CCUP PRODUCT IN BOUNDED COHOMOLOGY OF THE FREE GROUP
NICOLAUS HEUER
Abstract.
The theory of bounded cohomology of groups has many applications. A key openproblem is to compute the full bounded cohomology H nb ( F, R ) of a non-abelian free group F with trivial real coefficients. It is known that H nb ( F, R ) is trivial for n = 1 and uncountabledimensional for n = 2 ,
3, but H nb ( F, R ) remains unknown for any n ≥
4. For n = 4, onemay construct classes by taking the cup product α (cid:94) β ∈ H b ( F, R ) between two 2-classes α, β ∈ H b ( F, R ). However, we show that all such cup products are trivial if α and β are classesinduced by the quasimorphisms defined by Brooks or Rolli. Introduction
Bounded cohomology of groups was originally studied by Gromov in [Gro82]. Since thenbounded cohomology emerged as an indepenedent research field with many applications. Theseinclude stable commutator length ([Cal09]), circle actions ([BFH16]) and the Chern Conjecture.See [Mon06] for a survey and [Fri17] for a book on bounded cohomology.However, the bounded cohomology of a group G is notoriously hard to explicitly compute,even for trivial real coefficients. On the one hand, it is known that H nb ( G, R ) is trivial for all n ∈ N if G is amenable. On the other hand, H nb ( G, R ) is uncountable dimensional if G is anacylindrically hyperbolic group and n = 2 or n = 3; see [HO13], [Som97] and [FPS15].This paper will exclusively focus on the bounded cohomology of non-abelian free groups F with trivial real coefficients, denoted by H nb ( F, R ). We note that H nb ( F, R ) is fully unknown forany n ≥
4. Free groups play a distinguished rˆole in constructing non-trivial classes on otheracylindrically hyperbolic groups. Due to a result by Frigerio, Pozzetti and Sisto, any non-trivialalternating class in H nb ( F, R ) may be promoted to a non-trivial class in H nb ( G, R ) where G is anacylindrically hyperbolic group and n ≥
2; see Corollary 2 of [FPS15].All classes in the second bounded cohomology of a non-abelian free group F with trivial realcoefficients arise as coboundaries of quasimorphisms (see Subsection 2.2) i.e. for any ω ∈ H b ( F, R )there is a quasimorphism φ : F → R such that [ δ φ ] = ω . There are many explicit constructionsof quasimorphisms φ : F → R , most prominently the one defined by Brooks [Bro81] and Rolli[Rol09]; see Subsection 2.2. One may hope to construct non-trivial classes in H b ( F, R ) by takingthe cup product [ δ φ ] (cid:94) [ δ ψ ] ∈ H b ( F, R ) between two such quasimorphisms φ, ψ : F → R . Wewill show that this approach fails. Theorem A.
Let φ, ψ : F → R be two quasimorphisms on a non-abelian free group F whereeach of φ and ψ is either Brooks counting quasimorphisms on a non self-overlapping word orquasimorphisms in the sense of Rolli. Then [ δ φ ] (cid:94) [ δ ψ ] ∈ H b ( F, R ) is trivial.We note that Michelle Bucher and Nicolas Monod have independently proved the vanishingof the cup product between the classes induced by Brooks quasimorphisms with a differenttechnique; see [BM18].Theorem A will follow from a more general vanishing Theorem. For this, we will first define decompositions (see Definition 3.1) which are certain maps ∆ that assign to each element g ∈ F a Date : December 18, 2018. a r X i v : . [ m a t h . G R ] D ec nite sequence ( g , . . . , g n ) of arbitrary length with g j ∈ F and such that g = g · · · g n and thereis no cancellation between the g j . We then define two new classes of quasimorphisms, namely∆ -decomposable quasimorphisms (Definition 3.5) and ∆ -continuous quasimorphisms (Definition3.11). Each Brooks and Rolli quasimorphism will be both ∆-decomposable and ∆-continuouswith respect to some decomposition ∆. We will show: Theorem B.
Let ∆ be a decomposition of F and let φ, ψ : F → R be quasimorphisms such that φ is ∆-decomposable and ψ is ∆-continuous. Then [ δ φ ] (cid:94) [ δ ψ ] ∈ H b ( F, R ) is trivial.We will prove Theorem B by giving an explicit bounded coboundary in terms of φ and ψ in Theorem C. Let φ and ψ be as in Theorem B. A key observation of this paper is that thefunction ( g, h, i ) (cid:55)→ φ ( g ) δ ψ ( h, i ) “behaves like a honest cocycle with respect to ∆”. The idea ofthe proof of Theorem C is to mimic the algebraic proof that honest cocycles on free groups havea coboundary; see Subsection 4.1.It was shown by Grigorchuk [Gri95] that Brooks quasimorphisms are dense in the vector spaceof quasimorphisms in the topology of pointwise convergence. In light of Theorem A one wouldlike to deduce from this density that the cup product between all bounded 2-classes vanishes.However, this does not seem straightforward. The topology needed for such a deduction is thestronger defect topology . Brooks cocycles are not dense in this topology, in fact the space of2-cocycles is not even separable in this topology. We therefore ask: Question . Let F be a non-abelian free group. Is the cup product (cid:94) : H b ( F, R ) × H b ( F, R ) → H b ( F, R )trivial?Note that it is unknown if nontrivial classes in H b ( F, R ) exist. We mention that the cupproduct on bounded cohomology for other groups need not be trivial. Let G be a group withnon-trivial second bounded cohomology. Then G × G admits a non-trivial cup product (cid:94) : H b ( G × G, R ) × H b ( G × G, R ) → H b ( G × G, R )induced by the factors. See [L¨oh17] for results and constructions in bounded cohomology usingthe cup product. Organisation.
This paper is organised as follows: Section 2 introduces notation and recalls basicfacts about bounded cohomology. Section 3 defines and studies decompositions ∆ of non-abelianfree groups F mentioned above as well as ∆-decomposable and ∆-continuous quasimorphisms.In Section 4 we will introduce and prove Theorem C, which will provide the explicit boundedprimitives for the cup products studied in this paper. The key ideas of the proof are illustratedin Subsection 4.1. Theorems A and B will be corollaries of Theorem C and proved in Subsection4.4. Acknowledgements.
I would like to thank Roberto Frigerio, Clara L¨oh, Michelle Bucher andMarco Moraschini for helpful discussions and detailed comments and my supervisor Martin Brid-son for his helpful comments and support. I would further like to thank the anonymous referee formany helpful remarks which substantially improved the paper. The author would like to thankthe Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitalityduring the programme
Non-Positive Curvature Group Actions and Cohomology where work onthis paper was undertaken. This work was supported by EPSRC grant no EP/K032208/1. Theauthor is also supported by the Oxford-Cocker Scholarship.2.
Preliminaries
In Subsection 2.1 we will introduce notations which will be used throughout the paper. InSubsection 2.2 we define (bounded) cohomology of groups and the cup product. In Subsection2.3 we will define quasimorphisms, in particular the quasimorphisms defined by Brooks and Rolli.2.1.
Notation and conventions.
A generic group will be denoted by G and a non-abelianfree group will be denoted by F . The generating set S of F will consist of letters in code-font(” a , b “). The identity element of a group will be denoted by ”1“. Small Roman letters (” a, b “)typically denote elements of groups. 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 “). We stick to thisnotation unless it is mathematical convention to do otherwise.2.2. (Bounded) cohomology and the cup product. We recall the definition of discrete(bounded) cohomology with trivial real coefficients using the inhomogeneous resolution. Let G be a group, let C k ( G, R ) = { φ : G k → R } and let C kb ( G, R ) ⊂ C k ( G, R ) be the subset ofbounded functions with respect to the supremum norm. We define the coboundary operator δ n : C n ( G, R ) → C n +1 ( G, R ) via δ n ( α )( g , . . . , g n +1 )= α ( g , . . . , g n +1 )+ n (cid:88) j =1 ( − j α ( g , . . . , g j g j +1 , . . . , g n +1 )+( − n +1 α ( g , . . . , g n )and note that it restricts to δ n : C nb ( G, R ) → C n +1 b ( G, R ). The homology of the cochain complex( C ∗ ( G, R ) , δ ∗ ) is called the cohomology of the group G with trivial real coefficients and denotedby H ∗ ( G, R ). Similarly the homology of ( C ∗ b ( G, R ) , δ ∗ ) is called the the bounded cohomology of G with trivial real coefficients and denote it by H ∗ b ( G, R ). The inclusion C nb ( G, R ) (cid:44) → C n ( G, R ) is achain map which induces a map c n : H nb ( G, R ) → H n ( G, R ) called the comparison map . Cocyclesin the kernel of c n are called exact classes and will correspond to quasimorphisms if n = 2; seeSubsection 2.3. For a detailed discussion see [Mon06] for a survey and [Fri17] for a book onbounded cohomology.The cup product is a map (cid:94) : H n ( G, R ) × H m ( G, R ) → H n + m ( G, R ) defined by setting([ ω ] , [ ω ]) (cid:55)→ [ ω ] (cid:94) [ ω ] where [ ω ] (cid:94) [ ω ] ∈ H n + m ( G, R ) is represented by the cocycle ω (cid:94) ω ∈ C n + m ( G, R ) defined via ω (cid:94) ω : ( g , . . . , g n , g n +1 , . . . , g n + m ) (cid:55)→ ω ( g , . . . , g n ) · ω ( g n +1 , . . . , g n + m ) . It is easy to check that this map induces a well-defined map (cid:94) : H nb ( G, R ) × H mb ( G, R ) → H n + mb ( G, R ) . Quasimorphisms. A quasimorphism is a map α : G → R such that there is a constant D > g, h ∈ G , | α ( g ) − α ( gh ) + α ( h ) | < D and hence δ α ∈ C b ( G, R ). Hence theexact 2-classes of H b ( G, R ) are exactly the coboundaries of quasimorphisms. A quasimorphism α : G → R will be called symmetric if α satisfies in addition that α ( g ) = − α ( g − ) for all g ∈ G .In this case, we call its coboundary δ α ∈ C b ( G, R ) symmetric as well. It is easy to see thateach exact 2-class is represented by a symmetric cocycle. On a non-abelian free group F thereare several constructions of non-trivial quasimorphisms. Example 2.1.
In [Bro81], Brooks gave the first example of an infinite family of linearly indepen-dent quasimorphisms on the free group. Let F be a non-abelian free group on a fixed generatingset S . Let w, g ∈ F be two elements which are represented by reduced words w = y · · · y n and g = x · · · x m , where x j , y j are letters of F . We say that w is a sub-word of g if n ≤ m and there3s an s ∈ { , . . . , m − n } such that y j = x j + 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. For w a non self-overlapping word we define the function ν w : F → Z by setting ν w : g (cid:55)→ { w is a subword of g } . Then the Brooks counting quasimorphism on theword w is the function φ w = ν w − ν w − . It is easy to see that this defines a symmetric quasimorphism.
Example 2.2.
In [Rol09], Rolli gave a different example of an infinite family of linearly inde-pendent quasimorphisms. Suppose F is generated by S = { x , . . . , x n } . Let λ , . . . , λ n ∈ (cid:96) ∞ alt ( Z )be bounded functions λ j : Z → R that satisfy λ j ( − n ) = − λ j ( n ). Each non-trivial element g ∈ F may be uniquely written as g = x m n · · · x m k n k where all m j are non-zero and no consecutive n j arethe same. Then we can see that the map φ : F → R defined by setting φ : g (cid:55)→ k (cid:88) j =1 λ n j ( m j )is a symmetric quasimorphism called Rolli quasimorphism .We will generalise Brooks and Rolli quasimorphisms in the next section.3.
Decomposition
The aim of this section is to introduce decompositions of F in Subsection 3.2. Let F be anon-abelian free group with a fixed set of generators. Crudely, a decomposition ∆ is a wayof assigning a finite sequence ( g , . . . g k ) of elements g j ∈ F to an element g ∈ F such that g = g · · · g k as a reduced word and such that that this decomposition behaves well on geodesictriangles in the Cayley graph. We will see that any decomposition ∆ induces a quasimorphism(Proposition 3.6), called ∆-decomposable quasimorphism in Subsection 3.3. We will introducespecial decompositions, ∆ triv , ∆ w and ∆ rolli and see that ∆ triv -decomposable quasimorphismsare exactly the homomorphisms F → R , that Brooks quasimorphisms on a non self-overlappingword w are ∆ w -decomposable and that the quasimorphisms in the sense of Rolli are ∆ rolli -decomposable. In Subsection 3.4 we introduce ∆-continuous cocycles.3.1. Notation for sequences.
A set
A ⊂ F will be called symmetric if a ∈ A implies that a − ∈ 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 ( a , . . . , a k ) where k ∈ N is arbitrary and a j ∈ A . We will denote the element ( a , . . . , a k ) ∈ A ∗ by ( a ) and k will becalled the length of ( a ) where we set k = 0 if ( a ) is the empty sequence. For a sequence ( a ), wedenote by ( a − ) the sequence ( a − k , . . . , a − ) ∈ A ∗ and the element ¯ a ∈ F denotes the product a · · · a k ∈ F . We will often work with multi-indexes: The sequences ( a ) , ( a ) , ( a ) ∈ A ∗ willcorrespond to the sequences ( a j ) = ( a j, , . . . , a j,n j ), where n j is the length of ( a j ) for j = 1 , , a ) = ( a , . . . , a k ) and ( b ) = ( b , . . . , b l ) we define the common sequence of ( a ) and ( b ) to be the empty sequence if a (cid:54) = b and to be the sequence ( c ) = ( a , . . . , a n ) where n is the largest integer with n ≤ min { k, l } such that a j = b j for all j ≤ n . Moreover, ( a ) · ( b ) willdenote the sequence ( a , . . . , a k , b , . . . , b l ).3.2. Decompositions of F . We now define the main tool of this paper, namely decompositions .As mentioned in the introduction we will restrict our attention to non-abelian free groups F ona fixed generating set S . 4 igure 1. ∆( g ), ∆( h ) and ∆( h − g − ) have sides which can be identified. Definition 3.1.
Let
P ⊂ F be a symmetric set of elements of F called pieces and assume that P does not contain the identity. A decomposition of F into the pieces P is a map ∆ : F → P ∗ assigning to every element g ∈ F a finite sequence ∆( g ) = ( g , . . . , g k ) with g j ∈ P such that:(1) For every g ∈ F and ∆( g ) = ( g , . . . , g k ) we have g = g · · · g k as a reduced word (nocancelation). Also, we require that ∆( g − ) = ( g − k , . . . , g − ).(2) For every g ∈ F with ∆( g ) = ( g , . . . , g k ) we have ∆( g i · · · g j ) = ( g i , . . . , g j ) for 1 ≤ i ≤ j ≤ k . We refer to this property as ∆ being infix closed .(3) There is a constant R > g, h ∈ F and let • ( c ) ∈ P ∗ be such that ( c − ) is the common sequence of ∆( g ) and ∆( gh ), • ( c ) ∈ P ∗ be such that ( c − ) is the common sequence of ∆( g − ) and ∆( h ) and • ( c ) ∈ P ∗ be such that ( c − ) is the common sequence of ∆( h − ) and ∆( h − g − ).It is not difficult to see that there are ( r ) , ( r ) , ( r ) ∈ P ∗ such that∆( g ) = ( c − ) · ( r ) · ( c )∆( h ) = ( c − ) · ( r ) · ( c ) and∆( h − g − ) = ( c − ) · ( r ) · ( c ) . Then the length of ( r ), ( r ) and ( r ) is bounded by R . See Figure 1 for a geometric in-terpretation and Subsection 3.1 for the notation of common sequences and concatenationof sequences.For such a pair ( g, h ) we will call ( c ) , ( c ) , ( c ) the c -part of the ∆ -triangle of ( g, h ) and( r ) , ( r ) , ( r ) the r -part of the ∆ -triangle of ( g, h ). A sequence ( g , . . . , g k ) such that∆( g · · · g k ) = ( g , . . . , g k )will be called a proper ∆ sequence . Example 3.2.
Let S = { x , . . . , x n } be an alphabet generating F . Every word w ∈ F may beuniquely written as a word w = y · · · y k without backtracking where y i ∈ S ± . Set P triv = S ± and define the map ∆ triv : F → P ∗ triv by setting∆ triv : w (cid:55)→ ( y , . . . , y k )for w as above. Then we see that ∆ triv is indeed a decomposition. Let g, h ∈ G and let c , c , c be such that g = c − c , h = c − c and gh = c − c as reduced words. Then the c -part of the∆ triv -triangle of ( g, h ) is ∆ triv ( c ) , ∆ triv ( c ) , ∆ triv ( c ) and the r -part of the ∆ triv -triangle of( g, h ) is ( ∅ ) , ( ∅ ) , ( ∅ ) where ( ∅ ) denotes the empty sequence.We call the map ∆ triv the trivial decomposition .5 xample 3.3. Let w ∈ F be a non self-overlapping word (see Example 2.1). Every word g ∈ F may be written as g = u w (cid:15) u · · · u k − w (cid:15) k − u k , where the u j ’s may be empty, (cid:15) j ∈ {− , +1 } and no u j contains w or w − as subwords. It is not hard to show that this expression isunique. Observe that a reduced word in the free group does not overlap with its inverse. Set P w = { u ∈ F | neither w nor w − are subwords of u } ∪ { w, w − } .We define the Brooks-decomposition on the word w as the map ∆ w : F → P ∗ w by setting∆ w : g → ( u , w (cid:15) , u , · · · , u k − , w (cid:15) k − , u k )for g as above. It is easy to check that this is indeed a decomposition. Example 3.4.
As in Example 2.2, suppose that F is generated by S = { x , . . . , x n } and observethat every non-trivial element g ∈ F may be uniquely written as g = x m n · · · x m k n k where all m j are non-zero and no consecutive n j are the same. Set P rolli = { x mj | j ∈ { , . . . , n } , m ∈ Z } . Wedefine the Rolli-decompostion as the map ∆ rolli : F → P ∗ rolli via∆ rolli : g (cid:55)→ ( x m n , . . . , x m k n k )for g as above. It is easy to check that this is indeed a decomposition.Often we just talk about the decomposition without specifying the pieces P explicitly. Froma decomposition ∆ we derive the notion of two sorts of quasimorphisms: ∆ -decomposable quasi-morphisms (Definition 3.5) and ∆ -continuous quasimorphisms (Definition 3.11).3.3. ∆ -decomposable quasimorphisms. Each decomposition ∆ of F induces many differentquasimorphisms on F . Definition 3.5.
Let ∆ be a decomposition with pieces P and let λ ∈ (cid:96) ∞ alt ( P ) be a symmetricbounded map on P , i.e. λ ( p − ) = − λ ( p ) for every p ∈ P . Then the map φ λ, ∆ : F → R definedvia φ λ, ∆ : g (cid:55)→ k (cid:88) j =1 λ ( g j )where ∆( g ) = ( g , . . . , g k ) is called a ∆ -decomposable quasimorphism.We may check that such a φ λ, ∆ is indeed a quasimorphism. Proposition 3.6.
Let ∆ and λ be as in Definition 3.5. Then φ λ, ∆ is a symmetric quasimor-phism. If g, g (cid:48) ∈ F are such that ∆( g · g (cid:48) ) = (∆( g )) · (∆( g (cid:48) )) then δ φ ( g, g (cid:48) ) = 0 . In particular, forall g ∈ G with ∆( g ) = ( g , . . . , g k ) we have that δ φ λ, ∆ ( g j , g j +1 · · · g k ) = 0 for j = 1 , . . . , k − .Proof. Symmetry is immediate from the assumptions on ∆( g − ) and λ . Let g, h ∈ F and let( c j ) , ( r j ), j ∈ { , , } be as in the definition of the decomposition. We compute φ λ, ∆ ( g ) = − n (cid:88) j =1 λ ( c ,j ) + m (cid:88) j =1 λ ( r ,j ) + n (cid:88) j =1 λ ( c ,j ) φ λ, ∆ ( h ) = − n (cid:88) j =1 λ ( c ,j ) + m (cid:88) j =1 λ ( r ,j ) + n (cid:88) j =1 λ ( c ,j ) φ λ, ∆ ( gh ) = − n (cid:88) j =1 λ ( c ,j ) − m (cid:88) j =1 λ ( r ,j ) + n (cid:88) j =1 λ ( c ,j )and hence δ φ λ, ∆ ( g, h ) = φ λ, ∆ ( g ) + φ λ, ∆ ( h ) − φ λ, ∆ ( gh ) = m (cid:88) j =1 λ ( r ,j ) + m (cid:88) j =1 λ ( r ,j ) + m (cid:88) j =1 λ ( r ,j )6nd hence | δ φ λ, ∆ ( g, h ) | ≤ R (cid:107) λ (cid:107) ∞ . Note that from this calculation we also see that δ φ λ, ∆ ( g, h )only depends on the r -part of the ∆-triangle for ( g, h ) and not on the c -part. The second partfollows immediately from property (2) of a decomposition. (cid:3) Both Brooks and Rolli quasimorphisms are ∆-decomposable quasimorphisms with respect tosome ∆ as the following examples show:
Example 3.7.
Let ∆ triv be the trivial decomposition of Example 3.2. It is easy to see that the∆ triv -decomposable quasimorphisms are exactly the homomorphisms φ : F → R . Example 3.8.
Let P w be as in Example 3.3 and define λ : P w → R by setting λ : p (cid:55)→ p = w, − p = w − , . Then it we see that the induced decomposable quasimorphism φ λ, ∆ w is the Brooks countingquasimorphism on w ; see Example 2.1. Example 3.9.
Let λ , . . . , λ n be as in Example 2.2 and let P rolli be as in Example 3.4. Define λ : P rolli (cid:55)→ R by setting λ : x mj (cid:55)→ λ j ( m ) . Then we see that the induced quasimorphism φ λ, ∆ rolli is a Rolli quasimorphism; see Example2.2.3.4. ∆ -continuous quasimorphisms and cocycles. We will define ∆-continuous cocycles.Crudely, a cocycle ω is ∆-continuous, if the value ω ( g, h ) depends “mostly” on the neighbourhoodof the midpoint of the geodesic triangle spanned by e, g, gh in the Cayley graph of F . For this, wewill first establish a notion of when two pairs ( g, h ) and ( g (cid:48) , h (cid:48) ) of elements in F define triangleswhich are “close”.For this we define the function N ∆ : F × F → N ∪ ∞ as follows. Let ( g, h ) ∈ F and( g (cid:48) , h (cid:48) ) ∈ F be two pairs of elements of F . Let ( c j ) , ( r j ) for j = 1 , , g, h )where ( c j ) has length n j and let ( c (cid:48) j ) , ( r (cid:48) j ) for j = 1 , , g (cid:48) , h (cid:48) ) where ( c (cid:48) j )has length n (cid:48) j .We set N ∆ (( g, h ) , ( g (cid:48) , h (cid:48) )) = 0 if there is a j ∈ { , , } such that r j (cid:54) = r (cid:48) j and N ∆ (( g, h ) , ( g (cid:48) , h (cid:48) )) = ∞ if ( g, h ) = ( g (cid:48) , h (cid:48) ). Else, let N ∆ (( g, h ) , ( g (cid:48) , h (cid:48) )) be the largest integer N which satisfies that N ≤ min { n j , n (cid:48) j } and c j,k = c (cid:48) j,k for every k ≤ N and j ∈ { , , } such that c j (cid:54) = c (cid:48) j .Observe that N ∆ (( g, h ) , ( g (cid:48) , h (cid:48) )) = ∞ if and only if ( g, h ) = ( g (cid:48) , h (cid:48) ). This is because if( g, h ) (cid:54) = ( g (cid:48) , h (cid:48) ) then either there is some j such that r j (cid:54) = r (cid:48) j , in which case N ∆ (( g, h ) , ( g (cid:48) , h (cid:48) )) = 0or there is some j such that c j (cid:54) = c (cid:48) j in which case N ∆ (( g, h ) , ( g (cid:48) , h (cid:48) )) ≤ min { n j , n (cid:48) j } . Crudely, N ∆ measures how much the triangle corresponding to ( g, h ) agrees with the triangle correspondingto ( g (cid:48) , h (cid:48) ) arround the “centre” of the triangle; see Figure 2. To illustrate N ∆ we will give anexample for ∆ the trivial decomposition. Example 3.10.
Let ∆ be the trivial decomposition and let g, h, i ∈ F be such that ghi has nocancellation and assume that g is not-trivial. Then we claim that N ∆ (( gh, i ) , ( h, i )) = | h | , where | h | is the word-length of h . To see this observe that the r -part of the ∆ triangles of ( gh, i ) and( h, i ) agrees (it’s both ( ∅ , ∅ , ∅ ). Moreover, the c -part of the ∆-triangles ( gh, i ) and ( h, i ) is(∆( h ) − · ∆( g ) − , ∅ , ∆( i )) = ( c , c , c )and (∆( h ) − , ∅ , ∆( i )) = ( c (cid:48) , c (cid:48) , c (cid:48) ) . igure 2. The ∆-triangle for ( g, h ) vs. the ∆-triangle for ( g (cid:48) , h (cid:48) ) and N = N ∆ (( g, h ) , ( g (cid:48) , h (cid:48) ))We see that c = c (cid:48) and c = c (cid:48) but c (cid:54) = c (cid:48) . Observe that the length of c is | h | + | g | and the length of c (cid:48) is | h | . Moreover, c ,k = c (cid:48) ,k for every k ≤ | h | . This shows that indeed N ∆ (( gh, i ) , ( h, i )) = | h | . Definition 3.11.
Let ∆ be a decomposition of F and let N ∆ be as above. A quasimorphism φ is called ∆-continuous if φ is symmetric (i.e. φ ( g − ) = − φ ( g ) for every g ∈ G ) and ω = δ φ satisfies that there is a constant S ω, ∆ > s j ) j ∈ N with (cid:80) ∞ j =0 s j = S ω, ∆ such that for all ( g, h ) , ( g (cid:48) , h (cid:48) ) ∈ F we have that either ( g, h ) = ( g (cid:48) , h (cid:48) ) or, | ω ( g, h ) − ω ( g (cid:48) , h (cid:48) ) | ≤ s N where N = N ∆ (( g, h ) , ( g (cid:48) , h (cid:48) )). In this case we call ω ∆-continuous as well.Crudely, a cocycle ω is ∆-continuous if its values depend mostly on the parts of the decom-position which lies close to the centre of the triangle g, h, h − g − .Many quasimorphisms are ∆-continuous as the following proposition shows. Proposition 3.12.
Let ∆ be a decomposition of F .(1) Every ∆ -decomposable quasimorphism is ∆ -continuous.(2) Every Brooks quasimorphism φ : F → R is ∆ -continuous.Proof. To see (1) observe that the proof of Proposition 3.6 shows that δ φ ( g, h ) does not dependon the c -part of the ∆-triangle of ( g, h ). Hence if N ∆ (( g, h ) , ( g (cid:48) , h (cid:48) )) ≥
1, then δ φ ( g, h ) = δ φ ( g (cid:48) , h (cid:48) ).For (2), suppose that δ φ is a bounded cocycle induced by a Brooks quasimorphism φ on aword w and suppose that the length of w is m . The value of the Brooks cocycle δ φ ( g, h ) justdepends on the m -neighbourhood of the midpoint of the tripod with endpoints e, g, gh in theCayley graph. Hence, whenever N ∆ (( g, h ) , ( g (cid:48) , h (cid:48) )) ≥ m we have that δ φ ( g, h ) = δ φ ( g (cid:48) , h (cid:48) ).Note that this implies that Brooks quasimorphisms are ∆-continuous for any decomposition∆. (cid:3) Triangles and quadrangles in a tree.
Let g, h ∈ F . It is easy to see that there areunique elements t , t , d ∈ F such that g = t − d and h = d − t as reduced words and that t , t and d are the paths of the tripod with endpoints 1 , g, gh in the Cayley graph of F . We will call d the common 2-path of ( g, h ).For three elements g, h, i ∈ F there are three different cases how the geodesics between thepoints 1 , g, gh, ghi in the Cayley graph of F can be aligned. See Figure 3.(1) (Figure 3a): There are elements t , . . . , t such that g = t t , h = t − t t , i = t − t asreduced words. 8 a) (b) (c) Figure 3.
Different cases how g , h and i are aligned(2) (Figure 3b): There are elements t , . . . , t such that g = t t t , h = t − t , i = t − t − t as reduced words.(3) (Figure 3c): There are elements t , . . . , t and c such that g = t − ct , h = t − c − t , i = t − ct as reduced words.We will say that the common-3-path of ( g, h, i ) is empty in the first two cases and c in the thirdcase. 4. Constructing the bounded primitive
Recall that F is a non-abelian group and let ∆ be a decomposition of F ; see Definition3.1. Moreover, let φ : F → R be a ∆-decomposable quasimorphism (see Definition 3.5) and let ω ∈ C b ( F, R ) be a ∆-continuous symmetric 2-cocycle (see Definition 3.11). We define the map ζ ∈ C ( F, R ) by setting ζ : ( g, h, i ) (cid:55)→ k (cid:88) j =1 φ ( g j ) ω ( g j +1 · · · g k h, i )for ∆( g ) = ( g , . . . , g k ). Moreover, define the maps η, γ ∈ C ( F, R ) by setting • η : ( g, h ) (cid:55)→ ζ ( g, , h ) and • γ : ( g, h ) (cid:55)→ (cid:16) ζ ( d, d − , d ) + ζ ( d − , , d ) (cid:17) for d the common 2-path of ( g, h ); see Subsec-tion 3.5.We will show the following theorem: Theorem C.
Let φ be a ∆-decomposable quasimorphism and let ω be a symmetric, ∆-continuous2-cocycle. Moreover, let γ and η be as above. Then β ∈ C ( F, R ) defined by setting β : ( g, h, i ) (cid:55)→ φ ( g ) ω ( h, i ) + δ γ ( g, h, i ) + δ η ( g, h, i )is bounded, i.e. β ∈ C b ( F, R ).We will see in Subsection 4.4 that β will be the bounded primitive for the cup products studiedin this paper. Before we prove this theorem in Subsection 4.3, we will give an idea of the proofin Subsection 4.1. This will be inspired by a construction of coboundaries to 3-cocycles whichwe recall in Subsection 4.2. 9.1. Idea of the proof of Theorems B and C.
Theorem B states that [ δ φ (cid:94) ω ] = 0 inH b ( F, R ) for φ a ∆-decomposable quasimorphism and ω a ∆-continuous cocycle. Equivalently,there is a bounded primitive of the map δ τ ∈ C ( F, R ), where τ is given by τ : ( g, h, i ) (cid:55)→ φ ( g ) ω ( h, i ) since δ τ = δ φ (cid:94) ω . Note that τ is a priori not an interesting function for boundedcohomology: It is neither bounded nor is it a cocycle.Recall that a map α ∈ C ( F, R ) satisfies the cocycle condition if and only if for all g, g (cid:48) , h, i ∈ F we have that δ α ( g, g (cid:48) , h, i ) = 0 . As H ( F, R ) = 0, we know that there is some (cid:15) ∈ C ( F, R ) such that δ (cid:15) = α . We will givea purely algebraic construction of such an (cid:15) in terms of α , provided α satisfies certain weakconditions stated in Subsection 4.2, Equation 1.Observe that τ does not satisfy the cocycle condition for all g, g (cid:48) , h, i ∈ F . However, τ satisfies the cocycle condition in certain cases: Proposition 3.6 shows that if g, g (cid:48) ∈ F satisfythat ∆( g · g (cid:48) ) = (∆( g )) · (∆( g (cid:48) )) then δ τ ( g, g (cid:48) , h, i ) = 0for all h, i ∈ F . Following the techniques of Subsection 4.2 we will construct an (cid:15) ∈ C ( F, R )such that δ (cid:15) is boundedly close to τ . This is, such that the map β = τ − δ (cid:15) is bounded, i.e. β ∈ C b ( F, R ). This will imply that δ β = δ τ − δ δ (cid:15) = δ φ (cid:94) ω and hence the cup product has a bounded primitive and is trivial in bounded cohomology.4.2. Constructing -coboundaries from -cocycles. Let α ∈ C ( F, R ) be a 3-cocycle i.e. amap such that δ α = 0. We will show how to construct a map (cid:15) ∈ C ( F, R ) such that δ (cid:15) = α .We emphasize that this subsection just motivates the strategy of the proof of Theorem C. Thistheorem will be proved in detail in Subsection 4.3 and the proof can be understood withoutreading this subsection. In both subsections, the η and the ζ term will play analogous rˆole.To simplify our calculations we will assume that α is a cocycle and moreover satisfies α ( g, h,
1) = α ( g, , h ) = α (1 , g, h ) = α ( g, g − , h ) = 0 for all g, h ∈ F . (1)We note that alternating cochains in the sense of Subsection 4.10 of [Fri17] satisfy (1) and thatsuch maps may be used to fully compute H ( F, R ).Let α be as above and recall that the cocycle condition implies that for all g, g (cid:48) , h, i ∈ F wehave that(2) δ α ( g, g (cid:48) , h, i ) = α ( g (cid:48) , h, i ) − α ( gg (cid:48) , h, i ) + α ( g, g (cid:48) h, i ) − α ( g, g (cid:48) , hi ) + α ( g, g (cid:48) , h ) = 0 . In a first step we see how α may be rewritten as a sum of elements of the form α ( x , g (cid:48) , h (cid:48) ),where x is a letter and g (cid:48) , h (cid:48) ∈ F . Define ζ ∈ C ( F, R ) by setting ζ : ( g, g (cid:48) , h ) (cid:55)→ k (cid:88) j =1 α ( x j , x j +1 · · · x k g (cid:48) , h )where g = x · · · x k is the reduced word representing g . We claim that Claim 4.1.
Let α ∈ C ( F, R ) be a cocycle satisfying (1). Then α ( g, h, i ) = ζ ( g, h, i ) − ζ ( g, , hi ) + ζ ( g, , h ) for all g, h, i ∈ F .Proof. direct computation. (cid:3) η ∈ C ( F, R ) by setting η : ( g, h ) (cid:55)→ ζ ( g, , h ) . We then see that α ( g, h, i ) + δ η ( g, h, i ) = ζ ( g, h, i ) + ζ ( h, , i ) − ζ ( gh, , i )for all g, h, i ∈ F . Claim 4.2.
We have that ζ ( g, h, i ) + ζ ( h, , i ) − ζ ( gh, , i ) = ζ ( d, h, i ) + ζ ( d − , dh, i ) for all g, h, i ∈ F , where d is the common -path of ( g, h ) .Proof. We will prove this by an explicit calculation. Observe that it is immediate that if u, v ∈ F are such that uv is reduced then(3) ζ ( uv, g (cid:48) , h ) = ζ ( u, vg (cid:48) , h ) + ζ ( v, g (cid:48) , h ) . Now rewrite g = t − d and h = d − t , where d is the common 2-path of ( g, h ); see Subsection3.5. Then by (3) we see that • ζ ( g, h, i ) = ζ ( t − , dh, i ) + ζ ( d, h, i ) • ζ ( h, , i ) = ζ ( d − , dh, i ) + ζ ( t , , i ) • ζ ( gh, , i ) = ζ ( t − , dh, i ) + ζ ( t , , i )Hence ζ ( g, h, i ) + ζ ( h, , i ) − ζ ( gh, , i ) = ζ ( d, h, i ) + ζ ( d − , dh, i ) . (cid:3) Claim 4.3.
We have that ζ ( g, h, i ) + ζ ( h, , i ) − ζ ( gh, , i ) = 0 and hence that α ( g, h, i ) = δ (cid:15) for (cid:15) = − η .Proof. Let d be the common 2-path of ( g, h ) as above. Moreover, suppose that d · · · d l is the wordrepresenting d . By the previous claim, ζ ( g, h, i ) + ζ ( h, , i ) − ζ ( gh, , i ) = ζ ( d, h, i ) + ζ ( d − , dh, i ).We calculate ζ ( d, h, i ) + ζ ( d − , dh, i ) = k (cid:88) j =1 (cid:16) α ( d j , d j +1 · · · d l h, i ) + α ( d − j , d j · · · d l h, i ) (cid:17) . By evaluating δ α ( d j , d − j , d j · · · d l h, i ) using property (1) we have that α ( d − j , d j · · · d l h, i ) + α ( d j , d j +1 · · · d l h, i ) = 0. (cid:3) Together with Claim 4.2 the previous claim implies that α + δ η = α − δ (cid:15) = 0.4.3. Proof of Theorem C.
Let ∆ be a decomposition of F (Definition 3.1), let φ be a ∆-decomposable quasimorphism (Definition 3.5) and let ω be a ∆-continuous cocycle (Definition3.11). See the previous subsection for a brief discussion on the classical computations thatinspired our construction here. Analogously to Claim 4.1, we will first rewrite the function( g, h, i ) (cid:55)→ φ ( g ) ω ( h, i ) as sum of terms φ ( g j ) ω ( g (cid:48) , h (cid:48) ) where g j will be a piece of a fixed de-composition ∆. We will construct a map (cid:15) ∈ C ( F, R ) such that δ (cid:15) is boundedly close to( g, h, i ) (cid:55)→ φ ( g ) ω ( h, i ) by “treating” this function as a cocycle on the pieces of ∆ and thenperforming the calculations of Subsection 4.2. For this, define ζ ∈ C ( F, R ) by setting ζ ( g, g (cid:48) , h ) := k (cid:88) j =1 φ ( g j ) ω ( g j +1 · · · g k g (cid:48) , h )for ∆( g ) = ( g , . . . , g k ). Analogous to Claim 4.1 we show:11 roposition 4.4. The term φ ( g ) ω ( h, i ) is equal to ζ ( g, h, i ) − ζ ( g, , hi ) + ζ ( g, , h ) for ζ ∈ C ( F, R ) are as above.Proof. Let ∆( g ) = ( g , . . . , g k ). Observe that for all j ∈ { , . . . , k } by Proposition 3.6 we havethat0 = δ φ ( g j , g j +1 · · · g k ) ω ( h, i ) = φ ( g j +1 · · · g k ) ω ( h, i ) − φ ( g j g j +1 · · · g k ) ω ( h, i ) + φ ( g j ) ω ( g j +1 · · · g k h, i ) + . . . − φ ( g j ) ω ( g j +1 · · · g k , hi ) + φ ( g j ) ω ( g j +1 · · · g k , h )Rearranging terms we see that φ ( g j · · · g k ) ω ( h, i ) − φ ( g j +1 · · · g k ) ω ( h, i ) = φ ( g j ) ω ( g j +1 · · · g k h, i ) − φ ( g j ) ω ( g j +1 · · · g k , hi )+ φ ( g j ) ω ( g j +1 · · · g k , h ) . Summing for j = 1 , . . . , k − φ ( g · · · g k ) ω ( h, i ) − φ ( g k ) ω ( h, i ) = k − (cid:88) j =1 (cid:16) φ ( g j ) ω ( g j +1 · · · g k h, i ) − φ ( g j ) ω ( g j +1 · · · g k , hi )+ φ ( g j ) ω ( g j +1 · · · g k , h ) (cid:17) . As ω was supposed to be symmetric we have that ω (1 , h ) = ω (1 , hi ) = 0 and hence φ ( g ) ω ( h, i ) = ζ ( g, h, i ) − ζ ( g, , hi ) + ζ ( g, , h ) . (cid:3) As in Subsection 4.2 define η ∈ C ( F, R ) by setting η : ( g, h ) (cid:55)→ ζ ( g, , h )and note that δ η ( g, h, i ) = η ( h, i ) − η ( gh, i ) + η ( g, hi ) − η ( g, h )= ζ ( h, , i ) − ζ ( gh, , i ) + ζ ( g, , hi ) − ζ ( g, , h ) . Using Proposition 4.4 we see that φ ( g ) ω ( h, i ) + δ η ( g, h, i )is equal to ζ ( g, h, i ) + ζ ( h, , i ) − ζ ( gh, , i ) . We will need the following properties of ζ . Proposition 4.5.
The function ζ defined as above has the following properties.(1) If u , u , v ∈ F are such that u u is reduced then ζ ( u u , , v ) − ζ ( u , u , v ) − ζ ( u , , v ) is uniformly bounded.(2) Let u , u , u , u ∈ F be elements such that u u , u u and u u are reduced and u and u do not start with the same letter. Then ζ ( u − , u u u , u − u ) + ζ ( u , u u , u − u ) is uniformly bounded.(3) Let u, v , v ∈ F such that v uv is reduced. Then(a) ζ ( u, u − v − , v uv ) − ζ ( u, u − , u ) and(b) ζ ( u − , v − , v uv ) − ζ ( u − , , u ) are uniformly bounded.Proof. In the proof of items (1)-(3) we will frequently use the following claim:
Claim 4.6.
Let u, v , v ∈ F be such that v uv is reduced, let ∆( u ) = ( u , . . . , u n ) and let R beas in Definition 3.1. Then, there are sequences ( v (cid:48) ) , ( v (cid:48) ) such that
1) for every ≤ j ≤ n − R we have that ∆( u j · · · u n v ) = ( u j , . . . , u n − R ) · ( v (cid:48) ) and(2) for every R ≤ j ≤ n we have that ∆( v · u · · · u j ) = ( v (cid:48) ) · ( u R , . . . , u j ) .Proof. For (1) let ( c ) , ( c ) , ( c ) be the c -part of the ∆-triangle of ( u, v ) and let ( r ) , ( r ) , ( r )be the r -part of the ∆-triangle of ( u, v ). Then, as uv is reduced we see that ( c ) = ∅ . Hence∆( u ) = ( c ) − · ( r ) and ∆( uv ) = ( c ) − · ( r ) · ( c ). Moreover, observe that the length of ( r ) isbounded by R . Hence all of ( u , . . . , u n − R ) lie in ( c ) − . Comparing ∆( uv ) with ∆( u ) and usingthat decompositions are infix closed (see Definition 3.1) yields (1). Item (2) can be deduced bythe same argument. (cid:3) We first show (1) of Proposition 4.5. Let u , u ∈ F be such that u u is reduced. Let the c -part of the ∆-triangle of ( u , u ) be ( c ) , ( c ) , ( c ) and let the r -part of the ∆-triangle of ( u , u )be ( r ) , ( r ) , ( r ). As u u is reduced, ( c ) has to be empty. Hence • ∆( u ) = (( c ) − · ( r )), • ∆( u ) = (( r ) · ( c )) and • ∆( u u ) = (( c ) − · ( r ) − · ( c )).Suppose that ( c i ) = ( c i, , . . . c i,n i ) and that ( r i ) = ( r i, , . . . r i,m i ) for i = 1 , ,
3. Then ζ ( u u , , v ) = n (cid:88) j =1 φ ( c − ,j ) ω ( c − ,j − · · · c − , ¯ r − ¯ c , v ) + m (cid:88) j =1 φ ( r − ,j ) ω ( r − ,j − · · · r − , ¯ c , v ) ++ n (cid:88) j =1 φ ( c ,j ) ω ( c ,j +1 · · · c ,n , v ) ζ ( u , u , v ) = n (cid:88) j =1 φ ( c − ,j ) ω ( c − ,j − · · · c − , ¯ r u , v ) + m (cid:88) j =1 φ ( r ,j ) ω ( r ,j +1 · · · r ,n u , v ) ζ ( u , , v ) = m (cid:88) j =1 φ ( r ,j ) ω ( r ,j +1 · · · r ,m ¯ c , v ) + n (cid:88) j =1 φ ( c ,j ) ω ( c ,j +1 · · · c ,n , v )and hence ζ ( u u , , v ) − ζ ( u , u , v ) − ζ ( u , , v ) = m (cid:88) j =1 φ ( r − ,j ) ω ( r − ,j − · · · r − , ¯ c , v ) − m (cid:88) j =1 φ ( r ,j ) ω ( r ,j +1 · · · r ,n u , v ) − m (cid:88) j =1 φ ( r ,j ) ω ( r ,j +1 · · · r ,m ¯ c , v )which is indeed uniformly bounded, as m , m , m ≤ R (see Definition 3.1). Since φ is ∆-decomposable, φ is uniformly bounded on pieces and as ω is a bounded function.To see (2), let u , u , u , u be as in the proposition and suppose that ∆( u ) = ( u , , . . . , u ,n ). Claim 4.7.
We have that the r -part of the ∆ -triangles of ( u ,j · · · u ,n u u , u − u ) are the samefor any j ≤ n − R and that the c -part of ( u ,j · · · u ,n u u , u − u ) is ( c (cid:48) ) · ( u − ,n , · · · , u − ,j ) , ( c (cid:48) ) , ( c (cid:48) ) for appropriate sequences ( c (cid:48) ) , ( c (cid:48) ) , ( c (cid:48) ) . In particular there is a C ∈ N such that N ∆ (( u ,j · · · u ,n u u , u − u ) , ( u ,j +1 · · · u ,n u u , u − u )) = j + C for all j ≤ n − R . roof. It follows by comparing the sequences ∆( u ,j · · · u ,n u u ) and ∆( u ,j · · · u ,n u u ) usingClaim 4.6. (cid:3) For (2) of Proposition 4.5, we calculate ζ ( u − , u u u , u − u ) + ζ ( u , u u , u − u ) = n (cid:88) j =1 φ ( u − ,j ) ω ( u ,j · · · u ,n u u , u − u ) . . . + n (cid:88) j =1 φ ( u ,j ) ω ( u ,j +1 · · · u ,n u u , u − u )= n (cid:88) j =1 φ ( u ,j ) (cid:16) ω ( u ,j +1 · · · u ,n u u , u − u ) . . . − ω ( u ,j · · · u ,n u u , u − u ) (cid:17) . Hence we conclude that ζ ( u − , u u u , u − u ) + ζ ( u , u u , u − u ) is uniformly close to n − R (cid:88) j =1 φ ( u ,j ) (cid:16) ω ( u ,j +1 · · · u ,n u u , u − u ) − ω ( u ,j · · · u ,n u u , u − u ) (cid:17) as R just depends on ∆ and φ is uniformly bounded on pieces. Now let ( s j ) j ∈ N be the sequencein Definition 3.11. By Claim 4.7, | ω ( u ,j +1 · · · u ,n u u , u − u ) − ω ( u ,j · · · u ,n u u , u − u ) | < s n + C . and hence n − R (cid:88) j =1 | ω ( u ,j +1 · · · u ,n u u , u − u ) − ω ( u ,j · · · u ,n u u , u − u ) | < S ω, ∆ . Putting those estimations together we see that ζ ( u − , u u u , u − u ) + ζ ( u , u u , u − u ) isindeed uniformly bounded.To see (3a), let u, v , v be as in the proposition and suppose that ∆( u ) = ( u , . . . , u n ). ByClaim 4.6, we see that for n − R ≤ j and R ≤ j the r -part of the ∆-triangle of ( u − j · · · u − v − , v uv )is trivial and that there are sequences ( v (cid:48) ) , ( v (cid:48) ) such that the c -part of the ∆-triangle is ∅ , ( u − j , . . . , u − ) · ( v (cid:48) ) , ( u j − , . . . u n ) · ( v (cid:48) ). Hence there are integers C , C such that N ∆ (cid:16) ( u − j · · · u − v − , v uv ) , ( u − j · · · u − , u ) (cid:17) ≥ min { j − R + C , n − j − R + C } and hence n − R (cid:88) j = R | ω ( u − j · · · u − v − , v uv ) − ( u − j · · · u − , u ) | ≤ S ω, ∆ . Finally, observe that ζ ( u, u − v − , v uv ) − ζ ( u, u − , u ) = n (cid:88) j =1 φ ( u j ) (cid:16) ω ( u − j · · · u − v − , v uv ) − ( u − j · · · u − , u ) (cid:17) and is hence uniformly close to n − R (cid:88) j = R φ ( u j ) (cid:16) ω ( u − j · · · u − v − , v uv ) − ( u − j · · · u − , u ) (cid:17) ζ ( u, u − v − , v uv ) − ζ ( u, u − , u ) may be uniformlybounded. The proof of item (3b) is analogous to the proof for item (3a). (cid:3) Analogously to Claim 4.2 we show:
Proposition 4.8.
The term φ ( g ) ω ( h, i )+ δ η ( g, h, i ) is uniformly close to ζ ( d, h, i )+ ζ ( d − , dh, i ) where d is the common -path of ( g, h ) .Proof. Let g, h, i ∈ F . Furthermore write g = t − d and h = d − t where d is the common 2-pieceof ( g, h ). We know that φ ( g ) ω ( h, i ) + δ η ( g, h, i ) is equal to ζ ( g, h, i ) + ζ ( h, , i ) − ζ ( gh, , i ) . Using Proposition 4.5, (1) we see that • ζ ( g, h, i ) is uniformly close to ζ ( t − , t , i ) + ζ ( d, d − t , i ), • ζ ( h, , i ) is uniformly close to ζ ( d − , t , i ) + ζ ( t , , i ) and • ζ ( gh, , i ) is uniformly close to ζ ( t − , t , i ) + ζ ( t , , i ).Combining these estimates we see that φ ( g ) ω ( h, i )+ δ η ( g, h, i ) is uniformly close to ζ ( d, d − t , i )+ ζ ( d − , t , i ). (cid:3) Proposition 4.9.
We have that φ ( g ) ω ( h, i ) + δ η ( g, h, i ) is uniformly close to ζ ( c, c − , c ) + ζ ( c − , , c ) where c is the common -path of ( g, h, i ) .Proof. We consider the three different cases described in Subsection 3.5 of how three elements g, h, i ∈ F can be aligned.Case A : There are elements t , . . . , t such that g = t t , h = t − t t , i = t − t as reducedwords. Then the common 2-path of ( g, h ) is t . Hence φ ( g ) ω ( h, i ) + δ η ( g, h, i ) is uni-formly close to ζ ( t , t − t t , t − t ) + ζ ( t − , t t , t − t ) . Using Proposition 4.5, (2) for u = t − , u = t , u = t , u = t we see that in this case φ ( g ) ω ( h, i ) + δ η ( g, h, i ) is uniformly bounded.Case B : There are elements t , . . . , t such that g = t t t , h = t − t , i = t − t − t as re-duced words. Then the common 2-path of ( g, h ) is t . Hence φ ( g ) ω ( h, i ) + δ η ( g, h, i ) isuniformly close to ζ ( t , t − t , t − t − t ) + ζ ( t − , t , t − t − t ) . Using Proposition 4.5, (2) for u = t − , u = ∅ , u = t , u = t − t we see that in thiscase, φ ( g ) ω ( h, i ) + δ η ( g, h, i ) is uniformly bounded.Case C : There are elements t , . . . , t and c such that g = t − ct , h = t − c − t , i = t − ct asreduced words. Then the common 2-path of ( g, h ) is ct . Hence φ ( g ) ω ( h, i ) + δ η ( g, h, i )is uniformly close to ζ ( ct , t − c − t , t − ct ) + ζ ( t − c − , t , t − ct )Using Proposition 4.5 (1) we see that • ζ ( ct , t − c − t , t − ct ) is uniformly close to ζ ( c, c − t , t − ct )+ ζ ( t , t − c − t , t − ct )and • ζ ( t − c − , t , t − ct ) is uniformly close to ζ ( t − , c − t , t − ct ) + ζ ( c − , t , t − ct ).15ence φ ( g ) ω ( h, i ) + δ η ( g, h, i ) is uniformly close to ζ ( c, c − t , t − ct ) + ζ ( c − , t , t − ct ) + (cid:16) ζ ( t , t − c − t , t − ct ) + ζ ( t − , c − t , t − ct ) (cid:17) . Using Proposition 4.5 (2) for u = t − , u = c − , u = t , u = ct we see that (cid:16) ζ ( t , t − c − t , t − ct ) + ζ ( t − , c − t , t − ct ) (cid:17) is uniformly bounded. Using item (3a)of the same proposition for u = c, v = t − , v = t we see that ζ ( c, c − t , t − ct ) isuniformly close to ζ ( c, c − , c ) and by item (3b) again for u = c, v = t − , v = t we seethat ζ ( c − , t , t − ct ) is uniformly close to ζ ( c − , , c ). Putting the above estimationstogether we see that φ ( g ) ω ( h, i )+ δ η ( g, h, i ) is uniformly close to ζ ( c, c − , c )+ ζ ( c − , , c ). (cid:3) Proposition 4.10.
The map θ : F → R defined by setting θ : g (cid:55)→ ζ ( g, g − , g ) + ζ ( g − , , g ) is a symmetric quasimorphism.Proof. We will first show the following claim:
Claim 4.11. If v, w ∈ F are such that vw is reduced then θ ( vw ) is uniformly close to θ ( v )+ θ ( w ) .Proof. Note that θ ( vw ) = ζ ( vw, w − v − , vw ) + ζ ( w − v − , , vw ). Using Proposition 4.5 (1) wesee that θ ( vw ) is uniformly close to ζ ( w, w − v − , vw ) + ζ ( v, v − , vw ) + ζ ( w − , v − , vw ) + ζ ( v − , , vw ) . By item (3) of the same proposition we see that • ζ ( w, w − v − , vw ) is uniformly close to ζ ( w, w − , w ), for u = w, v = v, v = ∅ , • ζ ( v, v − , vw ) is uniformly close to ζ ( v, v − , v ), for u = v, v = ∅ , v = w , • ζ ( w − , v − , vw ) is uniformly close to ζ ( w − , , w ) for u = w, v = v, v = ∅ and • ζ ( v − , , vw ) is uniformly close ζ ( v − , , v ) for u = v, v = ∅ , v = w .Putting things together we see that θ ( vw ) is uniformly close to (cid:16) ζ ( v, v − , v ) + ζ ( v − , , v ) (cid:17) + (cid:16) ζ ( w, w − , w ) + ζ ( w − , , w ) (cid:17) = θ ( v ) + θ ( w ) . (cid:3) Claim 4.12.
The map θ : F → R is symmetric i.e. θ ( g ) = − θ ( g − ) for all g ∈ F .Proof. We first need two easy properties of ω . Note that ω is induced by a symmetric quasimor-phism, say ω = δ ρ for some quasimorphism ρ : F → R . We have that for all u, v ∈ F ,(4) ω ( u, u − v ) = ρ ( u ) + ρ ( u − v ) − ρ ( v ) = − ρ ( u − ) − ρ ( v ) + ρ ( u − v ) = − ω ( u − , v ) . and(5) ω ( u, v ) = ρ ( u ) + ρ ( v ) − ρ ( uv ) = − ρ ( u − ) − ρ ( v − ) − ρ ( v − u − ) = − ω ( v − , u − ) . Fix g ∈ F such that ∆( g ) = ( g , . . . , g k ). Recall that in this case ∆( g − ) = ( g − k , . . . , g − ). Then • ζ ( g, g − , g ) = (cid:80) kj =1 φ ( g j ) ω ( g − j · · · g − , g ) = (cid:80) kj =1 φ ( g j ) ω ( g · · · g j , g j +1 · · · g k ) using (4)for u = g − j · · · g − and v = g j +1 · · · g k . Similarly we see that • ζ ( g − , , g ) = (cid:80) kj =1 φ ( g − j ) ω ( g · · · g j − , g j · · · g k ) = − (cid:80) kj =1 φ ( g j ) ω ( g · · · g j − , g j · · · g k )using that φ is symmetric, • ζ ( g − , g, g − ) = (cid:80) kj =1 φ ( g − j ) ω ( g − k · · · g − j , g − j − · · · g − ) = (cid:80) kj =1 φ ( g j ) ω ( g · · · g j − , g j · · · g k )where we used that φ is symmetric and (5) and16 ζ ( g, , g − ) = (cid:80) kj =1 φ ( g j ) ω ( g − k · · · g − j , g − j − · · · g − ) = − (cid:80) kj =1 φ ( g j ) ω ( g · · · g j , g j +1 · · · g k ),where we used once more (5).We hence see that θ ( g ) + θ ( g − ) = ζ ( g, g − , g ) + ζ ( g − , , g ) + ζ ( g − , g, g − ) + ζ ( g, , g − ) = 0and θ is symmetric. (cid:3) We can now prove that θ is a quasimorphism. Let g, h ∈ F and suppose that d is the common2-path of ( g, h ) i.e. g = t − d , h = d − t as reduced words for some appropriate t , t ∈ F . Then,by Claim 4.11 we have that θ ( g ) + θ ( h ) is uniformly close to θ ( t − ) + θ ( d ) + θ ( d − ) + θ ( t )and by Claim 4.12, θ ( g ) + θ ( h ) is uniformly close to θ ( t − ) + θ ( t ). By Claim 4.11 again, θ ( t − ) + θ ( t ) is uniformly close to θ ( t − t ) = θ ( gh ). Hence θ ( g ) + θ ( h ) is uniformly close to θ ( gh ) and hence θ is a quasimorphism. (cid:3) We will need the following Lemma:
Lemma 4.13.
Suppose ρ : F → R is a symmetric quasimorphism. Define κ ∈ C ( F, R ) by κ ( g, h ) = ρ ( d ) where d is the common -path of ( g, h ) . Then δ κ ( g, h, i ) is uniformly close to − ρ ( c ) where c is the common -path of ( g, h, i ) .Proof. We have to evaluate δ 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 Cayleygraph of F as seen in Figure 3.(1) (see Figure 3a): In this case there are elements t , . . . , t such that g = t t , h = t − t t , i = t − t as reduced words. It follows that • t is the common 2-path of ( h, i ), • t is the common 2-path of ( gh, i ), • t is the common 2-path of ( g, hi ) and • t is the common 2-path of ( g, h ).Hence δ κ ( g, h, i ) = ρ ( t ) − ρ ( t ) + ρ ( t ) − ρ ( t ) = 0.(2) (see Figure 3b): In this case there are elements t , . . . , t such that g = t t t , h = t − t , i = t − t − t as reduced words. It follows that • t is the common 2-path of ( h, i ), • t t is the common 2-path of ( gh, i ), • t t is the common 2-path of ( g, hi ) and • t is the common 2-path of ( g, h ).Hence δ κ ( g, h, i ) = ρ ( t ) − ρ ( t t ) + ρ ( t t ) − ρ ( t ) which is uniformly bounded as ρ isa quasimorphism.(3) (see Figure 3c): In this case there are elements t , . . . , t and c such that g = t − ct , h = t − c − t , i = t − ct as reduced words. It follows that • c − t is the common 2-path of ( h, i ), • t is the common 2-path of ( gh, i ), • t is the common 2-path of ( g, hi ) and • ct is the common 2-path of ( g, h ).Hence δ κ ( g, h, i ) = ρ ( c − t ) − ρ ( t ) + ρ ( t ) − ρ ( ct ) which is uniformly close to − ρ ( c ).This shows Lemma 4.13. (cid:3) φ ( g ) ω ( h, i ) + δ η ( g, h, i ) is uniformlyclose to ζ ( c, c − , c ) + ζ ( c − , , c ) = θ ( c ) where c is the common 3-path of ( g, h, i ) and θ : F → R islike in Proposition 4.10. Define γ ∈ C ( F, R ) via γ ( g, h ) = θ ( d ) / d is the common 2-pathof ( g, h ). Observe that ρ : g (cid:55)→ θ ( g ) / δ γ ( g, h, i ) is uniformly close to − θ ( c ) where c is the common 3-pathof ( g, h, i ). Hence φ ( g ) ω ( h, i ) + δ η ( g, h, i ) + δ γ ( g, h, i ) is uniformly bounded.4.4. Proof of Theorems A and B.
Here we will prove Theorems A and B by providing anexplicit bounded primitive for the respective cup products.
Theorem B.
Let ∆ be a decomposition of F , let φ be a ∆ -decomposable quasimorphism and let ψ be ∆ -continuous. Then [ δ φ ] (cid:94) [ δ ψ ] ∈ H b ( F, R ) is trivial. The bounded primitive is given by β , as in Theorem C for ω = δ ψ .Proof. By Theorem C we know that β defined by setting β : ( g, h, i ) (cid:55)→ φ ( g ) δ ψ ( h, i )+ δ η ( g, h, i )+ δ γ ( g, h, i ) is bounded, as δ ψ ( h, i ) is a symmetric ∆-continuous cocycle. Then we calculate δ β ( g, h, i, j ) = δ φ ( g, h ) (cid:94) δ ψ ( i, j ) . Hence β is a bounded primitive for the cup product. (cid:3) Finally, we can prove Theorem A.
Theorem A.
Let φ, ψ : F → R be two quasimorphisms on a non-abelian free group F whereeach of φ and ψ is either Brooks counting quasimorphisms on a non self-overlapping word orquasimorphisms in the sense of Rolli. Then [ δ φ ] (cid:94) [ δ ψ ] ∈ H b ( F, R ) is trivial.Proof. First suppose that both φ and ψ are Brooks quasimorphisms. Suppose that φ is countingthe non-overlapping word w ∈ F . Let ∆ w be the decomposition described in Example 3.3.By Example 2.2, we have that φ is ∆ w -decomposable. Moreover, by Proposition 3.12, ψ is∆ w -continuous. We conclude by Theorem B.If not both φ and ψ are Brooks quasimorphisms then assume without loss of generality that φ isa quasimorphism in the sense of Rolli and ψ is either a Brooks quasimorphism or a quasimorphismin the sense of Rolli. Let ∆ rolli be the decomposition described in Example 3.4. Note that φ is∆ rolli -decomposable. If ψ is a quasimorphism in the sense of Rolli, then ψ is ∆ rolli -decomposableand hence ∆ rolli -continuous by Proposition 3.12. If ψ is a Brooks quasimorphism then by thesame proposition we see that ψ is also ∆ rolli -continuous. Again we may conclude by applyingTheorem B. (cid:3) References [BFH16] Michelle Bucher, Roberto Frigerio, and Tobias Hartnick. A note on semi-conjugacy for circle actions.
Enseign. Math. , 62(3-4):317–360, 2016.[BM18] Michelle Bucher and Nicolas Monod. The cup product of Brooks quasimorphisms.
Forum Math. ,30(5):1157–1162, 2018.[Bro81] Robert Brooks. Some remarks on bounded cohomology. In
Riemann surfaces and related topics: Proceed-ings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) , volume 97of
Ann. of Math. Stud. , pages 53–63. Princeton Univ. Press, Princeton, N.J., 1981.[Cal09] Danny Calegari. scl , volume 20 of
MSJ Memoirs . Mathematical Society of Japan, Tokyo, 2009.[FPS15] R. Frigerio, M. B. Pozzetti, and A. Sisto. Extending higher-dimensional quasi-cocycles.
J. Topol. ,8(4):1123–1155, 2015.[Fri17] Roberto Frigerio.
Bounded cohomology of discrete groups , volume 227 of
Mathematical Surveys andMonographs . American Mathematical Society, Providence, RI, 2017.[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. Gro82] Michael Gromov. Volume and bounded cohomology.
Inst. Hautes ´Etudes Sci. Publ. Math. , (56):5–99(1983), 1982.[HO13] Michael Hull and Denis Osin. Induced quasicocycles on groups with hyperbolically embedded subgroups.
Algebr. Geom. Topol. , 13(5):2635–2665, 2013.[L¨oh17] Clara L¨oh. A note on bounded-cohomological dimension of discrete groups.
J. Math. Soc. Japan ,69(2):715–734, 2017.[Mon06] Nicolas Monod. An invitation to bounded cohomology. In
International Congress of Mathematicians.Vol. II , pages 1183–1211. Eur. Math. Soc., Z¨urich, 2006.[Rol09] P. Rolli. Quasi-morphisms on Free Groups.
ArXiv e-prints, 0911.4234 , November 2009.[Som97] Teruhiko Soma. Bounded cohomology and topologically tame Kleinian groups.
Duke Math. J. , 88(2):357–370, 1997.
Department of Mathematics, University of Oxford
E-mail address , N. Heuer: [email protected]@maths.ox.ac.uk