Cosets of monodromies and quantum representations
CCOSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS
RENAUD DETCHERRY AND EFSTRATIA KALFAGIANNI
Abstract.
We use geometric methods to show that given any 3-manifold M , and g a sufficiently large integer, the mapping class group Mod(Σ g, ) contains a coset of anabelian subgroup of rank (cid:98) g (cid:99) , consisting of pseudo-Anosov monodromies of open-bookdecompositions in M. We prove a similar result for rank two free cosets of Mod(Σ g, ) . These results have applications to a conjecture of Andersen, Masbaum and Ueno aboutquantum representations of surface mapping class groups. For surfaces with boundary,and large enough genus, we construct cosets of abelian and free subgroups of theirmapping class groups consisting of elements that satisfy the conjecture. The mappingtori of these elements are fibered 3-manifolds that satisfy a weak form of the Turaev-Viroinvariants volume conjecture. Introduction
It has been known since Alexander [1] that any closed, oriented 3-manifold admitsan open book decomposition, and Myers [32] proved that there is one with connectedbinding. In recent years open book decompositions received attention as they are closelyrelated to contact geometry through the work of Giroux [20].For a compact oriented surface Σ := Σ g,n with genus g and n boundary components,let Mod(Σ) denote the mapping class group of Σ. For f ∈ Mod(Σ) and a subgroup H ofMod(Σ), we will say that f H is a rank k abelian coset if H is abelian of rank k . Similarly,we will say that f H is a free coset if H is free and non-abelian. Moreover in the wholetext, we use the term ”free group” for free non-abelian group.Given a closed, oriented 3-manifold M , it makes sense to ask how large is the set ofmapping classes in Mod(Σ g, ) that occur as monodromies of open book decompositionsof M . One way to quantify this size would be by the rank of abelian subgroups whosecosets occur as monodromies in M . The main result of this article, Theorem 1.1 below,gives a lower bound on the size. It says that for g > g , there is at least an abelian cosetof rank (cid:39) g/ , for any given M . At this writing we do not know of an upper-bound ofthis size other than the maximal rank of any abelian subgroup of Mod(Σ g, ) which isknown to be 3 g − . [30] Theorem 1.1.
Let M be a closed, orientable 3-manifold. There is an integer g = g ( M ) > , such that for any genus g (cid:62) g , there is a rank (cid:98) g (cid:99) abelian coset of Mod(Σ g, ) Date : October 16, 2020.Kalfagianni’s research was partially supported by NSF grants DMS-1708249 and DMS-2004155 anda grant from the Institute for Advanced Study School of Mathematics.Detcherry’s research was supported by a postdoctoral fellowship from the Max Planck Institute forMathematics. a r X i v : . [ m a t h . G T ] O c t RENAUD DETCHERRY AND EFSTRATIA KALFAGIANNI consisting of pseudo-Anosov mapping classes all of which occur as monodromies of openbook decompositions in M. Moreover, for any g (cid:62) g , there is a free non-abelian coset of Mod(Σ g +4 , ) , that containsinfinitely many pseudo-Anosov mapping classes, consisting of monodromies of fiberedknots in M. It has been long known that any closed, oriented 3-manifold admits open book decom-positions with connected binding and pseudo-Anosov monodromy. This is the startingpoint of our proof of Theorem 1.1. First, inspired by a construction of Colin and Honda[9], we show that any open book decomposition with pseudo-Anosov monodromy can bestabilized to one with pseudo-Anosov monodromy and so that the pages of the decompo-sition support several
Stallings twists (see Theorem 4.10). To find these stabilizations weuse techniques from the study of the geometry of curve complexes of surfaces. Other keyingredients of our proof are a refined version of a result of Long-Morton and Fathi (seeTheorem 3.4) on composing pseudo-Anosov mapping classes with products of powers ofDehn twists, and a result of Hamidi-Tehrani [22] on generating free subgroups of mappingclass groups.Given an abelian coset f H k in Mod(Σ g, ) one can ask in which 3-manifolds can theelements of f H k occur as monodromies of open book decompositions of M . In thisdirection we have the following contribution. Corollary 1.2.
Given k (cid:62) , there is a constant C = C ( k ) with the following property:For any g (cid:62) k , we have a rank k abelian coset f H k in Mod(Σ g, ) , consisting ofmapping classes that cannot occur as monodromies of open book decompositions in any3-manifold with Gromov norm larger than C . The author’s interest in Theorem 1.1 and Corollary 1.2 is partly motivated by openquestions in quantum topology. Indeed as we will explain next these results have appli-cations to a conjecture of Andersen, Masbaum and Ueno [3] that relate certain quantummapping class groups representations to Nielsen-Thurston theory. The results also pro-vide constructions of fibered manifolds that satisfy a weak version of the Turaev-Viroinvariants volume conjecture [8].1.1.
Quantum topology applications.
Given an odd integer r , a primitive 2 r -th rootof unity and a coloring c : | ∂ Σ | → { , , . . . , r − } , the SO(3)-Witten-Reshetikhin-TuraevTQFT [7, 34, 39] gives a projective representation ρ r,c : Mod(Σ) → PGL d r,c ( C )called the SO(3)- quantum representation . The dimensions of the representations d r,c canbe computed using the so-called Verlinde formula, see for example [7].The quantum representations have led to many interesting applications and haveturned out to be one of the most fruitful tools coming out of quantum topology [2,19, 24, 28]. However, the geometric content of the representations is still not understood.The AMU conjecture predicts how the Nielsen-Thurston classification of mapping classesshould be reflected in the quantum representations. OSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS 3
Conjecture 1.3. (AMU conjecture [3])
A mapping class φ ∈ Mod(Σ) has pseudo-Anosovparts if and only if, for any big enough r, there is a choice of colors c of the componentsof ∂ Σ , such that ρ r,c ( φ ) has infinite order. To clarify the hypothesis of Conjecture 1.3 recall that, by the Nielsen-Thurston classi-fication, a mapping class φ of infinite order is either reducible or pseudo-Anosov. In theformer case, a power of φ fixes a collection of disjoint simple closed curves on Σ and actson the components of S cut along these curves. If the induced map is pseudo-Anosov onat least one component, then φ is said to have pseudo-Anosov parts. One direction ofConjecture 1.3 is known, that is if ρ r,c ( φ ) has infinite order, then f has pseudo-Anosovparts. It is also known that the conjecture 1.3 is true for a mapping class φ if and onlyif it is true for its pseudo-Anosov parts [3, 4]. Theorem 1.4.
For any k (cid:62) , there is a rank k abelian coset of Mod(Σ k, ) , consistingentirely of mapping classes that satisfy the AMU conjecture. Furthermore, the cosetcontains infinitely many pseudo-Anosov mapping classes. In [3] Andersen, Masbaum and Ueno verified the conjecture for Σ , . Later, San-tharoubane proved it for Σ , [35] and Egsgaard and Jorgensen [15] have partial results forpseudo-Anosov maps on Σ , n . In both cases the quantum representations can be asymp-totically connected to homological representations, and this connection is the main toolused in those results. Santharoubane [36] gave additional evidence to the conjecturefor Σ , n , by relating the asymptotics of quantum representations to MacMullen’s braidgroup representations. However, for higher genus surfaces, there is no known connec-tion between quantum and homological representations. For g (cid:62)
2, the first examplesof mappings classes that satisfy the AMU conjecture, were given by March´e and San-tharoubane in [27]. Their method allows the construction of finitely many conjugacyclasses of pseudo-Anosov examples in Mod(Σ g, ) by choosing suitable kernel elements ofthe corresponding Birman exact sequence.In [13] we presented a new approach to Conjecture 1.3 that seems to be the mostpowerful and promising strategy currently available. Our approach is to relate the AMUconjecture to a weak version of the volume conjecture stated by Chen and Yang [8]which we first addressed in [12]. Using this approach, we gave the first infinite families ofpseudo-Anosov examples that satisfy the AMU conjecture. More specifically, we showedthat, for g (cid:62) n (cid:62) n = 2 and g (cid:62) , in Mod(Σ g,n ) there exists infinitely manynon-conjugate, non power of each other, pseudo-Anosov maps that satisfy the conjecture.Kumar [25], also using the approach of [13], constructed infinite families of such examplesin Mod(Σ g, ), for for g >
2. Moreover, for any n >
8, he gave explicit families of elementsin Σ ,n that satisfy the AMU conjecture.In this paper, we turn our attention to the mapping class group Mod(Σ g, ) of compactsurfaces with one boundary component, and construct entire cosets of abelian and freesubgroups that satisfy Conjecture 1.3. Our method for proving Theorem 1.4 is flexible andcan be adapted to construct similar cosets for surfaces with more boundary components.To describe the approach of [13] in more detail, given a manifold M, let T V r ( M ) bethe SO(3)-Turaev-Viro invariant of M at odd r ≥ q = e iπr . Also, let
RENAUD DETCHERRY AND EFSTRATIA KALFAGIANNI us define the
Turaev-Viro limit lT V ( M ) by(1) lT V ( M ) = lim inf r →∞ ,r odd πr log T V r ( M )To facilitate our exposition, we define a compact oriented 3-manifold M to be q-hyperbolic iff we have lT V ( M ) > . An important open problem in quantum topology is the volume conjecture of [8] as-serting that for any finite volume hyperbolic 3-manifold M we have lT V ( M ) = Vol( M ) , which in particular implies that M is q-hyperbolic. A perhaps more robust conjecture,supported by the computations of [8] and the results of [5, 11, 12, 14], is the following. Conjecture 1.5. (Exponential growth conjecture)
For Σ a compact orientable surfaceand f ∈ Mod(Σ) , let M f denote the mapping torus of f and let || M f || denote the Gromovnorm of M f . Then, M f is q-hyperbolic if and only if || M f || > . By the work of Thurston, f has non-trivial pseudo-Anosov parts if and only if theJSJ-decomposition of the mapping torus M f contains hyperbolic parts or equivalently iffwe have || M f || > . In particular, M f is hyperbolic precisely when f is pseudo-Anosov,and in this case Conjecture 1.5 is implied by the volume conjecture of [8].A result of the authors [12, Theorem 1.1] implies that for every f ∈ Mod(Σ) , we have lT V ( M f ) (cid:54) C · || M f || , for some universal constant C >
0. This in turn implies that if M f is q-hyperbolic, then we have || M f || > , which gives the “if” direction of Conjecture1.5. On the other hand, in [13] we also proved that if M f is q-hyperbolic, then f satisfiesthe AMU conjecture. Hence Conjecture 1.5 implies Conjecture 1.3.Although the Chen-Yang volume conjecture is wide open, there are vast families ofq-hyperbolic manifolds and of manifolds that satisfy Conjecture 1.5. For example, wehave families of closed q-hyperbolic 3-manifolds of arbitrarily large Gromov norm [12]and we know that 3-manifolds obtained by drilling out links from q-hyperbolic manifoldsare also q-hyperbolic [12]. A more detailed list of 3-manifolds known to be q-hyperbolicwill be described in Section 2.2. Applying Theorem 1.1 to q-hyperbolic manifolds M, wehave the following. Corollary 1.6.
Suppose that M is q-hyperbolic. Then the mapping classes in the abelianand free cosets given by Theorem 1.1 satisfy the AMU conjecture and the correspondingmapping tori are q-hyperbolic. Note that the method of our proof of Theorem 1.1, combined with hyperbolic Dehnfilling techniques, allows the construction of pairs of cosets that satisfy Conjecture 1.3and are independent in the sense that no mapping class in one coset is a conjugate of amapping class in the other. See Corollary 5.6.1.2.
Organization.
In Section 2 we lay out known definitions and results that we willuse in the remaining of the paper. In Section 3 we define a notion of independence ofmapping classes and we give the proof of a refined version of a result of Long-Mortonand Fathi that we need for the proof of Theorem 1.1. We also discuss the existence ofinfinitely many independent pseudo-Anosov classes in certain mapping class group cosets(Corollary 3.7). In Section 4 we prove Theorem 1.1 and Corollary 1.6. In Section 5we give constructions of fibered knots in certain 3-manifolds obtained by surgery on the
OSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS 5 figure eight knot. The main result of the section is Theorem 5.3 which, in particular,implies Corollary 1.2 and Theorem 1.4.We note that, although our results here are partly motivated by questions in quantumtopology, the techniques and tools we use are from geometric topology and hyperbolicgeometry. In particular, one needs no additional knowledge of quantum topology, thanwhat is given in this Introduction, to read the paper.
Acknowledgements.
We thank J. Andersen, Matt Hedden, Slava Krushkal, FengLuo and Filip Misev for their interest in this work and for useful conversations. Wealso thank Dave Futer for bringing to our attention the article [22]. This work is basedon research done while Kalfagianni was on sabbatical leave from MSU and supported byNSF grants DMS-1708249 and DMS-2004155 and a grant from the Institute for AdvancedStudy School of Mathematics. Detcherry was supported by the Max Planck Institute forMathematics during this work and thanks the institute for its hospitality and support.2.
Preliminaries
In this section we summarize some definitions and results that we will use in this paper.2.1.
Open book decompositions, fibrations and Stallings twist.
We start by de-scribing a more 2-dimensional way of thinking of fibered links in a closed compact oriented3-manifold M, or equivalently, open book decompositions of M. We will keep track of anopen book decomposition of a 3-manifold M as a pair (Σ , h ) , where Σ is the fiber surfaceand h is the monodromy. The mapping torus M h = Σ × [0 , / ( x, ∼ ( h ( x ) , , with fiber Σ :=Σ × { } , is homeomorphic to a link complement in M . The 3-manifold M is determined,up to homeomorphism, as the relative mapping torus of h : that is, M is homeomorphicto the quotient of M h under the identification ( x, t ) ∼ ( x, t (cid:48) ) , for all x ∈ ∂ Σ , t, t (cid:48) ∈ [0 , . The quotient of ∂M h under this identification, is the fibered link K , called the bindingof the open-book decomposition. We will slightly abuse the setting and consider M f embedded in M so that it is the complement of a neighborhood of K . It is known sinceAlexander that any compact oriented 3-manifold has an open book decomposition andwork of Myers [32] shows that there is one with connected binding and pseudo-Anosovmonodromy.Suppose that there is a homotopically non-trivial curve c on Σ that bounds an em-bedded disk D in M , which intersects Σ transversally, such that we have lk ( c, c + ) = 0 , where c + is the curve c pushed along the normal of Σ in Σ × { t } , for small t >
0. Wewill refer to this as pushing off in the positive direction. The disk D can be enclosed ina 3-ball B that intersects Σ transversally. In this ball we can perform a twist of order m along D. Definition 2.1.
This operation above is called a
Stallings twist of order m , along c . Wewill say that the curve c supports a Stallings twist. A curve that supports a Stallingstwist will be called a Stallings curve .We will consider M, Σ and K oriented so that the orientation of Σ (resp. K ) is inducedby that of M (resp. K ). A Stallings twist can also be thought of as performing 1 /m surgery along the unknotted curve c inside B , where the framing of c is induced by Σ RENAUD DETCHERRY AND EFSTRATIA KALFAGIANNI on c . This operation will not change the ambient manifold M but it changes M f to amapping torus with fiber Σ and monodromy f ◦ τ mc . See [37] for more details.2.2. Families of q-hyperbolic -manifolds. We recall from Section 1 that a q-hyperbolic3-manifold was defined to be a 3-manifold M such that lT V ( M ) > . Such manifoldswill be the starting pieces in our constructions in Section 5 and Section 4.The following theorem will sum up the known examples of q-hyperbolic manifolds:
Theorem 2.2.
The following compact oriented -manifolds are q-hyperbolic: (1) The figure-eight knot and the Borromean rings complements in S . (2) Any 3-manifold obtained by Dehn surgery on the figure-eight knot with slopesdetermined by integers p with | p | (cid:62) . (3) The fundamental shadow-link complements of [10] . This is an infinite family ofcusped hyperbolic manifolds with arbitrarily large volumes. (4)
Suppose that L is a link in S with q-hyperbolic complement. Then any linkobtained from L by replacing a component with any number of parallel copies orby a (2 n + 1 , -cable has q-hyperbolic complement. (5) Any -manifold M (cid:48) that can be obtained from a q-hyperbolic manifold M by drillingsolid tori. In fact we have LT V ( M (cid:48) ) (cid:62) LT V ( M ) > . (6) Any link complement in any q-hyperbolic 3-manifold. (7)
Any closed orientable 3-manifold DM obtained by gluing two oppositely orientedcopies of a q-hyperbolic 3-manifold M with toroidal boundary along ∂M. Proof.
The Turaev-Viro invariants volume conjecture is known for the classes of mani-folds in (1)-(3) above: For figure-eight knot and the Borromean rings complements theconjecture was proved by Detcherry-Kalfagianni-Yang [14], for the integral surgeries onthe figure-eight knot by Ohtsuki [33] and for fundamental shadow link complements byBelletti-Detcherry-Kalfagianni-Yang [5]. Part (4) follows by [12, Corollary 8.4] and bythe main result of [11]. Part (5) follows from [12, Corollary 5.3] and part (6) is justa reformulation of part (5). Finally, part (7) follows by [14, Theorem 3.1 and Remark3.4]. (cid:3)
As a corollary of Theorem 2.2 we have the following.
Corollary 2.3.
There are closed q -hyperbolic 3-manifolds of arbitrary large Gromovnorm.Proof. Given an oriented q -hyperbolic 3-manifold M with toroidal boundary let D ( M )denote the double. By Theorem 2.2, it follows that D ( M ) is q -hyperbolic. On theother hand, the doubles of fundamental shadow-link complements have arbitrarily largeGromov norm. (cid:3) Finally, we need to recall the following theorem.
Theorem 2.4.
Let M be a closed q-hyperbolic 3-manifold. Then any mapping class f ∈ Mod(Σ g,n ) that occurs as monodromy of a fibered link in M satisfies the AMUconjecture. OSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS 7
Proof.
Let L be an n -component fibered link in M . By Theorem 2.2 (4), the complementof L is q-hyperbolic. On the other hand, if we choose a fibration of this complement anda monodromy f , we can realize it as a mapping torus M f = Σ g,n × [0 , / ( x, ∼ ( f ( x ) , , where g is the genus of the fiber. Now the result follows by [13, Theorem 1.2] (cid:3) Abelian and free subgroups of mapping class groups.
It is known that themapping class group Mod(Σ) of any compact, connected, oriented surface Σ = Σ g,n satisfies a “Tits alternative property”. That is, any subgroup of Mod(Σ) either containsan abelian group of finite index or it contains a free non-abelian group [30]. Thus abelianand free subgroups are abundant in Mod(Σ). Furthermore it is known that any abeliansubgroup of Mod(Σ) is finitely generated and has rank at most 3 g − n [6].Let Γ be a set of k essential, simple, closed, disjoint curves on Σ that are non-boundaryparallel and no pair of which is parallel to each other. Then the Dehn twists along thecurves in Γ generate a free abelian subgroup of rank k . All the abelian subgroups ofMod(Σ) that we will consider in this paper are of this type. To produce free subgroupswe will use the following theorem of Hamidi-Tehrani. Theorem 2.5. ([22, Theorem 3.5])
Let a, b be essential, simple closed curves on Σ andlet i ( a, b ) denote their intersection number. Let τ a , τ b denote the positive Dehn twistsalong a, b , respectively. Suppose that i ( a, b ) (cid:62) . Then, the subgroup (cid:104) τ a , τ b (cid:105) < Mod(Σ) generated by τ a and τ b is a free group of rank two. Dehn filling and cosets of monodromies
Independance of mapping classes.
We start this section by clarifying the notionof independent mapping classes we mentioned in Section 1.
Definition 3.1.
Let Σ be a compact oriented surface and let f, g ∈ Mod(Σ). We say that f and g are independent if there is no h ∈ Mod(Σ) so that both f and g are conjugatedto non-trivial powers of h. Remark 3.2.
Definition 3.1 is motivated by the following observation: if f and g arenot independent, then f satisfies the AMU conjecture if and only if g does. Indeed, let f be conjugated to h k and g to h l , for some integers k, l (cid:54) = 0 . For any representation ρ ofMod(Σ) , the images ρ ( f ) and ρ ( g ) will have infinite order if and only if ρ ( h ) does.With the following lemma we can show the independence of two elements using thevolumes of the corresponding mapping tori. Lemma 3.3.
Let Σ be a compact oriented surface. For f ∈ Mod(Σ) let M f be themapping torus of f. There is a constant (cid:15) > such that for any pseudo-Anosov elements f, g ∈ Mod(Σ) , if vol( M f ) (cid:54) = vol( M g ) and | vol( M f ) − vol( M g ) | < (cid:15) then f and g areindependent. Note that although [13, Theorem 1.2] is stated for pseudo-Anosov maps, the proof given does notuse this hypothesis.
RENAUD DETCHERRY AND EFSTRATIA KALFAGIANNI
Proof. If f is conjugated to h k and g to h l for some h ∈ Mod(Σ) and k, l ∈ Z(cid:114) { } , thenvol( M f ) = k vol( M h ) and vol( M g ) = l vol( M h ) . As M f and M g have different volumes,their volume differ at least by vol( M h ) . By Jorgensen-Thurston [38] theory the set ofvolumes of hyperbolic manifolds is well ordered, thus it has a minimal element. Thus wecan simply take (cid:15) to be smaller than the minimal volume. (cid:3)
We need the following theorem which for k = 1 is proved by Long and Morton [26,Theorem 1.2]. Theorem 3.4.
Let f ∈ Mod(Σ) is a pseudo-Anosov map and let γ , . . . , γ k be homo-topically essential, non-boundary parallel curves. Suppose moreover that no pair of thecurves are parallel on Σ and that i ( f ( γ i ) , γ j ) (cid:54) = 0 , for any (cid:54) i, j (cid:54) k . Then, there is n ∈ N such that for all ( n , . . . , n k ) ∈ Z k with all | n i | > n, the mapping class g = f ◦ τ n γ ◦ . . . ◦ τ n k γ k is pseudo-Anosov. Furthermore, the family { f ◦ τ n γ ◦ . . . ◦ τ n k γ k , with ( n , . . . , n k ) ∈ Z k and | n i | > n } contains infinitely many, pairwise independent pseudo-Anosov mapping classes. The argument we give below follows the line of the proof given in [26], however for k > M f (cid:114) ( γ ∪ . . . ∪ γ k ) . The statement for k >
Proof.
Consider the mapping torus M f = Σ × [0 , / ( x, ∼ ( f ( x ) , , which is hyperbolic since f is pseudo-Anosov [38]. Now take numbers 1 / i < i <. . . < i k < γ , . . . , γ k lying on the fiber F := Σ × { i } ⊂ M f .For s = 2 , . . . , k , consider the annulus γ s × [ i , i s ] from γ s on F to δ s := γ s × { i s } whichis a curve lying on the fiber F i s := Σ × { i s } ⊂ M f . Let δ := γ . Now L = δ ∪ . . . ∪ δ k is a link in M f . Claim.
The manifold N := M h (cid:114) L is hyperbolic. Proof of Claim.
Since the curves γ , . . . , γ k are homotopically essential and M f ishyperbolic, the manifold N is irreducible and boundary irreducible. We need to showthat N contains no essential embedded tori. Let T be a torus embedded in N . If T is boundary parallel in M f it will also be in N otherwise one would be able to isotopesome of the γ i to be boundary parallel in Σ contradicting our assumptions. The torus T must intersect some of the fibers where the curves δ i lie since otherwise it wouldcompress in the handlebody outside of those fibers. Suppose, without loss of generality,it intersects F . Then after isotopy, F ∩ T is a collection of simple closed curves that arehomotopically essential in F (cid:114) δ . The intersection of T with the handlebody M F is acollection of properly embedded annuli in the handlebody M F := M f \\ F . After isotopyevery annulus in M F is either vertical with respect to the I -product or it has both itsboundary components on the same copy of F on ∂M F in which case the components OSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS 9 are parallel on F . Annuli of the later type can be eliminated by isotopy in N unlessthey intersect some fiber F i s containing a curve δ s , in which case the intersection of theannulus with F s is two parallel curves running on opposite sides of δ s .Suppose now that A is a component of T ∩ M F that is vertical with respect to the I -product of M F and let θ denote a component of F ∩ T . There are two cases to consider. Case 1.
The vertical annulus A connects to an annulus A (cid:48) of M F ∩ T such that thetwo components of ∂A (cid:48) run parallel to some δ j on opposite sides of δ j on the fiber thatcontains it. The vertical annulus runs from θ, which must be parallel to δ j now, to f ( θ )on the other copy of the fiber on ∂M F . Since we assumed that the original curves arepairwise non-parallel on Σ, and that the intersection number f ( γ i ) and γ j , is non-zero forany 1 (cid:54) i, j (cid:54) k , the curve θ cannot be parallel to any of the curves δ , . . . δ s . It followsthat the torus T cannot contain a second annulus whose boundary curves run parallel tosome δ s (cid:54) = δ j on a fiber. Thus T must be boundary parallel in N . Case 2.
The vertical annulus A eventually connects to the curve θ . Since f is pseudo-Anosov, and θ is essential on F , we cannot have f k ( θ ) = θ , for any integer k (cid:54) = 0.In fact it is known that the intersection number of θ with f k ( θ ) grows linearly in k .See Proposition 4.3 and Remark 4.4 below for the precise statements and references. Itfollows that such an annulus cannot close to become a torus in M f . Thus, this case willnot happen. This implies that N is atoroidal.Now an atoroidal manifold, with toroidal boundary, can only contain essential annuliif it is a Seifert fibered manifold. But since M f is hyperbolic it has non-zero Gromovnorm and since this norm doesn’t increase under filling the cusps with tori, we concludethat || N || (cid:62) || M f || >
0, which implies that it cannot be a Seifert fibered space. Thus N cannot have essential annuli and thus, by Thurston’s work N is hyperbolic .To continue we use the fact that the mapping torus of g = f ◦ τ n γ ◦ τ n k γ k is obtainedby N by doing Dehn filling along slopes say s , . . . , s k , respectively. The length of theslope s i on the corresponding cusp of N is an increasing function of | n j | . Let λ denotethe length the shortest slope. By Thurston’s hyperbolic Dehn Filling theorem [38] thereis n ∈ N such that for all n := ( n , . . . , n k ) ∈ Z k with all | n i | > n , the resulting manifold N ( n ) obtained by filling N , where the cusp corresponding to δ i is filled as above, ishyperbolic. The volumes of the filled manifolds N ( n ) approach the volume of N frombelow. To make things more concrete we use an effective form of the theorem proved in[16, Theorem 1.1], which states that provided that λ > π , then N ( n ) is hyperbolic andwe have (cid:32) − (cid:18) πλ (cid:19) (cid:33) / vol( N ) (cid:54) vol( N ( n ))) (cid:54) vol( N ) . Using this it becomes clear than one can find infinite sequences of k -tuples { n i } i ∈ N such that the manifolds N ( n i ) have strictly increasing volumes and such that for any i (cid:54) = j ∈ N we have | vol( N ( n j )) − vol( N ( n i )) | < (cid:15) , where (cid:15) is the constant of Lemma3.3. Now the monodromies of the manifolds N ( n i ) above form an infinite sequence ofindependent pseudo-Anosov mapping classes. (cid:3) Note that if k = 1 in the statement of Theorem 3.4, then in fact Fathi has shown thatthere can be at most seven integers n ∈ Z for which f ◦ τ n is not pseudo-Anosov. Wealso note that for the following corollary, which is [26, Theorem 1.3], we do not need theextra hypotheses on the curves γ i given in the statement of Theorem 3.4. Corollary 3.5.
Let f ∈ Mod(Σ) be a pseudo-Anosov map and let γ , . . . , γ k be homo-topically essential, non-boundary parallel curves. Then, for any k > , the family ofmaps g = f ◦ τ n γ ◦ . . . ◦ τ n k γ k , with ( n , . . . , n k ) ∈ Z − { } k , contains infinitely manypseudo-Anosov elements.Proof. By Theorem 3.4, the family of maps { f ◦ τ n γ } n ∈ N , contains infinitely many pseudo-Anosov elements. Again by Theorem 3.4, for each of these pseudo-Anosov elements, thefamily { f ◦ τ n γ ◦ τ n γ } n ∈ N contains infinitely many pseudo-Anosov elements. That isgiven m ≥ n , there is some m , depending on m , such that for any | n | > m , themap f ◦ τ m γ ◦ τ n γ is pseudo-Anosov. Continuing inductively we arrive at the desiredconclusion. (cid:3) Abelian and free elementary cosets.
To prove Theorems 1.4 and 1.1, we willconstruct cosets of abelian and free subgroups of Mod(Σ) generated by Dehn twistson curves as in Section 2. To facilitate clarity of exposition let us make the followingdefinition.
Definition 3.6.
Let Σ a compact oriented surface and f ∈ Mod(Σ) . • If H is a subgroup of Mod(Σ) generated by k (powers of) Dehn twists alongdisjoint curves on Σ, we say that f H is a rank k abelian elementary coset . • If H is a rank two free subgroup of Mod(Σ) generated by (powers of) Dehn twistsalong two curves on Σ we say that f H is a free elementary coset .Furthermore, in both cases, if f H contains a pseudo-Anosov element, we say that f H isa pseudo-Anosov elementary coset .An immediate corollary of Theorem 3.4 is the following. Corollary 3.7.
Any pseudo-Anosov abelian or free elementary coset of
Mod(Σ) containsinfinitely many independent pseudo-Anosov mapping classes.Proof.
Given a pseudo-Anosov abelian or free elementary coset gH of Mod(Σ) , we canwrite f H = gH , where f is a pseudo-Anosov mapping class. Recall that in both cases, H contains a free abelian group generated by a Dehn twist τ γ on a simple closed nonnon-peripheral curve γ . We can look at mapping classes of the form f ◦ τ nγ and applyTheorem 3.4. (cid:3) Abelian and free elementary cosets from 3-manifolds
The main results in this section are Theorem 4.15 and Theorem 4.16 which implyTheorem 1.1 and Corollary 1.6 stated in the Introduction. To prove these results, firstwe show that any open book decomposition of a 3-manifold, with pseudo-Anosov mon-odromy, can be stabilized to one with pseudo-Anosov monodromy and so that the pages
OSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS 11 of the decomposition support Stallings twists. See Theorem 4.10 and subsequent discus-sion. Roughly speaking, the key point is to choose stabilization arcs on the fiber surfacethat are parts of curves that are “complicated” in the sense of the geometry of the curvecomplex Σ g, . See Lemma 4.6 and Propositions 4.8 and 4.9. To prove Theorems 4.15 and4.16 we combine Theorem 4.10 with Theorems 3.4 and 2.5, respectively.4.1. Moves on open book decompositions.
An open book decomposition (Σ , h ) of M can be modified into another open book decomposition of M by some elementarymoves called stabilizations and destabilizations: Definition 4.1.
Let (Σ , h ) be an open book decomposition of M, and ( γ, ∂γ ) ⊂ (Σ , ∂ Σ)a properly embedded arc. We will say that the surface Σ (cid:48) is obtained by a positive (resp. negative ) stabilization along γ , if Σ (cid:48) is obtained from Σ by plumbing a positive (resp.negative) Hopf band H (resp. H − ) along γ . The intersection of the H ± with Σ is aneighborhood of γ and the arc γ becomes part of the core curve, say c , of H ± . Thecurve c bounds a disk in M, whose intersection with Σ is exactly the arc γ. Let τ c denote the corresponding Dehn twist on c ⊂ Σ (cid:48) and let h be extended to Σ (cid:48) by the identity on Σ (cid:48) (cid:114) Σ. Then the pair (Σ (cid:48) , h (cid:48) ) where h (cid:48) = h ◦ τ ± c is an open bookdecomposition of M which we call a stabilization of (Σ , h ) . Finally, (Σ , h ) is a destabilization of (Σ (cid:48) , h (cid:48) ) . Harer proved [23] that any two open book decompositions of the same 3-manifold M arerelated by stabilizations, destabilizations and a third elementary move called a doubletwist move. Harer conjectured that stabilizations and destabilizations are sufficient,which was later proved by Giroux and Goodman [21]. Despite Giroux and Goodman’sresult, it is in practice difficult to explicitly express a double twist move in terms ofstabilizations and destabilizations. Some particular examples of double twist moves arethe Stallings twists, which we already introduced in Definition 2.1.4.2. The curve graph and Penner systems.
A compact oriented surface Σ = Σ g,n iscalled non-sporadic if 3 g − n >
0. Given a non-sporadic surface Σ, the curve graph C (Σ) is defined as follows:The set of vertices, denoted by C (Σ), is the set of isotopy classes of homotopicallyessential, non-boundary parallel simple closed curves and two vertices a, b ∈ C (Σ) areconnected by an edge if they can be represented by disjoint curves, that is they haveintersection number zero ( i ( a, b ) = 0). The space C (Σ) becomes a geodesic metric spacewith the path metric that assigns length 1 to each edge of the graph, and elements of themapping class of Σ act as isometries on the space. Now for the curve complex C (Σ) oneadds to C (Σ) cells to collections of disjoint curves on Σ.For a, b ∈ C (Σ) the distance between a, b will be denoted by d ( a, b ). The geometryof the space was studied by Masur and Minsky [29]. Lemma 4.2. [29, Lemma 2.1]
Given a, b ∈ C (Σ) we have d ( a, b ) (cid:54) i ( a, b ) + 1 Proposition 4.3. [29, Proposition 4.6]
For any non-sporadic surface Σ there is a con-stant C > such that given a pseudo-Anosov h ∈ Mod(Σ) we have d ( a, h n ( a )) (cid:62) C | n | , for every a ∈ C (Σ) and n ∈ Z . Remark 4.4.
Proposition 4.3 implies that, if h is pseudo-Anosov and we have h k ( a ) = a ,for some a ∈ C (Σ), then k = 0. For, if k (cid:54) = 0, then we have infinitely many n ∈ Z such h n ( a ) = a , namely all the multiples of k . Thus we have d ( a, h n ( a )) (cid:54)
1, for infinitelymany n ∈ Z . On the other hand, by Proposition 4.3 we have d ( a, h n ( a )) (cid:62) C | n | , whichis a contradiction.We will need the following lemma. Lemma 4.5.
Let a, b ∈ C (Σ) and let h ∈ Mod(Σ) be pseudo-Anosov. Suppose that wehave d ( a, h ( a )) (cid:62) N and d ( b, h ( b )) (cid:62) N , for some N (cid:29) i ( a, b ) + 1 . Then, we have h ( a ) (cid:54) = b .Proof. We have d ( b, h ( a )) (cid:62) d ( b, h ( b )) − d ( h ( b ) , h ( a )) (cid:62) d ( b, h ( b )) − d ( a, b ) (cid:62) N − (2 i ( a, b ) + 1) (cid:29) h is an isometry, and the thirdinequality follows from Lemma 4.2. (cid:3) A Penner system of curves on a surface Σ = Σ g, is a collection C of 2 g simple closed,essential, non boundary parallel curves, that splits into two disjoint sets A , B with thefollowing properties:(1) The curves in each of A = { a , . . . , a g } and B = { b , . . . , b g } are mutually disjoint.(2) We have i ( a i , b j ) = 1 if i = j and zero otherwise.(3) Σ cut along C = A ∪ B is a 4 g -gon with a disk removed. Lemma 4.6.
Let Σ be a non-sporadic surface and let h ∈ Mod(Σ) be a pseudo-Anosovmapping class. For every N (cid:29) , there is a Penner system of curves C such that for any c ∈ C we have d ( c, h ( c )) (cid:62) N/ . That is, there are systems C for which the distance d ( c, h ( c )) is arbitrarily large for any c ∈ C .Proof. We begin with the following claim that is attributed to Minsky in [9].
Claim.
Given N (cid:29) c on Σ such that d ( c, h ( c )) = N . Proof of Claim.
We use the notation, terminology and setting used, for instance inthe proof of [29, Proposition 4.6]. The pseudo-Anosov representative h : Σ → Σ acts onthe space of projective measured laminations of S and in there it determines exactly twofixed laminations µ and ν . Now pick a geodesic lamination on S that is minimal (i.e. itcontains no proper sub laminations) and whose class, say λ , in the space of projectivelaminations is not µ or ν . Now we have h ( λ ) (cid:54) = λ .Let { a n ∈ C (Σ) } n ∈ N be a sequence that converges to λ in the space of projectivelaminations. Then it is known that h ( a n ) converges to h ( λ ).We claim that d ( a n , h ( a n )) → ∞ as n → ∞ . For otherwise, and after passing to asub-sequence if necessary, we can assume that d ( a n , h ( a n )) = m < ∞ . Now a n and h ( a n ) OSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS 13 are connected by geodesics in the curve complex of S . Thus we can find a sequence { b n ∈ C (Σ) } n ∈ N , with d ( b n , h ( a n )) = m − d ( b n , a n ) = 1. Since a n is disjoint from b n , after taking a subsequence if necessary, we can assume that b n converges to a classrepresented by a lamination, say λ (cid:48) , on S that has intersection number zero with λ , and h ( b n ) converges to a geometric lamination that has intersection number zero with h ( λ ).The minimality assumption implies that λ (cid:48) = λ . Thus b n converges to λ and we have d ( b n , h ( b n )) (cid:54) m −
1. Inductively, we will arrive at a sequence { c n ∈ C (Σ) } n ∈ N thatconverges to λ and we have d ( c n , h ( c n )) = 0. This would imply that h ( λ ) = λ . which isa contradiction since we assumed that h ( λ ) (cid:54) = λ . This finishes the proof of the claim.To continue, with the construction of the desired Penner system of curves, start with N (cid:29) c with d ( c, h ( c )) = N . If c is non-separating set a := c . Otherwise,since g ≥
2, we can find a non separating essential non-boundary parallel curve a , with i ( a , c ) = 0. Then N = d ( c, h ( c )) (cid:54) d ( c, a ) + d ( a , h ( a )) + d ( h ( a ) , h ( c )) . Since d ( c, a ) = d ( h ( a ) , h ( c )) = 1 we obtain d ( a , h ( a )) (cid:62) N − b so that i ( a , b ) = 1 and complete { a , b } into a Penner system of curves C . Now for any c ∈ C we have N − (cid:54) d ( a , h ( a )) (cid:54) d ( a , c ) + d ( c, h ( c )) + d ( h ( a ) , h ( c )) . Since i ( a, c ) (cid:54)
1, by Lemma 4.2, d ( a , c ) = d ( h ( a ) , h ( c )) (cid:54)
3. Thus d ( c, h ( c )) > N − >N/ N (cid:29) (cid:3) Stabilizing to pseudo-Anosov monodromies.
For our applications to the AMUconjecture, we want the open-book decompositions we construct to have pseudo-Anosovmonodromies, and we would like the genus of the fiber surface F to take any arbitrary highvalue. In this subsection we discuss a technique to stabilize open book decompositionsthat preserve the pseudo-Anosov property of monodromies while producing Stallingscurves on the pages of the decomposition. Our construction is inspired by an argumentof Colin and Honda used in the proof of [9, Theorem 1.1].Let (Σ , h ) be an open book decomposition, with connected binding ∂ Σ , of an oriented,closed 3-manifold M. Let c be a simple closed, homotopically essential, non-peripheralcurve c on Σ and let (cid:15) be an embedded arc that runs from a point P ∈ c to a pointon ∂ Σ. Consider a closed neighborhood of (cid:15) on Σ, which is a rectangle with one sidea closed neighborhood of P , say d ⊂ c , containing P . Replacing d with the rest of theboundary of this rectangle leads to a curve that is isotopic to c on Σ. This new curve isthe union of two properly embedded arcs γ ∪ δ (cid:48) , where δ (cid:48) is an arc on ∂ Σ that connectsthe endpoints of γ and contains the endpoint of (cid:15) on ∂ Σ. See Figure 1. To facilitate ourexposition we give the following definition.
Definition 4.7.
With the setting as above, we will say that γ is obtained by pushing c on ∂ Σ along the point P . We will also say that the arc δ (cid:48) ⊂ ∂ Σ complements γ .We need the following. γc Figure 1.
The curve γ is obtained by pushing c on ∂ Σ along the point P, the endpoint of the dashed line on c . Proposition 4.8.
Let
Σ := Σ g, and let (Σ , φ ) be an open book decomposition of M such that φ is pseudo-Anosov. Suppose that Σ is non-sporadic and that c is a simpleclosed, homotopically essential, non-peripheral curve c on Σ , such that d ( c, φ ( c )) = N ,for N (cid:29) . Let γ be an arc obtained by pushing c on ∂ Σ along any point, and let (Σ (cid:48) , φ (cid:48) ) denote an open book decomposition obtained by stabilizing (Σ , φ ) along γ . Then φ (cid:48) ispseudo-Anosov.Proof. Let c (cid:48) the closed curve in Σ (cid:48) which the core of the Hopf-band used in the stabiliza-tion, hence γ = c ∩ Σ. Recall that φ (cid:48) = φ ◦ τ ± c (cid:48) , according to whether we made a positiveor a negative stabilization. The proof of this proposition is essentially given in the proofof [9, Theorem 1.1]. We note that although the statement of [9, Theorem 1.1] contains thehypothesis that the starting monodromy φ is right-veering and the stabilization positive,this is actually only used to show that the new monodromy is also right-veering.For the sake of completeness, we sketch the proof of [9, Theorem 1.1], while referringto their paper for full details.First we argue that φ (cid:48) = φ ◦ τ ± c (cid:48) is not reducible, by arguing that φ (cid:48) ( δ ) (cid:54) = δ for anycollection δ of disjoint simple curves on Σ (cid:48) . Case 1.
First, suppose that δ ⊂ Σ ⊂ Σ (cid:48) . If i Σ ( φ ( δ ) , c ) = i Σ (cid:48) ( φ ( δ ) , c ) = 0 , in which case δ can not be parallel to ∂ Σ, then φ (cid:48) ( δ ) = φ ( δ ) (cid:54) = δ as φ is pseudo-Anosov. Suppose nowthat i Σ ( φ ( δ ) , c ) = i Σ (cid:48) ( φ ( δ ) , c (cid:48) ) = m > . Then we claim that i Σ (cid:48) ( φ (cid:48) ( δ ) , a ) = m , where a isthe co-core of the stabilization band. Indeed, there is an obvious representative of φ (cid:48) ( δ )with m intersection points with a. If i Σ (cid:48) ( φ (cid:48) ( δ ) , a ) < m, it means that there is a bigonconsisting of a subarc of a and a subarc of φ (cid:48) ( δ ) . One checks that this bigon should beobtained by the gluing of a bigon in δ ∪ c and a band that runs along c (cid:48) , but no suchbigon exists, otherwise we would i Σ (cid:48) ( φ ( δ ) , c ) < m ; a contradiction. Case 2.
Assume that δ (cid:42) Σ , that is i Σ (cid:48) ( δ, a ) = k > . Let B = Σ (cid:48) (cid:114) Σ be thestabilization band, the intersection of φ ( δ ) with B consists of “vertical arcs”, and arcsthat intersect c (cid:48) . The later arcs may be either all of positive or all of negative slopesin B. See [9, Figures 1, 2]. Up to isotopy, we may assume that there is no trianglewith boundary an arc of c (cid:48) , an arc of δ and an arc of ∂B. Under this hypothesis let m be the number of arcs of δ ∩ B of positive/negative slopes. Let n be the number ofother intersections of c (cid:48) and φ ( δ ) , so that i Σ (cid:48) ( φ ( δ ) , c (cid:48) ) = m + n. Then by a similar bigonchasing argument, Colin and Honda show that i Σ (cid:48) ( φ (cid:48) ( δ ) , a ) = k ± m + n, where the sign OSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS 15 depends on the sign of the stabilization and the sign of the slopes of the m arcs. Thus if ± m + n (cid:54) = 0 we have φ (cid:48) ( δ ) (cid:54) = δ as i Σ (cid:48) ( δ, a ) = k (cid:54) = k ± m + n = i Σ (cid:48) ( φ (cid:48) ( δ ) , a ) . We would like to stress that the above step is essentially the only place where allowingpositive as well as negative stabilizations plays any role: it just exchanges the two caseswhere i Σ (cid:48) ( φ (cid:48) ( δ ) , a ) = k + m + n and i Σ (cid:48) ( φ (cid:48) ( δ ) , a ) = k − m + n. In the case ± m + n = 0 , they show that i Σ ( δ, c (cid:48) ) (cid:54) = i Σ (cid:48) ( φ (cid:48) ( δ ) , c (cid:48) ) . Indeed, i Σ (cid:48) ( φ (cid:48) ( δ ) , c (cid:48) ) (cid:54) m + n (cid:54) k. If i Σ (cid:48) ( δ, c (cid:48) ) (cid:54) k, Colin and Honda use the fact that d ( c, φ ( c )) (cid:29) i Σ (cid:48) ( φ (cid:48) ( δ ) , c (cid:48) ) (cid:29) k, a contradiction. This finishes the proofthat φ (cid:48) is not reducible.Finally, in the proof of [9, Theorem 1.1], it is shown that φ (cid:48) is not periodic by consid-ering the curve ∂ Σ on Σ (cid:48) , and arguing that i Σ (cid:48) (( φ (cid:48) ) n ( ∂ Σ) , a ) −→ | n |→∞ ∞ . Note that, if φ (cid:48) were periodic, then ( φ (cid:48) ) m ( ∂ Σ) = ∂ Σ for some m ∈ N and thus above intersection numberwould be bounded. (cid:3) Next we turn our attention to surfaces with two boundary components and stabilizationalong an arc that connects the two boundary components: Suppose that Σ = Σ g, andlet β be a properly embedded arc on Σ connecting the two boundary components of ∂ Σ.Thicken γ into a rectangle R that intersects the two components into sub-arcs of ∂ Σ thatcontain the endpoints of γ . The union of the complementary arcs on ∂ Σ, together withthe arcs of ∂R that are parallel to γ , give us a simple closed curve c β ⊂ Σ. Proposition 4.9.
Let
Σ := Σ g, be non-sporadic and let (Σ , ψ ) be an open book de-composition of M such that ψ is pseudo-Anosov. Let β be a properly embedded arc on Σ connecting the two boundary components of ∂ Σ , and let (Σ (cid:48) , ψ (cid:48) ) denote an open bookdecomposition obtained by stabilizing (Σ , φ ) along β . If d ( c β , ψ ( c β )) = N , for N (cid:29) ,then ψ (cid:48) is pseudo-Anosov.Proof. Let c (cid:48) be the closed curve in Σ (cid:48) which consist of the arc β and the core of theHopf-band coming from the stabilization. Now ψ (cid:48) = ψ ◦ τ ± c (cid:48) , according to whether wemade a positive or a negative stabilization. The proof also follows from [9, Subcase 2of Theorem 1.1]. Again the statement of [9, Theorem 1.1] contains the hypothesis thatthe starting monodromy ψ is right-veering and the stabilization positive but a similaranalysis as this in the proof of Proposition 4.8 show that the conclusion holds withoutthese hypotheses. (cid:3) Start with a genus g open book decomposition (Σ , h ) of M such that h is pseudo-Anosov, and such that ∂ Σ has one component. Let C = A ∪ B be a Penner system ofcurves on Σ = Σ g, where A = { a , . . . , a g } and B = { b , . . . , b g } as earlier.For i = 1 , . . . , g , let P i denote the intersection point of a i and b i . Let α i and β i denotethe arcs obtained by pushing a i and b i on ∂ Σ along P i . Let Γ C denote the set of thesearcs. By Definition 4.7, this process involves choosing arcs (cid:15) i from P i to ∂S . We canchoose these arcs so that they are mutually disjoint. For i = 1 , . . . , g , we can arrange sothat we have,(1) the arcs in Γ are pairwise disjoint and splits into two sets { α , . . . , α g } and { β , . . . , β g } .(2) the endpoints α i on ∂ Σ are separated by these of β j , precisely when i = j ; (3) the intersection of the arcs complementing α i and β j on ∂ Σ is an interval when i = j and empty otherwise.Now let (Σ (cid:48) , h C ) denote the open book decomposition obtained by (Σ , h ) by a positivestabilization along each arc α i and a negative stabilization along β j . Note that Σ (cid:48) hasconnected boundary and genus 2 g, indeed, each pair of stabilizations along the arcs α i and β leaves the boundary connected while it raises the genus by 1 . For g = 2 thesituation is illustrated in Figure 2. Theorem 4.10.
Let (Σ , h ) be a genus g open book decomposition of M with h pseudo-Anosov. There is a Penner system of curves C , so that the monodromy of the stabilizedopen book decomposition (Σ (cid:48) , h C ) is also pseudo-Anosov.Proof. For any N = 4 g N ≫ , we can use Lemma 4.6 to pick a Penner system of curves C so that for any c ∈ C , d ( c, h ( c )) > g N . Then we stabilize along the correspondingarcs in Γ C = { α , β , . . . , α g , β g } , by a positive stabilization along each arc α i and anegative stabilization along each β j .The new monodromy h C is expressed as h C = h ◦ τ a (cid:48) ◦ τ − b (cid:48) ◦ . . . ◦ τ a (cid:48) g ◦ τ − b (cid:48) g , where each curve contains the corresponding stabilization arc in Γ C and is the core ofthe corresponding Hopf-band.Let us consider the stabilization process done in steps; one Hopf-band at a time.First, we stabilize along α to get an open book decomposition (Σ , h ), where h := h ◦ τ a (cid:48) and Σ has two boundary components. By Proposition 4.8 the monodromy h is pseudo-Anosov. By construction now β has its endpoints on different components of ∂ Σ . Consider the curve c β constructed out of β with the process described before thestatement of Proposition 4.9. We claim that we have d ( c β , h ( c β )) > g − N (cid:29) g. To see this claim first we note that, by construction, we have an arc δ ⊂ ∂ Σ such that(i) δ connects the endpoints of β ; (ii) the interior of δ contains exactly one endpointof α ; (iii) the curve β ∪ δ is isotopic to b on Σ. We have d ( b , h ( b )) (cid:54) d ( b , h ( b )) + d ( h ( b ) , h ( b )) . Using Lemma 4.2, we have d ( h ( b ) , h ( b )) = d ( τ a ( b ) , b ) (cid:54)
3, since i ( a , b ) = 1.Thus d ( b , h ( b )) > g N − . The intersection number of c β and b on Σ is 0 (see Figure 2). Therefore d ( c β , h ( c β )) (cid:62) d ( c β , b ) − d ( b , h ( b )) − d ( h ( b ) , h ( c β )) > g N − d ( c β , h ( c β )) > g − N . We can apply Proposition 4.9, to conclude that h = h ◦ τ − b (cid:48) is pseudo-Anosov.For 1 < k ≤ g , let (Σ k , h k ) be the genus g + k open-book decomposition obtained bystabilizing on the first k pairs of arcs ( α i , β i ) in Γ C , where h k = h ◦ τ a (cid:48) ◦ τ − b (cid:48) ◦ . . . ◦ τ a (cid:48) k ◦ τ − b (cid:48) k . OSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS 17
Proceeding inductively as in the case k = 1 , we can show that for any 1 (cid:54) k (cid:54) g, themap h k is pseudo-Anosov and that for any c ∈ C , d ( c, h k ( c )) > g − k N (cid:29) , where thelast distance is taken in C (Σ k ) . Note that for k = g, we have h g = h C and d ( c, h C ( c )) > N (cid:29) , for any c ∈ C . (cid:3) As a corollary of the proof of Theorem 4.10 we have the following which is a specialcase of Theorem 1.1 of [9].
Corollary 4.11.
For any -manifold M, there exists a genus g = g ( M ) such that forany g (cid:62) g , there is an open book decomposition (Σ , h ) of M, such that ∂ Σ is connected, Σ has genus g, and h is pseudo-Anosov.Proof. As said earlier M contains open book decompositions (Σ , h ), with connected bind-ing and pseudo-Anosov monodromy h . Let g = g ( M ) the smallest number that occursas the genus of such an open-book decomposition. The result follows by induction ap-plying Theorem 4.10. (cid:3) The proof of Theorem 4.10 shows that if we let Γ k = { α , β , . . . , α k , β k } ⊂ Γ C , for1 ≤ k ≤ g , the open book decomposition obtained by stabilizing along Γ k has pseudo-Anosov monodromy. More specifically, we have the following. Corollary 4.12.
Let (Σ , h ) be a genus g open book decomposition of M such that h ispseudo-Anosov. There is a Penner system of curves C so that, for ≤ k ≤ g , (Σ k , h k ) has genus g + k and the monodromy h k = h ◦ τ a (cid:48) ◦ τ − b (cid:48) ◦ . . . ◦ τ a (cid:48) k ◦ τ − b (cid:48) k is pseudo-Anosov. Stallings twists and mapping class cosets.
Let (Σ , h ) be a genus g ≥ g openbook decomposition of M with h pseudo-Anosov. In subsection 4.3 we showed that wecan choose a Penner system of curves C ⊂ Σ that leads to stabilized genus g + k openbook decomposition (Σ k , h k ), for any 1 ≤ k ≤ g . Next we argue that the fibers of allthese stabilized decompositions support Stallings twists. Lemma 4.13.
For ≤ k ≤ g , the genus g + k fiber Σ k contains at least k non-boundaryparallel and non-parallel to each other Stallings curves.Proof. We will argue that k disjoint Stallings curves can be obtained on the stabilizedsurface Σ k by taking appropriate band sums of the core curves of the stabilization bands.Note that while the curves a i and b i intersect in Σ , any two of the stabilization arcs sets { α , β . . . , α g , β g } are disjoint. Thus the cores of the stabilization bands a (cid:48) , b (cid:48) . . . , a (cid:48) k , b (cid:48) k are disjoint on Σ k , and they bound disjoint disks E , E ∗ . . . , E k , E ∗ k each intersecting thesurface Σ k transversally.As shown in Figure 2, we can choose disjoint arcs γ i , where i = 1 , . . . , k, so that γ i connects a (cid:48) i to b (cid:48) i +1 ( γ k connecting a (cid:48) k to b (cid:48) ). Now the disks D i = E i γ i E ∗ i +1 obtainedby band sum along γ i for i = 1 , . . . , k are mutually disjoint, and the boundary curves t i = ∂D i satisfy lk ( t i , t + i ) = 0 . So the curves t , . . . , t k are disjoint Stallings curves on Σ k . In H (Σ k , ∂ Σ k , Z ) they represent distinct non-zero elements, so they are non-parallel andnon boundary parallel. (cid:3) a b a b ∂ Σ a (cid:48) b (cid:48) a (cid:48) b (cid:48) γ γ a (cid:48) c β Figure 2.
The surface Σ is represented as a one-holed 4 g -gon (with g = 2here). From left to right: the Penner set of curves a i , b i , the curve c β onthe surface stabilized along α , and the curves a (cid:48) i , b (cid:48) i and arcs γ i on the fullystabilized surface Σ (cid:48) . Remark 4.14.
Our construction of the Stallings curves in the proof of Lemma 4.13guarantees that if we think of the Stallings curves t j as lying on different level fibersof M h k = Σ k / ( x, ∼ ( h k ( x ) , , we can assure that the Stallings disks become disjoint in themanifold M .We are now ready to prove Theorem 1.1 stated in the Introduction. The constructionof the desired cosets will be done in two steps. In Theorem 4.15 below we constructabelian cosets and then in Theorem 4.16 we deal with the construction of free cosets.As explained earlier, the constructions become relevant to the AMU conjecture whenapplied to q-hyperbolic manifolds. Theorem 4.15.
Let M be a closed, oriented 3-manifold. There is g = g ( M ) , sothat for any g (cid:62) g , and any ≤ k ≤ g , the following is true: Any genus g open-bookdecomposition (Σ , h ) of M with connected binding, and pseudo-Anosov h , can be stabilizedto a genus g + k decomposition (Σ k , h k ) such that h k represents a pseudo-Anosov elementin Mod(Σ g + k, ) . Furthermore, there is an abelian subgroup A k < Mod(Σ g + k, ) such that, (1) the rank of A k is k ; (2) every element in the coset h (cid:48) A k represents a pseudo-Anosov mapping class; and (3) every element in h (cid:48) A k occurs as monodromy of an open book decomposition of M. Proof.
Let g be the genus guaranteed by Corollary 4.11, which can be taken to bethe smallest number that occurs as genus of an open-book decomposition in M withconnected binding and pseudo-Anosov monodromy. Now for any g (cid:62) g , there is an openbook decomposition (Σ , h ) of M, such that ∂ Σ is connected, Σ has genus g, and h ispseudo-Anosov. Now fix g (cid:62) g and apply Theorem 4.10 and Corollary 4.12 to (Σ , h ).For any 1 ≤ k ≤ g , we get (Σ k , h k ) of genus g + k and h k pseudo-Anosov. By Lemma4.13, Σ k contains a collection T = { t , . . . , t k } of disjoint Stallings curves. Claim.
For any t i , t j ∈ T , we have i Σ k ( h k ( t i ) , t j ) (cid:54) = 0. Proof Claim.
Let C denote the Penner set of curves that comes from the applicationof Theorem 4.10 and Corollary 4.12. By the argument in the proof of Theorem 4.10 weconclude that d ( c, h k ( c )) (cid:29)
0, for any c ∈ C . By the construction of the curves in T , for any t ∈ T and c ∈ C we have i Σ k ( t, c ) (cid:54)
1. Hence, by Lemma 4.2 , we have
OSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS 19 d ( h k ( t ) , h k ( c )) = d ( t, c ) (cid:54) k . By triangle inequalities in thiscurve complex we get d ( t, h k ( t ) (cid:62) d ( c, h k ( c )) − (cid:29) . Since i Σ k ( t i , t j ) = 0, for t i (cid:54) = t j ∈ T , we can now apply Lemma 4.5 to conclude that i Σ k ( h k ( t i ) , t j ) (cid:54) = 0, which finishes the proof of the claim.Now we are in a situation where Theorem 3.4 applies. Note that after putting thecurves in T at different level fibers in M h k , we can assure that the Stallings disks boundedby them are disjoint, as we noted in Remark 4.14. We apply Theorem 3.4 to concludethat there is n ∈ N such that for all ( n , . . . , n k ) ∈ Z k with all | n i | > n, the map h k ◦ τ n t ◦ . . . ◦ τ n k t k is pseudo-Anosov. Furthermore, the family { h k ◦ τ n t ◦ . . . ◦ τ n k t k , with ( n , . . . , n k ) ∈ Z k and | n i | > n } contains infinitely many pairwise independent pseudo-Anosov mapping classes. Nowtaking A k to be the free abelian group generated by τ n i t i for some | n i | > n we are done. (cid:3) Next we construct the free cosets promised earlier.
Theorem 4.16.
Given a closed oriented 3-manifold M and g = g ( M ) as in Theorem4.15, the following is true: For g (cid:62) max( g , , any genus g open-book decomposition (Σ , h ) of M , with connected binding, can be stabilized to a genus g + 4 open book decom-position (Σ (cid:48) , h (cid:48) ) such that (1) h (cid:48) represents a pseudo-Anosov mapping class in Mod(Σ g +4 , ) ; and (2) there is a rank two free subgroup F <
Mod(Σ g +4 , ) such that every element in thecoset h (cid:48) F occurs as monodromy of an open book decomposition of M. Remark 4.17.
Note that a pseudo-Anosov rank two free coset always contains infinitelymany independent elements by Theorem 3.4, as it always contains (many) pseudo-Anosovrank one Abelian cosets in particular. Actually, repeated application of Theorem 3.4implies that ”most” elements, in the sense of Corollary 3.5 and its proof, in the free cosetare pseudo-Anosov.
Proof.
Recall that g is the genus guaranteed by Corollary 4.11. Now fix g (cid:62) max( g , , and apply Corollary 4.12 to obtain a genus g + 4 open book decomposition (Σ (cid:48) , h (cid:48) ) with h (cid:48) pseudo-Anosov. As in the proof of Lemma 4.13 , let a (cid:48) i , b (cid:48) i , for i = 1 , . . . , (cid:48) which bound mutually disjoint disks E i , E ∗ i ,each intersecting the surface Σ (cid:48) transversally. Now form the band sums D a = E γ E ∗ and D b = E δ E ∗ along arcs γ, δ that intersect exactly once and run from a to b andfrom a to b , respectively. The boundary curves a = ∂D a and b = ∂D b are Stallingscurves with their Stallings disks intersecting in a square on Σ (cid:48) . The intersection numberof a and b is four. See Figure 3.Since a, b have intersection four, the subgroup of Mod(Σ) generated by Dehn twists on a, b is a free group of rank 2. This is by Theorem 2.5. We claim that every word w ∈ F corresponds to an open book decomposition of M . Indeed we have w = τ m a τ n b . . . τ m k a τ n k b ,for some n i , m i ∈ Z . Take k copies of a ∪ b and place then on different level fibers ofthe open book decomposition as in the proof of Theorem 3.4. Now for each of these k copies place the copy of b on a different fiber than the copy of a that is slightly above b (cid:48) a (cid:48) a (cid:48) b (cid:48) Figure 3.
The choice of the arcs γ, δ illustrated in the case when Σ hasgenus 4 . the copy of a and below the next copy of a . We can put the copies of the Stalllingsdisks now so that all the 2 k curves bound disjoint disks. Since the open book to which f w = f τ m a τ n b . . . τ m k a τ n k b corresponds is obtained by surgery of along the curves above,and since these curves bound disjoint disks the result follows. (cid:3) Now we explain how to obtain Theorem 1.1 stated in the introduction: For g (cid:62) max(2 g , , Theorem 4.15 gives an abelian coset of rank (cid:98) g (cid:99) . and Theorems 4.16 give afree coset. Thus for g = max(2 g , , we have Theorem 1.1.5. Fibered knots in the Dehn fillings of the figure-eight knot
In this section, for p ∈ Z , let 4 ( p ) be the 3-manifold obtained by p -surgery on the figureeight knot. For g a large enough integer, we will produce infinite families of hyperbolicfibered knots in some manifolds 4 ( p ) with | p | (cid:62) g, . UsingTheorems 2.2 and 2.4 we will assure that the monodromies of these knots satisfy theAMU conjecture and they give infinite families of pseudo-Anosov mapping classes thatare independent in the sense defined in Section 3.1.5.1.
A family of two-component hyperbolic fibered links in S . We start byconsidering the family of two-component links { L l,m,k } l,m,k (cid:62) shown in Figure 4. As il-lustrated in the figure, a box labeled n denotes n successive positive crossings betweenthe two strings of the link the box involves. Similarly, a double box labeled k corre-sponds to stacking k -copies of pattern with two positive and two negative crossings onthe corresponding three strands of the link.The two-component link L l,m,k is the closure of an alternating braid. Furthermore, oneof its two components, shown in blue in Figure 4, is a figure-eight knot.We have the following. Lemma 5.1.
Let l, m, k (cid:62) . Then the complement of L l,m,k is hyperbolic and fibers withfiber of genus g ( l, m, k ) = 2 + m + l + 2 k .Proof. By [37], any alternating braid closure (actually, any homogeneous braid closure) isfibered. Thus the link L l,m,k is fibered. Moreover, one can concretely describe how a fiber OSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS 21 m l k k = ... (cid:41) k times...... n =... ... (cid:41) n times Figure 4.
The links L l,m,k . c c +1 c c +2 Figure 5.
Creating Stallings twist curves on the fiber F l,m,k .surface is built: first take a disk for each strand of the braid (so, as many as the indexof the braid) and connect them by twisted bands at crossings, where the orientation ofthe twisted bands agrees with the signs of the crossings.By this description, one can compute the Euler characteristic of the fiber surface F l,m,k of the closure of an homogeneous braid ˆ β by the formula χ ( F l,m,k ) = n ( β ) − c ( β ) where n ( β ) and c ( β ) are the braid index and number of crossings of β. In the case of the2-component link L l,m,k of the figure, we get χ (Σ) = 6 − (10 + 2 m + 2 l + 4 k ) = − − m − l − k. As F l,m,k has two boundary components, its genus is g ( l, m, k ) = − χ ( F l,m,k ) / m + l + 2 k. By a theorem of Menasco [31], any prime non-split alternating diagram of a link thatis not the standard diagram of the T ,q torus link represents a hyperbolic link. Here thediagram of L l,m,k in the figure is prime and non-split as long as both l, m and k are atleast 1 . (cid:3) Figure 6.
Crossing contributions to the framing determined by γ . Lemma 5.2.
Let L l,m,k be a fibered hyperbolic link as in lemma 5.1. The fiber F l,m,k contains k disjoint, pairwise non-parallel and boundary non-parallel Stallings curves.Proof. Recall that we have k blocks each of which contains a 3-braid with two positive andtwo negative crossings as illustrated in Figure 4. From each of these blocks we obtaina Stallings curve as follows: Consider a curve c ⊂ F l,m,k encircling the two negativecrossings of the block and intersecting each co-core of the corresponding half-twistedbands exactly once; similarly consider a curve c ⊂ F l,m,k encircling the two positivecrossings. The curve c i ( i = 1 ,
2) bounds an embedded disc E i in S that intersects the F l,m,k transversally and we have lk ( c , c +1 ) = − , while lk ( c , c +2 ) = 1 . For each block, wetherefore construct a Stallings curve by connecting the curves c , c by a band as shownin Figure 5. The disk bounded by this curve c is the band sum of c and c , and the curvesatisfies lk ( c, c + ) = 0 . Note that the Stalligns curves corresponding to different block aredisjoint, as they are supported on disjoint subsurfaces of Σ : the subsurfaces neighboringthe two positive and negative crossings of the block as in Figure 5.Note that Stallings curves corresponding to different blocks are non-parallel, since eachcurve intersects the co-cores of the half-twisted bands of its block exactly once and isdisjoint from the bands outside the block. Furthermore, none of Stallings curves can beparallel to a component of ∂F l,m,k since each of the two boundary components intersectsthe co-cores of every half-twisted band in the k -blocks an even number of times. (cid:3) Abelian cosets from S . We now prove the following theorem which, in particular,for m = 4 , l = 1 implies Theorem 1.4 stated in the Introduction. Theorem 5.3.
Let m, l, k (cid:62) , with l + m (cid:62) . In the mapping class group
Mod(Σ m + l +2 k, ) , there is a pseudo-Anosov abelian elementary coset of rank k consisting only of mappingclasses that satisfy the AMU conjecture.Proof. We begin by observing that the k Stallings curves and disks of Lemma 5.2 aredisjoint from the figure-eight component of L l,m,k . Thus these disks will survive in any3-manifold obtained by Dehn filling along the figure-eight component. For p ∈ Z , let4 ( p ) be the manifold obtained by p surgery on the figure-eight knot. We now want toperform a surgery on the figure-eight component of L l,m,k , so that the resulting link willbe a hyperbolic fibered knot L (cid:48) l,m,k in a manifold 4 ( p ) . Let γ be the curve on the fiber surface F l,m,k obtained by pushing the figure-eightcomponent inward along the fiber. If the slope p chosen for the surgery agrees with theslope determined by γ, then the link L (cid:48) l,m,k in 4 ( p ) will still be fibered. Moreover thefiber surface and the monodromy will now be obtained by simply capping off the relevant OSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS 23 boundary component of the fiber surface of L l,m,k , and the non-parallel Stallings curveswill remain non-parallel Stallings curves on the new fiber.To compute this framing we refer to Figure 6 where the strands of the figure-eightcomponent and the other component of L l,m,k are colored by blue and black respectivelyand the curve γ is shown in red. As illustrated in the right side panel of the figure,a positive crossing (resp. negative crossing) between two strands of the figure-eightcomponent contributes a − L l,m,k contribute a − . Hence the boxes of 2 m and 2 l successive positive crossings between the two-components of L l,m,k contribute − m and − l. In total, since the self linking number of the blue component is zero, the framingdetermined by γ is − m − l. When m + l (cid:62) , the new link L (cid:48) l,m,k is a link in a manifold 4 ( p ) with | p | (cid:62) . Thuswe know that lT V (4 ( p ) (cid:114) L (cid:48) l,m,k ) (cid:62) lT V (4 ( p )) > L (cid:48) l,m,k in 4 ( p ) is q-hyperbolic. The same is truefor all the 3-manifolds obtained from the complements of L (cid:48) l,m,k by (i) drilling out anynumber of the k Stallings curves described above; or (ii) performing any number of theStallings twists along these k curves. For all these manifolds are also link complementsin the same q-hyperbolic 3-manifold 4 ( p ) . These later links will be fibered in 4 ( p ) , and thus by Theorem 2.4, their monodromieswill satisfy the AMU conjecture. However, at this stage, we do not know whether themonodromy is pseudo-Anosov; they might just have some pseudo-Anosov parts. Nextwe will argue that this later case will not happen. Claim.
The link L (cid:48) l,m,k is hyperbolic and thus its monodromy is pseudo-Anosov. Proof of Claim.
Recall that a twist region in a link diagram consists of a maximalstring of bigons arranged end-to-end, so that the crossings alternate, and so that thereare no other bigons adjacent to the ends; the twist number of a diagram D is the numberof such twist regions, and is denoted by t ( D ). For instance, the twist number of thediagram of L l,m,k in Figure 4 is 10 + 2 k . The reader is referred to [16, Definition 2.4] formore details. A criterion of Futer-Purcell [18, Theorem 3.10], states that, if a component K of a hyperbolic link L visits at least 7 twist regions in a prime twist-reduced diagramof L, then any non-trivial slope on the cusp of the link complement corresponding to thecomponent K , has length bigger that 6. Then the “6-Theorem” of Agol-Lackenby [17,Theorem 3.13] implies that all non-trivial fillings along the complement K give hyperbolicmanifolds. As the figure-eight component of L l,m,k does visit 7 twist regions, we concludethat L (cid:48) l,m,k is hyperbolic, finishing thereby the proof of the claim.Continuing with the proof of the theorem, we can write the complement of L (cid:48) l,m,k asthe mapping torus of a pseudo-Anosov f ∈ Mod(Σ m + l +2 k, ). Considering all the othermonodromies we get after performing Stallings twists along the k curves of Lemma 5.2we will get a pseudo-Anosov abelian elementary coset of rank k of maps satisfying theAMU conjecture. More specifically, given m, l (cid:62)
1, let H k denote the free abelian groupof Mod(Σ m + l +2 k, ) generated by the k Stallings twists on the k curves. Now f H k isan elementary abelian pseudo-Anosov coset such that all the mapping classes in it are realized as monodromies of fibered links in 4 ( − l − m ). This concludes the proof of thetheorem. (cid:3) Applying Theorem 5.3 for m = 4 , l = 1 we get the following corollary which, byCorollary 3.7, gives Theorem 1.4 stated in the introduction. Corollary 5.4.
For any k (cid:62) , there is a rank k pseudo-Anosov abelian elementary cosetof Mod(Σ k, ) , consisting of mapping classes that satisfy the AMU conjecture. Let us point out that there is a lot of freedom in the construction described above andas a result we actually get many different pseudo-Anosov abelian elementary cosets. Forexample we can always change the values of m and l while keeping m + l (cid:62) Corollary 5.5.
Fix k (cid:62) . Then for every g (cid:62) k , the construction above gives apseudo-Anosov abelian elementary coset f H k in Mod(Σ g, ) , such that (1) all mapping classes in f H k satisfy the AMU Conjecture; and (2) for every pseudo-Anosov element in f H k the volume of the corresponding mappingtorus is bounded above by a constant that is independent of the genus.Proof. Fix k (cid:62)
1. Consider the knots L (cid:48) ,m,k constructed in the proof of Theorem 5.3,for l = 1. As discussed above for l = 1 and m (cid:62)
4, the knots L (cid:48) ,m,k are hyperbolic andfibered in 4 ( − − m ), with fiber a surface of genus g ( m ) = 3 + m + 2 k . Note that g ( m ) → ∞ as m → ∞ . Given m , we obtain a coset f H k that satisfies part (1) of thestatement of the corollary. It remains to show that the volumes of the correspondingmapping tori are bounded as claimed.By construction, the fibered manifolds with monodromies in f H k are obtained by Dehnfilling from S along the link J k := I k ∪ K , where K is the union of the disjoint Stallingscurves on the fiber of the link L ,m,k and I k the link obtained by augmenting the twistregion with the m crossings with a crossing circle and then removing all the 2 m crossings.Let || S (cid:114) J k || denote the Gromov norm of the complement of J k . The volume of anyfibered manifold with monodromy in f H k will be bounded above by B || S (cid:114) J k || where B is a universal constant. By construction, increasing m leaves B || S (cid:114) J k || unchanged,as all the crossings in the m -box of Figure 4 lie in a single twist region, while as notedearlier it changes the genus g ( m ) := 3 + m + 2 k of the link L (cid:48) ,m,k . (cid:3) We are now ready to prove the following, assuming Theorem 1.1, which we will provein the next section.
Corollary 5.6.
For any k > and g (cid:29) we have two rank k pseudo-Anosov abelian ele-mentary cosets in Mod(Σ g, ) that satisfy the AMU conjecture and such that no conjugatesof elements in one coset lies in the other.Proof. Given k > g (cid:62) k take a coset f A given by Corollary 5.5. With thenotation as in the proof of that corollary, let M be a closed q -hyperbolic 3-manifold with || M || (cid:29) B || S (cid:114) J k || (compare with Corollary 2.3), and apply Theorem 1.1 to M . Nowfor any g (cid:29) max { g ( M ) , k } take the second coset required by the statement of thecorollary to be that given by Theorem 1.1. By Theorem 2.4 the mapping classes in this OSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS 25 later coset, denoted by f (cid:48) H (cid:48) , satisfy the AMU conjecture. Recall that conjugate mappingclasses define homeomorphic mapping tori. Since the Gromov norm of mapping tori forelements in f (cid:48) H (cid:48) are bounded below by || M || while mapping tori for elements in f H arebounded above by || M || , the result follows. (cid:3) Finally to obtain Corollary 1.2 stated in the Introduction consider the coset constructedin Corollary 5.5 and take C = C ( k ) = B || S (cid:114) J k || , where this later quantity is defined inthe proof of Corollary 5.5. References [1] J. W. Alexander,
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Institut de Math´ematiques de Bourgogne, Facult´e des Sciences Mirande, 9 avenueAlain Savary, BP 47870, 21078 Dijon Cedex, France
Email address : [email protected] Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA
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