4The relative homological approach

IV Bounded Cohomology

4.2 Amenable actions
In Riemannian geometry, we have the Hodge decomposition theorem. It allows
us to understand the de Rham cohomology of a Riemannian manifold in terms
of the complex of harmonic forms, whose cohomology is the complex itself. In
bounded cohomology, we don’t have something like this, but we can produce a
complex such that the complex is equal to the cohomology in the second degree.
The setting is that we have a locally-compact second-countable group
G
with
a non-singular action on a standard measure space (
S, M, µ
). We require that
the action map
G × S S
which is measurable. Moreover, for any
g G
, the
measure
g
µ
is equivalent to
µ
. In other words, the
G
-action preserves the null
sets.
Example.
Let
M
be a smooth manifold. Then the action of
Diff
(
M
) on
M
is
non-singular.
We want to come up with a notion similar to amenability. This is what we
call conditional expectation.
Definition
(Conditional expectation)
.
A conditional expectation on
G × S
is a
linear map M : L
(G × S) L
(S) such that
(i) M(1) = 1;
(ii) If M 0, then M (f) 0; and
(iii) M is L
(S)-linear.
We have a left
G
-action on
L
(
G × S
) given by the diagonal action, and
also a natural on L
(S). We say M is G-equivariant if it is G-equivariant.
Definition
(Amenable action)
.
A
G
-action on
S
is amenable if there exists a
G-equivariant conditional expectation.
Note that a point (with the trivial action) is a conditional
G
-space if
G
is
amenable itself.
Example.
Let
H
be a closed subgroup of
G
, then the
G
action on
G/H
is
amenable iff H is amenable.
Theorem
(Burger–Monod, 2002)
.
Let
G × S S
be a non-singular action.
Then the following are equivalent:
(i) The G action is amenable.
(ii) L
(S) is an injective G-module.
(iii) L
(S
n
) for all n 1 is injective.
So any amenable
G
-space can be used to compute the bounded cohomology
of G.
Corollary.
If (
S, µ
) is an amenable
G
-space, then we have an isometric isomor-
phism H
·
(L
(S
n
, µ)
G
, d
n
)
=
H
·
(L
alt
(S
n
, µ)
G
, d
n
)
=
H
b
(G, R).
Example.
Let Γ
< G
be a lattice in
SL
(
n, R
), say. Let
P < G
be a parabolic
subgroup, e.g. the subgroup of upper-triangular matrices. We use
L
alt
((
G/P
)
n
)
Γ
to compute bounded cohomology of Γ, since the restriction of amenable actions
to closed subgroups is amenable. We have
0 L
(G/p)
Γ
L
alt
((G/P )
2
)
Γ
L
alt
((G/P )
3
)
Γ
· · ·
R 0
0
So we know that
H
2
b
, R
) is isometric to
Z
(
L
alt
((
G/P
)
3
)
Γ
). In particular, it is
a Banach space.