1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

5.2. Groups acting on trees 181

We observe that for each n there exists at most one x(n) such that {a :

xa(n) = x(n)} EU and such that (x(n))-,;;=O is a geodesic path. Thus, if N =

oo, then the boundary point represented by (x(n))~=O is an accumulation

point. On the other hand, if N < oo, then x(N) is an accumulation point.

Thus T is compact.

If s is an automorphism of T, then it naturally acts on T = T LJ 8T.

Namely, for every x E T, we define s .x E T to be the equivalence class of

the geodesic path (s.x(n))n, where (x(n))n is a representative of x. It is

routine to check that this is a homeomorphism. D

Lemma 5.2.6. Let T be a countable tree with fixed base point o. There

exists a sequence of Borel maps

(n: T ---+ Prob(T)

such that

II
sup s. ..,n ;-x - ;-s.xll i.,,n ::::; ----2d(s.o, o)
xET n
for every automorphism s on T.

Proof. As before, we identify every x E T with the unique geodesic path
(x(n))n connecting o to x. (Recall our convention that x(k) = x when x ET
and k 2: dist(x, o).) The maps (n, defined by

n-1

(~ = _!_ L 5x(k) E Prob(T),

n k=O

satisfy the desired inequality. Indeed, s.(~(p) > 0 if and only if s-^1 .p is one
of the first n points in the geodesic from o to x; equivalently, p is one of the
first n points in the geodesic from s.o to s.x. Similarly, (~·x(q) > 0 if and
only if q is one of the first n points in the geodesic from o to s.x. Hence, for
n > d(s.o, o) we have cancellation in the difference s.(;, - (~·x, because the
geodesics from s.o to s.x and o to s.x are equivalent.

Finally, it is easy to see that the (n's are Borel since the set { x E T :
x(k) = z} is clopen in T for every k 2: 0 and z ET. D

Theorem 5.2. 7. Let r be a countable group and T be a countable tree on
which r acts. If every vertex stabilizer rx of x E T is exact, then r is exact.
In particular, an amalgamated free product of exact groups is exact.

Proof. This follows from Proposition 5.2.1 and Lemma 5.2.6. D

We close this section with a result that won't be needed until much later
in the book - but it makes sense to prove it now.