182 5. Exact Groups and Related TopicsLemma 5.2.8. Let r be a group and T be a tree on which r acts. Let (sn)
be a net in r such that sn ~ sA eventually^8 for every s E r and every edge
stabilizer A. If sn.X ----+ z for some x E T and z E T, then Sn·Y ----+ z for
every y ET.Proof. We consider the open neighborhood of z given by a finite set F of
edges in T. It suffices to show that the geodesic paths [sn.x, sn·Yl between
Sn.X and Sn·Y do not cross F eventually. Since [sn.X, Sn·Y] = Sn.[x, y], this
reduces to showing that Sn.e =/= e' eventually for any edges e, e' in T. Take
s E r such that se = e'. (If there is no such s, then we are already done.)
Then Sn.e = e' if and only if Sn E sre' where re is the edge stabilizer of e.
Hence sn.e =/= e' eventually, by assumption. D
Exercises
Exercise 5.2.1. Let X be a compact space which has no isolated points.
Prove that the cardinality of X is at least c (cardinality of the continuum).
Exercise 5.2.2. Let X be a compact r-space whose cardinality is countable.
Prove that there is a point x E X whose stabilizer subgroup rx has finite
index in r. (This explains the fact that r is exact if each stabilizer subgroup
rx is exact, which follows from Proposition 5.2.1 with K = X and (: X 3
x H bx E Prob(K).)
Exercise 5.2.3. Let T be a countable tree which has no infinite geodesic
path. Prove that Aut(T) fixes either a point or an unoriented edge (i.e., a
pair of points).
5.3. Hyperbolic groups
In this section we study an important class of graphs, namely those which
are hyperbolic in the sense of Gromov. The main result is that groups which
act properly on such graphs are exact.
Let K be a connected graph. We view K as a discrete metric space
with the graph metric d (cf. Appendix E). As in the previous section, a
geodesic path a is a sequence of vertices such that d(a(m), a(n)) = Im -nl
for every m and n. Since K is connected, for every pair x, y E K, there
exists a (not necessarily unique) geodesic path connecting x to y. Though
not exactly well-defined, we often use [x, y] to denote a geodesic path from
x toy (multiple such geodesics may exist). For every subset A c K and
r > 0, we define
d(x,A) = inf{d(x,a): a EA} and Nr(A) = {x EK: d(x,A) < r}.
(^8) That is, Vs, VA there exists no such that Vn, n 2: no=? Sn rf: sA.