Trivial cup products in bounded cohomology of the free group via aligned chains
aa r X i v : . [ m a t h . G R ] F e b TRIVIAL CUP PRODUCTS IN BOUNDED COHOMOLOGY OFTHE FREE GROUP VIA ALIGNED CHAINS
SOFIA AMONTOVA AND MICHELLE BUCHER
Abstract.
We prove that the cup product of ∆-decomposable quasimor-phisms with any bounded cohomology class of arbitrary positive degree istrivial. As a corollary we obtain that this is also the case for Brooks quasimor-phisms (in particular on selfoverlapping words) and Rolli quasimorphisms. Introduction
Despite its wide range of uses in mathematics, bounded cohomology turns outto be hard to compute in general. Even for the case of a non-abelian free group F ,while something can be said about the bounded cohomology group with trivial realcoefficients H nb ( F, R ) up to degree 3, for degrees 4 and higher it is still not knownwhether H nb ( F, R ) vanishes or not. Expecting the former to hold, we consider thefollowing weaker question: Open Problem.
For all k > is the cup product ∪ : H b ( F, R ) × H kb ( F, R ) −→ H k +2 b ( F, R ) trivial? In recent years a whole variety of examples of trivial cup products have beenexhibited in the case k = 2, namely it holds that: Theorem 1.
The cup product ∪ : H b ( F, R ) × H b ( F, R ) −→ H b ( F, R )([ δ φ ] , [ δ ψ ]) [0] vanishes if(a) (Bucher, Monod, [BM18]) φ and ψ are Brooks quasimorphisms.(b) (Heuer, [Heu17]) φ is a ∆ -decomposable quasimorphism and ψ a ∆ -contin-uous quasimorphism.(c) (Heuer, [Heu17]) Each of φ and ψ is either a Brooks quasimorphism onnon-selfoverlapping words.(d) (Fournier-Facio, [Fac20]) φ and ψ are certain Calegari quasimorphisms. Although the publication dates of Theorem 1 (a) and (b)-(c) differ, these tworesults, which have a significant overlap, were announced at the same conference in2017.In the present paper we provide an intermediate result towards answering theOpen Problem for all k > Theorem A.
For any k > it holds that ∪ : H b ( F, R ) × H kb ( F, R ) −→ H k +2 b ( F, R )([ δ φ ] , α ) [0] Supported by the Swiss National Science Foundation, NCCR SwissMAP. We refer to [Fac20] for the precise statement and the definition of Calegari quasimorphisms. if φ is a ∆ -decomposable quasimorphism and α ∈ H kb ( F, R ) is arbitrary. Even in the case of k = 2, our Theorem is a priori stronger than Theorem 1, sinceit is not known whether every quasimorphism is ∆-decomposable, ∆-continuous ora Calegari quasimorphism. Moreover for k = 3, since both H b ( F, R ) and H b ( F, R )are known to be infinite dimensional (see [Bro81], [Gri95] as well as [Som97]), ourTheorem gives infinitely many examples of the triviality of the cup product of nontrivial classes.Important examples of ∆-decomposable quasimorphisms are the classical Brooksquasimorphisms on non-selfoverlapping words and Rolli quasimorphisms. For Brooksquasimorphisms on selfoverlapping words we are not aware of a construction as∆-decomposable quasimorphisms. However, in order to include this case also asa corollary of Theorem A, we shall modify the notion of ∆-decompositions from[Heu17] and adapt the proof of Theorem A accordingly to obtain: Corollary B.
For any k > it holds that ∪ : H b ( F, R ) × H kb ( F, R ) −→ H k +2 b ( F, R )([ δ φ ] , α ) [0] if φ is a Brooks quasimorphism or a Rolli quasimorphism and α ∈ H kb ( F, R ) isarbitrary. Besides, similar to the generalisation of Heuer’s vanishing result in Theorem 1(b)by Fournier-Facio in Theorem 1(d), one could generalize Theorem A to Fournier-Facio’s setting [Fac20] as well.Our strategy to prove Theorem A is to apply the geometric proof from [Heu17] tothe setting of aligned cochains from [BM19] instead of inhomogeneous cochains. Onthe one hand, Nicolaus Heuer introduced in [Heu17] the notion of ∆-decompositionwhich abstracts the geometric properties of both Brooks quasimorphisms on non-selfoverlapping words and Rolli quasimorphisms needed to prove Theorem 1(b). Onthe other hand, the aligned chain complex is a geometric construction introducedby Monod and the second author in [BM19] to prove the vanishing of the continuousbounded cohomology of automorphism groups of trees and also the vanishing of thecup product of Brooks quasimorphisms (Theorem 1(a)) in [BM18]. In our setting,the aligned cochain complex allows for consequent simplifications in the proof byHeuer to the point where we can relax the hypothesis on the second factor of thecup product in Theorem 1(b) significantly; that is, from being a ∆-continuous quasi-morphism, and hence an a priori strict subset of H b ( F ), to an arbitrary cohomologyclass in any degree k > Remark.
There was a general principle that the bounded cohomology of free groupsis a receptacle for the bounded cohomology of all finitely generated groups. Moreprecisely, the following question is attributed to Gromov: Let G be a finitely gener-ated group with k generators. Does the natural surjection from the free group F k of rank k onto G , induce an injection on bounded cohomology,H ∗ b ( G, R ) ֒ → H ∗ b ( F k , R )?Theorem A shows that this is not the case in arbitrary degrees greater or equal to4. Indeed, let G be a finitely generated group with k generators such that thereexists a non trivial class α ∈ H nb ( G, R ) in degree n >
2. Let further β ∈ H b ( F , R )be a nontrivial cohomology class coming from a ∆-decomposable quasimorphism.Then the surjection F k → F × G does not induce an injective mapH nb ( F × G, R ) −→ H nb ( F k +2 , R ) UP PRODUCTS IN BOUNDED COHOMOLOGY VIA ALIGNED CHAINS 3 in degree 2 + n , since the class 0 = α ∪ β ∈ H nb ( F × G, R ) is mapped to 0 byTheorem A. (For n = 2 and G a free group, this already follows from Theorem 1.)The paper is structured as follows: In Section 2 we recall a selection of ba-sic notions on bounded cohomology theory. Section 3 introduces the inhomo-geneous aligned cochain complex, where we follow [BM19]. Then to define ∆-decompositions, the induced ∆-decomposable quasimorphisms and their specialcases of Brooks quasimorphisms and Rolli quasimorphisms, we follow [Heu17] inSection 4. This culminates in Sections 5 and 6 where we prove Theorem A andCorollary B. Acknowledgements.
This paper is part of a master thesis project by the firstauthor under the supervision of the second author within the framework of theMaster Class program organized by NCCR SwissMAP. We are especially gratefulto Nicolas Monod for his interesting feedback while refereeing the thesis and toFrancesco Fournier-Facio for several useful comments.2.
Preliminaries
Inhomogeneous cochains and bounded cohomology.
We give the defi-nition of (bounded) cohomology of a group G with trivial real coefficients by usingthe inhomogeneous resolution. For all n > n ( G, R ) = { ϕ : G n → R } andC nb ( G, R ) = { ϕ ∈ C n ( G, R ) | || ϕ || ∞ < ∞} , where || ϕ || ∞ := sup {| ϕ ( g , . . . , g n ) | | ( g , . . . , g n ) ∈ G n } ∈ [0 , + ∞ ] , the set of cochains and bounded cochains respectively. The coboundary map δ n :C n ( G, R ) → C n +1 ( G, R ) is defined by δ n ( ϕ )( g , . . . , g n +1 ) = ϕ ( g , . . . , g n +1 )+ n X j =1 ( − j ϕ ( g , . . . , g j g j +1 , . . . , g n +1 )+ ( − n +1 ϕ ( g , . . . , g n )and it restricts to the bounded coboundary map δ n : C nb ( G, R ) → C n +1 b ( G, R ). Since δ n +1 ◦ δ n = 0 holds for all n >
0, we have that (cid:16) C ∗ ( b ) ( G, R ) , δ ∗ (cid:17) is a (bounded) cochain complex. Moreover, the (bounded) n -cochain ϕ is called a (bounded) n -cocycle if δ n +1 ϕ = 0, and a (bounded) n -coboundary if there exists a (bounded) ( n − -cochain ψ such that δ n − ψ = ϕ . The (bounded) cohomology of G is thendefined as the cohomology of the (bounded) cochain complex. More precisely: Definition 2.
The n -th (bounded) cohomology of G with trivial coefficients is de-fined as H n ( b ) ( G, R ) := Ker h C n ( b ) ( G, R ) → C n +1( b ) ( G, R ) i Im h C n − b ) ( G, R ) → C n ( b ) ( G, R ) i . To simplifiy the notation we will from now on denote by δ ∗ both the boundedand the unbounded coboundary maps. SOFIA AMONTOVA AND MICHELLE BUCHER
Cup product.
We introduce the operation that allows to obtain elementsof (bounded) cohomology groups of higher degrees from elements of (bounded)cohomology groups of lower degrees.
Definition 3.
The cup product is the map given by ∪ : H n ( G, R ) × H m ( G, R ) −→ H n + m ( G, R )([ f ] , [ g ]) [ f ] ∪ [ g ] , where [ f ] ∪ [ g ] is represented by the cocycle f ∪ g ∈ C n + m ( G, R ) that is defined asfollows f ∪ g : ( g , . . . , g n , g n +1 , . . . , g n + m ) f ( g , . . . , g n ) · g ( g n +1 , . . . , g n + m ) . This map induces a well defined map on bounded cohomology that we also call thecup product ∪ : H nb ( G, R ) × H mb ( G, R ) −→ H n + mb ( G, R ) . Quasimorphisms on a free group.
We recall that for any group G a quasi-morphism is a map φ : G → R for which there exists a constant D > g,h ∈ G | φ ( g ) + φ ( g ) − φ ( gh ) | < D. There is a straightforward connection to bounded cohomology: for any such φ thesupremum of || δ φ || ∞ is bounded by D and therefore δ φ ∈ C b ( G, R ) is a cocycleand represents a cohomology class in H b ( G, R ).For the case of a non-abelian free group F , the study of H b ( F, R ) boils down tostudying the set of quasimorphisms on F which we denote by QM( F ), since thefollowing holds H b ( F, R ) ∼ = QM( F )[Hom( F, R ) ⊕ ℓ ∞ ( F )] , where Hom( G, R ) and ℓ ∞ ( G ) denote the set of homomorphisms and real-valuedbounded maps on G respectively. For instance, with help of various explicit con-structions of certain families of quasimorphisms it was proved that H b ( F, R ) isnon-vanishing and moreover infinite-dimensional, e.g. Brooks quasimorphisms , dueto Brooks in [Bro81] (see also [Gri95]), and
Rolli quasimorphisms , due to Rolli in[Rol09].
Example 4.
Brooks quasimorphisms.
Let S be a free generating set of F and w beany reduced word in F . Then a Brooks quasimorphism on the word w is a function φ : F → Z defined by φ w ( g ) = { subwords w in g } − { subwords w − in g } , for all g ∈ F .
In accordance with the terminology in [Cal09], we say that φ w ( g ) is a • small Brooks quasimorphisms if only disjoint occurences of w and w − arecounted, • big Brooks quasimorphisms if overlapping occurences of w and w − are alsocounted.Now, any word w in the non-abelian free group F is either a selfoverlapping ornon-selfoverlapping word. In particular w is called a non-selfoverlapping word ifthere do not exist words s and m with s non-trivial such that w = sms as a reducedword. Otherwise we say that w is a selfoverlapping word . Note that if w is non-selfoverlapping, then the corresponding big and small Brook quasimorphisms areequal.To simplify the terminology, we refer to big Brooks quasimorphisms as “Brooksquasimorphisms”. UP PRODUCTS IN BOUNDED COHOMOLOGY VIA ALIGNED CHAINS 5
Example 5.
Rolli quasimorphisms . Let S = { x , . . . , x n } ∪ { x − , . . . , x − n } be afree symmetric generating set of F . A Rolli quasimorphism is a map φ : F → Z defined as g k X j =1 λ n j ( m j ) , where • λ , . . . , λ n are bounded alternating functions. • m , . . . , m k are the unique integers such that it holds that g = x m j · . . . · x m k j k , where no consecutive j i are the same.We encounter another viewpoint for these examples of quasimorphisms in Section4. 3. The inhomogeneous aligned cochain complex
In this section we briefly present the inhomogeneous version of aligned cochaincomplexes of non-abelian free groups first introduced in [BM19] in the homogeneouscontext to prove the vanishing of continuous bounded cohomology for certain auto-morphisms groups of a tree.Let again F = F n be a free group of rank n and S be a free and symmetricgenerating set of F .For any n > B n = { ( g , . . . , g n ) ∈ F n | g i g i +1 is reduced with respect to S, g i = id } . Geometrically, this means that in the Cayley graph of G with respect to S thecorresponding homogenous ( n + 1)-tuple (id , g , g g , . . . , g · . . . · g n ) of distinctvertices is ordered on the geodesic segment between id and g · . . . · g n : · · · id g g g n g g g g · . . . · g n − g · . . . · g n Definition 6.
The (inhomogeneous) aligned cochain complex is defined as the setof functions A n ( F, R ) = { ϕ : B n −→ R } . The bounded (inhomogeneous) aligned chain complex A nb ( F, R ) consists of boundedaligned cochains: A nb ( F, R ) = { ϕ ∈ A n ( F, R ) | k ϕ k ∞ < ∞} . Definition 7.
The (bounded) (inhomogeneous) aligned alternating cochain complex A n + , ( b ) ( F, R ) is defined as the set of functions ϕ ∈ A n ( b ) ( F, R ) such that ϕ ( g , . . . , g n ) = ( − ⌈ n/ ⌉ ϕ ( g − n , . . . , g − )for every ( g , . . . , g n ) ∈ B n . λ : F → R is alternating , if it holds that λ ( − n ) = − λ ( n ) for all n ∈ N . SOFIA AMONTOVA AND MICHELLE BUCHER
These four complexes are endowed with their inhomogenous coboundary δ n : A n (+) , ( b ) ( F, R ) −→ A n (+) , ( b ) ( F, R ) . There is a chain map A : A n ( b ) ( F, R ) −→ A n + , ( b ) ( F, R )defined as A ( ϕ )( g , . . . , g n ) = 12 (cid:16) ϕ ( g , . . . , g n ) + ( − ⌈ n/ ⌉ ϕ ( g − n , . . . , g − ) (cid:17) , for every ϕ ∈ A n ( b ) ( F, R ) and ( g , . . . , g n ) ∈ B n .Since B n is a subset of F n , we have by restriction natural maps r : C n ( b ) ( F, R ) −→ A n ( b ) ( F, R )which commute with the coboundary maps. It is proven in [BM18, Proposition 8]that the composition of the chain maps A ◦ r : C nb ( F, R ) −→ A n + ,b ( F, R )induces an isomorphism between the bounded cohomology of the free group F andthe cohomology of the cocomplex (cid:16) A ∗ ( b ) ( F, R ) , δ ∗ (cid:17) .Let now α , α ∈ H ∗ b ( F, R ) be represented by inhomogenous cocoycles ω ∈ C nb ( F, R ) and ω ∈ C mb ( F, R ). To show that the cup product α ∪ α vanishes it ishence sufficient to show that Ar ( ω ∪ ω ) = A ( r ( ω ) ∪ r ( ω ))is a coboundary in A n + n + ,b ( F, R ). This immediately follows if r ( ω ) ∪ r ( ω ) is acoboundary in A n + n b ( F, R ). Indeed, if r ( ω ) ∪ r ( ω ) = δ ϕ, for some ϕ ∈ A n + n − b ( F, R ) then Ar ( ω ) ∪ r ( ω ) = A δ ϕ = δ ( Aϕ ) . This is the strategy we will use to prove Theorem A, by proving the slightly strongerTheorem A’, stated at the beginning of Section 5.4. ∆ -decompositions and induced quasimorphisms of the free group
We recall the notion of ∆-decomposable quasimorphisms introduced by Heuer asa way of uniformizing the geometric properties of both Brooks quasimorphisms onnon-selfoverlapping words and Rolli quasimorphims. For more details, the readeris referred to [Heu17, Section 3]4.1. ∆ -decompositions.
Let F be a free group on a free and symmetric generatingset S . For any symmetric set P ⊂ F satisfying id
6∈ P we call its elements pieces .Furthermore, let P ∗ be the set of finite sequences of pieces, more precisely: P ∗ := { ( g , . . . , g K ) | ∀ g j ∈ P , j K ∞} ∪ { empty sequence } . For any g = ( g , . . . , g K ) in P ∗ , we shall write | g | for the length K of the tuple( g , . . . , g K ), not to be confused with the length of g with respect to the generatingset S . Definition 8.
A ∆ -decomposition of F into the pieces P is a map∆ : F −→ P ∗ g ( g , . . . , g K ) , that satisfies the following properties [A] for single words: For every g ∈ F with ∆( g ) = ( g , . . . , g K ) we have thefollowing: UP PRODUCTS IN BOUNDED COHOMOLOGY VIA ALIGNED CHAINS 7 • g = g · . . . · g K as a reduced word, • ∆( g − ) = ( g − K , . . . , g − ), • ∆( g i · . . . · g j ) = ( g i , . . . , g j ) for all 1 i j K . [B] for products of two words: There exists a constant
R > g, h ∈ F , the sequences ∆( g ), ∆( h ) and ∆( gh ) written in the followingform ∆( g ) = ∆( c − )∆( r )∆( c ) , ∆( h ) = ∆( c − )∆( r )∆( c ) and∆( gh ) = ∆( c − )∆( r − )∆( c ) , where ∆( c i ) , ∆( r i ) ∈ P ∗ so that ∆( c i ) is of maximal length, satisfy | ∆( r i ) | R, for i = 1 , , [B] of a given ∆-decomposition, we say for short that g and h form a ( g, h ) -triangle . This reflectsthe geometric picture of property [B] : c r r r g hgh c c Moreover, we refer to the part of the ( g, h )-triangle consisting of c , c and c asthe c -part and the part consisting of r , r and r as the (bounded) r -part .4.2. ∆ -decomposable quasimorphisms. Assigning a real value to each elementof the pieces P in an appropriate way, ensures that any ∆-decomposition of F induces quasimorphisms on F . This is done in the following way: Definition 9.
Let ∆ be a decomposition with pieces P and let λ ∈ ℓ ∞ alt ( P ), i.e. analternating bounded map on P . Then the map φ λ, ∆ : F → R defined by g K X j =1 λ ( g j ) , where ∆( g ) = ( g , . . . , g K ), is called a ∆ -decomposable quasimorphism . Remark.
For any p ∈ P and λ as in Definition 9, it holds by property [A] of the∆-decomposition, that φ λ, ∆ ( p ) = λ ( p ).This is indeed a legitimate name for this map which can be justified by usingproperty [B] of the ∆-decomposition: Proposition 10 (Heuer, [Heu17]) . The map φ λ, ∆ is a quasimorphism. SOFIA AMONTOVA AND MICHELLE BUCHER
Proof.
Let g, h ∈ F and let the r - and c -part of the ( g, h )-triangle be as in thedefinition of the ∆-decomposition. By using the fact that the map λ in Definition9 is alternating on pieces, we can check that φ λ, ∆ ( g ) = − φ λ, ∆ ( c ) + φ λ, ∆ ( r ) + φ λ, ∆ ( c ) ,φ λ, ∆ ( h ) = − φ λ, ∆ ( c ) + φ λ, ∆ ( r ) + φ λ, ∆ ( c ) and φ λ, ∆ ( gh ) = − φ λ, ∆ ( c ) − φ λ, ∆ ( r ) + φ λ, ∆ ( c ) . This implies that δ φ λ, ∆ ( g, h ) solely depends on the r -part of the ( g, h )-triangle.In particular we have that: (cid:12)(cid:12) δ φ λ, ∆ ( g, h ) (cid:12)(cid:12) = (cid:12)(cid:12) φ λ, ∆ ( g ) + φ λ, ∆ ( h ) − φ λ, ∆ ( gh ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) | r | X j =1 λ ( r ,j ) + | r | X j =1 λ ( r ,j ) + | r | X j =1 λ ( r ,j ) (cid:12)(cid:12)(cid:12)(cid:12) R || λ || ∞ , where ∆( r i ) = ( r i, , . . . , r i, | r i | ), for i = 1 , ,
3. Note that for the last inequality wemake use of the fact that the r -part is bounded (i.e | r | , | r , | r | K ) and that λ is bounded on pieces. (cid:3) By taking P := S and λ arbitrary, one can recover all homomorphisms on thefree group. We now give examples of ∆-decomposable quasimorphisms: Brooksquasimorphisms on non-selfoverlapping words and Rolli quasimorphisms. Example 11. ∆ w -decomposition and Brooks quasimorphisms on non-selfoverlap-ping words w . Every word g ∈ F can be represented uniquely as g = u w ǫ u · . . . · u k − w ǫ k − u k , where each possibly empty u j does not contain w or w − as subwords and ǫ j ∈{− , +1 } . In this setting, define the pieces to be P w = { w, w − } ∪ { u ∈ F r { id } | u does not contain w or w − as subwords } . Then the
Brooks-decomposition on the word w is the map∆ w : F → P ∗ w g ( u , w ǫ , u , . . . , u k − , w ǫ k − , u k ) , for g as above. The map λ : P w → R given by λ ( p ) := , if p = w, − , if p = w − , , else , induces the ∆ w -decomposable quasimorphism φ λ, ∆ w , namely precisely the Brooksquasimorphism on w . Example 12. ∆ Rolli -decomposition and Rolli quasimorphisms. If F is a free groupof rank n and S = { x , . . . , x n } ∪ { x − , . . . , x − n } then any non-trivial g ∈ F can beuniquely represented in the following form g = x m j · . . . · x m k j k , where all m j are non-zero and no consecutive j i ∈ { , . . . , n } are identical. If wedefine the pieces to be P Rolli = { x mj | j ∈ { , . . . , n } , m ∈ Z r { }} then we definethe ∆ Rolli -decomposition as the map∆
Rolli : F −→ P ∗ Rolli g ( x m j , . . . , x m k j k ) , UP PRODUCTS IN BOUNDED COHOMOLOGY VIA ALIGNED CHAINS 9 for g as above. Let λ , . . . , λ n ∈ ℓ ∞ alt ( Z ), namely bounded functions λ j : Z → R satisfying λ j ( − n ) = − λ j ( n ). Then the map λ : P Rolli −→ R x mj λ j ( m )induces the quasimorphism φ λ, ∆ Rolli , namely precisely the Rolli quasimorphism.5.
Proof of Theorem A
Due to the discussion in Section 3 we can prove Theorem A by showing thatthe cup product of the restriction to aligned cochains of cocycles representing thecohomology classes in consideration is a coboundary. Thus Theorem A is a directconsequence of
Theorem A’.
Let φ be a ∆-decomposable quasimorphism and for any k > ω ∈ A kb ( F, R ). Then there exists β ∈ A k +1 b ( F, R ) such that δ β = δ φ ∪ ω. Proof.
Let φ and ω be as in Theorem A’ and fix a ∆-decomposition on pieces P for which φ is ∆-decomposable. Further let η ∈ A k ( F, R ) be defined by(1) η : ( g, h , . . . , h k − ) K X j =1 φ ( g j ) ω ( z j ( g ) , h , . . . , h k − ) , where ∆( g ) = ( g , . . . , g K ) and z j ( g ) := g j +1 · . . . · g K for j = 1 , . . . , K , and β ∈ A k +1 b ( F, R ) by β = φ ∪ ω + δ η. By construction, δ β = δ φ ∪ ω , so that we only need to show that β is bounded. Let( g, h , . . . , h k ) in B n be arbitrary. Then by definition, we have that β ( g, h , . . . , h k ) = φ ( g ) ω ( h , . . . , h k )(2) + n η ( h , . . . , h k ) − η ( gh , h . . . , h k ) − k − X i =1 ( − i η ( g, h , . . . , h i h i +1 , . . . , h k ) − ( − k η ( g, h , . . . , h k − ) o . Let ∆( g ) = ( g , . . . , g K ) , ∆( h ) = ( h , , . . . , h ,L ) and ∆( gh ) = (( gh ) , . . . , ( gh ) M )be the ∆-decompositions of g , h and gh . By expanding η as in its definition (1)and by recalling that φ as a ∆-decomposable quasimorphism satisfies φ ( g ) = K X j =1 φ ( g j ) , we obtain β ( g, h , . . . , h k )(3) = K X j =1 φ ( g j ) ω ( h , . . . , h k )+ L X j =1 φ ( h ,j ) ω ( z j ( h ) , h , . . . , h k ) − M X j =1 φ (( gh ) j ) ω ( z j ( gh ) , h , . . . , h k ) − k − X i =1 ( − i K X j =1 φ ( g j ) ω ( z j ( g ) , h , . . . , h i h i +1 , . . . , h k ) − ( − k K X j =1 φ ( g j ) ω ( z j ( g ) , h , . . . , h k − ) . Using the cocycle relation ω ( z j ( g ) h , . . . , h k ) = ω ( h , . . . , h k )(4) − k − X i =1 ( − i ω ( z j ( g ) , h , . . . , h i h i +1 , . . . , h k ) − ( − k ω ( z j ( g ) , h , . . . , h k − ) , the expression in (3) reduces to β ( g, h , . . . , h k ) = K X j =1 φ ( g j ) ω ( z j ( g ) h , . . . , h k )(5) + L X j =1 φ ( h ,j ) ω ( z j ( h ) , h , . . . , h k ) − M X j =1 φ (( gh ) j ) ω ( z j ( gh ) , h , . . . , h k ) . We now decompose the ( g, h )-triangle into its c - and r -parts as in Definition 8.Observe that since we are working with aligned chains, we have c = id so that∆( g ) = ∆( c − )∆( r ) , ∆( h ) = ∆( r )∆( c ) and∆( gh ) = ∆( c − )∆( r − )∆( c ) . We set K ′ = | c | and L ′ = | r | and note that since the lengths of the r -parts arebounded by R we have | r | = K − K ′ R, | r | = L ′ R and | r | = M − K ′ − ( L − L ′ ) R. Furthermore, by comparing the c -parts, we see that ∆( g ) and ∆( gh ) agree atthe head, while ∆( h ) and ∆( gh ) agree at the tail of their corresponding ∆-decompositions (see figure below). More precisely, we have g j = ( gh ) j , ∀ j K ′ ,h j = ( gh ) j , ∀ L ′ j L. Finally, if 1 j K ′ , then z j ( g ) h = z j ( gh ), and if L ′ j L then z j ( h ) = z j ( gh ). This allows us to further reduce the expression for β in (5) UP PRODUCTS IN BOUNDED COHOMOLOGY VIA ALIGNED CHAINS 11 c r r r g h gh c · · · g g K ′ h ,L ′ +1 h ,L · · · ( gh ) = ( gh ) K ′ = ( gh ) L ′ +1 = ( gh ) L = to β ( g, h , . . . , h k ) = K X j = K ′ +1 φ ( g j ) ω ( z j ( g ) h , . . . , h k )+ L ′ X j =1 φ ( h ,j ) ω ( z j ( h ) , h , . . . , h k ) − M − L + L ′ X j = K ′ +1 φ (( gh ) j ) ω ( z j ( gh ) , h , . . . , h k ) , which is bounded since each of the three sums has at most R summands, eachof which is bounded since ω is bounded and φ is bounded on pieces of the ∆-decomposition. (cid:3) Proof of Corollary B
As we have seen in Section 4, Brooks quasimorphisms on non-selfoverlappingwords and Rolli quasimorphisms are special cases of ∆-decomposable quasimor-phisms, so that it just remains to prove Corollary B, which is in fact Theorem A’for Brooks quasimorphisms on selfoverlapping words.We fix a free and symmetric generating set of F and let w ∈ F be a selfover-lapping word. Then there exists s ∈ F non-trivial and m ∈ F such that w = sms where s is the prefix and suffix of w of maximal possible length. We can then definethe following pieces P sms = (cid:8) s, s − , m, m − (cid:9) ∪ u ∈ F r { id } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u does not contain( sm ) n s or h ( sm ) n s i − as a subword for all n > u = s, s − , m, m − . The ∆ sms -decomposition for the selfoverlapping word w = sms as above, is themap given by ∆ sms : F → P ∗ sms g ( g , . . . , g K ) , where g = g · . . . · g K is a reduced word and unique. We now show how thisdecomposition induces Brooks quasimorphisms on selfoverlapping words. For thisdefine new pieces consisting of triplets of words in P sms , namely P sms := { G = abc | a, b, c ∈ P sms } . Further, we set the map e λ : P sms → R to be given by e λ ( P ) := , if P = sms, − , if P = s − m − s − , , else , so that e λ ∈ ℓ ∞ ( P sms ), meaning that e λ is bounded and alternating. Finally, similarto Definition 9, we define the map φ : F → R by g K − X j =1 e λ ( g j g j +1 g j +2 ) , where ∆ sms ( g ) = ( g , . . . , g K ). We obtain the following proposition whose proof isimmediate from the construction above: Proposition 13.
The map φ : F → R is the Brooks quasimorphism on the self-overlapping word w = sms . To prove Theorem A’ (and hence Corollary B) for any Brooks quasimorphism φ over some self overlapping word w = sms , we modify slightly the definition of η ∈ A k ( F, R ) in the proof of Theorem A’ by η : ( g, h , . . . , h k − ) K − X j =1 φ ( g j g j +1 g j +2 ) ω ( z j +2 ( g ) , h , . . . , h k − ) , where ∆ sms ( g ) = ( g , . . . , g K ) and z j +2 ( g ) := g j +3 · . . . · g K for j = 1 , . . . , K − β ∈ A k +1 b ( F, R ) by β = φ ∪ ω + δ η. To show that β is bounded, we follow further the steps in Section 5: We expand β = φ ∪ ω + δ η as in (2) and use that the Brooks quasimorphism φ can be expressedas φ ( g ) = K − X j =1 φ ( g j g j +1 g j +2 ) , resulting in a slightly modified form of (3), which in turn allows to express β viathe cocycle relation (4) as follows: β ( g, h , . . . , h k ) = K − X j =1 φ ( g j g j +1 g j +2 ) ω ( z j ( g ) h , . . . , h k )(6) + L − X j =1 φ ( h ,j h ,j +1 h ,j +2 ) ω ( z j ( h ) , h , . . . , h k ) − M − X j =1 φ (( gh ) j ( gh ) j +1 ( gh ) j +2 ) ω ( z j ( gh ) , h , . . . , h k ) . In the same manner as in Section 5, we can decompose the ( g, h )-triangle into its c - and r -parts and observe that ∆ sms ( g ) and ∆ sms ( gh ) agree at the head, while∆ sms ( h ) and ∆ sms ( gh ) agree at the tail of their corresponding ∆ sms -decompositions. UP PRODUCTS IN BOUNDED COHOMOLOGY VIA ALIGNED CHAINS 13
The expression for β in (6) is thus equal to β ( g, h , . . . , h k ) = K − X j = K ′ − φ ( g j g j +1 g j +2 ) ω ( z j ( g ) h , . . . , h k )+ L ′ X j =1 φ ( h ,j h ,j +1 h ,j +2 ) ω ( z j ( h ) , h , . . . , h k ) − M − L + L ′ − X j = K ′ − φ (( gh ) j ( gh ) j +1 ( gh ) j +2 ) ω ( z j ( gh ) , h , . . . , h k ) , which is, as in the proof of Theorem A’, bounded. This concludes the proof ofCorollary B. References [Bro81] Robert Brooks,
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