12.2. The Haagerup property 357o and x. We leave it to the reader to check w(x,y) = li((y) - ((x)JJ^2.
Hence w is conditionally negative definite. The second assertion follows
from Theorem D .11. DLet us recall the definition of the wreath product Y ( A of a group Y
by another group A. To ~ase notation, denote by YA the alge9raic direct
product group E9 A Y and view an element x E YA as a finitely supported
function x: A--+ Y, where the support of xis supp(x) = {p EA: x(p) # e}.
We note that (xy)(p) = x(p)y(p) E Y for x, y E YA and p EA. Then, A acts
on YA by left translation: a 8 (x)(p) = x(s-^1 p). The wreath product Y (A is
defined to be the semidirect product YA ><la A.
Definition 12.2.10. We say a group r has a proper wall structure if there
is a family W of walls in r such that (r, W) is a space with walls on which
r acts properly by left multiplication.
Theorem 12.2.11. Let r = Y (A be the wreath product of Y by A. If Y is
finite and A has a proper wall structure, then r has a proper wall structure.
In particular, the group (Z/2Z) ( lF2 has the Haagerup property.Proof. Below, we'll lets and t denote elements of A, while x and y represent
elements of YA. Thus xs or yt will be generic elements of r.
Let WA be a proper wall structure of A and 1i be the corresponding
family of half spaces: 1i = {H: {H, He} EWA}. For HE 1i and a finitely
supported function μ : He --+ Y, we define
E(H,μ) = {xs Er: s EH and xlw = μ} c r.
Now, define a family W of walls in r by
W = { {E(H,μ),E(H,μ)e}: HE 1-l, μ:He--+ Y finitely supported}.We first claim that (r, W) is a space with walls. Let xs, yt E r be
given and we'll show that there are finitely many H's and μ's such that
xs E E(H, μ) and yt E E(H, μ)e. Indeed, xs E E(H, μ) means s E H and
xJHc = μ, while yt E E(H, μ)e means either t E He or Y!Hc # μ. It follows
that
Hen ({t} U supp(x-^1 y)) # 0
and hence the half space H separa.tes sand some element in {t}Usupp(x-^1 y).
Since the set {t} U supp(x-^1 y) is finite, there are finitely many such H's.
Note thatμ= x!Hc is uniquely determined by xs and H.
We next observe that r preserves the wall structure; that is, sE(H, μ) =
E(sH, s.μ) for s EA and xE(H, μ) = E(H, (x-^1 !Hc )μ) for x E YA.
Finally, we check that the action of r on (r, W) is proper. Let wr(xs)
(resp. WA(s)) bethenumberofwallsinr (resp. inA) separatingxs Er (resp.