1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

12.2. The Haagerup property 355

ll - 'Pn(sk)I < 2-n for every k and n 2:: k. By the GNS construction (see

Theorem 2.5.11), for each n we have a unitary representation (Kn, 'Hn) and

a unit vector en E 'Hn such that 'Pn(s) = (7rn(s)en, en)· Let'}{= EB 'Hn and

1f = EB 1f n. We define a 1-cocycle b with coefficients in ( 1f, 'H) by

00

b: r 3 s i--+ (en - Kn(s)en)~=l E E3j'Hn = 'H.

n=l

The infinite sum converges because

00 00

n=l n=l

for every s E r. Moreover, this equation implies that b is proper because

llb(s)ll^2 :::; N implies Jcpn(s)l 2:: 1/2 for some n E {1, ... ,N}. D

In order to give examples of groups with the Haagerup property, we now
show how to construct a 1-cocycle on a group which acts properly on a tree.
(See Appendix E for our convention on trees.)
Let T be a tree and fix a base point o in T. We view every edge
e = (x, y) E E as a path of length one from x toy and denote its reverse
path bye. For each vertex x in T, we denote by [o, x] the unique geodesic
path that connects o to x. We define b 0 (x) E .€^2 (E) by

bo(x)(e) ~ { ~1

A computation confirms that

if e is on [o, x],

if e is on [o, x],

otherwise.

JJba(x)-bo(Y)JJ^2 = 2d(x,y),

where d is the graph metric. It follows that the graph metric d on the tree
Tis a conditionally negative definite kernel. (See Section D.)

Now let r be a group which acts on T. Denote by 1f the associated
unitary representation of r on .€^2 (E). We set b(s) = b 0 (so). It is not hard
to check that b is a 1-cocycle on r with coefficients in ( 1f, .€^2 (E)) such that
llb(s)Jl^2 = 2d(o, so) (see Figure 1).
Since every finitely generated free group lFr acts properly on its Cayley
graph (which is a tree), the associated 1-cocycle is proper on lFr· Hence free
groups have the Haagerup property. (Infinitely generated free groups also
have the Haagerup property because they are increasing unions of finitely
generated free groups.) The group SL(2, Z) = (Z/4Z)*z; 2 z(Z/6Z) is another
example which acts properly on a tree (being an amalgamated free product
of finite groups, its action on the Bass-Serre tree is proper - see Appendix E).