1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

356 12. Approximation Properties for Groups

sto

,

~t)

b( so

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0

Figure 1. b(st) = b(s) +7r(s)b(t)

In summary, we obtain the following theorem of Haagerup. (See Theorem
D.11.)

Theorem 12.2.5. The graph metric on a tree is conditionally negative def-

inite. A group which acts properly on a tree has the Haagerup property. In

particular, free groups and SL(2, Z) have the Haagerup property.

A tree is an example of a space with walls.
Definition 12.2.6. A space with walls is a pair (X, W) consisting of a set
X and a family of partitions of X into two classes, called walls, such that
the number w(x, y) of walls separating x and y is finite for every x, y EX.
Here, a wall w = { H, He} separates x and y if either x E H and y E He or
x E He and y E H.
Example 12.2. 7. Let T be a tree, X be the vertex set and W be the edge
set. Each edge w E W of the tree T can be considered as a partition of X
into two connected components. Hence, (X, W) is a space with walls such
that w(x, y) = d(x, y), the graph metric.
Example 12.2.8. A finite Cartesian product of spaces with walls is nat-
urally a space with walls. In particular, (zn, hyperplanes) is a space with
walls, such that w is again the graph metric on zn.

We say a group r acts on a space (X, W) with walls if r acts on X
as permutations and preserves the wall structure. The action is said to be
proper if lim 8 _,. 00 w ( x, s .x) = oo for every/ some x E X. The following result
generalizes Theorem 12.2.5.

Theorem 12:.2.9. For any space (X, W) with walls, the function w: X x

X -+ Z;:::o is conditionally negative definite. A group which acts properly on

a space with walls has the Haagerup property.

Proof. Let o E X be a fixed base point. For every x E X, we define
( ( x) E £^2 (W) to be the characteristic function of the set of walls separating