CUP PRODUCT IN BOUNDED COHOMOLOGY OF THE FREE GROUP
NICOLAUS HEUER

arXiv:1710.03193v1 [] 9 Oct 2017

Abstract. The bounded cohomology Hbn (F, R) of a non-abelian free group F has uncountable
dimension for n = 2, 3 but it is unkown for n ≥ 4. The aim of this paper is to show that
the cup product between many known bounded 2-cocycles does not yield non-trivial classes
in degree 4. Those classes include the counting quasimorphisms of Brooks.

1. Introduction
The discrete bounded cohomology of groups is notoriously hard to compute. It is unknown if
the bounded cohomology Hbn (F, R) of a non-abelian free group F with real coefficients is trivial
or not for any n ≥ 4. One way of constructing non-trivial classes in degree 2 is to count certain
non-overlapping subwords. Indeed this was the first construction of infinitely many non-trivial 2cocycles of the free group in bounded cohomology by Brooks [Bro81]. The idea of defining classes
by counting subwords has been vastly generalised, see for example [EF97] for a generalisation
to hyperbolic groups. In this paper we propose a new generalisation of Brooks-cocycles using a
decomposition ∆; see Definition 3.1. These cocycles will be called ∆-decomposable cocycles.
The main corollary of this paper is the following:
Corollary 4.2. Let φ1 , φ2 ∈ Q(F ) be two quasimorphisms on the free group F where each φj is
either a Brooks counting quasimorphism on a non self-overlapping word or a quasimorphism in
the sense of Rolli. Then
[δ 1 φ1 ^ δ 1 φ2 ] = 0
as cocycles in Hb4 (F, R). Here, ^ denotes the cup product.
This will follow from the following theorem:
Theorem 4.1. Let ∆ be a decomposition of F , let φ be a ∆-decomposable quasimorphism and
let ω be a bounded 2-cocycle which is alternating and ∆-continuous. Then
[δ 1 φ ^ ω] = 0
as cocycles in Hb4 (F, R).
See Section 3 for the definitions involved in this statement. We will prove Theorem 4.1 by
providing an explicit bounded coboundary for the cup product; see Theorem 4.3.
Michelle Bucher and Nicolas Monod have proved the vanishing of the cup product between
Brooks-classes with a different technique. Both approaches provide an explicit bounded coboundary for the cup product, but whereas we use the bar-resolution, they use a resolution built from
“aligned chains”; see [BM17].
It was shown by Grigorchuk [Gri95] that Brooks quasimorphisms are dense in the vectorspace
of quasimorphisms in the topology of pointwise convergence. In light of Corollary 4.2 one would
like to deduce from this density that the cup product vanishes for all bounded 2-classes, but
one can not: The topology needed for such a deduction is the stronger defect topology. Brooks
Date: October 10, 2017.
1

cocycles are not dense in this topology, in fact the space of 2-cocycles is not even seperable in
this topology. We therefore ask:
Question 1.1. Does the cup product Hb2 (F, R)×Hb2 (F, R) → Hb4 (F, R) between arbitrary bounded
2-classes vanish?
We mention that the cup product on bounded cohomology for other groups need not be trivial.
For any group G with non-trivial second bounded cohomology it can be seen that the group G×G
admits a non-trivial cup product ^ : Hb2 (G × G, R) × Hb2 (G × G, R) → Hb4 (G × G, R), induced
by the factors. See [Löh15] for results and constructions in bounded cohomology using the cup
product.
This note is organised as follows: Section 2 collects basic results and notation on both the cup
product and bounded cohomology. Section 3 defines a crucial map of this note: the decomposition
∆. It will “decompose” elements g ∈ F into “pieces” such that ∆-decomposable quasimorphisms
“behave” like homomorphisms on these pieces. Section 4 will collect the main results of this note.
A key theorem will be Theorem 4.3, which asserts that for a ∆-decomposable quasimorphism φ
and an alternating ∆-continuous 2-cocycle ω, the function (g, h, i) 7→ φ(g)ω(h, i) is boundedly
close to an (unbounded) coboundary. In Section 5 we will give a proof of this theorem.
Acknowledgements. I would like to thank Roberto Frigerio, Clara Löh, Michelle Bucher and
Marco Moraschini for helpful discussions and detailed comments and my supervisor Martin Bridson for his comments and support. Further, I would like to thank the Isaac Newton Institute
for Mathematical Sciences, Cambridge, for support and hospitality during the programme NonPositive Curvature Group Actions and Cohomology where work on this paper was undertaken.
This work was supported by EPSRC grant no EP/K032208/1. The author is supported by the
Oxford-Cocker Scholarship.
2. Bounded Cohomology, Notations and Conventions
We briefly recall the definition of discrete (bounded) cohomology with trivial real coefficients.
For what follows we will work in the inhomogeneous resolution (or bar resolution) of bounded
cohomology. Let G be a group, C k (G, R) be the set of functions on k-tuples in G to R and
let Cbk (G, R) ⊂ C k (G, R) be subset of bounded functions. We define the coboundary operator
δ n : C n (G, R) → C n+1 (G, R) via
δ n (α)(g1 , . . . , gn+1 )=α(g2 , . . . , gn+1 )
n
X
+
(−1)j α(g1 , . . . , gj gj+1 , . . . , gn+1 )
j=1

+(−1)n+1 α(g1 , . . . , gn )
and note that it restricts to δ n : Cbn (G, R) → Cbn+1 (G, R). It is well known that δ ∗ is indeed
a coboundary operator. The cohomology of the cochain complex (C ∗ (G, R), δ ∗ ) is called the
cohomology of the group G with trivial real coefficients and denoted by H ∗ (G, R). Similarly we
call the cohomology of (Cb∗ (G, R), δ ∗ ) the bounded cohomology of G with real trivial coefficients
and denote it by Hb∗ (G, R). The inclusion Cbn (G, R) → C n (G, R) is a chain map which induces
a map cn : Hbn (G, R) → H n (G, R) called the comparison map. Cocycles in the kernel of cn
are called exact classes. It can be seen that exact 2-classes correspond to the coboundary of
quasimorphisms. A quasimorphism is a map α : G → R so that there is a constant C(α) such
that for every g, h ∈ G, |α(g) − α(gh) + α(h)| ≤ C(α). In other words, the coboundary ∂ 1 α is
an element of Cb2 (G, R).
For a detailed discussion see [Mon06] for a survey and [Fri16] for a book on bounded cohomology.
2

The cup product is a map ^ : H n (G, R) × H m (G, R) → H n+m (G, R) defined via ([ω1 ], [ω2 ]) 7→
[ω1 ] ^ [ω2 ] where [ω1 ] ^ [ω2 ] ∈ H n+m (G, R) is represented by the cocycle ω1 ^ ω2 ∈
C n+m (G, R) defined via
(g1 , . . . , gn , gn+1 , . . . , gn+m ) 7→ ω1 (g1 , . . . , gn ) · ω2 (gn+1 , . . . , gn+m ).
It is easy to check that this map preserves boundedness and that the cup product is also well
defined for bounded cohomology, i.e. ^ : Hbn (G, R) × Hbm (G, R) → Hbn+m (G, R). Note that
the cup product gives the (bounded) cohomology the structure of a graded ring. As mentioned
earlier it is notoriously difficult to find non-trivial elements in Hb4 (F, R), if they exist in the first
place. On the contrary there is a relative abundance of non-trivial classes in Hb2 (F, R) and it
may hence be tempting to construct non-trivial classes in Hb4 (F, R) using the cup product. In
short, this paper shows that this idea fails for many known classes.
A quasimorphism α : G → R will be called alternating if α satisfies α(g) = −α(g −1 ) for all
g ∈ G. In this case, we call its coboundary δ 1 α ∈ Cb2 (G, R) alternating as well. Note that
alternating is a property of cycles not of classes and it is easy to see that each exact 2-class is
represented by an alternating cocycle.
We collect two easy identities for alternating exact 2-cocycles.
Lemma 2.1. Suppose ω ∈ C 2 (F, R) is an alternating exact 2-cocycle. Then for all x, y ∈ G,
ω(x−1 , y) = −ω(x, x−1 y) and ω(x, y) = −ω(y −1 , x−1 ).
Proof. We calculate:
ω(x−1 , y) + ω(x, x−1 y) = f (x−1 ) + f (y) − f (x−1 y) + f (x) + f (x−1 y) − f (y) = 0
and similarly
ω(x, y) + ω(y −1 , x−1 ) = f (x) + f (y) − f (xy) + f (y −1 ) + f (x−1 ) − f (y −1 x−1 ) = 0

Remark 2.2. We will make the following notational conventions: F will always denote the nonabelian free group on some generating set S, where S consists of letters in code-font (“a, b“).
The identity element of F will be denoted by ”1“. Small Roman letters (”a, b“) typically denote
elements of F . Curly capitals (”A, B“), denote sets, typically subsets of F . Functions (typically
from F k to R) will be denoted by Greek letters (”α, β“). The set of all functions from F k to R
will be denoted by C k (F, R) and the subset of bounded functions by Cbk (F, R). Here ”bounded“
means bounded with respect to the supremum norm k · k∞ on C ∗ (F, R).
A set A ⊂ F will be called symmetric if a ∈ A implies that a−1 ∈ A. For such a symmetric set
A ⊂ F , we denote by A∗ the set of finite sequences in A including the empty sequence. This is,
the set of all expressions (a1 , . . . , ak ) where k ∈ N0 is arbitrary and aj ∈ A. We will denote the
element (a1 , . . . , ak ) ∈ A∗ by (a) and k will be called the length of (a) where we set k = 0 if (a) is
−1
∗
the empty sequence. For a sequence (a), (a−1 ) denotes the sequence (a−1
k , . . . , a1 ) ∈ A and the
element ā ∈ F denotes the product a1 · · · ak ∈ F . We will often work with multi-indexes: The
sequences (a1 ), (a2 ), (a3 ) ∈ A∗ will correspond to the sequences (aj ) = (aj,1 , . . . , aj,nj ), where nj
is the length of (aj ). As ”i“ will denote an element in F , ”j“ will be the generic index used.
Remark 2.3. In what follows we will often perform estimations. The statement “the term T1 is
uniformly close to the term T2 ” means that the terms T1 , T2 are functions to R which depend
on variables (usually elements g, h, i of the group F ) and that there is a constant C such that
|T1 − T2 | ≤ C independent of the explicit values of g, h, i in F . A term is uniformly bounded if
it is uniformly close to a constant function. For example, if c ∈ F is a group element which just
depends on the choice of g, h, i ∈ F , the statement “T1 (c) is boundedly close to T2 (c)” means
3

Figure 1. ∆(g), ∆(h) and ∆(h−1 g −1 ) have sides which can be matched up.
that there is a C > such that for all g, h, i ∈ F , |T1 (c(g, h, i)) − T2 (c(g, h, i))| < C, but we might
not make explicit how c depends on g, h, i.
3. Decomposition
We now define the main operator of this paper, the decomposition, denoted by ∆. For the
rest of the paper, we restrict out attention to a non-abelian free group F . See Example 3.3 for
the motivating examples of this definition.
Definition 3.1. Let P ⊂ F be a symmetric set of elements of F called pieces each represented
by a reduced word. Assume further that P does not contain the empty word. A decomposition
∆ of F into the pieces P is a map ∆ : F → P ∗ satisfying the following conditions:
(1) For every g ∈ F and ∆(g) = (g1 , . . . , gk ) we have g ≡ g1 · · · gk as a reduced word (no
cancelation). Also, we require that ∆(g −1 ) = (gk−1 , . . . , g1−1 ).
(2) For every g ∈ F with ∆(g) = (g1 , . . . , gk ) we have ∆(gi · · · gj ) = (gi , . . . , gj ) for 1 ≤ i ≤
j ≤ k. We refer to this property as ∆ being infix closed.
(3) There is a constant R > 0 such that for every g, h ∈ F there are sequences (c1 ), (c2 ), (c3 ) ∈
P ∗ of length n1 , n2 and n3 and sequences (r1 ), (r2 ), (r3 ) ∈ P ∗ of length m1 , m2 and m3
such that mj ≤ R and
−1
∆(g) = (c−1
1,n1 , . . . , c1,1 , r1,1 , . . . , r1,m1 , c2,1 , . . . , c2,n2 )
−1
∆(h) = (c−1
2,n2 , . . . , c2,1 , r2,1 , . . . , r2,m1 , c3,1 , . . . , c3,n3 )
−1
∆(h−1 g −1 ) = (c−1
3,n3 , . . . , c3,1 , r3,1 , . . . , r3,m3 , c1,1 , . . . , c1,n1 )

For such elements g, h ∈ F , we will call these sequences (cj ), (rj ) for j = 1, 2, 3 the ∆-triangle of
(g, h) or simply the triangle of (g, h) if there is no danger of ambiguity for ∆. See Figure 1. A
sequence (g1 , . . . , gk ) such that ∆(g1 · · · gk ) = (g1 , . . . , gk ) will be called a proper ∆ sequence.
Often we just talk about the decomposition without specifying the pieces P explicitly. From
a decomposition ∆ we derive the notion of two sorts of quasimorphisms: ∆-decomposable quasimorphisms and ∆-continuous quasimorphisms.
3.1. Decomposable Quasimorphisms. Each decomposition ∆ induces many different quasimorphisms on F and in fact most combinatorially known quasimorphisms are induced by some
∞
∆. Let λ ∈ lalt
(P) be a bounded, alternating function on the pieces P. Here, alternating means
that λ(g) = −λ(g −1 ). Define the map
φλ : g 7→

k
X
j=1

4

λ(gj )

for ∆(g) = (g1 , . . . , gk ).
∞
Proposition 3.2. Let ∆ be a decomposition into pieces P and let λ ∈ lalt
(P). Then the map
φλ : F → R is an alternating quasimorphism.
Moreover, for all g ∈ F with ∆(g) = (g1 , . . . , gk ) we have that δ 1 φλ (gj , gj+1 · · · gk ) = 0 for
j = 1, . . . , k − 1.

Proof. Alternating is immediate from the assumption on ∆(g −1 ) and λ. Let g, h ∈ F and let
(cj ), (rj ), j ∈ {1, 2, 3} be as in the definition of the decomposition. By definition,
φλ (g) = −
φλ (h) = −
φλ (gh) = −

n1
X
j=1
n2
X
j=1
n1
X

λ(c1,j ) +
λ(c2,j ) +
λ(c1,j ) −

j=1

m1
X
j=1
m2
X
j=1
m3
X

λ(r1,j ) +
λ(r2,j ) +
λ(r3,j ) +

j=1

n2
X
j=1
n3
X
j=1
n3
X

λ(c2,j )
λ(c3,j )
λ(c3,j )

j=1

and hence
δ 1 φ(g, h) = φ(g) + φ(h) − φ(gh) =

m1
X
j=1

λ(r1,j ) +

m2
X
j=1

λ(r2,j ) +

m3
X

λ(r3,j )

j=1

and hence |δ 1 φ(g, h)| ≤ 3Rkλk∞ . Note that from this calculation we also see that δ 1 φλ (g, h)
does only depend on the rj and not at all on the cj . The second part follows immediately from
property (2) of a decomposition.

We will call each quasimorphism φ which arises in such a way ∆-decomposable or simply a
decomposable quasimorphism when there is no danger of ambiguity for ∆. The property that
δ 1 (gj , gj+1 · · · gk ) = 0 in the last proposition emphasises that decomposable quasimorphisms
”behaves like homomorphisms on the decomposition” which will be crucial in what follows.
Example 3.3. The motivating examples for the above definition are the quasimorphisms both
in the sense of Rolli and Brooks. Suppose first that F is freely generated by the letters a and
b. Each element g ∈ F can be uniquely written as an1 bm1 · · · ank bmk where nj , mj ∈ Z and all
but possibly n1 , mk are non-zero. For such an element, set ∆(g) = (an1 , bm1 , . . . , ank , bmk ). It
is easy to check that the map ∆ : F → P ∗ defined like this is indeed a decomposition as above
∞
with the pieces P being powers of the generators a and b. Note that for any λ ∈ lalt
(P), the
quasimorphism φλ is exactly a quasimorphism defined in the sense of Rolli [Rol09].
Another example of a decomposition comes from the quasimorphisms of Brooks [Bro81]. Let
w = y1 · · · yn and v = x1 · · · xm be two reduced words where xi and yi denote letters of the words
i.e. generating elements or their inverses. We say that w is a sub-word of v if n ≤ m and there
is an s ∈ {0, . . . , m − n} such that yj = xj+s for all j = 1, . . . , n.
Let w be a reduced non self-overlapping word, i.e. a word w such that there are no words x
and y with x non-trivial such that w = xyx as a reduced word. Then, each element g ∈ F can
be written uniquely as g = u1 w1 u2 w2 · · · wk−1 uk where uj are (possibly empty) words in F
not containing w or w−1 as sub-words and j ∈ {−1, +1}. We may define a decomposition via
∆(g) = (u1 , w1 , u2 , w2 , . . . , wk−1 , uk ). The pieces P of this decomposition are the elements in
F which correspond to reduced words which do not contain w and w−1 as sub-words and w, w−1
5

Figure 2. The triangle g, h, h−1 g −1 vs. the triangle g 0 , h0 , h0−1 g 0−1
∞
itself. Define λ ∈ lalt
(P) via



g=w
1,
λ(g) = −1, g = w−1


0,
else
Then it is easy to see that φλ is the Brooks-quasimorphism on the word w.
3.2. ∆-Continuous Quasimorphisms and Cocycles. Another and much more general type
of quasimorphisms (or cocycles) which will be crucial are ∆-continuous cocycles.
Definition 3.4. Fix a decomposition ∆. Let g, g 0 , h, h0 ∈ F be elements and let (cj ), (rj ) for
j = 1, 2, 3 be the ∆-triangle for g, h and similarly let (c0j ), (r0j ) for j = 1, 2, 3 be the ∆-triangle
for g 0 , h0 . If there is an j ∈ {1, 2, 3} such that (rj ) 6= (r0j ) set N (g, g 0 , h, h0 ) = 0. Else, let
N (g, g 0 , h, h0 ) be the smallest integer N such that cj,N 6= c0j,N for one j ∈ {1, 2, 3}.
A bounded cocycle ω is said to be ∆-continuous
if there is a constant S > 0 and a nonP∞
0
0
negative summable sequence (sj )j∈N with
j=1 sj ≤ S such that for all g, g , h, h ∈ F and
0
0
N = N (g, g , h, h ) defined as above,
|ω(g, h) − ω(g 0 , h0 )| ≤ sN
See Figure 2.
Crudely, a cocycle ω is ∆-continuous if its values depend mostly on the parts of the decomposition which lies close to the centre of the triangle g, h, h−1 g −1 . Note that this assumption is
natural if one thinks of real (bounded) cohomology as (hyperbolic) volume. Indeed, the area of
a geodesic triangle in hyperbolic space does mostly depend on the behaviour at the centre and
less on the exact position of the vertices of a triangle.
Example 3.5. Many cocycles are ∆-continuous. For example every quasimorphism φ which is
∆-decomposable is ∆-continuous. Indeed, the proof of Proposition 3.2 shows that δ 1 φ(g, h) does
not at all depend on the cj ’s, hence if N (g, g 0 , h, h0 ) ≥ 1, then δ 1 φ(g, h) = δ 1 φ(g 0 , h0 ).
On the other hand, the cocycles of Brooks only depend on a finite neighbourhood of the
midpoint of triangles. Hence it is easy to see that if N is sufficiently large then δ 1 φ(g, h) =
δ 1 φ(g 0 , h0 ) for φ a Brooks quasimorphism as we assumed that the pieces of the decomposition
are non-empty. Hence Brooks cocycles are ∆-continuous for any decomposition ∆.
We collect two easy properties of ∆-continuous cocycles here, emphasising that the value of
ω(g, h) just depends on the behaviour around the centre of the geodesic triangle with endpoints
1, g, gh in the Cayley graph.
6

Proposition 3.6. Let ∆ be a decomposition, let g, h ∈ F with triangle (cj ), (rj ), for j = 1, 2, 3
where (cj ) has length nj and (rj ) has length mj ≤ R. Further, let ω be a ∆-continuous cocycle.
Then:
(1)
n1
X
−1
−1
−1
|ω(c−1
1,j · · · c1,1 r̄1 c̄2 , h) − ω(c1,j−1 · · · c1,1 r̄1 c̄2 , h)|
j=2

is uniformly bounded.
(2)
n1
X

−1 −1
−1
−1
−1 −1
−1
|ω(c−1
1,n1 · · · c1,j , c1,j−1 · · · c1,1 r̄1 c̄2 ) − ω(c1,n1 · · · c1,j , c1,j−1 · · · c1,1 )|

j=2

is uniformly bounded.
(3) Let C ∈ N, c ∈ F and suppose that ∆(c) = (c1 , . . . , cl ) where l ≥ 2C. Then
l−C−1
X

|ω(c1 · · · cj , cj+1 · · · ck ) − ω(cC · · · cj , cj+1 · · · ck−C )|

j=C+1

is uniformly bounded.
Proof. We first show (1). Note that:
−1
−1
−1
N (c−1
1,j · · · c1,1 r̄1 c̄2 , c1,j−1 · · · c1,1 r̄1 c̄2 , h, h) = j − 1

where N is as in Definition 3.4. To see this note that since ∆ is infix closed, the triangle of
−1
−1
−1
c−1
1,j−1 · · · c1,1 r̄1 c̄2 and h shares the same sides as the triangle of c1,j · · · c1,1 r̄1 c̄2 and h.
Hence, in the notion of Definition 3.4,
−1
−1
−1
|ω(c−1
1,j · · · c1,1 r̄1 c̄2 , h) − ω(c1,j−1 · · · c1,1 r̄1 c̄2 , h)| ≤ sj−1

and so
n1
X
j=2

−1
−1
|ω(c1,j −1 · · · c−1
1,1 r̄1 c̄2 , i) − ω(c1,j−1 · · · c1,1 r̄1 c̄2 , i)| ≤

nX
1 −1

sj ≤ S

j=1

where S is a constant as in Definition 3.4, which does not depend on the choices of g, h, i made
above which justifies ”uniform boundedness“ as in Remark 2.3.
To see the second estimation, note that, as ∆ is infix closed and hence
−1 −1
−1 −1
−1
−1
−1
N (c−1
1,n1 · · · c1,j , c1,n1 · · · c1,j , c1,j−1 · · · c1,1 r̄1 c̄2 , c1,j−1 · · · c1,1 ) = j − 1

and hence
n1
n1
X
X
−1 −1
−1
−1
−1 −1
−1
|ω(c−1
·
·
·
c
,
c
·
·
·
c
r̄
c̄
)
−
|ω(c
·
·
·
c
,
c
·
·
·
c
)|
≤
sj−1 < S
1,n1
1,1 1 2
1,n1
1,1
1,j 1,j−1
1,j 1,j−1
j=2

j=2

and hence the expression is also uniformly bounded. The proof of the last expression is just
analogous.

3.3. Triangles and Quadrangles in a tree. Let g, h ∈ F . It is easy to see that there are
−1
unique elements t1 , t2 and c such that g = t−1
t2 as reduced words and that t1 , t2
1 c and h = c
and c are the sides of the triangle with endpoints 1, g, gh in the Cayley graph of F . We will call
c the common 2-path of (g, h) and denote it by c(g, h) := c.
For three elements g, h, i ∈ F there are three different cases how the geodesics of the points
1, g, gh, ghi in the Cayley graph of F are aligned. See Figure 3.
7

(a)

(b)

(c)

Figure 3. Different cases how g, h and i are aligned
−1
(1) (Figure 3a): There are elements t1 , . . . , t5 such that g = t1 t2 , h = t−1
2 t3 t4 , i = t4 t5 as
reduced words.
−1 −1
(2) (Figure 3b): There are elements t1 , . . . , t5 such that g = t1 t2 t3 , h = t−1
3 t4 , i = t4 t2 t5
as reduced words.
−1 −1
(3) (Figure 3c): There are elements t1 , . . . , t4 and c such that g = t−1
t3 ,
1 ct2 , h = t2 c
−1
i = t3 ct4 as reduced words.
In the first two cases we say that the common-3-path of (g, h, i) is empty and write c(g, h, i) = 1
and in the third case we set c(g, h, i) = c.

3.4. Close Sequences. To ease notation later, we will need the notion of closeness of sequences.
Definition 3.7. Let C ∈ N. Two sequences (g1 , · · · , gk ) and (h1 , · · · , hl ) are C-close, if there is
a D ∈ Z for |D| ≤ C such that gj = hj+D for all j where this equality makes sense apart from at
most C terms on the beginning and C terms at the end of the sequences. If the two sequences are
a function of other variables and there is a constant C such that both are C-close independent of
the variables we say that the two sequences are uniformly close. A sequence which has uniform
bounded length will be called uniformly bounded.
Lemma 3.8. Let g, h ∈ F and let c = c(g, h) be the common 2-piece of (g, h) with g = t−1
1 c and
h = c−1 t2 as reduced words. Let further (cj ), (rj ) for j = 1, 2, 3 be the triangle for g, h. Then
• ∆(t1 ) is uniformly close to the sequence (c1 )
• ∆(c) is uniformly close to the sequence (c2 )
• ∆(t2 ) is uniformly close to the sequence (c3 )
Proof. This follows immediately from the definitions. For example to see the first two statements
compare the side ∆(g) in the triangle g, h to the side in the triangle t−1

1 ,c
Lemma 3.9. Let (g1 , . . . , gk ) and (h1 , . . . , hl ) be proper ∆-sequences that are C-close. Let D be
as in the definition of ”close“ sequences. Further let ω be a continuous cocycle with respect to
∆. Then
k−1−C
X
|ω(g1 · · · gj , gj+1 · · · gk ) − ω(h1 · · · hj+D , hj+D+1 · · · hl )|
j=C

can be bounded as a function of C and S for S as in the definition of a ∆-continuous cocycle.
Proof. Analogous to the proof of part (3) of Proposition 3.6.
8



4. Vanishing Theorems
Here, we prove the main Theorem 4.1. It will follow from the Technical Theorem 4.3 which we
also state in the section. However, for the sake of readability, we postpone the proof of Theorem
4.3 to the next section.
Theorem 4.1. Let ∆ be a decomposition of F , let φ be a ∆-decomposable quasimorphism and
let ω be a bounded 2-cocycle which is alternating and ∆-continuous. Then
[δ 1 φ ^ ω] = 0
as cocycles in Hb4 (F, R).
Proof. According to Theorem 4.3 there are functions η, γ ∈ C 2 (F, R) such that β(g, h, i) =
φ(g)ω(h, i) − δ 2 γ(g, h, i) + δ 2 η(g, h, i) is bounded, i.e. β ∈ Cb3 (F, R). Moreover, we see that
δ 3 β(g, h, i, j) = δ 1 φ(g, h)ω(i, j) − δ 3 δ 2 γ(g, h, i, j) + δ 3 δ 2 η(g, h, i, j)
= (δ 1 φ ^ ω)(g, h, i, j)
and hence β is an explicit bounded coboundary of δ 1 φ ^ ω and hence trivial in Hb4 (F, R).



Corollary 4.2. Let φ1 , φ2 ∈ Q(F ) be two quasimorphisms on the free group F where each φi is
either a Brooks counting quasimorphism on a non self-overlapping word or a quasimorphism in
the sense of Rolli. Then
[δ 1 φ1 ^ δ 1 φ2 ] = 0
as cocycles in Hb4 (F, R).
Proof. Suppose that at least one of φ1 , φ2 is a quasimorphism in the sense of Rolli, say without
loss of generality φ1 . By Example 3.3, there is a decomposition ∆ such that φ1 is ∆-decomposable.
By Example 3.5 any quasimorphism φ2 which is either a quasimorphism in the sense of Rolli or
a Brooks-quasimorphism on a non-self overlapping word is ∆-continuous.
Similarly, if both φ1 , φ2 are Brooks quasimorphisms on a non self-overlapping word we may use
the decomposition ∆ as in Example 3.3 for φ1 and see by Example 3.5 that δ 1 φ2 is ∆-continuous.
In both cases we see that (after possibly changing the indexes) there is a decomposition ∆
such that φ1 is ∆-decomposable and δ 1 φ2 is alternating and ∆-continuous. By Theorem 4.1,
[δ 1 φ1 ^ δ 1 φ2 ] is trivial in Hb4 (F, R).

To state the Technical Theorem 4.3, we have to define auxiliary functions η, γ ∈ C 2 (F, R).
Define η ∈ C 2 (F, R) via
k−1
X
η(g, h) :=
φ(gj )ω(gj+1 · · · gk , h)
j=1

for ∆(g) = (g1 , . . . , gk ). Further, define γ ∈ C 2 (F, R) via
l−1

γ(g, h) := −


1X
φ(cj ) ω(c1 · · · cj−1 , cj · · · cl ) − ω(c1 · · · cj , cj+1 · · · cl )
2 j=2

where c = c(g, h) the common 2-path defined as in Section 3.3 and ∆(c) = (c1 , . . . , cl ).
Theorem 4.3. (Technical Theorem) Let φ be ∆-decomposable and let ω be an alternating, ∆continuous cocycle. Then δ 2 γ(g, h, i) − δ 2 η(g, h, i) is uniformly close to φ(g)ω(h, i).
The next section will be dedicated to prove Theorem 4.3.
9

5. Proof of Theorem 4.3
Using the decomposition ∆ and the cocycle condition, we will first rewrite the term φ(g)ω(h, i)
as a sum of terms φ(gj )ω(·, ·) where gj is a piece of ∆ i.e. a sum of terms where φ only evaluates
pieces of the decomposition ∆. For this, define ζ ∈ C 3 (F, R) via
ζ(g, g 0 , h) :=

k−1
X

φ(gj )ω(gj+1 · · · gk g 0 , h)

j=1

for ∆(g) = (g1 , . . . , gk ). Note that by definition η(g, h) ≡ ζ(g, 1, h).
Proposition 5.1. The function φ(g)ω(h, i) + δ 2 η(g, h, i) is uniformly close to
ζ(g, h, i) + ζ(h, 1, i) − ζ(gh, 1, i)
for η ∈ C 2 (F, R), ζ ∈ C 3 (F, R) are as above.
Proof. Let ∆(g) = (g1 , . . . , gk ). We know that δ 1 φ(gj , gj+1 · · · gk )ω(h, i) = 0, see Proposition
3.2. On the other hand we know that
0=δ 1 φ(gj , gj+1 · · · gk )ω(h, i)
=φ(gj+1 · · · gk )ω(h, i) − φ(gj · · · gk )ω(h, i) + φ(gj )ω(h, i)
=φ(gj+1 · · · gk )ω(h, i) − φ(gj · · · gk )ω(h, i) + φ(gj ) ω(gj+1 · · · gk h, i)

−ω(gj+1 · · · gk , hi) + ω(gj+1 · · · gk , h)
where we used that ω(h, i) = ω(gj+1 · · · gk h, i) − ω(gj+1 · · · gk , hi) + ω(gj+1 · · · gk , h) as ω is a
2-cocycle. Hence, we see that
φ(gj · · · gk )ω(h, i) − φ(gj+1 · · · gk )ω(h, i)= φ(gj )ω(gj+1 · · · gk h, i) − φ(gj )ω(gj+1 · · · gk , hi)
+φ(gj )ω(gj+1 · · · gk , h)
summing over both sides for j = 1, . . . , k − 1 yields
φ(g)ω(h, i) − φ(gk )ω(h, i) = ζ(g, h, i) − ζ(g, 1, hi) + ζ(g, 1, h).
and hence
φ(g)ω(h, i) − φ(gk )ω(h, i) + δ 2 η(g, h, i) = ζ(g, h, i) + ζ(h, 1, i) − ζ(gh, 1, i)
which finishes the proof of Proposition 5.1 as φ(gk )ω(h, i) is uniformly bounded. In the last step
we used that
δ 2 η(g, h, i) = η(h, i) − η(gh, i) + η(g, hi) − η(g, h)
= ζ(h, 1, i) − ζ(gh, 1, i) + ζ(g, 1, hi) − ζ(g, 1, h).

Note what we have achieved! Using the decomposition ∆ we managed to express the quasicocycle φ(g)ω(h, i) in terms of a sum over pieces.
Proposition 5.2. The function φ(g)ω(h, i) + δ 2 η(g, h, i) is uniformly close to
l−1
X

φ(ci ) (ω(c1 · · · cj−1 , cj · · · cl ) − ω(c1 · · · cj , cj+1 · · · cl ))

j=2

where c = c(g, h, i) the common 3-piece of (g, h, i) and ∆(c) = (c1 , . . . , cl ).
Proof. We compare the ζ terms of φ(g)ω(h, i) + δ 2 η(g, h, i) from Proposition 5.1. For this, let
g, h ∈ F and let (cj ), (rj ) for j = 1, 2, 3 be the triangle for g, h. Recall that c̄j and r̄j denote the
product of the elements of the corresponding sequences. We claim that:
10

• ζ(g, h, i) is uniformly close to
nX
1 −1

−1
−1
φ(c−1
1,j )ω(c1,j−1 · · · c1,1 r̄1 c̄2 h, i) +

nX
2 −1

j=2

φ(c2,j )ω(c2,j+1 · · · c2,n2 h, i).

j=2

To see this, note that by the definition of the decomposition ∆
−1
∆(g) = (c−1
1,n1 , . . . , c1,1 , r1,1 , . . . , r1,m1 , c2,1 , . . . , c2,n2 )

and hence
−1
−1
ζ(g, h, i)=φ(c−1
1,n1 )ω(c1,n1 −1 · · · c1,1 r̄1 c̄2 h, i)

+

nX
1 −1

−1
−1
−1
φ(c−1
1,j )ω(c1,j−1 · · · c1,1 r̄1 c̄2 h, i) + φ(c1,1 )ω(r̄1 c̄2 h, i)

j=2

+

m
1 −1
X

φ(r1,j )ω(r1,j+1 · · · r1,m1 c̄2 h, i) + φ(r1,m1 )ω(c̄2 h, i)

j=1

+φ(c2,1 )ω(c2,2 · · · c2,n2 h, i) +

nX
2 −1

φ(c2,j )ω(c2,j+1 · · · c2,n2 h, i)

j=2

Pn1 −1
Pn2 −1
and note that all terms but the terms of j=2
and j=2
can be uniformly bounded.
To explain what this means we recall our convention of Remark 2.3. For example, the
term
m
1 −1
X
φ(r1,j )ω(r1,j+1 · · · r1,m1 c̄2 h, i)
j=1

above is implicitly a function of g, h, i ∈ F , as the sequences (r1 ) and (c2 ) are fully
determined by g and h. We know that there is a constant R > 0 such that m1 (also an
implicit function of g, h, i ∈ F !) is bounded by R. Hence,
|

m
1 −1
X

φ(r1,j )ω(r1,j+1 · · · r1,m1 c̄2 h, i)| ≤ R · supp∈P |φ(p)| · kωk∞

j=1

independent of the exact values of g, h, i ∈ F . Here, P denotes the pieces of the decomposition ∆.
With the same sort of arguments we see:
• ζ(h, 1, i) is uniformly close to
nX
2 −1

−1
φ(c−1
2,j )ω(c2,j−1

· · · c−1
2,1 r̄2 c̄3 , i)

+

nX
3 −1

j=2

φ(c3,j )ω(c3,j+1 · · · c3,n3 , i)

j=2

• ζ(gh, 1, i) is uniformly close to
nX
1 −1

−1
−1 −1
φ(c−1
1,j )ω(c1,j−1 · · · c1,1 r̄3 c̄3 , i) +

j=2

nX
3 −1

φ(c3,j )ω(c3,j+1 · · · c3,n3 , i)

j=2

Note that r̄1 c̄2 h = r̄1 r̄2 c3 = r̄3−1 c̄3 where we used that h = c̄−1
2 r̄2 c̄3 and r̄1 r̄2 r̄3 = 1. Hence,
comparing the summands above we see that ζ(g, h, i) + ζ(h, 1, i) − ζ(gh, 1, i) is uniformly close
to
nX
2 −1
−1
−1
φ(c2,j )ω(c2,j+1 · · · c2,n2 h, i) + φ(c−1
2,j )ω(c2,j−1 · · · c2,1 r̄2 c̄3 , i)
j=2

11

−1
using that h = c̄−1
) = −φ(x) we obtain that ζ(g, h, i) + ζ(h, 1, i) − ζ(gh, 1, i)
2 r̄2 c̄3 and that φ(x
is uniformly close to
nX
2 −1


−1
−1
−1
φ(c2,j ) ω(c−1
2,j · · · c2,1 r̄2 c̄3 , i) − ω(c2,j−1 · · · c2,1 r̄2 c̄3 , i) .

j=2

Next we consider how the triangle g, h is aligned with respect to the triangle h, i to be able to use
that ω is ∆-continuous. Consider Figure 3 for the different cases of alignments of triangles. Let
(c0j ), (r0j ) j = 1, 2, 3 be the triangle h, i. This yields two different ways of writing the elements
of ∆(h):
∆(h) = (c2,n2 −1 , . . . , c2,1 −1 , r2,1 , . . . , r2,m2 , c3,1 , . . . c3,n3 )
0
0
0
0
= (c01,n01 −1 , . . . , c01,1 −1 , r1,1
, . . . , r1,m
0 , c2,1 , . . . c2,n0 )
1
2

which we will exploit in what follows.
Using Proposition 3.6 (1) we note that
0

(*)

n1
X

|ω(c01,j −1 · · · c01,1 −1 r¯0 1 c¯0 2 , i) − ω(c01,j−1 −1 · · · c01,1 −1 r¯0 1 c¯0 2 , i)|

j=1

is uniformly bounded.
Finally, we can prove Proposition 5.2. We will distinguish between different cases depending
on the length of (c2 ), (c01 ) and (r01 ), i.e. n2 , n01 and m01 . To avoid confusion with the triangles
(cj ) of (g, h), the common 3-path of (g, h, i) will be denoted by c̃ and not by c.
(1) n2 ≤ n01 : In this case the length the sequence ∆(c̃), with c̃ = c(g, h, i) is uniformly
bounded. To see this, suppose that c̃ 6= 1. This means that there are elements t1 , . . . , t4 ∈
−1 −1
F such that g = t−1
t3 and i = t−1
1 c̃t2 , h = t2 c̃
3 c̃t4 . By Lemma 3.8 (c2 ) is uniformly
0
close to ∆(c̃t2 ) and (c1 ) is uniformly close to ∆(t2 ). Finally, comparing the c̃, t2 -triangle
we see that the length of ∆(c̃) has to be uniformly bounded. We therefore have to show
that ζ(g, h, i) + ζ(h, 1, i) − ζ(gh, 1, i) is uniformly bounded. Using that ∆(h) arises as a
side both in the g, h and in the h, i-triangle we see that
c2,n2 = c01,n0 ; . . . ;
1

c2,1 = c01,n0 −n2 +1
1

Hence, ζ(g, h, i) + ζ(h, 1, i) − ζ(gh, 1, i) is uniformly close to
0

n1
X

φ(c01,j ) ω(c01,n01 −1 · · · r̄10 c̄02 , i) − ω(c01,n01 −1 · · · r̄10 c̄02 , i)



j=n01 −n2 +2

Using (∗) we see that in this case, ζ(g, h, i) + ζ(h, 1, i) − ζ(gh, 1, i) is uniformly bounded.
(2) n01 < n2 ≤ n01 + m01 : Analogous to (1), we see that ζ(g, h, i) + ζ(h, 1, i) − ζ(gh, 1, i) is
uniformly bounded by adding at most R more bounded terms. Similarly, the length of
∆(c) can be uniformly bounded and hence the same argument as in (1) applies.
(3) n01 + m01 < n2 : Realising ∆(h) as a side of both the (h, i) and the (g, h) triangle as above
we see that
0
c2,n2 = c01,n0 ;
...;
c2,n2 −n01 +1 = c01,1 ;
...;
c2,n2 −n01 = r1,1
;
1
0
−1
0
c2,n2 −n01 −m01 +1 = r1,m01 ; . . . ; c2,n2 −n01 −m01 = c2,1 ; . . . ; c2,1 = c2,n2 −n0 −m0 −1
1
1
12

using (∗) for the c01,j -terms once more and using that there is just a bounded number of
0
r1,j
-terms, we see that ζ(g, h, i) + ζ(h, 1, i) − ζ(gh, 1, i) is uniformly close to
n2 −n01 −m01 −1

X



φ(c02,j −1 ) ω(c02,j · · · c02,n02 , i) − ω(c02,j+1 · · · c02,n02 , i) .

j=2

Since ω is alternating we have the identity ω(x, x−1 y) = −ω(x−1 , y) for all x, y ∈ F ; see
Lemma 2.1. Substituting x = c02,j · · · c02,n0 and x−1 y = i = c02,n0 −1 · · · c02,1 −1 r¯0 2 c¯0 3 and
2
2
using that φ(c02,j −1 ) = −φ(c02,j ) we see that ζ(g, h, i) + ζ(h, 1, i) − ζ(gh, 1, i) is uniformly
close to
n2 −n01 −m01 −1

X

φ(c02,j ) ω(c02,n02 −1 · · · c02,j −1 , c02,j−1 −1 · · · c02,1 −1 r¯0 2 c¯0 3 ) · · ·

j=2


−ω(c02,n02 −1 · · · c02,j+1 −1 , c02,j −1 · · · c02,1 −1 r¯0 2 c¯0 3 )
Using Proposition 3.6 (2) we see that ζ(g, h, i) + ζ(h, 1, i) − ζ(gh, 1, i) is uniformly
close to
n2 −n01 −m01 −1

X

φ(c02,j ) ω(c02,n2 −n01 −m01 −1 · · · c02,j −1 , c02,j−1 −1 · · · c02,1 −1 ) · · ·

j=2

−ω(c02,n2 −n01 −m01 −1 · · · c02,j+1 −1 , c02,j −1 · · · c02,1 −1 )



Using Claim 5.3 the above expression is indeed uniformly close to
l−1
X

φ(c̃i ) (ω(c̃1 · · · c̃j−1 , c̃j · · · c̃l ) − ω(c̃1 · · · c̃j , c̃j+1 · · · c̃l ))

j=2

where c̃ = c(g, h, i), the common 3-path of (g, h, i). This concludes Proposition 5.2.

We are left to prove the claim:
Claim 5.3. Let c̃ = c(g, h, i) be the common 3-piece of g, h, i and let (c02 ), n2 , n01 and m01 be as
above in the part ”n01 + m01 < n2 ” of the proof of Proposition 5.2. Then the sequences
(c02,n2 −n01 −m01 −1 , . . . , c02,1 −1 ) and ∆(c̃) = (c̃1 , . . . , c̃l )
are uniformly close as in Definition 3.7.
Proof. We will check the different alignments of the triple (g, h, i) as in Figure 3.
• Alignment of Figure 3a: There are elements t1 , . . . , t5 such that g = t1 t2 , h = t−1
2 t3 t4
and i = t−1
t
as
reduced
words.
As
c(g,
h,
i)
is
trivial
in
this
case,
we
have
to
show
5
4
that n2 − n01 − m01 is uniformly bounded. In light of Lemma 3.8 we see that (c2 ) is
uniformly close to ∆(t2 ) and that (c01 ) is uniformly close to ∆(t−1
3 t2 ). By considering
the t3−1 , t2 -triangle, we see that the sequence ∆(t−1
t
)
ends
with
a sequence which is
2
3
uniformly close to ∆(t2 ). Hence, n2 − n01 is uniformly bounded and hence n2 − n01 − m01
is uniform bounded as m01 ≤ R.
• Alignment of Figure 3b: Analogous to the previous case (Figure 3a).
13

• Alignment of Figure 3c: There are elements t1 , t2 , t3 , t3 and c̃ such that g = t−1
1 c̃t2 ,
−1
−1
h = t−1
c
t
and
i
=
t
c̃t
.
Using
Lemma
3.8
for
the
(g,
h)
triangle,
we
see
that
∆(c̃t
3
4
2)
2
3
is uniformly close to (c2 ). Applying the same lemma to the (h, i) triangle, we see that
∆(t2 ) = (c01 ). Using the triangle condition of a decomposition for the (c̃, t2 )-triangle,
we see that ∆(c) is uniformly close to (c2,1 , . . . , c2,n2 −n01 ). Comparing once more the
different ways h occurs in the (g, h) and the (h, i) triangle yields the claim.

Lemma 5.4. Suppose θ : F → R is an arbitrary alternating quasimorphism i.e. not necessarily
decomposable. Define κ ∈ C 2 (F, R) by κ(g, h) = θ(c) where c = c(g, h), the common 2-piece.
Then δ 2 κ(g, h, i) is uniformly close to −2θ(c̃) where c̃ = c(g, h, i) is the common 3-path of (g, h, i).
Proof. We have to evaluate
δ 2 κ(g, h, i) = κ(h, i) − κ(gh, i) + κ(g, hi) − κ(g, h)
For what follows we will use the different cases of how g, h and i can be aligned in the Cayley
graph of F as seen in Figure 3.
(1) (see Figure 3a): In this case there are elements t1 , . . . , t5 such that g = t1 t2 , h = t−1
2 t3 t4 ,
i = t−1
4 t5 as reduced words. It follows that c(h, i) = t4 , c(gh, i) = t4 , c(g, hi) = t2 and
c(g, h) = t2 . Hence δ 2 κ(g, h, i) = θ(t4 ) − θ(t4 ) + θ(t2 ) − θ(t2 ) = 0. Note that in this case
c(g, h, i) = 1.
(2) (see Figure 3b): In this case there are elements t1 , . . . , t5 such that g = t1 t2 t3 , h = t−1
3 t4 ,
−1
i = t−1
t
t
as
reduced
words.
It
follows
that
c(h,
i)
=
t
,
c(gh,
i)
=
t
t
,
c(g,
hi)
=
t2 t3
5
4
4
2
4 2
and c(g, h) = t3 . Hence δ 2 κ(g, h, i) = θ(t4 ) − θ(t4 t2 ) + θ(t2 t3 ) − θ(t3 ) which is uniformly
bounded. Note that in this case c(g, h, i) = 1.
(3) (see Figure 3c): In this case there are elements t1 , . . . , t4 and c̃ such that g = t−1
1 c̃t2 ,
−1
−1
−1
h = t−1
c̃
t
,
i
=
t
c̃t
as
reduced
words.
It
follows
that
c(h,
i)
=
c̃
t
,
c(gh,
i)
= t3 ,
3
4
3
2
3
c(g, hi) = t2 and c(g, h) = c̃t2 . Hence δ 2 κ(g, h, i) = θ(c̃−1 t3 ) − θ(t3 ) + θ(t2 ) − θ(c̃t2 ) which
is uniformly close to −2θ(c̃) and of course in this case c(g, h, i) = c̃. This shows Lemma
5.4.

Lemma 5.5. The map θ : F → R defined via
l−1

θ(g) = −

1X
φ(gj ) (ω(g1 · · · gj−1 , gj · · · gl ) − ω(g1 · · · gj , gj+1 · · · gl ))
2 j=2

for ∆(g) = (g1 , . . . , gl ) is a quasimorphism.
Note that the map θ is not a ∆-decomposable quasimorphism as its value not only depend on
the individual pieces but also on how they are aligned.
Proof. Let g, h ∈ F and let (cj ), (rj ), for j = 1, 2, 3 be the corresponding sequences of the
decomposition. Then by once more dropping boundedly many terms we see that θ(g) is uniformly
close to
n1 −1

1 X
−1
−1
−1
−1
−1
−1 −1
−1
−
φ(c−1
1,j ) ω(c1,n1 · · · c1,j+1 , c1,j · · · c1,1 r̄1 c̄2 ) − ω(c1,n1 · · · c1,j , c1,j−1 · · · c1,1 r̄1 c̄2 )
2 j=2
n2 −1
1 X
−
φ(c2,j ) (ω(c̄1 r̄1 c2,1 · · · c2,j−1 , c2,j · · · c2,n2 ) − ω(c̄1 r̄1 c2,1 · · · c2,j , c2,j+1 · · · c2,n2 ))
2 j=2

14

Again, we use continuity to simplify the above expression. By applying Proposition 3.6 (2) we
see that θ(g) is uniformly close to
−

n1 −1

1 X
−1
−1
−1
−1
−1
−1 −1
−1
φ(c−1
1,j ) ω(c1,n1 · · · c1,j+1 , c1,j · · · c1,1 ) − ω(c1,n1 · · · c1,j , c1,j−1 · · · c1,1 )
2 j=2

−

n2 −1
1 X
φ(c2,j ) (ω(c2,1 · · · c2,j−1 , c2,j · · · c2,n2 ) − ω(c2,1 · · · c2,j , c2,j+1 · · · c2,n2 ))
2 j=2

Since ω is alternating we have that for all x, y ∈ F , ω(x, y) = −ω(y −1 , x−1 ) and hence θ(g) is
indeed uniformly close to
+

n1 −1
1 X
φ(c1,j ) (ω(c1,1 · · · c1,j , c1,j+1 · · · c1,n1 ) − ω(c1,1 · · · c1,j−1 , c1,j · · · c1,n1 ))
2 j=2

−

n2 −1
1 X
φ(c2,j ) (ω(c2,1 · · · c2,j−1 , c2,j · · · c2,n2 ) − ω(c2,1 · · · c2,j , c2,j+1 · · · c2,n2 ))
2 j=2

With the same line of reasoning for the other sides of the (g, h)-triangle, we see that θ(h) is
uniformly close to
+

n2 −1
1 X
φ(c2,j ) (ω(c2,1 · · · c2,j , c2,j+1 · · · c2,n2 ) − ω(c2,1 · · · c2,j−1 , c2,j · · · c2,n2 ))
2 j=2

−

n3 −1
1 X
φ(c3,j ) (ω(c3,1 · · · c3,j−1 , c3,j · · · c3,n3 ) − ω(c3,1 · · · c3,j , c3,j+1 · · · c3,n3 ))
2 j=2

and that θ(gh) is uniformly close to
+

n1 −1
1 X
φ(c1,j ) (ω(c1,1 · · · c1,j , c1,j+1 · · · c1,n1 ) − ω(c1,1 · · · c1,j−1 , c1,j · · · c1,n1 ))
2 j=2

−

n3 −1
1 X
φ(c3,j ) (ω(c3,1 · · · c3,j−1 , c3,j · · · c3,n3 ) − ω(c3,1 · · · c3,j , c3,j+1 · · · c3,n3 ))
2 j=2

Comparing the terms we see that θ(g)+θ(h)−θ(gh) uniformly close to zero, i.e. a quasimorphism.

Finally, we can give a proof to the Technical Theorem 4.3.
Proof. (of Theorem 4.3). According to Proposition 5.1, φ(g)ω(h, i) + δ 2 η(g, h, i) is uniformly
close to ζ(g, h, i) + ζ(h, 1, i) − ζ(gh, 1, i). which in turn by Proposition 5.2 is uniformly close to
l−1
X

φ(cj ) (ω(c1 · · · cj−1 , cj · · · cl ) − ω(c1 · · · cj , cj+1 · · · cl )) = −2θ(c)

j=2

On the other hand observe that
γ(g, h) = θ(c)
where c = c(g, h), the common 2-path of (g, h). Hence by Lemma 5.4, δ 2 γ(g, h, i) is uniformly
close to −2θ(c) as θ is a quasimorphism by Lemma 5.5. We conclude that φ(g)ω(h, i)+δ 2 η(g, h, i)
is uniformly close to δ 2 γ(g, h, i), which concludes the proof of the Technical Theorem 4.3.

15

References
[BM17] M. Bucher and N. Monod. The cup product of Brooks quasimorphisms. ArXiv e-prints, 1710.02392,
October 2017.
[Bro81] Robert Brooks. Some remarks on bounded cohomology. In Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume 97
of Ann. of Math. Stud., pages 53–63. Princeton Univ. Press, Princeton, N.J., 1981.
[EF97] David B. A. Epstein and Koji Fujiwara. The second bounded cohomology of word-hyperbolic groups.
Topology, 36(6):1275–1289, 1997.
[Fri16] R. Frigerio. Bounded cohomology of discrete groups. ArXiv e-prints, 1610.08339, October 2016.
[Gri95] R. I. Grigorchuk. Some results on bounded cohomology. In Combinatorial and geometric group theory
(Edinburgh, 1993), volume 204 of London Math. Soc. Lecture Note Ser., pages 111–163. Cambridge Univ.
Press, Cambridge, 1995.
[Löh15] Clara Löh. A note on bounded-cohomological dimension of discrete groups. ArXiv e-prints, 1504.05760,
April 2015.
[Mon06] Nicolas Monod. An invitation to bounded cohomology. In International Congress of Mathematicians.
Vol. II, pages 1183–1211. Eur. Math. Soc., Zürich, 2006.
[Rol09] P. Rolli. Quasi-morphisms on Free Groups. ArXiv e-prints, 0911.4234, November 2009.
Department of Mathematics, University of Oxford
E-mail address, N. Heuer: heuer@maths.ox.ac.uk

16

