1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

5.3. Hyperbolic groups 185

distance) to an honest geodesic. To make this precise, we must define "close

to being geodesic." For C ~ 1 and r > 0, we say that a finite sequence^10 a

in K is ( C, r )-quasigeodesic if

c-^1 d(a(m), a(n)) - r::::; Jm - nJ ::::; Cd(a(m), a(n)) + r

for every m, n.

Proposition 5.3.5. Let K be a hyperbolic graph, C ~ 1 and r > 0. Then,

there exists D > 0 with the following property: For any (C, r)-quasigeodesic

sequence a and any geodesic path (3 having the same origin and terminal

point as a, one has dH(a, (3) < D.

In particular, a graph is hyperbolic if it quasi-isometrically embeds into

some hyperbolic graph.^11

Proof. Let a and (3 be given. We set Do= max{d(p, a): p on /3} (hence (3
is contained in the Do tubular neighborhood of a).
Now suppose that qo is a point on a such that d(qo, (3) ~ D 0. By
maximality of Do, for every point u on (3, there exists a point u' on a such
that d(u, u') ::::; Do. Since the endpoints of a and (3 are the same, "the
intermediate value theorem" implies the existence of consecutive u 0 , u 1 on
(3 such that Ub is on the origin (or "left") side and ul is on the terminus (or
"right") side of qo. (That is, u' is on the left of qo when u is the starting
point of (3, and it is on the right when u is the endpoint - hence there is a
place where u' jumps over qo.) Since d(ub, ul.)::::; 2Do + 1, the length of the
subsequence of a from Ub to ul is at most C(2Do + 1) + r. It follows that
d(uo, qo) ::::; d(uo, ub) + d(ub, qo)::::; Do+ C(C(2Do + 1) + 2r).
Therefore, we have dH(a, (3) ::::; D for D =Do+ C(C(2Do + 1) + 2r). Hence,
we must show that Do is bounded above by a function depending only on
o, C and r, where o is the constant satisfying Definition 5.3.2.
Choose a point Po on (3 such that d(po, a) = Do. Choose two points
bo and b 1 on (3, one coming before Po and one after, such that d(bo,po) =
2Do = d(b1,po) - or, if this isn't possible, take an endpoint of (3. Let ak,
k = 1, 2, be points on a such that d(bk, ak) = d(bk, a) and choose geodesic
paths /o and /1 connecting bo to ao and bi to a1, respectively. (Nate that
if bk is an endpoint, then ak = bk, so we take /k to be the single point
ak = bk in this case.) Maximality of Do implies that d(bk, ak) ::::; Do, and
hence d(po, /k) ~Do. Let a' be the subsequence of a connecting ao to a1.
(It may flow backward.) Since
d(ao, a1) ::::; d(ao, bo) + d(bo, b1) + d(b1, a1) ::::; 6Do
lOit is important that we don't require a to be a path in this definition, because quasi-
isometric embeddings don't always map paths to paths - i.e., neighbors need not map to neighbors.
llThink about the 8-slim condition and this is easily deduced.