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

(Γ

, R

) is isometric to

Z

(

L

∞

alt

((

G/P

)

3

)

Γ

). In particular, it is

a Banach space.