1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

232 6. Amenable Traces

Proof. Let T1, ... ,Tn E GL(n,K) be given and let r be the group they
generate. Let R c K be the ring generated by all the entries of the matrices
T 1 , ... , Tn, T;-^1 , ... , T,;;^1. Evidently R is finitely generated, abelian and I' C
G L( n, R). It suffices to show that R is residually finite; that is, there exist
ideals In <1 R such that [R/ Inf < oo, for all n, and n In = 0. Indeed, if this
is true, then the maps GL(n, R) -----+ GL(n, R/ In) will provide a separating
family of finite quotients of r.

So, fix a finite set F c R. Let R c K be the (finitely generated,

abelian) ring generated by Rand {(s - t)-^1 : s, t E F, s i-t}. Let M <l R

be a (nontrivial) maximal ideal; hence, R/M is a field, which is finitely

generated as a ring; hence, it is a finite field, by the previous theorem. Note

that M can't contain any of the elements { s - t: s, t E F, s -=f. t} since they

are all invertible in R (and M -=f. R). Thus I= Rn Mis a finite-index ideal

which separates points in F. 0

Theorem 6.4.14. Let I' be a property (T) group. Then I' has the factor-

ization property if and only if r is residually finite.

Proof. Let T>.. be the trace on C* (I') coming from the left regular represen-

tation. Since we already know residually finite groups have the factorization

property (Proposition 3.7.10), we may assume that T>.. is amenable (cf. The-

orem 6.4.3). By Corollary 6.4.10, there exist representations 1rn: C*(I') -----+

Mk(n) (C) such that for each nonneutral element g E r we have

tr(7rn(g))-----+ T;...(g) = 0.

In particular, 7rn is a separating family of representations and thus Malcev's

theorem implies that r is residually finite. 0

Remark 6.4.15. In [70] it was shown that there are infinite property (T)
groups which are simple. Such groups are the only known examples which
fail the factorization property.

In [129] it was shown that every hyperbolic group can be embedded into

a hyperbolic group with property (T). Since residual finiteness obviously

passes to subgroups, we get the following corollary.

Corollary 6.4.16. Every hyperbolic group is residually finite if and only if

every hyperbolic group has the factorization property.

Whether or not every hyperbolic group is residually finite has been a
long-standing open problem in geometric group theory. The corollary above
says that this is the case if and only if the natural *-homomorphism

C*(r) 0 C*(r)-----+ JB(£^2 (r))