1Quasi-homomorphisms

IV Bounded Cohomology

1.2 Relation to commutators
A lot of interesting information about quasi-homomorphisms 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 quasi-homomorphisms, we can bound
the value of f by the defect:
Lemma. If f is a homogeneous quasi-homomorphism and x, y G, then
|f([x, y])| D(f ).
For non-homogeneous 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 quasi-homomorphism, 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 quasi-homomorphism is a homomorphism.
Note that if
cl
is bounded, then we have
scl
0. There exists interesting
groups with bounded
cl
, such as nilpotent finitely-generated 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 finitely-generated group and let
µ
be a symmetric probability measure on Γ whose support generates Γ. Then
every class in
QH
, R
)
/`
, R
) has a unique
µ
-harmonic representative. In
addition, this harmonic representative f satisfies the following:
kdfk
kdgk
for any g f + `
, R).
This is somewhat like the Hodge decomposition theorem.