1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

E. Groups and Graphs 475

is the combinatorial Laplacian of Xn (modulo identification of £^2 (r n) and
L^2 (rn)). Let £^2 (rn)^0 c £^2 (rn) be the orthogonal complement of the con-
stant functions and 1!"~ be the restriction of 1l"n to the 1l"n-invariant sub-
space £^2 (r n)^0. We set 1l"o = EB 1!"~ and observe that 1l" does not have a
nonzero invariant vector. Suppose by contradiction that inf .A1(Xn) = 0.
Let ~n E £^2 (r n)o be a unit eigenvector of .6..n with eigenvalue .A1(Xn)· Pass-
ing to a subsequence, we may assume that lim ll.6..n~nll = 0. By uniform
convexity of Hilbert spaces, this implies that lim ll~n - 1T"n(s)~nll = 0 for
every s E S and hence for all s E r. This contradicts property (T). For a
qualitative proof, refer to Lemma 12.1.8. 0

Amalgamated free products. In this section, we state the definitions
of amalgamated free products and HNN extensions (that they exist is a
theorem which we do not prove).

Definition E.8 (Amalgamated free products). Let ri, i = 1, 2, be groups
and A be a common subgroup (i.e., A comes with an injective homomorphism
into each ri). Then, the amalgamated free product r = r1 *Ar2 is the group
satisfying the following properties:

(1) r contains r1 and r2 as subgroups and r is generated by r1 and

r2;

(2) r1 n r2 =A in r;

(3) s1 · · · sna i-e whenever n 2".: 1, a E A and Sk E rik \A with ik i-ik+l

for 1 ::; k < n;

( 4) if we choose systems Si of representatives of r i I A and let sf =

Si \ { e} (we always assume that the representative of the coset A

is e), then any element s in r can be uniquely written as

s = s1 · · · Sna,

where a E A and sk E Sfk such that ik i-ik+l for 1 ::; k < n.

The expression in condition ( 4) is called the normal form of s E r. By
definition, for any homomorphisms 'Pi from ri into a group r' which agree
on A, there exists a unique homomorphism r.p: r--+ r' such that 'PJri ='Pi·

Since we do not use it, we do not give the normal form theorem for an
HNN extension. But here's the definition.

Definition E.9 (HNN extension). Let r be a group, A::; r be a subgroup
and (): A --+ r be an injective homomorphism. Then, the HNN extension

r* = (r, z I z-^1 az = e(a) for a EA)

is the group satisfying the following properties: