1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

15.3. Examples 415

Theorem 15.3.4. Let ri, ... , r n be hyperbolic groups and Ni, ... , Nm be

noninjective II1 -factors. If there exists an embedding

Ni @···@Nm '---t pL(r1 x · · · x r n)P,

for some projection p E L(r1 x · · · x r n), then m :Sn.

Proof. By Theorem 15.1.5 and Lemma 15.3.3, after permuting indices, one

has

eiN1e1 ® · · · ® Nm-1 '---t PoL(r1 x · · · x r n-1)Po

for some nonzero projections ei E Ni and Po E L(r1 x · · · x r n- 1 ). By

induction, we are done. D

Semidirect products and wreath products.

Lemma 15.3.5. Let r = Y ><1 A be a semidirect product of discrete groups.
Let 9A be a family of subgroups of A and set g = {Y ><I Ao : Ao E 9A}· If Y
is amenable and A is bi-exact relative to g A, then r is bi-exact relative to g.

Proof. Letμ: A- Prob(A) be a map as in Definition 15.1.2. It is not hard
to see that the composition of μ with the quotient r - A = r /Y satisfies
the conditions of Lemma 15.2.6. D

This is not so interesting unless 9A is very small (e.g., if A is hyperbolic).
So, we consider another example, namely the wreath product, which was
defined right before Definition 12.2.10.

In what follows, we denote YI A by r and agree that p, s and t represent
elements of A, while x, y and z represent elements of YA (the group EBAY
of finitely supported functions from A into Y). Hence a typical element of
r will be denoted by xs or yt. In particular, sx = a 8 (x)s, where a is the
left translation action of A on YA.

Proposition 15.3.6. Let r =YI A be the wreath product and let g ={A}.
If Y is amenable and A is exact, then r is bi-exact relative to g.

The proof of this proposition requires several steps. We fix a proper
length function I · IA on A:

(1) islA = ls-^1 IA E IR~o for s EA and lslA = 0 if and only ifs= e;
(2) [stiA :S lslA + [t[A for every s, t E A;
(3) the subset BA(R) = {s EA: lslA :SR} is finite for every R > 0.
(See Proposition 5.5.2 for the existence of such a function.) Likewise, fix a
length function on Y. For yt Er, we define ((yt) E £^1 (A) by

((yt)(p) = { min{IPIA, IC^1

0

PIA} + jy(p)[r if p E supp(y),

if p ~ supp(y).