1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

194 5. Exact Groups and Related Topics

define a Borel map jj,: X----+ Prob(G) by P,x = a-(x)v. One checks that

max sup llsP,x - P,sxll =max sup lla-(sx)-^1 sa-(x)v - vii< E.

sEE xEX sEE xEX

Composingμ with Prob(G)----+ Prob(r), we are done.

5.5. Coarse metric spaces

D

Let (X, d) be a metric space. The associated topology on X reflects the
"small scale structure" of the metric space - i.e., what X looks like under a
microscope. In this section we take a step back. .. way back. Coarse geometry
is the study of "large scale structure" of a (metric) space. For simplicity,
we confine ourselves to countable discrete spaces X equipped with a proper
metric d. (Recall that a metric d is said to be proper if any closed bounded
subset of Xis compact, i.e., finite because Xis a discrete space.) We first
consider a countable group r equipped with a proper metric d which is right
invariant, meaning d(s, t) depends only on st-^1. We will show that such a
metric is intrinsic to r, but first we need a suitable notion of equivalence.

Definition 5.5.1. Let (X, dx) and (Y, dy) be metric spaces. We say a map
f: X ----+ Y is coarse if the inverse image under f of any bounded subset
in Y is bounded in X and if for any R > 0, there exists S > 0 such that
dx(x, x') < R implies dy(f (x), f(x')) < S for any x, x' EX.
Let f': X----+ Y be another map. We say f is close to f' if dy(f(x), f'(x))
is bounded on X. (You may want to verify that if f is close to a coarse map,
then f is also coarse.) A map f is a coarse isomorphism if there exists a
coarse map g: Y ----+ X such that go f and fog are close to the identity maps
on X and Y, respectively. Metric spaces (X,dx) and (Y,dy) are coarsely
isomorphic if there exists a coarse isomorphism f: X----+ Y. Finally, we say
metrics d and d' on X are coarsely equivalent if the identity map from (X, d)
to (X, d') is a coarse isomorphism.

We loosely refer to a space X as a coarse space when a distinguished
coarse equivalence class of metrics on X is chosen. The following result says
that the coarse space structure of a countable discrete group r is unique.

Proposition 5.5.2. Let r be a countable discrete group. Then, there exists

a proper right invariant metric d on r. Moreover, such a metric d is unique

up to coarse equivalence.

Proof. Since r is countable, there exists a sequence

{ e} = Ea c Ei c E2 c · · · c r