5.3. Hyperbolic groups 191Since (D(2n + d))-^2 d/n----+ 1 as n----+ oo, this proves the claim.
Now we fix a base point o E K, set (~ = 'IJn(o, z) and observe that
the maps (n: oK ----+ Prob(K) are Borel. Since we have (s.'IJn)(x, z)
'IJn(s.x, s.z) for every s Er and (x, z) E K x oK, it follows that
lim sup lls.G - (~zll = lim sup ll'IJn(s.o, s.z) - 1J(o, s.z)ll = 0.
n-+oo zEBK n-+oo zEBK
Finally, for x EK we set (;;'. = 8x, and one can check that the Borel maps
(n satisfy the hypotheses of Proposition 5.2.1 -hence the action of r on K
is amenable. DWe close this section with a few results which will be extremely impor-
tant for later applications to von Neumann algebras (cf. Chapter 15).
Definition 5.3.16. A compactification of a group r is a compact topological
spacer = r u or containing r as an open dense subset. We assume that
a compactification is (left) equivariant in the sense that the left translation
action of r on r extends to a continuous action on r. The compactification
fi is said to be small at infinity if for every net {Sn} C r converging to a
boundary point x E or and every t E r, one has that Snt----+ x.
By Gelfand duality, there is a one-to-one correspondence between com-
pactifications r and C*-algebras C(fi), where c 0 (r) c C(r) c .e^00 (r) is
left-translation invariant. The proof of the following lemma is a good exer-
cise.
Lemma 5.3.17. Let r be a group and r = r U or be a compactification.
The following are equivalent:
(1) the compactification r is small at infinity;
(2) the right translation action extends to a continuous action on r in
such a way that it is trivial on or;
(3) one has ft - f E co(r) for every f E C(r) and t E r, where
P(s) = f(sr^1 ) for f E .e^00 (r).
Proposition 5.3.18. For any hyperbolic group r, the Gromov compactifi-
cation r is small at infinity.
Proof. Let a sequence {sn} converging to a boundary point x E or and
t E r be given. Let f3 be a geodesic path converging to x. Since d( snt, sn) =
d(t, e) for every n, we have
m,n---+oo liminf (smt, f3(n))e 2:: m,n---+oo liminf (sm, /3(n))e - d(t, e) = oo.This means that snt----+ x. D