1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

5.1. Exact groups 175

4.1.4). Thus we have an embedding (A><lrr)0C~(r) c JE(7i'.0£^2 (r)0£^2 (r)).

Letting V: £^2 (r) --+ £^2 (r) 0 £^2 (r) be the isometry such that V ( Ot) = Ot 0 Ot

for all t Er, an unenlightening (read: unpleasant) calculation shows that

(lrt 0 V*)(a>.s 0 >.g)(lrt: 0 V) = a>. 8 EA ><lr r c JE(h'. 0 £^2 (r))

whenever s = g, and it equals zero otherwise. D

Much more can be found in [108] and [109]. Interestingly, it isn't yet

known whether the "only if' direction of the theorem above extends to

locally compact groups.

Exercises

Exercise 5.1.1. Prove the following permanence properties:

(1) subgroups of exact groups are exact;

(2) exactness is preserved by amalgamated free products;

(3) an increasing union of exact groups is exact. (Warning: Arbitrary

inductive limits of exact groups apparently need not be exact. Nei-

ther of the authors understand the proof, but Gromov claims -and

experts seem to agree - that one can construct a nonexact group

as an inductive limit of hyperbolic groups.)

Exercise 5.1.2. It is also easy to show that extensions of exact groups are

exact.

Just kidding. The first three permanence properties really are trivial

consequences of what we know about C*-algebras. The extension problem

is not.

Proposition 5.1.11. Letr be a discrete group, A be a normal subgroup and

f' = r /A. If X is a compact amenable f'-space and Y is a compact r-space

which is amenable as a A-space, then Xx Y (with the diagonal r-action) is

an amenable r-space. In particular, an extension of exact groups is exact.^6

Proof. Let a finite subset E c r and c > 0 be given. For s E r, we
denote by s the corresponding element in f'. We can find a continuous map
e: x --+ Prob(f') such that F = UxEX supp e is finite and
max sup [[s.e-e·x[l < c.
sEE xEX
Choose a cross section O": f' --+ r (i.e., O"(p) = p for all p E f'). Since
O"(p )-^1 s-^1 0"(sp) E A for every s E r and p E f', we can find a continuous

6Let r be a group and A :::; r be a subgroup. If A is exact, then the left multiplication action

of A on (3r is amenable. Indeed, there exists a A-equivariant continuous map from (3r onto f3A.