5.3. Hyperbolic groups 189Now, it is easy to check that {U(z, R)}R>O defines a (not necessarily
open) neighborhood basis and the resulting topology is Hausdorff.
Definition 5.3.13. We equip K = KU 8K with a topology by declaring
that a subset 0 c K is open if and only if for every z E 8Kn0, there exists
R > 0 such that U(z, R) c 0. We note that for every x EK, the singleton
set {x} is open in K.It is clear that this topology is independent of the choice of the base
point o. Moreover, for a hyperbolic group r, the Gromov compactifica-
tion r is independent of the choice of finite generating subset (thanks to
Proposition 5.3.5).
Theorem 5.3.14. Let K be a locally finite hyperbolic graph. Then the
topological space K defined above is compact and contains K as a dense
open subset. Every automorphism (i.e., isometric bijection) on K extends
uniquely to a homeomorphism on K.
Proof. The proof is similar to that of Proposition 5.2.5. We only prove
compactness; the rest is trivial. It suffices to show that an arbitrary net
(xi)iEI in K has an accumulation point (Theorem A.8). For every i, choose
a geodesic path ai connecting o to Xi· For convenience, we set ai(n) = Xi
when n 2:: d(o, xi)· Let Ube a cofinal ultrafilter on the directed set I. Since
K is locally finite, for every n, there exists a unique point a( n) E K such
that {i : ai(n) = a(n)} EU. Since each ai is a geodesic path, a is also a
geodesic path (or perhaps a path which is eventually constant). It is not
too hard to see that a+ E K is an accumulation point. D
Here is the exactness result we have been after.
Theorem 5.3.15. Let K be a uniformly locally finite hyperbolic graph and
r be a group acting properly^12 on it. Then the action of r on the Gromov
compactification K is amenable. In particular, every hyperbolic group is
exact (since it acts properly on its Cayley graph).
Proof. For x, y E K, we denote by T(x, y) the set of z E 8K such that
there exists a geodesic path connecting x to z which passes through y. It
is not hard to see that T(x, y) is a closed subset of 8K. For every x E K,
z E {)Kand integers l, k with l 2:: k, we define a subset S(x, z, l, k) CK by
declaring
S(x, z, l, k) = { a(l) : a a geodesic path in K
such that d(a-,x)::::; k and a+= z}.
12In this case, being proper is equivalent to saying every vertex stabilizer is finite.