354 12. Approximation Properties for GroupsExercise l~!.1.3. Let I' be an ICC^7 group and let x: I'----? {z: lzl = 1} be
a nontrivial character. Prove that Ox E Aut(L(I')), defined by Ox(>.(s)) =
x(s)>.(s), is not inner.
12.2. The Haagerup property
Haagerup's property is, in many ways, the opposite of property (T). Indeed,
compare the definition below with condition (3) in Theorem 12.1.7.
Definition 12.2.1. A discrete group I' has the Haagerup property if there
exists a net ( cpi) of positive definite functions on r with cpi( e) = 1, such that
each cpi vanishes at infinity (i.e., for any c: > 0, the set { s E r : lcpi( s) I > c:}
is finite) and cpi ----? 1 pointwise.
Example 12.2.2. Amenable groups have the Haagerup property (since the
left regular representation weakly contains the trivial representation). We'll
soon prove that free groups and SL(2, Z) also enjoy this property. In addi-
tion, the class of groups having the Haagerup property is closed under taking
subgroups, direct products, free products and increasing unions. (Most of
these are easy exercises; the proof of the free product result requires the fact
that the free product of positive definite functions is positive definite -see
Theorem 4.8.5.)
The next proposition is a trivial consequence of the definitions.Proposition 12.2.3. If I' has the Haagerup property, then any infinite
subgroup A C r does not have relative property (T). In particular, a group
with the H aagerup property and property (T) is finite.
We say a 1-cocycle b: r----? 1t is (metrically) proper if for any R > 0, the
set {s EI': llb(s)ll :SR} is finite (see Appendix D).
Theorem 12.2.4. Let I' be a countable discrete group. Then the following
are equivalent:
(1) I' has the Haagerup property;
(2) r admits a proper 1-cocycle;
(3) r admits a proper affine isometric action on a (real) Hilbert space.
Proof. The equivalence (2) {::} (3) is tautological (see Appendix D). The
implication (2) =?-(1) is a consequence of Schoenberg's Theorem D.11. To
prove the converse, let cpn be a sequence of positive definite functions on
I' satisfying the conditions in Definition 12.2.1. Let I' = { sn}~=I be an
enumeration. Passing to a subsequence if necessary, we may assume that
(^7) Meaning {sts- (^1) : s Er} is infinite for every nonneutral element t Er.