1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

180 5. Exact Groups and Related Topics

The lemma above is really a special case. Indeed, essentially the same

proof yields the following fundamental fact (left to the reader): If x, y E T,

then there exists a geodesic connecting x with y (since T is connected), and

it is unique (since Tis a tree); this path will be denoted [x, y].

If [x, wo] is a finite geodesic path and [wo, y] is any other geodesic, then

we let [x, wo] U [w 0 , y] denote the concatenation of these two paths (which

is equal to [x, y], of course). The following important lemma will be used

repeatedly.

Lemma 5.2.4. Given x,y,z ET, [x,y] n [y,z] n [z,x] is a singleton (i.e.,

there exists a unique point wo E T such that [x, y] = [x, wo] U [wo, y], [x, z] =

[x, wo] U [wo, z] and [y, z] = [y, wo] U [wo, z]).

Proof. Again, the proof is trivial pictorially, so we only state the main idea.

First note that [x, y] and [z, y] are equivalent geodesics. Letting wo be the

first point of intersection of [x, y] and [z, y], the remainder of the proof is

routine. D

We're now ready to topologize T. For x ET and a finite subset F c T,

we define

U(x; F) = {x} U {y ET: [x, y] n F = 0}.

One checks that {U(x; F)}x,F forms a basis for a topology (if x E U(xi, Fi)n

U(x2, F2), then Lemma 5.2.4 implies that U(x, Fi U F2) c U(xi, Fi) n

U(x2, F2)) and that the resulting topology is Hausdorff (given x, y, and

any point zo =J x, yon the geodesic [x, y], Lemma 5.2.4 implies U(x, {zo}) n

U(y, {zo}) = 0; and if x and y are adjacent, then U(x, {y}) n U(y, {x}) = 0).

This topology is very visual: cut finitely many edges in T and the connected

components of Tare open (first verify this when only one edge is cut). Fi-

nally, it is worth checking that for a point x E T, the set { x} is open if and

only if x has finite degree.

Proposition 5.2.5. The topological space T is compact and any automor-

phism (i.e., isometric bijection) of the tree T extends to a homeomorphism

ofT.

Proof. We must show that any net (xa)aEA in 'i' has an accumulation point
(by Theorem A.8).
Fix a base point o E T and identify every Xa E T with the unique
geodesic path connecting o to Xa· Let N be the largest integer (possibly 0
or oo) such that there exist x(O), ... , x(N) satisfying

N

n {a EA: xa(n) = x(n)} EU.

n=O