200 5. Exact Groups and Related Topics
Remark 5 .. 5,10. Gromov's infamous construction of nonexact groups yields
examples whose C:ayley graphs satisfy the hypotheses of the previous result
- thus they aren't coarsely embeddable into Hilbert space and consequently
can't be exact (cf. [71]).
ExerciseE.xerdse 5 .. 5,.1. Formulate and prove an analogue of Lemma 5.5.6 for spaces
which are <ioarsely embeddable into Hilbert space. (Hint: The "supported in
tubes" condition should be replaced with a "co-off the diagonal" condition.)
5.6. Gr.oupoidsG rou poids (and their associated C* -algebras) generalize almost everything
we have discussed so far.: topological spaces, groups, group actions on spaces,
coarse metric spaces, graphs, and many other things. Thus they provide a
unifying framework for everything in this chapter.
To a groupoid G one can associate a reduced groupoid C* -algebra C~ ( G).
There is a natural notion of amenability for groupoids and the main result
of this section states that C~ ( G) is nuclear if and only if G is amenable.
Definition 5.6.1. A groupoid is a small category, in which every morphism
is invertible. More specifically, a groupoid consists of a set G of morphisms
and a distinguished subset Q(O) CG of objects (often called units), together
with source and range maps s, r: G-+ Q(O), and a composition map
Q(^2 ) = {(o:,,8) E G x G: s(o:) = r(,8)} :3 (o:,,8) f--t o:,8 E G,
such that
(1) s(a:,8) = s(,8) and r(o:,8) = r(o:) for every (a:, ,8) E Q(^2 ),
(2) s(x) = x = r(x) for every x E Q(O),
(3) rys(1) = / = r(r)r for every/ E G,
(4) (o:,B)r = 0:(,81), and
(5) every/ has an inverse ,-1, with ,,-^1 = r(r) and ,-^1 , = s(r).A topological groupoid G is a groupoid together with a topology on G such
that all structure maps are continuous. We note that Q(^2 ) inherits th.e
relative topology from G x G.
A topological groupoid G is said to be etale (or r-discrete) ifs and r are
local homeomorphisms.^14
We only consider ( etale) locally compact groupoids and always assume
that the Hausdorff property is included in the definition of local compactness
14 Actually, ifs is locally homeomorphic, then so is r.