 0 for every s ES. By

Lemma 12.1.8, it suffices to show

o-(h) c [-1, .\1^1 -1] u {1},

where o-(h) is the spectrum of h. Let ~ E 1-{ be given and define f: S --+ 1-{

by f(s) = 1T'(s)~. Notice that the mean offish~.

Now view fas an element of L^2 (S, v)@H. By a Poincare-type inequality

(Lemma E.5), we have

.\1(11!11

2

• llh~ll
  2
  ) :S ((.6. 0 Irt)f, f) = 2 l~I L llf(t) - f(s)ll^2 ·
  (s,t)EE
  Also note that llJll^2 - llh~ll^2 = 11~11^2 - llh~ll^2 = ((1 - h^2 )~, ~). On the other
  hand, since v(r) = l{(s, t) EE: C^1 s = r}I for every r ES, we have
  1 1
  2IEI L llJ(t) - J(s)ll2 = 2IEI L 111T'(t)~ -1T'(s)~ll2
  (s,t)EE (s,t)EE
  1
  = ~ L v(r)ll~ -1T'(r)~ll^2
  rES
  = ((1-h)~, ~).

It follows that .\1(1-h^2 ) :S 1-h, or equivalently, o-(h) n(.\1^1 -1, 1) = 0. D