Part IV — Bounded Cohomology
Based on lectures by M. Burger
Notes taken by Dexter Chua
Easter 2017
These notes are not endorsed by the lecturers, and I have modified them (often
significantly) after lectures. They are nowhere near accurate representations of what
was actually lectured, and in particular, all errors are almost surely mine.
The cohomology of a group or a topological space in degree
k
is a real vector space
which describes the “holes” bounded by k dimensional cycles and enco des their relations.
Bounded cohomology is a refinement which provides these vector spaces with a (semi)
norm and hence topological objects acquire mysterious numerical invariants. This
theory, introduced in the beginning of the 80’s by M. Gromov, has deep connections
with the geometry of hyperbolic groups and negatively curved manifolds. For instance,
hyperbolic groups can be completely characterized by the “size” of their bounded
cohomology.
The aim of this course is to give an introduction to the bounded cohomology of groups,
and treat more in detail one of its important applications to the study of groups acting
by homeomorphisms on the circle. More precisely we will treat the following topics:
(i)
Ordinary and bounded cohomology of groups: meaning of these objects in low
degrees, that is, zero, one and two; relations with quasimorphisms. Proof that
the bounded cohomology in degree two of a non ab elian free group contains an
isometric copy of the Banach space of bounded sequences of reals. Examples
and meaning of bounded cohomology classes of geometric origin with non trivial
coefficients.
(ii)
Actions on the circle, the bounded Euler class: for a group acting by orientation
preserving homeomorphisms of the circle, Ghys has introduced an invariant, the
b ounded Euler class of the action, and shown that it characterizes (minimal)
actions up to conjugation. We will treat in some detail this work as it leads to
important applications of bounded cohomology to the question of which groups
can act non trivially on the circle: for instance
SL
(2
, Z
) can, while lattices in
“higher rank Lie groups”, like SL(n, Z) for n at least 3, can’t.
(iii)
Amenability and resolutions: we will set up the abstract machinery of resolutions
and the notions of injective modules in ordinary as well as bounded cohomology;
this will provide a powerful way to compute these objects in important cases. A
fundamental role in this theory is played by various notions of amenability; the
classical notion of amenability for a group, and amenability of a group action on
a measure space, due to R. Zimmer. The goal is then to describe applications of
this machinery to various rigidity questions, and in particular to the theorem
due, independently to Ghys, and Burger–Monod, that lattices in higher rank
groups don’t act on the circle.
Prerequisites
Prerequisites for this course are minimal: no prior knowledge of group cohomology of
any form is needed; we’ll develop everything we need from scratch. It is however an
advantage to have a “zoo” of examples of infinite groups at one’s disposal: for example
free groups and surface groups. In the third part, we’ll need basic measure theory;
amenability and ergodic actions will play a role, but there again everything will be
built up on elementary measure theory.
The basic reference for this course is R. Frigerio, “Bounded cohomology of discrete
groups”, arXiv:1611.08339, and for part 3, M. Burger & A. Iozzi, “A useful formula
from bounded cohomology”, available at:
https://people.math.ethz.ch/
~
iozzi/
publications.html.
Contents
1 Quasihomomorphisms
1.1 Quasihomomorphisms
1.2 Relation to commutators
1.3 Poincare translation quasimorphism
2 Group cohomology and bounded cohomology
2.1 Group cohomology
2.2 Bounded cohomology of groups
3 Actions on S
1
3.1 The bounded Euler class
3.2 The real bounded Euler class
4 The relative homological approach
4.1 Injective modules
4.2 Amenable actions
1 Quasihomomorphisms
1.1 Quasihomomorphisms
In this chapter,
A
will denote
Z
or
R
. Let
G
be a group. The usual definition of
a group homomorphism f : G → A requires that for all x, y ∈ G, we have
f(xy) = f(x) + f(y).
In a quasihomomorphism, we replace the equality with a weaker notion, and
allow for some “errors”.
Definition
(Quasihomomorphism)
.
Let
G
be a group. A function
f : G → A
is a quasihomomorphism if the function
df : G × G → A
(x, y) 7→ f(x) + f(y) − f(xy)
is bounded. We define the defect of f to be
D(f) = sup
x,y∈G
df(x, y).
We write QH(G, A) for the Amodule of quasihomomorphisms.
Example.
Every homomorphism is a quasihomomorphism with
D
(
f
) = 0.
Conversely, a quasihomomorphism with D(f) = 0 is a homomorphism.
We can obtain some “trivial” quasihomomorphisms as follows — we take
any homomorphism, and then edit finitely many values of the homomorphism.
Then this is a quasihomomorphism. More generally, we can add any bounded
function to a quasihomomorphism and still get a quasihomomorphism.
Notation. We write
∞
(G, A) = {f : G → A : f is bounded}.
Thus, we are largely interested in the quasihomomorphisms modulo
∞
(
G, A
).
Often, we also want to quotient out by the genuine homomorphisms, and obtain
QH(G, A)
∞
(G, A) + Hom(G, A)
.
This contains subtle algebraic and geometric information about G, and we will
later see this is related to the second bounded cohomology H
2
b
(G, A).
We first prove a few elementary facts about quasihomomorphisms. The
first task is to find canonical representatives of the classes in the quotient
QH(G, R)/
∞
(G, R).
Definition
(Homogeneous function)
.
A function
f : G → R
is homogeneous if
for all n ∈ Z and g ∈ G, we have f(g
n
) = nf(g).
Lemma. Let f ∈ QH(G, A). Then for every g ∈ G, the limit
Hf(g) = lim
n→∞
f(g
n
)
n
exists in R. Moreover,
(i) Hf : G → R is a homogeneous quasihomomorphism.
(ii) f − Hf ∈
∞
(G, R).
Proof. We iterate the quasihomomorphism property
f(ab) − f(a) − f(b) ≤ D(f).
Then, viewing g
mn
= g
m
· · · g
m
, we obtain
f(g
mn
) − nf(g
m
) ≤ (n − 1)D(f).
Similarly, we also have
f(g
mn
) − mf(g
n
) ≤ (m − 1)D(f).
Thus, dividing by nm, we find
f(g
mn
)
nm
−
f(g
m
)
m
≤
1
m
D(f)
f(g
mn
)
nm
−
f(g
n
)
n
≤
1
n
D(f).
So we find that
f(g
n
)
n
−
f(g
m
)
m
≤
1
m
+
1
n
D(f). (∗)
Hence the sequence
f(g
n
)
n
is Cauchy, and the limit exists.
The fact that
Hf
is a quasihomomorphism follows from the second assertion.
To prove the second assertion, we can just take
n
= 1 in (
∗
) and take
m → ∞
.
Then we find
f(g) − Hf(g) ≤ D(f).
So this shows that f − Hf is bounded, hence Hf is a quasihomomorphism.
The homogeneity is left as an easy exercise.
Notation.
We write
QH
h
(
G, R
) for the vector space of homogeneous quasi
homomorphisms G → R.
Then the above theorem gives
Corollary. We have
QH(G, R) = QH
h
(G, R) ⊕
∞
(G, R)
Proof.
Indeed, observe that a bounded homogeneous quasihomomorphism must
be identically zero.
Thus, if we want to study
QH
(
G, R
), it suffices to just look at the ho
mogeneous quasihomomorphisms. It turns out these have some very nice
perhapsunexpected properties.
Lemma. Let f : G → R be a homogeneous quasihomomorphism.
(i) We have f (xyx
−1
) = f(y) for all x, y ∈ G.
(ii) If G is abelian, then f is in fact a homomorphism. Thus
QH
h
(G, R) = Hom(G, R).
Thus, quasihomomorphisms are only interesting for nonabelian groups.
Proof.
(i) Note that for any x, the function
y 7→ f(xyx
−1
)
is a homogeneous quasihomomorphism. It suffices to show that the
function
y 7→ f(xyx
−1
) − f(y)
is a bounded homogeneous quasihomomorphism, since all such functions
must be zero. Homogeneity is clear, and the quasihomomorphism property
follows from the computation
f(xyx
−1
) − f(y) ≤ f (x) + f(y) + f(x
−1
) − f(y) + 2D(f ) = 2D(f ),
using the fact that f(x
−1
) = −f(x) by homogeneity.
(ii)
If
x
and
y
commute, then (
xy
)
n
=
x
n
y
n
. So we can use homogeneity to
write
f(xy) − f(x) − f(y) =
1
n
f((xy)
n
) − f(x
n
) − f(y
n
)
=
1
n
f(x
n
y
n
) − f(x
n
) − f(y
n
)
≤
1
n
D(f).
Since n is arbitrary, the difference must vanish.
The case of
QH
(
G, Z
)
/
∞
(
G, Z
) is more complicated. For example, we have
the following nice result:
Example. Given α ∈ R, define the map g
α
: Z → Z by
g
α
(m) = [mα].
Then this is a homomorphism, and one can check that the map
R −→
QH(Z, Z)
∞
(Z, Z)
α 7−→ g
α
is an isomorphism. This gives a further isomorphism
R/Z
∼
=
QH(Z, Z)
∞
(Z, Z) + Hom(Z, Z)
.
We next turn to the case
G
=
F
2
, the free group on two generators
a, b
. We
will try to work out explicitly a lot of nontrivial elements of
QH
(
F
2
, R
). In
general, when we try to construct quasihomomorphisms, what we manage to
get are not homogeneous. So when we construct several quasihomomorphisms,
it takes some nontrivial work to show that they are distinct. Our construction
will be one such that this is relatively easy to see.
Consider the vector space:
∞
odd
(Z) = {α: Z → R : α bounded and α(−n) = −α(n)}.
Note that in particular, we have α(0) = 0.
Given
α, β ∈
∞
odd
(
Z
), we define a quasihomomorphisms
f
α,β
: F
2
→ R
as
follows — given a reduced word w = a
n
1
b
m
1
· · · a
n
k
b
m
k
, we let
f
α,β
(w) =
k
X
i=1
α(n
i
) +
k
X
j=1
β(m
j
).
Allowing for
n
1
= 0 or
m
k
= 0, this gives a welldefined function
f
α,β
defined on
all of F
2
.
Let’s see what this does on some special sequences.
Example. We have
f
α,β
(a
n
) = α(n), f
α,β
(b
m
) = β(m),
and these are bounded functions of n, m.
So we see that f
α,β
is never homogeneous unless α = β = 0.
Example. Pick k
1
, k
2
, n 6= 0, and set
w = a
nk
1
b
nk
2
(b
k
2
a
k
1
)
−n
= a
nk
1
b
nk
2
a
−k
1
b
−k
2
· · · a
−k
1
b
−k
2

{z }
n times
.
This is now in reduced form. So we have
f
α,β
(w) = α(nk
1
) + β(nk
2
) − nα(k
1
) − nβ(k
2
).
This example is important. If
α
(
k
1
) +
β
(
k
2
)
6
= 0, then this is an unbounded
function as
n → ∞
. However, we know any genuine homomorphisms
f : F
2
→ R
must factor through the abelianization, and
w
vanishes in the abelianization. So
this suggests our
f
α,β
is in some sense very far away from being a homomorphism.
Theorem
(P. Rolli, 2009)
.
The function
f
α,β
is a quasihomomorphism, and
the map
∞
odd
(Z) ⊕
∞
odd
(Z) →
QH(F
2
, R)
∞
(F
2
, R) + Hom(F
2
, R)
is injective.
This tells us there are a lot of nontrivial elements in QH(F
2
, R).
The advantage of this construction is that the map above is a linear map.
So to see it is injective, it suffices to see that it has trivial kernel.
Proof.
Let
α, β ∈
∞
odd
(
Z, R
), and define
f
α,β
as before. By staring at it long
enough, we find that
f(xy) − f(x) − f(y) ≤ 3 max(kαk
∞
, kβk
∞
),
and so it is a quasihomomorphism. The main idea is that
f(b
n
) + f(b
−n
) = f(a
n
) + f(a
−n
) = 0
by oddness of
α
and
β
. So when we do the word reduction in the product, the
amount of error we can introduce is at most 3 max(kαk
∞
, kβk
∞
).
To show that the map is injective, suppose
f
α,β
= ϕ + h,
where
ϕ: F
2
→ R
is bounded and
h: F
2
→ R
is a homomorphism. Then we
must have
h(a
`
) = f(a
`
) − ϕ(a
`
) = α() − ψ(a
`
),
which is bounded. So the map
7→ h
(
a
`
) =
h
(
a
) is bounded, and so
h
(
a
) = 0.
Similarly, h(b) = 0. So h ≡ 0. In other words, f
α,β
is bounded.
Finally,
f((a
`
1
b
`
2
)
k
) = k(α(
1
) + β(
2
)) = 0.
Since this is bounded, we must have
α
(
1
) +
β
(
2
) = 0 for all
1
,
2
6
= 0. Using
the fact that
α
and
β
are odd, this easily implies that
α
(
1
) =
β
(
2
) = 0 for all
1
and
2
.
More generally, we have the following theorem, which we shall not prove, or
even explain what the words mean:
Theorem (Hull–Osin 2013). The space
QH(G, R)
∞
(G, R) + Hom(G, R)
is infinitedimensional if G is acylindrically hyperbolic.
1.2 Relation to commutators
A lot of interesting information about quasihomomorphisms can be captured by
considering commutators. Recall that we write
[x, y] = xyx
−1
y
−1
.
If
f
is a genuine homomorphisms, then it vanishes on all commutators, since the
codomain is abelian. For homogeneous quasihomomorphisms, we can bound
the value of f by the defect:
Lemma. If f is a homogeneous quasihomomorphism and x, y ∈ G, then
f([x, y]) ≤ D(f).
For nonhomogeneous ones, the value of
f
on a commutator is still bounded,
but requires a bigger bound.
Proof. By definition of D(f), we have
f([x, y]) − f(xyx
−1
) − f(y
−1
) ≤ D(f).
But since
f
is homogeneous, we have
f
(
xyx
−1
) =
f
(
y
) =
−f
(
y
−1
). So we are
done.
This bound is in fact the best we can obtain:
Lemma (Bavard, 1992). If f is a homogeneous quasihomomorphism, then
sup
x,y
f([x, y]) = D(f).
We will neither prove this nor use this — it is merely for amusement.
For a general element
a ∈
[
G, G
], it need not be of the form [
x, y
]. We can
define
Definition
(Commutator length)
.
Let
a ∈
[
G, G
]. Then commutator length
cl(a) is the word length with respect to the generators
{[x, y] : x, y ∈ G}.
In other words, it is the smallest n such that
a = [x
1
, y
1
][x
2
, y
2
] · · · [x
n
, y
n
]
for some x
i
, y
i
∈ G.
It is an easy inductive proof to show that
Lemma. For a ∈ [G, G], we have
f(a) ≤ 2D(f) cl(a).
By homogeneity, it follows that
f(a) =
1
n
f(a
n
) ≤ 2D(f)
cl(a
n
)
n
.
Definition
(Stable commutator length)
.
The stable commutator length is defined
by
scl(a) = lim
n→∞
cl(a
n
)
n
.
Then we have
Proposition.
f(a) ≤ 2D(f)scl(a).
Example. Consider F
2
with generators a, b. Then clearly we have
cl([a, b]) = 1.
It is not hard to verify that we also have
cl([a, b]
2
) = 2.
But interestingly, this “pattern” doesn’t extend to higher powers. By writing it
out explicitly, we find that
[a, b]
3
= [aba
−1
, b
−1
aba
−2
][b
−1
ab, b
2
].
In general, something completely mysterious can happen as we raise the
power, especially in more complicated groups.
Similar to the previous result by Bavard, the bound of
f
(
a
)

by
scl
(
a
) is
sharp.
Theorem (Bavard, 1992). For all a ∈ [G, G], we have
scl(a) =
1
2
sup
φ∈QH
h
(G,R)
φ(a)
D(φ)
,
where, of course, we skip over those
φ ∈ Hom
(
G, R
) in the supremum to avoid
division by zero.
Example. It is true that
scl([a, b]) =
1
2
.
However, showing this is not very straightforward.
Corollary.
The stable commutator length vanishes identically iff every homo
geneous quasihomomorphism is a homomorphism.
Note that if
cl
is bounded, then we have
scl ≡
0. There exists interesting
groups with bounded
cl
, such as nilpotent finitelygenerated groups, and so
these have
QH
h
(
G, R
) =
Hom
(
G, R
). We might think that the groups with
cl
bounded are “almost abelian”, but it turns out not.
Theorem (Carder–Keller 1983). For n ≥ 3, we have
SL(n, Z) = [SL(n, Z), SL(n, Z)],
and the commutator length is bounded.
More generally, we have
Theorem
(D. Witte Morris, 2007)
.
Let
O
be the ring of integers of some number
field. Then
cl:
[
SL
(
n, O
)
, SL
(
n, O
)]
→ R
is bounded iff
n ≥
3 or
n
= 2 and
O
×
is infinite.
The groups
SL
(
n, O
) have a common property — they are lattices in real
semisimple Lie groups. In fact, we have
Theorem
(Burger–Monod, 2002)
.
Let Γ
< G
be an irreducible lattice in a
connected semisimple group
G
with finite center and rank
G ≥
2. Then every
homogeneous quasimorphism Γ → R is ≡ 0.
Example.
If Γ
< SL
(
n, R
) is a discrete subgroup such that Γ
\SL
(
n, R
) is
compact, then it falls into the above class, and the rank condition is n ≥ 3.
It is in fact conjectured that
– The commutator length is bounded.
–
Γ is boundedly generated, i.e. we can find generators
{s
1
, · · · , s
k
}
such
that
Γ = hs
1
ihs
2
i · · · hs
k
i.
There is another theorem that seems completely unrelated to this, but actually
uses the same technology.
Theorem
(Burger–Monod, 2009)
.
Let Γ be a finitelygenerated group and let
µ
be a symmetric probability measure on Γ whose support generates Γ. Then
every class in
QH
(Γ
, R
)
/
∞
(Γ
, R
) has a unique
µ
harmonic representative. In
addition, this harmonic representative f satisfies the following:
kdfk
∞
≤ kdgk
∞
for any g ∈ f +
∞
(Γ, R).
This is somewhat like the Hodge decomposition theorem.
1.3 Poincare translation quasimorphism
We will later spend quite a lot of time studying actions on the circle. Thus, we
are naturally interested in the homeomorphism group of the sphere. We are
mostly interested in orientationpreserving actions only. Thus, we need to define
what it means for a homeomorphism ϕ: S
1
→ S
1
to be orientationpreserving.
The topologist will tell us that ϕ induces a map
ϕ
∗
: H
1
(S
1
, Z) → H
1
(S
1
, Z).
Since the homology group is generated by the fundamental class [
S
1
], invertibility
of
ϕ
∗
implies
ϕ
∗
([
S
1
]) =
±
[
S
1
]. Then we say
ϕ
is orientationpreserving if
ϕ
∗
([S
1
]) = [S
1
].
However, this definition is practically useless if we want to do anything with
it. Instead, we can make use of the following definition:
Definition
(Positivelyoriented triple)
.
We say a triple of points
x
1
, x
2
, x
3
∈ S
1
is positivelyoriented if they are distinct and ordered as follows:
x
1
x
2
x
3
More formally, recall that there is a natural covering map
π : R → S
1
given by
quotienting by
Z
. Formally, we let
˜x
1
∈ R
be any lift of
x
1
. Then let
˜x
2
, ˜x
3
be
the unique lifts of
x
2
and
x
3
respectively to [
˜x
1
, ˜x
1
+ 1). Then we say
x
1
, x
2
, x
3
are positivelyoriented if ˜x
2
< ˜x
3
.
Definition
(Orientationpreserving map)
.
A map
S
1
→ S
1
is orientation
preserving if it sends positivelyoriented triples to positivelyoriented triples.We
write
Homeo
+
(
S
1
) for the group of orientationpreserving homeomorphisms of
S
1
.
We can generate a large collection of homeomorphisms of
S
1
as follows — for
any x ∈ R, we define the translation map
T
x
: R → R
y 7→ y + x.
Identifying
S
1
with
R/Z
, we see that this gives a map
T
x
∈ Homeo
+
(
S
1
). Of
course, if n is an integer, then T
x
= T
n+x
.
One can easily see that
Proposition.
Every lift
˜ϕ: R → R
of an orientation preserving homeomorphism
ϕ: S
1
→ S
1
is a monotone increasing homeomorphism of
R
, commuting with
translation by Z, i.e.
˜ϕ ◦ T
m
= T
m
◦ ˜ϕ
for all m ∈ Z.
Conversely, any such map is a lift of an orientationpreserving homeomor
phism.
We write
Homeo
+
Z
(
R
) for the set of all monotone increasing homeomorphisms
R → R
that commute with
T
m
for all
m ∈ Z
. Then the above proposition says
there is a natural surjection
Homeo
+
Z
(
R
)
→ Homeo
+
(
S
1
). The kernel consists of
the translationby
m
maps for
m ∈ Z
. Thus,
Homeo
+
Z
(
R
) is a central extension
of Homeo
+
(S
1
). In other words, we have a short exact sequence
0 Z Homeo
+
Z
(R) Homeo
+
(S
1
) 0
i
p
.
The “central” part in the “central extension” refers to the fact that the image of
Z is in the center of Homeo
+
Z
(R).
Notation.
We write
Rot
for the group of rotations in
Homeo
+
(
S
1
). This
corresponds to the subgroup T
R
⊆ Homeo
+
Z
(R).
From a topological point of view, we can see that
Homeo
+
(
S
1
) retracts to
Rot
. More precisely, if we fix a basepoint
x
0
∈ S
1
, and write
Homeo
+
(
S
1
, x
0
) for
the basepoint preserving maps, then every element in
Homeo
+
(
S
1
) is a product
of an element in
Rot
and one in
Homeo
+
(
S
1
, x
0
). Since
Homeo
+
(
S
1
, x
0
)
∼
=
Homeo
+
([0, 1]) is contractible, it follows that Homeo
+
(S
1
) retracts to Rot.
A bit more fiddling around with the exact sequence above shows that
Homeo
+
Z
(
R
)
→ Homeo
+
(
S
1
) is in fact a universal covering space, and that
π
1
(Homeo
+
(S
1
)) = Z.
Lemma.
The function
F : Homeo
+
Z
(
R
)
→ R
given by
ϕ 7→ ϕ
(0) is a quasi
homomorphism.
Proof. The commutation property of ϕ reads as follows:
ϕ(x + m) = ϕ(x) + m.
For a real number x ∈ R, we write
x = {x} + [x],
where 0 ≤ {x} < 1 and [x] = 1. Then we have
F (ϕ
1
ϕ
2
) = ϕ
1
(ϕ
2
(0))
= ϕ
1
(ϕ
2
(0))
= ϕ
1
({ϕ
2
(0)} + [ϕ
2
(0)])
= ϕ
1
({ϕ
2
(0)}) + [ϕ
2
(0)]
= ϕ
1
({ϕ
2
(0)}) + ϕ
2
(0) − {ϕ
2
(0)}.
Since 0 ≤ {ϕ
2
(0)} < 1, we know that
ϕ
1
(0) ≤ ϕ
1
({ϕ
2
(0)}) < ϕ
1
(1) = ϕ
1
(0) + 1.
Then we have
ϕ
1
(0) + ϕ
2
(0) − {ϕ
2
(0)} ≤ F (ϕ
1
ϕ
2
) < ϕ
1
(0) + 1 + ϕ
2
(0) − {ϕ
2
(0)}.
So subtracting, we find that
−1 ≤ −{ϕ
2
(0)} ≤ F (ϕ
1
ϕ
2
) − F (ϕ
1
) − F (ϕ
2
) < 1 − {ϕ
2
(0)} ≤ 1.
So we find that
D(f) ≤ 1.
Definition
(Poincare translation quasimorphism)
.
The Poincare translation
quasimorphism T : Homeo
+
Z
(R) → R is the homogenization of F .
It is easily seen that T (T
x
) = x. This allows us to define
Definition
(Rotation number)
.
The rotation number of
ϕ ∈ Homeo
+
(
S
1
) is
T ( ˜ϕ) mod Z ∈ R/Z, where ˜ϕ is a lift of ϕ to Homeo
+
Z
(R).
This rotation number contains a lot of interesting information about the
dynamics of the homeomorphism. For instance, minimal homeomorphisms of
S
1
are conjugate iff they have the same rotation number.
We will see that bounded cohomology allows us to generalize the rotation
number of a homeomorphism into an invariant for any group action.
2 Group cohomology and bounded cohomology
2.1 Group cohomology
We can start talking about cohomology. Before doing bounded cohomology, we
first try to understand usual group cohomology. In this section,
A
will be any
abelian group. Ultimately, we are interested in the case
A
=
Z
or
R
, but we can
develop the theory in this generality.
The general idea is that to a group Γ, we are going to associate a sequence
of abelian groups H
k
(Γ, A) that is
– covariant in A; and
– contravariant in Γ.
It is true, but we will not prove or use, that if
X
=
K
(Γ
,
1), i.e.
X
is a CW
complex whose fundamental group is Γ and has a contractible universal cover,
then there is a natural isomorphism
H
k
(Γ, A)
∼
=
H
k
sing
(X, A).
There are several ways we can define group cohomology. A rather powerful
way of doing so is via the theory of derived functors. However, developing the
machinery requires considerable effort, and to avoid scaring people off, we will
use a more downtoearth construction. We begin with the following definition:
Definition
(Homogeneous
k
cochain)
.
A homogeneous
k
cochain with values
in A is a function f : Γ
k+1
→ A. The set C(Γ
k+1
, A) is an abelian group and Γ
acts on it by automorphisms in the following way:
(γ
∗
f)(γ
0
, · · · , γ
m
) = f(γ
−1
γ
0
, · · · , γ
−1
γ
k
).
By convention, we set C(Γ
0
, A)
∼
=
A.
Definition
(Differential
d
(k)
)
.
We define the differential
d
(k)
: C
(Γ
k
, A
)
→
C(Γ
k+1
, A) by
(d
(k)
f)(γ
0
, · · · , γ
k
) =
k
X
j=0
(−1)
j
f(γ
0
, · · · , ˆγ
j
, · · · , γ
k
).
In particular, we set d
(0)
(a) to be the function that is constantly a.
Example. We have
d
(1)
f(γ
0
, γ
1
) = f(γ
1
) − f(γ
0
)
d
(2)
f(γ
0
, γ
1
, γ
2
) = f(γ
1
, γ
2
) − f(γ
0
, γ
2
) + f(γ
0
, γ
1
).
Thus, we obtain a complex of abelian groups
0 A C(Γ, A) C(Γ
2
, A) · · ·
d
(0)
d
(1)
d
(2)
.
The following are crucial properties of this complex.
Lemma.
(i) d
(k)
is a Γequivariant group homomorphism.
(ii) d
(k+1)
◦ d
(k)
= 0. So im d
(k)
⊆ ker d
(k+1)
.
(iii) In fact, we have im d
(k)
= ker d
(k+1)
.
Proof.
(i) This is clear.
(ii) You just expand it out and see it is zero.
(iii) If f ∈ ker d
(k)
, then setting γ
k
= e, we have
0 = d
(k)
f(γ
0
, · · · , γ
k−1
, e) = (−1)
k
f(γ
0
, · · · , γ
k−1
)
+
k−1
X
j=0
(−1)
j
f(γ
0
, · · · , ˆγ
j
, · · · , γ
k−1
, e).
Now define the following (k − 1)cochain
h(γ
0
, · · · , γ
k−2
) = (−1)
k
f(γ
0
, · · · , γ
k−2
, e).
Then the above reads
f = d
(k−1)
h.
We make the following definitions:
Definition (kcocycle and kcoboundaries).
– The kcocycles are ker d
(k+1)
.
– The kcoboundaries are im d
(k)
.
So far, every cocycle is a coboundary, so nothing interesting is happening.
To obtain interesting things, we use the action of Γ on C(Γ
k
, A). We denote
C(Γ
k
, A)
Γ
= {f : Γ
k
→ A  f is Γinvariant}.
Since the differentials
d
(k)
commute with the Γaction, it restricts to a map
C(Γ
k
, A)
Γ
→ C(Γ
k+1
, A)
Γ
. We can arrange these into a new complex
0 A C(Γ, A)
Γ
C(Γ
2
, A)
Γ
· · ·
0 A C(Γ, A) C(Γ
2
, A) · · ·
d
(0)
d
(1)
d
(2)
d
(0)
d
(1)
d
(2)
.
We are now in a position to define group cohomology.
Definition
(Group cohomology
H
k
(Γ
, A
))
.
We define the
k
th cohomology group
to be
H
k
=
(ker d
(k+1)
)
Γ
d
(k)
(C(Γ
k
, A)
Γ
)
=
(d
(k)
(C(Γ
k
, A)))
Γ
d
(k)
(C(Γ
k
, A)
Γ
)
.
Before we do anything with group cohomology, we provide a slightly different
description of group cohomology, using inhomogeneous cochains. The idea is to
find a concrete description of all invariant cochains.
Observe that if we have a function
f :
Γ
k+1
→ A
that is invariant under the
action of Γ, then it is uniquely determined by the value on
{
(
e, γ
1
, · · · , γ
k
) :
γ
i
∈
Γ
}
. So we can identify invariant functions
f :
Γ
k+1
→ A
with arbitrary functions
Γ
k
→ A
. So we have one variable less to worry about, but on the other hand,
the coboundary maps are much more complicated.
More explicitly, we construct an isomorphism
C(Γ
k
, A)
Γ
C(Γ
k−1
, A)
ρ
(k−1)
τ
(k)
,
by setting
(ρ
(k−1)
f)(g
1
, · · · , g
k−1
) = f(e, g
1
, g
2
, · · · , g
1
· · · g
k−1
)
(τ
(k)
h)(g
1
, · · · , g
k
) = h(g
−1
1
g
2
, g
−1
2
g
3
, · · · , g
−1
k−1
g
k
).
These homomorphisms are inverses of each other. Then under this identifica
tion, we obtain a new complex
C(Γ
k
, A)
Γ
C(Γ
k+1
, A)
Γ
C(Γ
k−1
, A) C(Γ
k
, A)
d
(k)
ρ
(k)
τ
(k)
d
k
where
d
k
= ρ
k
◦ d
(k)
◦ τ
k
.
A computation shows that
(d
k
f)(g
1
, · · · , g
k
) = f(g
2
, · · · , g
k
) +
k−1
X
j=1
(−1)
j
f(g
1
, · · · , g
j
g
j+1
, · · · , g
k
)
+(−1)
k
f(g
1
, · · · , g
k−1
).
It is customary to denote
Z
k
(Γ, A) = ker
d+1
⊆ C(Γ
k
, A)
B
k
(Γ, A) = im d
k
⊆ C(Γ
k
, A),
the inhomogeneous
k
cocycles and inhomogeneous
k
coboundaries. Then we
simply have
H
k
(Γ, A) =
Z
k
(Γ, A)
B
k
(Γ, A)
.
It is an exercise to prove the following:
Lemma.
A homomorphism
f
: Γ
→
Γ
0
of groups induces a natural map
f
∗
:
H
k
(Γ
0
, Z
)
→ H
k
(Γ
, Z
) for all
k
. Moreover, if
g
: Γ
0
→
Γ
00
is another group
homomorphism, then f
∗
◦ g
∗
= (gf)
∗
.
Computation in degrees k = 0, 1, 2
It is instructive to compute explicitly what these groups mean in low degrees.
We begin with the boring one:
Proposition. H
0
(Γ, A)
∼
=
A.
Proof. The relevant part of the cochain is
0 A C(Γ, A)
d
1
=0
.
The k = 1 case is not too much more interesting.
Proposition. H
1
(Γ, A) = Hom(Γ, A).
Proof. The relevant part of the complex is
A C(Γ, A) C(Γ
2
, A)
d
1
=0 d
2
,
and we have
(d
2
f)(γ
1
, γ
2
) = f(γ
1
) − f(γ
1
γ
2
) + f(γ
2
).
The k = 2 part is more interesting. The relevant part of the complex is
C(Γ, A) C(Γ
2
, A) C(Γ
3
, A)
d
2
d
3
.
Here d
3
is given by
d
3
α(g
1
, g
2
, g
3
) = α(g
2
, g
3
) − α(g
1
g
2
, g
3
) + α(g
1
, g
2
g
3
) − α(g
1
, g
2
).
Suppose that
d
3
α
(
g
1
, g
2
, g
3
) = 0, and, in addition, by some magic, we managed
to pick
α
such that
α
(
g
1
, e
) =
α
(
e, g
2
) = 0. This is known as a normalized
cocycle. We can now define the following operation on Γ × A:
(γ
1
, a
2
)(γ
2
, a
2
) = (γ
1
γ
2
, a
1
+ a
2
+ α(γ
1
, γ
2
)).
Then the property that
α
is a normalized cocycle is equivalent to the assertion
that this is an associative group law with identity element (
e,
0). We will write
this group as Γ ×
α
A.
We can think of this as a generalized version of the semidirect product. This
group here has a special property. We can organize it into an exact sequence
0 A Γ ×
α
A Γ 0 .
Moreover, the image of
A
is in the center of Γ
×
α
A
. This is known as a central
extension.
Definition
(Central extension)
.
Let
A
be an abelian group, and Γ a group.
Then a central extension of Γ by A is an exact sequence
0 A
˜
Γ Γ 0
such that the image of A is contained in the center of
˜
Γ.
The claim is now that
Proposition. H
2
(Γ
, A
) parametrizes the set of isomorphism classes of central
extensions of Γ by A.
Proof sketch. Consider a central extension
0 A G Γ 0
i
p
.
Arbitrarily choose a section
s:
Γ
→ G
of
p
, as a function of sets. Then we know
there is a unique α(γ
1
, γ
2
) such that
s(γ
1
γ
2
)α(γ
1
, γ
2
) = s(γ
1
)s(γ
2
).
We then check that α is a (normalized) 2cocycle, i.e. α(γ
1
, e) = γ(e, γ
2
) = 0.
One then verifies that different choices of
s
give cohomologous choices of
α
,
i.e. they represent the same class in H
2
(Γ, A).
Conversely, given a 2cocycle
β
, we can show that it is cohomologous to a
normalized 2cocycle
α
. This gives rise to a central extension
G
= Γ
×
α
A
as
constructed before (and also a canonical section s(γ) = (γ, 0)).
One then checks this is a bijection.
Exercise. H
2
(Γ
, A
) has a natural structure as an abelian group. Then by the
proposition, we should be able to “add” two central extensions. Figure out what
this means.
Example. As usual, write F
r
for the free group on r generators. Then
H
k
(F
r
, A) =
A k = 0
A
r
k = 1
0 k = 2
.
The fact that
H
2
(
F
r
, A
) vanishes is due to the fact that
F
r
is free, so every short
exact sequence splits.
Example. Consider Γ
g
= π
1
(S
g
) for g > 0. Explicitly, we can write
Γ
g
=
(
a
1
, b
1
, · · · , a
g
, b
g
:
g
Y
i=1
[a
i
, b
i
] = e
)
Then we have H
1
(Γ
g
, Z) = Z
2g
and H
2
(Γ
g
, Z)
∼
=
Z.
We can provide a very explicit isomorphism for H
2
(Γ
g
, Z). We let
0 Z G Γ 0
i
p
be a central extension. Observe that whenever
γ, η ∈
Γ
g
, and
˜γ, ˜η ∈ G
are lifts,
then [
˜γ, ˜η
] is a lift of [
γ, η
] and doesn’t depend on the choice of
˜γ
and
˜η
. Thus,
we can pick ˜a
1
,
˜
b
1
, · · · , ˜a
g
,
˜
b
g
. Then notice that
g
Y
i=1
[˜a
i
,
˜
b
i
]
is in the kernel of p, and is hence in Z.
Alternatively, we can compute the group cohomology using topology. We
notice that
R
2
is the universal cover of
S
g
, and it is contractible. So we know
S
g
=
K
(Γ
g
,
1). Hence, by the remark at the beginning of the section (which
we did not prove), it follows that
H
1
(Γ
g
, Z
)
∼
=
H
1
sing
(
S
g
;
Z
), and the latter is a
standard computation in algebraic topology.
Finally, we look at actions on a circle. Recall that we previously had the
central extension
0 Z Homeo
+
Z
(R) Homeo
+
(S
1
) 0
i
p
.
This corresponds to the Euler class e ∈ H
2
(Homeo
+
(S
1
), Z).
We can in fact construct a representative cocycle of
e
. To do so, we pick
a section
s: Homeo
+
(
S
1
)
→ Homeo
+
Z
(
R
) by sending
f ∈ Homeo
+
(
S
1
) to the
unique lift
¯
f : R → R such that
¯
f(0) ∈ [0, 1).
Then we find that
s(f
1
, f
2
)T
c(f
1
,f
2
)
= s(f
1
)s(f
2
)
for some c(f
1
, f
2
) ∈ Z.
Lemma. We have c(f
1
, f
2
) ∈ {0, 1}.
Proof. We have f
1
f
2
(0) ∈ [0, 1), while
¯
f
2
(0) ∈ [0, 1). So we find that
¯
f
1
(
¯
f
2
(0)) ∈ [
¯
f
1
(0),
¯
f
1
(1)) = [
¯
f
1
(0),
¯
f
1
(0) + 1) ⊆ [0, 2).
But we also know that c(f
1
, f
2
) is an integer. So c(f
1
, f
2
) ∈ {0, 1}.
Definition
(Euler class)
.
The Euler class of the Γaction by orientation
preserving homeomorphisms of S
1
is
h
∗
(e) ∈ H
2
(Γ, Z),
where h : Γ → Homeo
+
(S
1
) is the map defining the action.
For example, if Γ
g
is a surface group, then we obtain an invariant of actions
valued in Z.
There are some interesting theorems about this Euler class that we will not
prove.
Theorem (Milnor–Wood). If h : Γ
g
→ Homeo
+
(S
1
), then h
∗
(e) ≤ 2g − 2.
Theorem
(Gauss–Bonnet)
.
If
h:
Γ
g
→ PSL
(2
, R
)
⊆ Homeo
+
(
S
1
) is the holon
omy representation of a hyperbolic structure, then
h
∗
(e) = ±(2g − 2).
Theorem
(Matsumoko, 1986)
.
If
h
defines a minimal action of Γ
g
on
S
1
and
h
∗
(e) = 2g − 2, then h is conjugate to a hyperbolization.
2.2 Bounded cohomology of groups
We now move on to bounded cohomology. We will take
A
=
Z
or
R
now. The
idea is to put the word “bounded” everywhere. For example, we previously had
C(Γ
k+1
, A) denoting the functions Γ
k+1
→ A. Likewise, we denote
C
b
(Γ
k+1
, A) = {f ∈ C(Γ
k+1
, A) : f is bounded} ⊆ C(Γ
k+1
, A).
We have
d
(k)
(
C
b
(Γ
k
, A
))
⊆ C
b
(Γ
k+1
, A
), and so as before, we obtain a chain
complexes
0 A C
b
(Γ, A)
Γ
C
b
(Γ
2
, A)
Γ
· · ·
0 A C
b
(Γ, A) C
b
(Γ
2
, A) · · ·
d
(0)
d
(1)
d
(2)
d
(0)
d
(1)
d
(2)
.
This allows us to define
Definition
(Bounded cohomology)
.
The
k
th bounded cohomology group of Γ
with coefficients in A Is
H
k
b
(Γ, A) =
ker(d
(k+1)
: C
b
(Γ
k+1
, A)
Γ
→ C
b
(Γ
k+2
, A)
Γ
)
d
(k)
(C
b
(Γ
k
, A)
Γ
)
.
This comes with two additional features.
(i)
As one would expect, a bounded cochain is bounded. So given an element
f ∈ C
b
(Γ
k+1
, A), we can define
kfk
∞
= sup
x∈Γ
k+1
f(x).
Then
k · k
∞
makes
C
b
(Γ
k+1
, A
) into a normed abelian group, and in the
case A = R, a Banach space.
Then for [f ] ∈ H
k
b
(Γ, A), we define
k[f]k
∞
= inf{kf + dgk
∞
: g ∈ C
b
(Γ
k
, A)
Γ
}.
This induces a seminorm on
H
k
b
(Γ
, A
). This is called the canonical semi
norm.
(ii) We have a map of chain complexes
C
b
(Γ, A)
Γ
C
b
(Γ
2
, A)
Γ
C
b
(Γ
3
, A)
Γ
· · ·
C(Γ, A)
Γ
C(Γ
2
, A)
Γ
C(Γ
3
, A)
Γ
· · ·
Thus, this induces a natural map
c
k
: H
k
b
(Γ
, A
)
→ H
k
(Γ
, A
), known as the
comparison map. In general, c
k
need not be injective or surjective.
As before, we can instead use the complex of inhomogeneous cochains. Then
we have a complex that looks like
0 A C
b
(Γ, A) C
b
(Γ
2
, A) · · ·
d
1
=0 d
2
d
3
In degree 0, the boundedness condition is useless, and we have
H
0
b
(Γ, A) = H
0
(Γ, A) = A.
For
k
= 1, we have
im d
1
= 0. So we just have to compute the cocycles. For
f ∈ C
b
(Γ
, A
), we have
d
2
f
= 0 iff
f
(
g
1
)
− f
(
g
1
g
2
) +
f
(
g
2
) = 0, iff
f ∈ Hom
(Γ
, A
).
But we have the additional information that
f
is bounded, and there are no
nonzero bounded homomorphisms to Γ or A! So we have
H
1
b
(Γ, A) = 0.
If we allow nontrivial coefficients, then
H
1
b
(Γ
, A
) may be always be zero. But
that’s another story.
The interesting part starts at
H
2
b
(Γ
, A
). To understand this, We are going to
determine the kernel of the comparison map
c
2
: H
2
b
(Γ, A) → H
2
(Γ, A).
We consider the relevant of the defining complexes, where we take inhomogeneous
cochains
C(Γ, A) C(Γ
2
, A) C(Γ
3
, A)
C
b
(Γ, A) C
b
(Γ
2
, A) C
b
(Γ
3
, A)
d
2
d
3
d
2
d
3
By definition, the kernel of
c
2
consists of the [
α
]
∈ H
2
b
(Γ
, A
) such that
α
=
d
2
f
for some
f ∈ C
(Γ
, A
). But
d
2
f
=
α
being bounded tells us
f
is a quasi
homomorphism! Thus, we have a map
¯
d
2
: QH(Γ, A) ker c
2
f [d
2
f].
Proposition. The map
¯
d
2
induces an isomorphism
QH(Γ, A)
∞
(Γ, A) + Hom(Γ, A)
∼
=
ker c
2
.
Proof.
We know that
¯
d
2
is surjective. So it suffices to show that the kernel is
∞
(Γ, A) + Hom(Γ, A).
Suppose
f ∈ QH
(Γ
, A
) is such that
¯
d
2
f ∈ H
2
b
(Γ
, A
) = 0. Then there exists
some g ∈ C
b
(Γ, A) such that
d
2
f = d
2
g.
So it follows that
d
2
(
f − g
) = 0. That is,
f − g ∈ Hom
(Γ
, A
). Hence it follows
that
ker
¯
d
2
⊆
∞
(Γ, A) + Hom(Γ, A).
The other inclusion is clear.
Since we already know about group cohomology, the determination of the
kernel can help us compute the bounded cohomology. In certain degenerate
cases, it can help us determine it completely.
Example.
For
G
abelian and
A
=
R
, we saw that
QH
(Γ
, A
) =
∞
(Γ
, A
) +
Hom(Γ, A). So it follows that c
2
is injective.
Example.
For
H
2
b
(
Z, Z
), we know
H
2
(
Z, Z
) = 0 since
Z
is a free group (hence,
e.g. every extension splits, and in particular all central extensions do). Then we
know
H
2
b
(Z, Z)
∼
=
QH(Z, Z)
∞
(Z, Z) + Hom(Z, Z)
∼
=
R/Z.
Example.
Consider
H
2
b
(
F
r
, R
). We know that
H
2
(
F
r
, R
) = 0. So again
H
2
b
(
F
r
, R
) is given by the quasihomomorphisms. We previously found many
such quasihomomorphisms — by Rollis’ theorem, we have an inclusion
∞
odd
(Z, R) ⊕
∞
odd
(Z, R) H
2
b
(F
r
, R)
(α, β) [d
2
f
α,β
]
Recall that
H
2
b
(
F
r
, R
) has the structure of a seminormed space, which we called
the canonical norm. One can show that
k[d
2
f
α,β
]k = max(kdαk
∞
, kdβk
∞
).
Returning to general theory, a natural question to ask ourselves is how the
groups
H
·
b
(Γ
, Z
) and
H
·
b
(Γ
, R
) are related. For ordinary group cohomology, if
A ≤ B
is a subgroup (we are interested in
Z ≤ R
), then we have a long exact
sequence of the form
· · · H
k−1
(Γ, B/A) H
k
(Γ, A) H
k
(Γ, B) H
k
(Γ, B/A) · · ·
β
,
where
β
is known as the Bockstein homomorphism. This long exact sequence
comes from looking at the short exact sequence of chain complexes (of inhomo
geneous cochains)
0 C(Γ
·
, B) C(Γ
·
, A) C(Γ
·
, B/A) 0 ,
and then applying the snake lemma.
If we want to perform the analogous construction for bounded cohomology,
we might worry that we don’t know what
C
b
(Γ
·
, R/Z
) means. However, if we
stare at it long enough, we realize that we don’t have to worry about that. It
turns out the sequence
0 C
b
(Γ
·
, Z) C
b
(Γ
·
, R) C(Γ
·
, R/Z) 0
is short exact. Thus, snake lemma tells us we have a long exact sequence
· · · H
k−1
(Γ, R/Z) H
k
b
(Γ, Z) H
k
b
(Γ, R) H
k
(Γ, R/Z) · · ·
δ
.
This is known as the Gersten long exact sequence
Example. We can look at the beginning of the sequence, with
0 = H
1
b
(Γ, R) Hom(Γ, R/Z) H
2
b
(Γ, Z) H
2
b
(Γ, R)
δ
.
In the case Γ =
Z
, from our first example, we know
c
2
:
H
2
b
(
Z, R
)
→ H
2
(
Z, R
) = 0
is an injective map. So we recover the isomorphism
R/Z = Hom(Z, R/Z)
∼
=
H
2
b
(Z, Z)
we found previously by direct computation.
We’ve been talking about the kernel of
c
2
so far. In Gersten’s Bounded
cocycles and combing of groups (1992) paper, it was shown that the image of
the comparison map
c
2
: H
2
b
(Γ
, Z
)
→ H
2
(Γ
, Z
) describes central extensions with
special metric features. We shall not pursue this too far, but the theorem is as
follows:
Theorem.
Assume Γ is finitelygenerated. Let
G
α
be the central extension of
Γ by
Z
, defined by a class in
H
2
(Γ
, Z
) which admits a bounded representative.
Then with any word metric, Γ
α
is quasiisometric to Γ
× Z
via the “identity
map”.
Before we end the chapter, we produce a large class of groups for which
bounded cohomology (with real coefficients) vanish, namely amenable groups.
Definition
(Amenable group)
.
A discrete group Γ is amenable if there is a
linear form m :
∞
(Γ, R) → R such that
– m(f ) ≥ 0 if f ≥ 0;
– m(1) = 1; and
– m is leftinvariant, i.e. m(γ
∗
f) = m(f), where (γ
∗
f)(x) = f(γ
−1
x).
A linear form that satisfies the first two properties is known as a mean, and
we can think of this as a way of integrating functions. Then an amenable group
is a group with a left invariant mean. Note that the first two properties imply
m(f) ≤ kf k
∞
.
Example.
– Abelian groups are amenable, and finite groups are.
– Subgroups of amenable groups are amenable.
– If
0 Γ
1
Γ
2
Γ
3
0
is a short exact sequence, then Γ
2
is amenable iff Γ
1
and Γ
3
are amenable.
–
Let Γ =
hSi
for
S
a finite set. Given a finite set
A ⊆
Γ, we define
∂A
to
be the set of all edges with exactly one vertex in A.
For example, Z
2
with the canonical generators has Cayley graph
Then if
A
consists of the red points, then the boundary consists of the
orange edges.
It is a theorem that a group Γ is nonamenable iff there exists a constant
c = c(S, Γ) > 0 such that for all A ⊆ Γ, we have ∂A ≥ cA.
– There exists infinite, finitely generated, simple, anemable groups.
–
If Γ
⊆ GL
(
n, C
), then Γ is amenable iff it contains a finiteindex subgroup
which is solvable.
– F
2
is nonamenable.
– Any nonelementary wordhyperbolic group is nonamenable.
Proposition. Let Γ be an amenable group. Then H
k
b
(Γ, R) = 0 for k ≥ 1.
The proof requires absolutely no idea.
Proof.
Let
k ≥
1 and
f :
Γ
k+1
→ R
a Γinvariant bounded cocycle. In other
words,
d
(k+1)
f = 0
f(γγ
0
, · · · , γγ
k
) = f(γ
0
, · · · , γ
k
).
We have to find ϕ : Γ
k
→ R bounded such that
d
(k)
ϕ = f
ϕ(γγ
0
, · · · , γγ
k−1
) = ϕ(γ
0
, · · · , γ
k−1
).
Recall that for η ∈ Γ, we can define
h
η
(γ
0
, · · · , γ
k−1
) = (−1)
k+1
f(γ
0
, · · · , γ
k+1
, η),
and then
d
(k+1)
f = 0 ⇐⇒ f = d
(k)
(h
η
).
However, h
η
need not be invariant. Instead, we have
h
η
(γγ
0
, · · · , γγ
k−1
) = h
γ
−1
η
(γ
0
, · · · , γ
k−1
).
To fix this, let
m:
∞
(Γ)
→ R
be a leftinvariant mean. We notice that the map
η 7→ h
η
(γ
0
, · · · , γ
k−1
)
is bounded by kfk
∞
. So we can define
ϕ(γ
0
, · · · , γ
k−1
) = m
n
η 7→ h
η
(γ
0
, · · · , γ
k−1
)
o
.
Then this is the ϕ we want. Indeed, we have
ϕ(γγ
0
, · · · , γγ
k−1
) = m
n
η 7→ h
γ
−1
η
(γ
0
, · · · , γ
k−1
)
o
.
But this is just the mean of a left translation of the original function. So this is
just ϕ(γ
0
, · · · , γ
k−1
). Also, by properties of the mean, we know kϕk
∞
≤ kf k
∞
.
Finally, by linearity, we have
d
(k)
ϕ(γ
0
, · · · , γ
k
) = m
n
η 7→ d
(k)
h
η
(γ
0
, · · · , γ
k
)
o
= m
n
f(γ
0
, · · · , γ
k
) · 1
Γ
o
= f(γ
0
, · · · , γ
k
)m(1
Γ
)
= f(γ
0
, · · · , γ
k
).
3 Actions on S
1
3.1 The bounded Euler class
We are now going going to apply the machinery of bounded cohomology to
understand actions on S
1
. Recall that the central extension
0 Z Homeo
+
Z
(R) Homeo
+
(S
1
) 0
defines the Euler class
e ∈ H
2
(
Homeo
+
(
S
1
)
, Z
). We have also shown that there
is a representative cocycle c(f, g) taking the values in {0, 1}, defined by
f ◦ g ◦ T
c(f,g)
=
¯
f ◦ ¯g,
where for any f, the map
¯
f is the unique lift to R such that
¯
f(0) ∈ [0, 1).
Since
c
takes values in
{
0
,
1
}
, in particular, it is a bounded cocycle. So we
can use it to define
Definition (Bounded Euler class). The bounded Euler class
e
b
∈ H
2
b
(Homeo
+
(S
1
), Z)
is the bounded cohomology class represented by the cocycle c.
By construction, e
b
is sent to e via the comparison map
c
2
: H
2
b
(Homeo
+
(S
1
), Z) H
2
(Homeo
+
(S
1
), Z) .
In fact, the comparison map is injective. So this
e
b
is the unique element that is
sent to
e
, and doesn’t depend on us arbitrarily choosing
c
as the representative.
Definition
(Bounded Euler class of action)
.
The bounded Euler class of an
action h : Γ → Homeo
+
(S
1
) is h
∗
(e
b
) ∈ H
2
b
(Γ, Z).
By naturality (proof as exercise),
h
∗
(
e
b
) maps to
h
∗
(
e
) under the comparison
map. The bounded Euler class is actually a rather concrete and computable
object. Note that if we have an element
ϕ ∈ Homeo
+
(
S
1
), then we obtain a
group homomorphism
Z → Homeo
+
(
S
1
) that sends 1 to
ϕ
, and vice versa. In
other words, we can identify elements of
Homeo
+
(
S
1
) with homomorphisms
h
:
Z → Homeo
+
(
S
1
). Any such homomorphism will give a bounded Euler class
h
∗
(e
b
) ∈ H
2
b
(Z, Z)
∼
=
R/Z.
Exercise.
If
h: Z → Homeo
+
(
S
1
) and
ϕ
=
h
(1), then under the isomorphism
H
2
b
(Z, Z)
∼
=
R/Z, we have h
∗
(e
b
) = Rot(ϕ), the Poincar´e rotation number of ϕ.
Thus, one way to think about the bounded Euler class is as a generalization
of the Poincar´e rotation number.
Exercise.
Assume
h:
Γ
→ Homeo
+
(
S
1
) takes values in the rotations
Rot
. Let
χ:
Γ
→ R/Z
the corresponding homomorphism. Then under the connecting
homomorphism
Hom(Γ, R/Z) H
2
b
(Γ, Z)
δ
,
we have δ(χ) = h
∗
(e
b
).
Exercise.
If
h
1
and
h
2
are conjugate in
Homeo
+
(
S
1
), i.e. there exists a
ϕ ∈
Homeo
+
(S
1
) such that h
1
(γ) = ϕh
2
(γ)ϕ
−1
, then
h
∗
1
(e) = h
∗
2
(e), h
∗
1
(e
b
) = h
∗
2
(e
b
).
The proof involves writing out a lot of terms explicitly.
How powerful is this bounded Euler class in distinguishing actions? We just
saw that conjugate actions have the same bounded Euler class. The converse
does not hold. For example, one can show that any action with a global fixed
point has trivial bounded Euler class, and there are certainly nonconjugate
actions that both have global fixed points (e.g. take one of them to be the trivial
action).
It turns out there is a way to extend the notion of conjugacy so that the
bounded Euler class becomes a complete invariant.
Definition
(Increasing map of degree 1)
.
A map
ϕ: S
1
→ S
1
is increasing of
degree 1 if there is some
˜ϕ: R → R
lifting
ϕ
such that
˜ϕ
is is monotonically
increasing and
˜ϕ(x + 1) = ˜ϕ(x) + 1
for all x ∈ R.
Note that there is no continuity assumption made on
ϕ
. On the other hand,
it is an easy exercise to see that any monotonic map
R → R
has a countable set
of discontinuities. This is also not necessarily injective.
Example.
The constant map
S
1
→ S
1
sending
x 7→
0 is increasing of degree 1,
as it has a lift ˜ϕ(x) = [x].
Equivalently, such a map is one that sends a positive 4tuple to a weakly
positive 4tuple (exercise!).
Definition
(Semiconjugate action)
.
Two actions
h
1
, h
2
:
Γ
→ Homeo
+
(
S
1
) are
semiconjugate if there are increasing maps of degree 1
ϕ
1
, ϕ
2
: S
1
→ S
1
such
that
(i) h
1
(γ)ϕ
1
= ϕ
1
h
2
(γ) for all γ ∈ Γ;
(ii) h
2
(γ)ϕ
2
= ϕ
2
h
1
(γ) for all γ ∈ Γ.
One can check that the identity action is semiconjugate to any action with a
global fixed point.
Recall the following definition:
Definition
(Minimal action)
.
An action on
S
1
is minimal if every orbit is dense.
Lemma.
If
h
1
and
h
2
are minimal actions that are semiconjugate via
ϕ
1
and
ϕ
2
, then ϕ
1
and ϕ
2
are homeomorphisms and are inverses of each other.
Proof. The condition (i) tells us that
h
1
(γ)(ϕ
1
(x)) = ϕ
1
(h
2
(γ)(x)).
for all
x ∈ S
1
and
γ ∈
Γ. This means
im ϕ
1
is
h
1
(Γ)invariant, hence dense in
S
1
. Thus, we know that
im ˜ϕ
1
is dense in
R
. But
˜ϕ
is increasing. So
˜ϕ
1
must
be continuous. Indeed, we can look at the two limits
lim
x%y
˜ϕ
1
(x) ≤ lim
x&y
˜ϕ
1
(x).
But since
˜ϕ
1
is increasing, if
˜ϕ
1
were discontinuous at
y ∈ R
, then the inequality
would be strict, and hence the image misses a nontrivial interval. So
˜ϕ
1
is
continuous.
We next claim that
˜ϕ
1
is injective. Suppose not. Say
ϕ
(
x
1
) =
ϕ
(
x
2
). Then
by looking at the lift, we deduce that
ϕ
((
x
1
, x
2
)) =
{x}
for some
x
. Then by
minimality, it follows that
ϕ
is locally constant, hence constant, which is absurd.
We can continue on and then decide that ϕ
1
, ϕ
2
are homeomorphisms.
Theorem
(F. Ghys, 1984)
.
Two actions
h
1
and
h
2
are semiconjugate iff
h
∗
1
(
e
b
) =
h
∗
2
(e
b
).
Thus, in the case of minimal actions, the bounded Euler class is a complete
invariant of actions up to conjugacy.
Proof.
We shall only prove one direction, that if the bounded Euler classes agree,
then the actions are semiconjugate.
Let
h
1
, h
2
:
Γ
→ Homeo
+
(
S
1
). Recall that
c
(
f, g
)
∈ {
0
,
1
}
refers to the
(normalized) cocycle defining the bounded Euler class. Therefore
c
1
(γ, η) = c(h
1
(γ), h
1
(η))
c
2
(γ, η) = c(h
2
(γ), h
2
(η)).
are representative cocycles of h
∗
1
(e
b
), h
∗
2
(e
b
) ∈ H
2
b
(Γ, Z).
By the hypothesis, there exists u: Γ → Z bounded such that
c
2
(γ, η) = c
1
(γ, η) + u(γ) − u(γη) + u(η)
for all γ, η ∈ Γ.
Let
¯
Γ = Γ ×
c
1
Z be constructed with c
1
, with group law
(γ, n)(η, m) = (γη, c
1
(γ, η) + n + m)
We have a section
s
1
: Γ →
¯
Γ
γ 7→ (γ, 0).
We also write
δ
= (
e,
1)
∈
¯
Γ
, which generates the copy of
Z
in
¯
Γ
. Then we have
s
1
(γη)δ
c
1
(γ,η)
= s
1
(γ)s
2
(η).
Likewise, we can define a section by
s
2
(γ) = s
1
(γ)δ
u(γ)
.
Then we have
s
2
(γη) = s
1
(γη)δ
u(γη)
= δ
−c
1
(γ,η)
s
1
(γ)s
1
(η)δ
u(γη)
= δ
−c
1
(γ,η)
δ
−u(γ)
s
2
(γ)δ
−u(η)
s
2
(η)δ
u(γη)
= δ
−c
1
(γ,η)−u(γ)+u(γη)−u(η)
s
2
(γ)s
2
(η)
= δ
−c
2
(γ,η)
s
2
(γ)s
2
(η).
Now every element in
¯
Γ
can be uniuely written as a product
s
1
(
γ
)
δ
n
, and the
same holds for s
2
(γ)δ
m
.
Recall that for
f ∈ Homeo
+
(
S
1
), we write
¯
f
for the unique lift with
¯
f
(0)
∈
[0, 1). We define
Φ
i
(s
i
(γ)δ
n
) = h
i
(γ) · T
n
.
We claim that this is a homomorphism! We simply compute
Φ
i
(s
i
(γ)δ
n
s
i
(η)δ
m
) = Φ
i
(s
i
(γ)s
i
(η)δ
n+m
)
= Φ
i
(s
i
(γη)δ
c
i
(γ,η)+n+m
)
= h
i
(γη)T
c
i
(γ,η)
T
n+m
= h
i
(γ)h
i
(η)T
n+m
= h
i
(γ)T
n
h
i
(η)T
m
= Φ
i
(s
i
(γ)δ
n
)Φ
i
(s
i
(η)δ
m
).
So we get group homomorphisms Φ
i
:
¯
Γ → Homeo
+
Z
(R).
Claim. For any x ∈ R, the map
¯
Γ → R
g 7→ Φ
1
(g)
−1
Φ
2
(g)(x)
is bounded.
Proof. We define
v(g, x) = Φ
1
(g)
−1
Φ(g)x.
We notice that
v(gδ
m
, x) = Φ
1
(gδ
m
)
−1
Φ
2
(gδ
m
)(x)
= Φ
1
(g)
−1
T
−m
T
m
Φ
2
(g)
= v(g, x).
Also, for all g, the map x 7→ v(g, x) is in Homeo
+
Z
(R).
Hence it is sufficient to show that
γ 7→ v(s
2
(γ), 0)
is bounded. Indeed, we just have
v(s
2
(γ), 0) = Φ
1
(s
2
(γ)
−1
Φ
2
(s
2
(γ))(0)
= Φ
1
(s
1
(γ)δ
u(γ)
)
−1
Φ
2
(s
2
(γ))(0)
= δ
−u(γ)
h
1
(γ)
−1
h
2
(γ)(0)
= −u(γ) + h
1
(γ)
−1
(h
2
(γ)(0)).
But u is bounded, and also
h
1
(γ)
−1
(h
2
(γ)(0)) ∈ (−1, 1).
So we are done.
Finally, we can write down our two quasiconjugations. We define
˜ϕ(x) = sup
g∈
¯
Γ
v(g, x).
Then we verify that
˜ϕ(Φ
2
(h)x) = Φ
1
(h)(ϕ(x)).
Reducing everything modulo Z, we find that
ϕh
2
(γ) = h
1
(γ)ϕ.
The other direction is symmetric.
3.2 The real bounded Euler class
The next thing we do might be a bit unexpected. We are going to forget that
the cocycle
c
takes values in
Z
, and view it as an element in the real bounded
cohomology group.
Definition
(Real bounded Euler class)
.
The real bounded Euler class is the
class e
b
R
∈ H
2
b
(Homeo
+
(S
1
), R) obtained by change of coefficients from Z → R.
The real bounded Euler class of an action
h:
Γ
→ Homeo
+
(
S
1
) is the pullback
h
∗
(e
b
R
) ∈ H
2
b
(Γ, R).
A priori, this class contains less information that the original Euler class.
However, it turns out the real bounded Euler class can distinguish between
very different dynamical properties. Recall that we had the Gersten long exact
sequence
0 Hom(Γ, R/Z) H
2
b
(Γ, Z) → H
2
b
(Γ, R)
δ
.
By exactness, the real bounded Euler class vanishes if and only if
e
b
is in
the image of
δ
. But we can characterize the image of
δ
rather easily. Each
homomorphism
χ
: Γ
→ R/Z
gives an action by rotation, and in a previous
exercise, we saw the bounded Euler class of this action is
δ
(
χ
). So the image of
δ
is exactly the bounded Euler classes of actions by rotations. On the other hand,
we know that the bounded Euler class classifies the action up to semiconjugacy.
So we know that
Corollary.
An action
h
is semiconjugate to an action by rotations iff
h
∗
(
e
b
R
) = 0.
We want to use the real bounded Euler class to classify different kinds of
actions. Before we do that, we first classify actions without using the real
bounded Euler class, and then later see how this classification is related to the
real bounded Euler class.
Theorem.
Let
h:
Γ
→ Homeo
+
(
S
1
) be an action. Then one of the following
holds:
(i) There is a finite orbit, and all finite orbits have the same cardinality.
(ii) The action is minimal.
(iii)
There is a closed, minimal, invariant, infinite, proper subset
K ( S
1
such
that any x ∈ S
1
, the closure of the orbit h(Γ)x contains K.
We will provide a proof sketch. More details can be found in Hector–Hirsch’s
Introduction to the geometry of foliations.
Proof sketch.
By compactness and Zorn’s lemma, we can find a minimal, non
empty, closed, invariant subset
K ⊆ S
1
. Let
∂K
=
K \
˚
K
, and let
K
0
be the
set of all accumulation points of
K
(i.e. the set of all points
x
such that every
neighbourhood of
x
contains infinitely many poin