1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

5.3. Hyperbolic groups 183

The set Nr(A) is called the r-tubular neighborhood of A in K. For subsets
A, B c K, the Hausdorff distance between A and Bis defined by
dH(A, B) = inf{r: Ac Nr(B) and B c Nr(A)}.
Definition 5.3.1. Let K be a connected graph. A geodesic triangle!:::. in K
consists of three points x, y, z in Kand three geodesic paths [x, y], [y, z], [z, x]
connecting them.
Definition 5.3.2 (Hyperbolic graph). For 8 > 0, we say a geodesic triangle
!:::. is 8-slim if each of its sides is contained in the open 8-tubular neigh-
borhood of the union of the other two - i.e., [x, y] c N 0 ([y, z] U [z, x]) and
similarly for the other two sides. We say that the graph K is hyperbolic if
there exists 8 > 0 such that every geodesic triangle in K is 8-slim.

Note that hyperbolicity makes sense for any geodesic metric space (i.e.,
metric space where geodesics always exist). To get a feel for this concept,
one should check that a tree is c:-hyperbolic (i.e., every geodesic triangle is
c:-slim) for every c: > 0.

A comparison tripod is a geodesic triangle in a tree. It is not too hard
to see that for every geodesic triangle !:::. in a graph K there exist a unique
tripod and a unique map f from !:::. into the tripod that is isometric on
all edges. Indeed, the lengths of the legs of the comparison tripod are
determined by the Gromov product

(y, z)x = ~(d(y, x) + d(z, x) - d(y, z)).^9

Definition 5.3.3. For 8 > 0, we say that a geodesic triangle !:::. is 8-thin if
u, v E !:::. and f(u) = f(v) imply that d(u, v) < 8, where f is the unique map
to L's comparison tripod.

y y

x x

f(u)

z z

Figure 3. Thin geodesic triangle Figure 4. Comparison tripod

It is clear that any 8-thin geodesic triangle is 8-slim. The converse almost
holds.

9This number is the distance from x to the intersection point in Figure 4. It is not an integer,
in general, of course.