1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

5.3. Hyperbolic groups 187

Lemma 5.3.8. There exists a constant C = C(K) > 0 with the following

property: For any two equivalent infinite geodesic paths a and f3 in K and

any m 2: d(a(O), /3(0)), there exists n with Im-nl ~ d(a(O), /3(0)) such that

d(a(m), f3(n)) < C.

In particular, a and /3 are equivalent if and only ifsupm d(a(m), /3(m)) <

oo (and this is clearly an equivalence relation).

Proof. Choose 8 > 0 so that every geodesic triangle is 8-thin. Let m 2:

d(a(O), /3(0)) be given. Let o = a(O) and find mi, ni E N such that

(a(m1), /3(n1)) 0 > m. Choose any geodesic path [o, /3(n1)] connecting o to

/3(n1). Let x be the vertex on [o, /3(n1)] such that d(o, x) = d(o, a(m)) = m.

Since (a(m1), /3(n1)) 0 > m, we have d(x, a(m)) < 8. Let n be such that

n < ni and d(x, /3(n1)) = d(/3(n), /3(n1)). Since d(o, x) 2: d(a(O), /3(0)), such

an n exists. Moreover Im -nl ~ d(a(O), /3(0)) and d(x, f3(n)) < 8. It follows.

that d(a(m),/3(n)) < 28. This proves the first assertion.

For the second assertion, let equivalent geodesic paths a and f3 and

m 2: d(a(O), /3(0)) be given. Then, by the first assertion, there is n with lm-

nl ~ d(a(O), /3(0)) such that d(a(m), f3(n)) < C. Hence, d(a(m), /3(m)) ~

d(a(m), f3(n)) + lm-nl ~ C+d(a(O), /3(0)). Conversely, suppose dH(a, /3) <

oo and take mo, no 2: 0 such that d(a(mo), f3(no)) ~ dH(a, /3) + 1. Then, for

any m 2: mo and n 2: no, one has

2(a(m), f3(n)) 0 = d(a(m), o) + d(/3(n), o) - d(a(m), f3(n))

2: m - d(a(O), o) + n - d(/3(0), o)

• ((m - mo)+ d(a(mo),/3(no)) + (n-no))
  2: mo+ no - (d(a(O), o) + d(/3(0), o) + dH(a, /3) + 1).
  This proves liminfm,n->oo(a(m),/3(n)) 0 = oo. D
  Definition 5.3.9. We define the Gromov boundary 8K of a hyperbolic
  graph K to be the set of all equivalence classes of infinite geodesic paths.
  We call K = KU8K the Gromov compactification of K (we soon describe the
  topology). For a hyperbolic group r, f' denotes the Gromov compactification
  of its Cayley graph.

Definition 5.3.10. For a finite or infinite geodesic path a= xox1 · · · in K,
we denote by a = x 0 its starting point and a+ its terminal point (i.e., the
boundary point a represents, in the infinite case). As before, we say that a
connects a with a+.

The Cayley graph of a finitely generated group is always uniformly locally
finite (or has bounded geometry) - i.e., there is a uniform bound on the
degree of vertices. This is a nice property for a graph to have, so from now
on, we assume that the hyperbolic graph K is uniformly locally finite and, in