6.4. Factorization and property (T) 229representations 7f(i): r -----+ JIB(Hi) such that each 7f(i) has no fixed vectors but
there is a net of unit vectors Vi E Hi such that 117f~i)(vi) - viii -----+ 0 for all
g Er. Taking the direct sum of the 7f(i)'s, we get a representation with no
fixed vectors but which does contain almost invariant vectors.
(2) =} (3): Let F c r and "' > 0 be as in the second statement above.
For an arbitrary representation 7f: r -----+ JIB(H) we let Ho c H be the (closed
subspace) of all fixed vectors (it is possible that Ho = {O}) and we let JC c H
be its orthogonal complement. Evidently Ho is an invariant subspace and
hence JC is as well. Since JC has no fixed vectors, it follows that for each
v E JC there exists g E F such that 111fg(v) - vii 2: "'llvll·
Let w EH be arbitrary and write w = vo EB v E Ho EB JC= H. Then
1. 1
llw - voJI = JJvJI:::; -111fg(v) K, - vii= -Jl7rg(w) K, - wJJ,for some g E F.
(3) =} (1) is obvious. DRemark 6.4.6. A finite set F as in statement (2) above is called a critical
set. The constant "'is called a Kazhdan constant for (r, F). The pair (F, "')
is called a Kazhdan pair. See [15, 84] for much more on property (T). (See
also Section 12.1.)
Corollary 6.4. 7. Property (T) groups are finitely generated.
Proof. Let F c r and "'> 0 be as in statement (3) of the last proposition
and let r 0 c r be the subgroup generated by F. Consider the left translation
action of r on the R^2 space of left cosets - i.e., s.(tr 0 ) = stro. Since every
element of F leaves the trivial coset invariant, it follows that every group
element leaves ro invariant; hence ro = r. D
With the help of Stinespring's Theorem, it is not too difficult to ex-
tend the representation-theoretic rigidity of property (T) groups from one-
dimensional subspaces to finite-dimensional u.c.p. maps. In C* -language,
rigidity means that a u.c.p. map which is nearly multiplicative on a critical
set is close (in trace) to an honest homomorphism.
Proposition 6.4.8. Let F c r be a critical set and "' be some Kazhdan
constant. If <p: C(r)-----+ Mn(q is a u.c.p. map such that
1
tr(l - cp(g )cp(g)) <
2
s^2 "'^2
for some s > 0 and all g E F, then there exists an integer m and a unital -
homomorphism 1f: C(r)-----+ Mm(C) such that J trocp(x)-tro7r(x)I < 5sllxll
for all x E C*(r).