Exact Groups and
Related Topics
Chapter 5
We begin this chapter by defining exact groups and proving that this concept
is equivalent to acting amenably on some compact space. The next three
sections deal with examples; that is, we show how to construct amenable
actions in several interesting cases. The last two sections contain a discussion
of two natural generalizations of groups: coarse metric spaces and groupoids.
We don't give proper introductions to these important topics; we only focus
on approximation issues.
5.1. Exact groups
Definition 5.1.1. A discrete group is exact if its reduced group C*-algebra
is exact.^1
Though the proof is beyond the scope of these notes (see [73]), the
following result shows that most familiar groups are known to be exact.
Recall that a linear group is any subgroup of the invertible matrices over
some field.
Theorem 5.1.2 (Guentner, Higson and Weinberger). Linear groups are
exact.
To begin our reformulations of exactness, we introduce some terminol-
ogy. Let r be a discrete group and E c r be a finite subset. The tube of
(^1) This definition is not the original one - see the end of this section.
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