Commuting symplectomorphisms on a surface and the flux homomorphism
Morimichi Kawasaki, Mitsuaki Kimura, Takahiro Matsushita, Masato Mimura
CCOMMUTING SYMPLECTOMORPHISMS ON A SURFACE ANDTHE FLUX HOMOMORPHISM
MORIMICHI KAWASAKI, MITSUAKI KIMURA, TAKAHIRO MATSUSHITA,AND MASATO MIMURA
Abstract.
Let (
S, ω ) be a closed orientable surface whose genus l is at leasttwo. Then we provide an obstruction for commuting symplectomorphisms interms of the flux homomorphism. More precisely, we show that for every non-negative integer n and for every homomorphism α : Z n → Symp c ( S, ω ), theimage of Flux ω ◦ α : Z n → H ( S ; R ) is contained in an l -dimensional real linearsubspace of H ( S ; R ).For the proof, we show the following two keys: a refined version of thenon-extendability of Py’s Calabi quasimorphism µ P on Ham c ( S, ω ), and anextension theorem of ˆ G -invariant quasimorphisms on G for a group ˆ G and anormal subgroup G with certain conditions.We also pose the conjecture that the cup product of the fluxes of commutingsymplectomorphisms is trivial. Contents
1. Introduction 1Organization of the paper 52. Preliminaries on symplectic geometry 53. Proof of Theorem 1.3 74. Proof of Theorem 1.4 155. Proof of the principal theorem 186. A result towards Conjecture 1.5 207. Continuity of extended quasimorphisms 20Acknowledgment 23References 231.
Introduction
Let (
M, ω ) be a symplectic manifold, and Symp c ( M, ω ) the group of symplec-tomorphisms with compact supports on M . Let Symp c ( M, ω ) denote the identitycomponent of Symp c ( M, ω ), and (cid:94)
Symp c ( M, ω ) its universal cover. The flux homo-morphism (cid:93)
Flux ω : (cid:94) Symp c ( M, ω ) → H c ( M ; R ) is defined by (cid:93) Flux ω ([ { ψ t } t ∈ [0 , ]) = (cid:90) [ ι X t ω ] dt, where X t = ˙ ψ t . It is known that the flux homomorphism is a well-defined grouphomomorphism. The image of π (Symp c ( M, ω )) with respect to (cid:94)
Flux ω is called the a r X i v : . [ m a t h . S G ] F e b M. KAWASAKI, M. KIMURA, T. MATSUSHITA, AND M. MIMURA flux group of ( M, ω ), and the flux homomorphism descends a homomorphismFlux ω : Symp c ( M, ω ) → H ( M ; R ) / Γ ω , which is also called the flux homomorphism. The flux homomorphism is a funda-mental object in symplectic geometry and theory of diffeomorphism groups, andhas been extensively studied by many authors such as [LMP98], [K¸ed00], [Ono06],[Buh14], and [KLM18].Let S be a closed orientable surface whose genus l is at least two, and ω a volumeform on S . In this case, since π (Symp c ( S, ω )) is trivial, the flux homomorphismgives a group homomorphism from Symp c ( S, ω ) to H ( S ; R ). The goal of thispaper is to reveal that there is a restriction of commuting elements in Symp c ( S, ω )in terms of the flux homomorphism. To state our result precisely, we introducesome terminology. For a subset Z of a real vector space V , we write (cid:104) Z (cid:105) R to meanthe ( R -)linear subspace of V spanned by Z . Then, our main result is formulated asfollows: Theorem 1.1 (Restriction on commuting symplectomorphisms in terms of theflux homomorphism) . Let S be a closed orientable surface whose genus l is atleast two and ω a symplectic form on S . Then, for any homomorphism A : Z n → Symp c ( S, ω ) , dim R (cid:104) Im(Flux ω ◦ A ) (cid:105) R ≤ l holds true. We can also rephrase the statement of Theorem 1.1 in the following manner. Asubset C of a group is said to commuting if every pair of its elements commutes.Theorem 1.1 asserts that if C is a commuting subset of Symp c ( M, ω ), then theimage Flux ω ( C ) of C with respect to the flux homomorphism is contained in some l -dimensional linear subspace of H ( M ; R ). This theorem has the following corollarywhich was recently proved by the first and second authors. Corollary 1.2 (Kawasaki–Kimura [KK19, Theorem 1.11]) . Let S be a closed ori-entable surface whose genus is at least two, and ω a symplectic form on S . Then,the flux homomorphism Flux ω : Symp c ( S, ω ) → H ( S ; R ) does not have a section homomorphism. Here we make some remarks on Theorem 1.1. It is easy to see that there exist l elements f , · · · , f l of Symp c ( M, ω ) such that Flux( f ) , · · · , Flux( f l ) are linearlyindependent (see Remark 3.5). Thus the inequality in Theorem 1.1 is tight. Also,despite that for every homomorphism A : Z n → Symp c ( S, ω ) the image of A iscontained in some l -dimensional vector space, the rank of the image as a Z -modulecan be arbitrarily large. To see this, let X be a vector field on S such that L X ω =0 and [ ι X ω ] is a non-trivial element of H ( S ; R ). Let { ϕ tX } t ∈ R denote the flowgenerated by X . For a positive integer n , let A : Z n → Symp c ( S, ω ) denote thehomomorphism defined by A ( k , . . . , k n ) = ϕ k √ p + k √ p + ··· + k n √ p n X , where p i is the i -th smallest prime. Then, we can easily confirmrank Z Im(Flux ω ◦ A ) = n. OMMUTING SYMPLECTOMORPHISMS 3
The proof of Theorem 1.1 is obtained by examining the extendability of cer-tain quasimorphisms on the group of Hamiltonian diffeomorphisms Ham c ( S, ω ) on(
S, ω ). Recall that a real-valued function φ : G → R on a group G is called a quasimorphism if there exists a non-negative number D ≥ | φ ( xy ) − φ ( x ) − φ ( y ) | ≤ D for every x, y ∈ G . The smallest D is called the defect of φ , and denoted by D ( φ ).A quasimorphism φ is said to be homogeneous if φ ( g n ) = n · φ ( g ) for every g ∈ G and for every n ∈ Z .Suppose that G is a normal subgroup of another group ˆ G , and that φ is ahomogeneous quasimorphism on G . We say that φ is ˆ G -invariant if φ ( gxg − ) = φ ( x )for every g ∈ G and for every x ∈ H . We say that φ is extendable to ˆ G if thereexists a homogeneous quasimorphism ˆ φ on ˆ G such that the restriction ˆ φ | G of ˆ φ to G coincides with φ . It is easy to see that if a homogeneous quasimorphism on G isextendable to ˆ G , then it must be ˆ G -invariant.As is expected, there exists a ˆ G -invariant homogeneous quasimorphism which isnot extendable. One such example is Py’s Calabi quasimorphism µ P (see [Py06])on Ham c ( M, ω ). It is known that µ P is a Symp c ( M, ω )-invariant homogeneousquasimorphism. Nevertheless, the first and second authors showed in [KK19] that µ P is not extendable to the whole group Symp c ( M, ω ). One of the keys to the proofof Theorem 1.1 is to strengthen this result in [KK19] in the following manner.
Theorem 1.3.
Let S be a closed orientable surface whose genus l is at least two, ω a symplectic form on S and V a linear subspace of H ( S ; R ) . If dim R ( V ) > l ,then there does not exist homogeneous quasimorphism ˆ µ on Flux − ω ( V ) such that ˆ µ | Ham c ( S,ω ) = µ P . We note that there exists an l -dimensional linear subspace V of H ( S ; R ) suchthat µ P is extendable to Flux − ω ( V ) (see Remark 5.4).Let ˆ G be a group, and G a normal subgroup of ˆ G . As was noted in Ishida(see Lemma 3.1 of [Ish14]), every ˆ G -invariant homogenous quasimorphism on G is extendable to ˆ G , provided that Q = ˆ G/G is finite. The authors [KKMM21,Proposition 1.6] extend this to the case where the short exact sequence 1 → G → ˆ G → Q → Q which admits a section homomorphism Λ → ˆ G . Our second key to theproof of Theorem 1.1 is a further generalization of Ishida’s argument. To state itprecisely, we need some terminology. Let ˆ G be a group and G a normal subgroupof ˆ G . A quasimorphism φ : G → R is said to be ˆ G -quasi-invariant if there exists anon-negative number D (cid:48) such that ˆ g ∈ ˆ G and x ∈ G imply | φ (ˆ gx ˆ g − ) − φ ( x ) | ≤ D (cid:48) . Let D (cid:48) ( φ ) denote the smallest number D (cid:48) which satisfies the above inequality. Thenour third main theorem is formulated as follows. Here, for a group action Λ (cid:121) Q ,a strict fundamental domain B of it is a subset of Q which satisfies Q = (cid:71) λ ∈ Λ λB. M. KAWASAKI, M. KIMURA, T. MATSUSHITA, AND M. MIMURA
Theorem 1.4 (Extension theorem for uniform lattices) . Let ˆ G be a topologicalgroup, G a topological subgroup of ˆ G , and Q a locally compact Hausdorff group. Let → G → ˆ G q −→ Q → be an exact sequence of ( abstract ) groups. Let Λ be a discrete subgroup of Q whichhas a group homomorphism s : Λ → ˆ G satisfying q ◦ s ( x ) = x for every x ∈ Λ .Assume that there exists a relatively compact strict fundamental domain B of theaction Λ (cid:121) Q , which satisfies the following property: ( ∗ ) There exists a finite partition of B into Borel sets B = p (cid:97) i =1 B i , and for each i there exists a continuous ( set-theoretic ) section ¯ s i : B i → ˆ G of the map q | q − ( B i ) : q − ( B i ) → B i . Here, B i is the closure of B i in Q .Then for each ˆ G -quasi-invariant continuous quasimorphism φ : G → R , there existsa quasimorphism ˆ φ : ˆ G → R such that ˆ φ | G = φ and D ( ˆ φ ) ≤ D ( φ ) + 3 D (cid:48) ( φ ) . Under the assumption of Theorem 1.4, Λ is a uniform lattice in Q , namely, Λ is adiscrete subgroup of Q such that Λ \ Q is compact. Theorem 1.4 recovers [KKMM21,Proposition 1.6]. Indeed, if Q is discrete, then every subgroup Λ of finite index of Q fulfills condition ( ∗ ). We remark that if the locally compact group Q is moreoversecond countable, then there always exists a strict fundamental domain of the actionΛ (cid:121) Q which is a Borel subset of Q ; see for instance [BdlHV08, Proposition B.2.4].See Remark 4.9 for the homogeneous version of Theorem 1.4.Note that we do not assume that the group homomorphism q : ˆ G → Q is contin-uous. Consequently, the continuity of the extension ˆ φ is not necessarily guaranteedin Theorem 1.4. We apply Theorem 1.4 to the case that q is the flux homomor-phism, and its continuity is a quite subtle problem. We will discuss the case that q iscontinuous in Section 7: in this case, we show that we can take ˆ φ to be continuous.See Theorem 7.1 for the precise statement.We will apply Theorem 1.4 to the case of G = Ham c ( M, ω ) and ˆ G = Flux − ω ( V )with C -topology, where V is a linear subspace of H ( S ; R ) of dimension strictlygreater than l . However, µ P is known to be dis continuous on Ham c ( M, ω ) with C -topology, and hence we can not apply Theorem 1.4 directly to µ P . Nevertheless, bytaking the difference of µ P and some extendable quasimorphism µ B constructed byBrandenbursky [Bra15], we are able to show from a result of Entov-Polterovich-Py[EPP12] that µ P − µ B is continuous. Thus, Theorem 1.4 can be applied to µ P − µ B .In contrast to µ P , it is known that µ B is extendable to Symp c ( M, ω ). Thismeans that µ P − µ B is again non -extendable to Flux − ω ( V ) by Theorem 1.3. Thisis a contradiction, and we therefore conclude that dim R V ≤ l . This is an outlineof the proof of Theorem 1.1.We end this section by stating a conjecture, which can be regarded as a strength-ening of Theorem 1.1: Conjecture 1.5.
Let S be a closed orientable surface whose genus l is at leasttwo and ω a symplectic form on S . Then, for every h , h ∈ Symp c ( S, ω ) with h h = h h , Flux ω ( h ) (cid:94) Flux ω ( h ) = 0 . OMMUTING SYMPLECTOMORPHISMS 5
Here ‘ (cid:94) ’ means the cup product in cohomology.Note that Conjecture 1.5 implies Theorem 1.1. As supporting evidence, we willprove Conjecture 1.5 when Flux ω ( h ) lies in the Sp(2 l, Z )-orbit of a certain linearsubspace W β of H ( S ; R ); see Theorem 6.1 for the precise statement. Organization of the paper.
In Section 2, we review some concepts in symplecticgeometry. In Section 3, we will prove Theorem 1.3. In Section 4, we will proveTheorem 1.4. In Section 5, we will prove our main theorem, Theorem 1.1. In Section6, we will prove Theorem 6.1, which is an intermediate step towards Conjecture 1.5.In Section 7, we will prove Theorem 7.1, which ensures that we can take ˆ φ to becontinuous under the extra assumption that q is continuous.2. Preliminaries on symplectic geometry
In this section, we review some concepts in symplectic geometry which we willneed in the subsequent sections. For a more comprehensive introduction to thissubject, we refer to [Ban97, MS17, Pol01].Let (
M, ω ) be a symplectic manifold. Let Symp c ( M, ω ) denote the group ofsymplectomorphism with compact support and Symp c ( M, ω ) denote the identitycomponent of Symp c ( M, ω ). In this section, we endow Symp c ( M, ω ) with the C ∞ -topology.For a smooth function H : M → R , we define the Hamiltonian vector field X H associated with H by ω ( X H , V ) = − dH ( V ) for any V ∈ X ( M ) , where X ( M ) is the set of smooth vector fields on M .Let S denote R / Z . For a (time-dependent) smooth function H : S × M → R with compact support and for t ∈ S , we define a function H t : M → R by H t ( x ) = H ( t, x ). Let X tH denote the Hamiltonian vector field associated with H t and let { ϕ tH } t ∈ R denote the isotopy generated by X tH such that ϕ = id. We set ϕ H = ϕ H and ϕ H is called the Hamiltonian diffeomorphism generated by H . For a symplecticmanifold ( M, ω ), we define the group of Hamiltonian diffeomorphisms byHam c ( M, ω ) = { ϕ ∈ Diff( M ) | ∃ H ∈ C ∞ ( S × M ) such that ϕ = ϕ H } . Then, Ham c ( M, ω ) is a normal subgroup of Symp c ( M, ω ).Let (cid:94)
Symp c ( M, ω ) denote the universal covering of Symp c ( M, ω ). We define the(symplectic) flux homomorphism (cid:93)
Flux ω : (cid:94) Symp c ( M, ω ) → H c ( M ; R ) by (cid:93) Flux ω ([ { ψ t } t ∈ [0 , ]) = (cid:90) [ ι X t ω ] dt, where { ψ t } t ∈ [0 , is a path in Symp c ( M, ω ) with ψ = 1 and [ { ψ t } t ∈ [0 , ] is theelement of the universal covering (cid:94) Symp c ( M, ω ) represented by the path { ψ t } t ∈ [0 , .It is known that (cid:93) Flux ω is a well-defined homomorphism.We also define the (descended) flux homomorphism. We setΓ ω = Flux ω ( π (Symp c ( M, ω ))) , which is called the symplectic flux group . Then, (cid:93) Flux ω : (cid:94) Symp c ( M, ω ) → H c ( M ; R )induces a homomorphism Symp c ( M, ω ) → H c ( M ; R ) / Γ ω , which is denoted byFlux ω . M. KAWASAKI, M. KIMURA, T. MATSUSHITA, AND M. MIMURA
Proposition 2.1 ([Ban78, Ban97]) . Let ( M, ω ) be a closed symplectic manifold.Then, the following hold. (1) The flux homomorphism
Flux ω : Symp c ( M, ω ) → H ( M ; R ) / Γ ω is surjective. (2) Ker(Flux ω ) = Ham c ( M, ω ) . If S is a closed orientable surface whose genus is at least two, then the fluxgroup Γ ω of ( S, ω ) is trivial. Thus in this case, the flux homomorphism Flux ω is ahomomorphism from Symp c ( S, ω ) to H ( S ; R ).We review the definition of Calabi quasimorphisms. Recall that a real-valuedfunction φ : G → R on a group G is called a quasimorphism if D ( φ ) := sup x,y ∈ G | φ ( xy ) − φ ( x ) − φ ( y ) | < + ∞ . The constant D ( φ ) is called the defect of φ . A quasimorphism φ is said to be homogeneous if φ ( g n ) = n · φ ( g ) for every g ∈ G and for every n ∈ Z . The followingproperties are fundamental. Lemma 2.2.
Let φ be a homogenous quasimorphism on a group G . Then, forevery x, y ∈ G , the following hold true: (1) φ ( yxy − ) = φ ( x ) , (2) if xy = yx , then φ ( xy ) = φ ( x ) + φ ( y ) . Indeed, for x, y ∈ G and n ∈ N , we have ( yxy − ) n = yx n y − ; if xy = yx , thenwe in addition have ( xy ) n = x n y n . Lemma 2.2 immediately follows from theseequalities.A subset X of a symplectic manifold ( M, ω ) is said to be displaceable if thereexists φ ∈ Ham c ( M, ω ) satisfying φ ( X ) ∩ X = ∅ . Here, X is the topological closureof X .For a 2 n -dimensional exact symplectic manifold ( M, ω ) (meaning that the sym-plectic form ω is exact), we recall that the Calabi homomorphism is a functionCal M : Ham c ( M, ω ) → R defined byCal M ( ϕ F ) = (cid:90) (cid:90) M F t ω n dt. It is known that the Calabi homomorphism is a well-defined group homomorphism(see [Cal70, Ban78, Ban97, MS17, Hum11]).
Definition 2.3.
Let µ : Ham c ( M, ω ) → R be a homogeneous quasimorphism. Anon-empty open subset U of M has the Calabi property with respect to µ if ω is exact on U and the restriction of µ to Ham c ( U, ω ) coincides with the Calabihomomorphism Cal U .In terms of subadditive invariants, the Calabi property corresponds to the asymp-totically vanishing spectrum condition in [KO, Definition 3.5]. Definition 2.4 ([EP03, PR14]) . A Calabi quasimorphism is a homogeneous quasi-morphism µ : Ham c ( M, ω ) → R such that every non-empty displaceable open exactsubset of M has the Calabi property with respect to µ .For examples of Calabi quasimorphisms, see [EP03, McD10, FOOO19, Bra15,Cas17, LZ18, BKS18, Py06]. OMMUTING SYMPLECTOMORPHISMS 7 Proof of Theorem 1.3
Recall that S is a closed orientable surface whose genus l is at least two. Wetake curves α , . . . , α l , β , . . . , β l : [0 , → S on S as depicted in Figure 1. Let[ α ] ∗ , . . . , [ α l ] ∗ , [ β ] ∗ , . . . , [ β l ] ∗ ∈ H ( S ; R ) be the dual basis of [ α ] , . . . , [ α l ] , [ β ] , . . . , [ β l ] ∈ H ( S ; R ). Figure 1. α , . . . , α l , β , . . . , β l : [0 , → S The goal in this section is to prove the following theorem, which immediatelyimplies Theorem 1.3.
Theorem 3.1 (Precise form of Theorem 1.3) . Let S be a closed orientable surfacewhose genus l is at least two, and ω a symplectic form on S . Let V be a linear sub-space of H ( S ; R ) such that there exist vectors v = (cid:80) li =1 a i [ α i ] ∗ + (cid:80) lj =1 b (cid:48) j [ β j ] ∗ ∈ V and v = (cid:80) li =1 b i [ β i ] ∗ ∈ V with (cid:80) li =1 a i b i (cid:54) = 0 . Then, there does not exist homoge-neous quasimorphism ˆ µ on Flux − ω ( V ) such that ˆ µ | Ham c ( S,ω ) = µ P . Before proceeding to the proof of Theorem 3.1, we first deduce Theorem 1.3 fromTheorem 3.1.
Proof of Theorem . modulo Theorem . . Consider H ( S ; R ) as a symplectic vec-tor space whose symplectic form (cid:104)− , −(cid:105) is defined by (cid:104) [ α i ] ∗ , [ α j ] ∗ (cid:105) = (cid:104) [ β i ] ∗ , [ β j ] ∗ (cid:105) = 0 , (cid:104) [ α i ] ∗ , [ β j ] ∗ (cid:105) = (cid:40) i = j, . For a linear subspace U ⊂ H ( S ; R ), we write U ⊥ to mean the subspace { x ∈ H ( S ; R ) | (cid:104) x, y (cid:105) = 0 for every y ∈ U } . Since the bilinear form (cid:104)− , −(cid:105) is non-degenerate, we have dim U + dim U ⊥ =dim H ( S ; R ) = 2 l .Let W β be the linear subspace of H ( S ; R ) generated by [ β ] ∗ , · · · , [ β l ] ∗ . Supposedim V > l . To deduce Theorem 1.3 from Theorem 3.1, it suffices to show that V (cid:54)⊂ ( V ∩ N ) ⊥ . Suppose V ⊂ ( V ∩ N ) ⊥ . Since N ⊂ ( V ∩ N ) ⊥ , we have V + N ⊂ ( V ∩ N ) ⊥ .Therefore we havedim V − dim( V ∩ N ) + l = dim( V + N ) ≤ dim( V ∩ N ) ⊥ = 2 l − dim( V ∩ N ) . This implies dim V ≤ l , a contradiction. (cid:3) The rest of this section is devoted to the proof of Theorem 3.1. To prove thenon-extendability of µ P , we use the following simple argument: M. KAWASAKI, M. KIMURA, T. MATSUSHITA, AND M. MIMURA
Lemma 3.2 ([KK19, Lemma 4.8]) . Let ˆ G be a group, G a normal subgroup of ˆ G ,and µ a homogeneous ˆ G -invariant quasimorphism on G . Assume that there exist ˆ f , ˆ g ∈ ˆ G satisfying the following three conditions: • ˆ f (ˆ g ˆ f − ˆ g − ) = (ˆ g ˆ f − ˆ g − ) ˆ f , • [ ˆ f , ˆ g ] ∈ G , • µ ([ ˆ f , ˆ g ]) (cid:54) = 0 .Then, there does not exist a homogeneous quasimorphism ˆ µ on ˆ G such that ˆ µ | G = µ . Note that the second condition is automatically satisfied if Q = ˆ G/G is abelian.It is the case for ( ˆ
G, G, Q ) = (Symp c ( S, ω ) , Ham c ( S, ω ) , H ( S ; R )).Therefore, to prove Theorem 3.1, it suffices to find a pair of elements ˆ f andˆ g in Flux − ω ( V ) such that ˆ f , ˆ g , and µ P satisfy the three conditions in Lemma3.2. We construct such symplectomorphisms ˆ f and ˆ g on S by combining severalsymplectomorphisms on 1-punctured tori embedded into S . Hence we start withthe construction of certain symplectomorphisms on a 1-punctured torus.Let p : R → R / Z be the natural projection. For a positive number r with r < /
2, we set D r = ([0 , × [0 , \ ([ r, − r ] × [ r, − r ]) and P r = p ( D r ).We note that P r is diffeomorphic to the 1-punctured torus. We consider P r as asymplectic manifold with the symplectic form ω = dx ∧ dy , where ( x, y ) is thestandard coordinates on P r ⊂ R / Z . We define the curves α, β : [0 , → P r on P r by α ( t ) = p (0 , t ), β ( t ) = p ( t, (cid:15) be a real number with 0 < (cid:15) < / ι , . . . , ι l : ( P (cid:15) , ω ) → ( S, ω ) satisfying the following conditions:(1) ι i ◦ α = α i , ι i ◦ β = β i for i = 1 , . . . , l .(2) ι i ( P (cid:15) ) ∩ ι j ( P (cid:15) ) = ∅ if i (cid:54) = j .For a real number a with 0 < | a | < (cid:15) , we define I a to be the open interval(min { , a } , max { , a } ). Lemma 3.3.
For every real number a with | a | < (cid:15) , there exists a vector field Y a on P (cid:15) with compact support such that L Y a ω = 0 and ( Y a ) p ( x,y ) = a ∂∂x for any ( x, y ) ∈ ([ −| a | , | a | ] × R ) ∪ ( R × Z ) . Here, L Y a is the Lie derivative with respect to Y a .Proof. Let G a be a smooth function on R satisfying the following conditions: • G a ( x + m, y + n ) = G a ( x, y ) − an for any ( x, y ) ∈ R . • G a ( x, y ) = − ay for any ( x, y ) ∈ ([ −| a | , | a | ] × R ) ∪ ( R × [ −| a | , | a | ]). • There exists an open neighborhood U of [ (cid:15), − (cid:15) ] × [ (cid:15), − (cid:15) ] such that G a ( x, y ) = − a for any ( x, y ) ∈ U .Let ˆ Y a be the Hamiltonian vector field generated by G a (Figure 2). Then, bythe first and third conditions on G a , ˆ Y a induces the vector field Y a with compactsupport on P (cid:15) . Since ˆ Y a is a Hamiltonian vector field, we have L Y a ω = 0. Thesecond condition on G a implies ( Y a ) p ( x,y ) = a ∂∂x for every ( x, y ) ∈ ( I − a × R ) ∪ ( R × Z ).This completes the proof. (cid:3) For every real numbers a with | a | < (cid:15) , let ˆ g a be the time-1 map of the flowgenerated by Y a and we set ˆ g = id. Since L Y a ω = 0, we have ˆ g a ∈ Symp c ( P (cid:15) , ω ).We note that ˆ g a ( p ( x, y )) = p ( x + a, y ) for any ( x, y ) ∈ ( I − a × R ) ∪ ( R × Z ). OMMUTING SYMPLECTOMORPHISMS 9
Figure 2.
The vector field Y a (when a > a with 0 < | a | < (cid:15) , let ρ a : [ − / , / → [ − ,
1] be a smoothfunction satisfying the following conditions:(1) Supp( ρ a ) ⊂ ( −| a | , | a | ),(2) ρ a ( x ) + ρ a ( x + a ) = a | a | for any x ∈ I − a .By the above conditions, we can see that ρ a ≡ a | a | in a neighborhood of 0. For a = 0, we set ρ a ≡ a with 0 < | a | < (cid:15) , let H a : P (cid:15) → R be the function defined by H a ( p ( x, y )) = − ρ a ( x ) for | x | ≤ /
2. We note that H a has compact support since | a | < (cid:15) .Then, for a real number b , we define ˆ f ( a, b ) ∈ Symp c ( P (cid:15) , ω ) by(3.1) ˆ f ( a, b )( z ) = (cid:40) ϕ bH a ( z ) if z ∈ p ( I − a × R ) ,z otherwise . We define ˆ f ba ∈ Symp c ( P (cid:15) , ω ) by ˆ f ba = ˆ f ( a, b ) if a (cid:54) = 0 and ˆ f b = ˆ f ( (cid:15)/ , b ) if a = 0.The values of the flux homomorphism at ˆ f ba and ˆ g a are determined as follows. Lemma 3.4.
For every real number a with | a | < (cid:15) , and for every real numbers b , Flux ω (cid:16) ˆ f ba (cid:17) = b [ β ] ∗ and Flux ω (ˆ g a ) = a [ α ] ∗ . Proof.
Let X ba be the vector field that generates the flow { ˆ f bta } t ∈ R (Figure 3).First, we assume a (cid:54) = 0. Then, we have (cid:0) X ba (cid:1) z = (cid:40) b · ( X H a ) z (if z ∈ p ( I − a × R )) , . Hence, (cid:0) ι X ba ω (cid:1) z = p ( x,y ) = − ( d ( b · H a )) z = b ∂ρ a ∂x ( x ) dx if z ∈ p ( I − a × R ) , . Figure 3.
The vector field X ba (when a > ρ a ≡ a | a | in a neighborhood of 0, we have ∂ρ a ∂x (0) = 0; hence (cid:2)(cid:0) ι X ba ω (cid:1)(cid:3) ([ α ]) = (cid:90) α ι X ba ω = (cid:90) b ∂ρ a ∂x (0) dy = 0 . Since ρ a (0) = a | a | and Supp( ρ a ) ⊂ ( −| a | , | a | ), (cid:2)(cid:0) ι X ba ω (cid:1)(cid:3) ([ β ])= (cid:90) β ι X ba ω = (cid:90) I − a b ∂ρ a ∂x ( x ) dx = (cid:40) b ( ρ a (0) − ρ a ( − a )) (if a > ,b ( ρ a ( − a ) − ρ a (0)) (if a < . = b. Since we set ˆ f b = ˆ f ( (cid:15)/ , b ), we haveFlux ω (cid:16) ˆ f ba (cid:17) = b [ β ] ∗ . Here, we start to calculate Flux ω (ˆ g a ). By the definition of Y a ,( ι Y a ω ) z = p ( x,y ) = ady for every ( x, y ) ∈ ( I − a × R ) ∪ ( R × Z ) . Therefore, [( ι Y a ω )] ([ α ]) = (cid:90) α ι Y a ω = (cid:90) ady = a, [( ι Y a ω )] ([ β ]) = (cid:90) β ι Y a ω = (cid:90) β ady = 0 . OMMUTING SYMPLECTOMORPHISMS 11
Hence, Flux ω (cid:0) ˆ g ba (cid:1) = a [ α ] ∗ , as desired. (cid:3) Remark . Set ϕ i = ι i ∗ (ˆ g ) for i = 1 , · · · , l . Then ϕ , · · · , ϕ l commute with eachother since their supports are disjoint. The image[ β ] ∗ = Flux ω ( ϕ ) , · · · , [ β l ] ∗ = Flux ω ( ϕ l )of ϕ , · · · , ϕ l with respect to Flux ω are linearly independent. Therefore the in-equality mentioned in Theorem 1.1 is tight. Lemma 3.6.
For every real number a with | a | < (cid:15) and for every real number b , [ ˆ f ba , ˆ g a ] = ϕ bH a . Proof.
Since ρ a ( x ) + ρ a ( | a | + x ) = a | a | for any x ∈ ( −| a | , g a ,(3.2) ˆ g a ( ˆ f ba ) − ˆ g − a ( z ) = (cid:40) ϕ bH a ( z ) if z ∈ p ( I a × R ) ,z otherwise . Figure 4.
The vector field generating the flow { ˆ g a ( ˆ f bta ) − ˆ g − a } t ∈ R (when a > H a is contained in p ([ −| a | , | a | ] × R ), we obtain[ ˆ f ba , ˆ g a ] = ˆ f ba (cid:16) ˆ g a ( ˆ f ba ) − ˆ g − a (cid:17) = ϕ bH a . (cid:3) Lemma 3.7.
For every real number a with | a | < (cid:15) and for every real number b , [ ˆ f ba , ˆ g a ] = [ˆ g a , ( ˆ f ba ) − ] . Proof. If a = 0, then, since ˆ g a = id, we have [ ˆ f ba , ˆ g a ] = [ˆ g a , ( ˆ f ba ) − ] = id. Hence, wemay assume a (cid:54) = 0.By (3.1) and (3.2), the supports of ˆ f ba and ˆ g a ( ˆ f ba ) − ˆ g − a are disjoint. Therefore,we have [ ˆ f ba , ˆ g a ] = ˆ f ba (cid:16) ˆ g a ( ˆ f ba ) − ˆ g − a (cid:17) = (cid:16) ˆ g a ( ˆ f ba ) − ˆ g − a (cid:17) ˆ f ba = [ˆ g a , ( ˆ f ba ) − ] . (cid:3) Lemma 3.8.
For every real number a with | a | < (cid:15) , (cid:90) P (cid:15) H a ω = − a. Proof.
First, if a = 0, then we have H a ≡ (cid:82) P (cid:15) H a ω = 0. Hence, we mayassume a (cid:54) = 0.By Supp( ρ a ) ⊂ ( −| a | , | a | ), (cid:90) P (cid:15) H a ω = (cid:90) p ([ −| a | , | a | ] × R ) H a ω . Since ρ a ( x ) + ρ a ( | a | + x ) = a | a | for any x ∈ ( −| a | , (cid:90) p ([ −| a | , | a | ] × R ) H a ω = (cid:90) p ([ −| a | , × R ) H a ω + (cid:90) p ([0 , | a | ] × R ) H a ω = (cid:90) p ([ −| a | , × R ) H a ω + (cid:90) p ([ −| a | , × R ) ( − a | a | − H a ) ω = (cid:90) p ([ −| a | , × R ) − a | a | ω = − a. (cid:3) Lemma 3.9.
For every real number a with | a | < (cid:15) and for every pair b and b (cid:48) ofreal numbers, [ ˆ f ba , ˆ g a ˆ f b (cid:48) a ] = [ ˆ f ba , ˆ g a ] , [ˆ g a ˆ f b (cid:48) a , ( ˆ f ba ) − ] = [ˆ g a , ( ˆ f ba ) − ] . Proof.
Since { ˆ f ta } t ∈ R is the flow generated by a time-independent vector field, wehave ˆ f ca ˆ f da = ˆ f c + da = ˆ f da ˆ f ca and ˆ f − da = ( ˆ f da ) − for every pair c and d of real numbers.Thus, we have [ ˆ f ba , ˆ g a ˆ f b (cid:48) a ] = ˆ f ba ˆ g a ˆ f b (cid:48) a ( ˆ f ba ) − (cid:16) ˆ g a ˆ f b (cid:48) a (cid:17) − = ˆ f ba ˆ g a ˆ f b (cid:48) a ( ˆ f ba ) − ( ˆ f − b (cid:48) a )ˆ g − a = ˆ f ba ˆ g a ( ˆ f ba ) − ˆ g − a = [ ˆ f ba , ˆ g a ] . OMMUTING SYMPLECTOMORPHISMS 13
Similarly, we have[ˆ g a ˆ f b (cid:48) a , ( ˆ f ba ) − ] = ˆ g a ˆ f b (cid:48) a ( ˆ f ba ) − (cid:16) ˆ g a ˆ f b (cid:48) a (cid:17) − ˆ f ba = ˆ g a ˆ f b (cid:48) a ( ˆ f ba ) − ( ˆ f b (cid:48) a ) − ˆ g − a ˆ f ba = ˆ g a ( ˆ f ba ) − ˆ g − a ˆ f ba = [ˆ g a , ( ˆ f ba ) − ] . This completes the proof. (cid:3)
We have completed the preparation of arguments on a local model, and we nowstart a global argument. Since ι i ◦ α = α i and ι i ◦ β = β i for i = 1 , . . . , l , thefollowing lemma follows from Lemma 3.4. Lemma 3.10.
For i = 1 , . . . , l , for every real number a with | a | < (cid:15) , and for everypair b and b (cid:48) of real numbers, Flux ω (cid:16) ι i ∗ (cid:16) ˆ f ba (cid:17)(cid:17) = b [ β i ] ∗ and Flux ω (cid:0) ι i ∗ (ˆ g a ) (cid:1) = a [ α i ] ∗ . As the next step, we will prove the following lemma. For i = 1 , . . . , l , let ι i ∗ : Symp c ( P (cid:15) , ω ) → Symp c ( S, ω ) denote the natural pushout induced by the sym-plectic embedding ι i : ( P (cid:15) , ω ) → ( S, ω ). Lemma 3.11.
For i = 1 , . . . , l , for every real number a with | a | < (cid:15) , and for everyreal number b , µ P (cid:16) ι i ∗ (cid:16) [ ˆ f ba , ˆ g a ] (cid:17)(cid:17) = − ab To prove Lemma 3.11, we use the following property of Py’s Calabi quasimor-phism.
Proposition 3.12 ([Py06, Th´eor`eme 2]) . Let S be a closed orientable surface whosegenus is larger than one, ω a symplectic form on S and U an open subset of S whichis homeomorphic to the annulus. Then U has the Calabi property with respect toPy’s Calabi quasimorphism µ P .Proof of Lemma . . First, if a = 0, since ˆ g a = id, we have [ ˆ f ba , ˆ g a ] = id. Since µ P is a homogeneous quasimorphism, µ P (cid:16) ι i ∗ (cid:16) [ ˆ f ba , ˆ g a ] (cid:17)(cid:17) = 0(= 0 · b ). Hence, we mayassume a (cid:54) = 0.By Lemma 3.6, we have µ P (cid:16) ι i ∗ (cid:16) [ ˆ f ba , ˆ g a ] (cid:17)(cid:17) = µ P (cid:0) ι i ∗ (cid:0) ϕ bH a (cid:1)(cid:1) . Since p ([ −| a | , | a | ] × R ) is homeomorphic to the annulus and Supp( H a ) ⊂ p ([ −| a | , | a | ] × R ), by Proposition 3.12, we have µ P (cid:0) ι i ∗ (cid:0) ϕ bH a (cid:1)(cid:1) = (cid:90) S ι i ∗ ( bH a ) ω = (cid:90) P (cid:15) bH a ω . Hence, by Lemma 3.8, we have µ P (cid:16) ι i ∗ (cid:16) [ ˆ f ba , ˆ g a ] (cid:17)(cid:17) = b (cid:90) P (cid:15) H a ω = − ab. (cid:3) Now, we are ready to prove Theorem 3.1.
Proof of Theorem . . Let v = (cid:80) li =1 a i [ α i ] ∗ + (cid:80) lj =1 b (cid:48) j [ β j ] ∗ ∈ V , v = (cid:80) li =1 b i [ β i ] ∗ ∈ V be vectors with (cid:80) li =1 a i b i (cid:54) = 0. Without loss of generality, we may assume | a i | < (cid:15) for i = 1 , . . . , l . Setˆ f = ι ∗ (cid:16) ˆ f b a (cid:17) ◦ ι ∗ (cid:16) ˆ f b a (cid:17) ◦ · · · ◦ ι l ∗ (cid:16) ˆ f b l a l (cid:17) ∈ Symp c ( S, ω ) , ˆ g = ι ∗ (cid:16) ˆ g a ˆ f b (cid:48) a (cid:17) ◦ ι ∗ (cid:16) ˆ g a ˆ f b (cid:48) a (cid:17) ◦ · · · ◦ ι l ∗ (cid:16) ˆ g a l ˆ f b (cid:48) l a l (cid:17) ∈ Symp c ( S, ω ) . By Lemma 3.10, we haveFlux ω ( ˆ f ) = l (cid:88) i =1 b i [ β i ] ∗ , Flux ω (ˆ g ) = l (cid:88) i =1 a i [ α i ] ∗ + l (cid:88) j =1 b (cid:48) j [ β j ] ∗ . This implies ˆ f , ˆ g ∈ Flux − ω ( V ).It suffices to prove that ˆ f , ˆ g , and µ P : Ham c ( S, ω ) → R satisfy the assumptionsof Lemma 3.2. Since Ker(Flux ω ) = Ham c ( S, ω ) (see Section 2), we have[ ˆ f , ˆ g ] ∈ Ham c ( S, ω ) . Since ι i ( P (cid:15) ) ∩ ι j ( P (cid:15) ) = ∅ if i (cid:54) = j , we haveˆ f (ˆ g ˆ f − ˆ g − )= [ ˆ f , ˆ g ]= (cid:104) ι ∗ (cid:16) ˆ f b a (cid:17) ◦ ι ∗ (cid:16) ˆ f b a (cid:17) ◦ · · · ◦ ι l ∗ (cid:16) ˆ f b l a l (cid:17) , ι ∗ (cid:16) ˆ g a ˆ f b (cid:48) a (cid:17) ◦ ι ∗ (cid:16) ˆ g a ˆ f b (cid:48) a (cid:17) ◦ · · · ◦ ι l ∗ (cid:16) ˆ g a l ˆ f b (cid:48) l a l (cid:17)(cid:105) = ι ∗ (cid:16) [ ˆ f b a , ˆ g a ˆ f b (cid:48) a ] (cid:17) ◦ ι ∗ (cid:16) [ ˆ f b a , ˆ g a ˆ f b (cid:48) a ] (cid:17) ◦ · · · ◦ ι l ∗ (cid:16) [ ˆ f b l a l , ˆ g a l ˆ f b (cid:48) l a l ] (cid:17) = ι ∗ (cid:16) [ ˆ f b a , ˆ g a ] (cid:17) ◦ ι ∗ (cid:16) [ ˆ f b a , ˆ g a ] (cid:17) ◦ · · · ◦ ι l ∗ (cid:16) [ ˆ f b l a l , ˆ g a l ] (cid:17) = ι ∗ (cid:16) [ˆ g a , ( ˆ f b a ) − ] (cid:17) ◦ ι ∗ (cid:16) [ˆ g a , ( ˆ f b a ) − ] (cid:17) ◦ · · · ◦ ι l ∗ (cid:16) [ˆ g a l , ( ˆ f b l a l ) − ] (cid:17) = ι ∗ (cid:16) [ˆ g a ˆ f b (cid:48) a , ( ˆ f b a ) − ] (cid:17) ◦ ι ∗ (cid:16) [ˆ g a ˆ f b (cid:48) a , ( ˆ f b a ) − ] (cid:17) ◦ · · · ◦ ι l ∗ (cid:16) [ˆ g a l ˆ f b (cid:48) l a l , ( ˆ f b l a l ) − ] (cid:17) = (cid:20) ι ∗ (cid:16) ˆ g a ˆ f b (cid:48) a (cid:17) ◦ ι ∗ (cid:16) ˆ g a ˆ f b (cid:48) a (cid:17) ◦ · · · ◦ ι l ∗ (cid:16) ˆ g a l ˆ f b (cid:48) l a l (cid:17) , (cid:16) ι ∗ (cid:16) ˆ f b a (cid:17) ◦ ι ∗ (cid:16) ˆ f b a (cid:17) ◦ · · · ◦ ι l ∗ (cid:16) ˆ f b l a l (cid:17)(cid:17) − (cid:21) = [ˆ g, ˆ f − ]= (ˆ g ˆ f − ˆ g − ) ˆ f Here, the fourth and sixth equalities follow from Lemma 3.9, the fifth equalityfollows from Lemma 3.7, and the third and seventh equalities follow from the factthat diffeomorphisms having disjoint supports commute.
OMMUTING SYMPLECTOMORPHISMS 15
Recall that ι i ( P (cid:15) ) ∩ ι j ( P (cid:15) ) = ∅ for i (cid:54) = j . Hence Lemmas 2.2, 3.9, 3.11, and theassumption (cid:80) li =1 a i b i (cid:54) = 0 imply µ P (cid:16) [ ˆ f , ˆ g ] (cid:17) = µ P (cid:16) ι ∗ (cid:16) [ ˆ f b a , ˆ g b a ˆ f b (cid:48) a ] (cid:17) ◦ ι ∗ (cid:16) [ ˆ f b a , ˆ g b a ˆ f b (cid:48) a ] (cid:17) ◦ · · · ◦ ι l ∗ (cid:16) [ ˆ f b l a l , ˆ g b l a l ˆ f b (cid:48) l a l ] (cid:17)(cid:17) = µ P (cid:16) ι ∗ (cid:16) [ ˆ f b a , ˆ g b a ] (cid:17) ◦ ι ∗ (cid:16) [ ˆ f b a , ˆ g b a ] (cid:17) ◦ · · · ◦ ι l ∗ (cid:16) [ ˆ f b l a l , ˆ g b l a l ] (cid:17)(cid:17) = l (cid:88) i =1 µ P (cid:16) ι i ∗ (cid:16) [ ˆ f b i a i , ˆ g b i a i ] (cid:17)(cid:17) = − l (cid:88) i =1 a i b i (cid:54) = 0 . Therefore, ˆ f , ˆ g , and µ P satisfy the assumptions of Lemma 3.2. This completes theproof of Theorem 3.1. (cid:3) Proof of Theorem 1.4
The goal in this section is to prove Theorem 1.4. Before proceeding to the proofof Theorem 1.4, we note the following facts.
Lemma 4.1.
Let Q be a locally compact Hausdorff group, and Λ a discrete sub-group. For every pair K and L of compact subsets of Q , the number of elements λ of Λ satisfying K ∩ ( λL ) (cid:54) = ∅ is finite.Proof. Define a map f : Q × Q → Q by f ( x, y ) = xy − . Then f ( K × L ) ∩ Λ isa compact discrete space and hence is finite. Since K ∩ λL (cid:54) = ∅ if and only if λ ∈ f ( K × L ), this completes the proof. (cid:3) Corollary 4.2.
Let K be a relatively compact subset of Q . Then there exist finitelymany elements λ , · · · , λ r ∈ Λ such that K ⊂ λ B ∪ · · · ∪ λ r B . In this section, we will prove Theorem 1.4 in the case of p = 1; the proof for ageneral case can be done without any essential change.Since Q has a lattice, the Haar measure ν of Q is bi-invariant (in other words, Q is unimodular; see [BdlHV08, Proposition B.2.2]). Since B is a Borel set andrelatively compact, we have ν ( B ) < ∞ . Now we show ν ( B ) >
0. Since Λ is adiscrete subgroup of Q , there exists an open neighborhood U of e Q in Q such that U ∩ λU = ∅ for every λ ∈ Λ − { e Q } . Since Q is locally compact, we can assume that U is relatively compact. Then, for each x ∈ U there exists exactly one λ x ∈ Λ suchthat λ x x ∈ B . Thus we have a map f : U → B defined by f ( x ) = λ x · x . We claimthat f preserves the measure ν . Indeed, by Corollary 4.2, the set Λ U := { λ x | x ∈ U } is finite. Observe that for each λ ∈ Λ U , the map U ∩ λ − B → Q , x (cid:55)→ λx preserves ν by the left-invariance of ν . By (finite-)additivity of ν , the claim above follows.Hence we have ν ( B ) ≥ ν ( f ( U )) = ν ( U ) >
0. Therefore, multiplying ν by aconstant, we may assume that ν ( B ) = 1.Let ¯ s : B → ˆ G be a continuous section of the map q | q − ( B ) : q − ( B ) → B . Since B is a strict fundamental domain of the action Λ (cid:121) Q , for each x ∈ Q there existsa unique pair λ ∈ Λ and b ∈ B satisfying x = λb . Define a map s : Q → ˆ G by s ( λb ) = s ( λ ) · ¯ s ( b ). We can easily prove the following lemma. Lemma 4.3.
The map s : Q → ˆ G is a set-theoretic section of q : ˆ G → Q and s ( λx ) = s ( λ ) · s ( x ) holds for every x ∈ Q and λ ∈ Λ . Lemma 4.4.
For every λ ∈ Λ , the restriction s | λB : λB → ˆ G can be extended to acontinuous section ¯ s λ : λB → ˆ G .Proof. Set ¯ s λ ( x ) = s ( λ )¯ s ( λ − x ) . (cid:3) Using s , we can define a map Φ : ˆ G × Q → G byΦ(ˆ g, x ) = ˆ g · s (cid:0) xq (ˆ g ) (cid:1) − s ( x ) . Indeed, the right-hand side of the above equality belongs to G because q (cid:16) ˆ g · s (cid:0) x · q (ˆ g ) (cid:1) − s ( x ) (cid:17) = q (ˆ g ) q (ˆ g ) − x − x = e Q . It follows from Lemma 4.3 that Φ(ˆ g, λx ) = Φ(ˆ g, x ) for every ˆ g ∈ ˆ G , λ ∈ Λ, and x ∈ Q . In other words, if we set Φ ˆ g ( x ) = Φ(ˆ g, x ), then Φ ˆ g : Q → G is a leftΛ-invariant function. Lemma 4.5.
For each ˆ g ∈ ˆ G , there exists a compact subset K of ˆ G such that Φ ˆ g ( B ) ⊂ K .Proof. Since Bq (ˆ g ) is relatively compact, Corollary 4.2 implies that there existfinitely many elements λ , · · · , λ r of Λ such that Bq (ˆ g ) ⊂ λ B ∪ · · · ∪ λ r B . Foreach i = 1 , · · · , r , set B i = { x ∈ B | xq (ˆ g ) ∈ λ i B } . Let ¯ s : B → ˆ G and ¯ s λ : λB → ˆ G be continuous sections extending s | B and s | Bλ ,respectively. Define a continuous map Φ i : B i → ˆ G byΦ i ( x ) = ˆ g · ¯ s λ i ( xq (ˆ g )) − ¯ s ( x ) . Then we have that Φ ˆ g | B i = Φ i | B i . Since Bq (ˆ g ) ⊂ λ B ∪ · · · ∪ λ r B , we have B ⊂ λ Bq (ˆ g ) − ∪ λ Bq (ˆ g ) − ∪ · · · ∪ λ r Bq (ˆ g ) − ⊂ B ∪ B ∪ · · · ∪ B r . Therefore we haveΦ ˆ g ( B ) ⊂ Φ ˆ g ( B ) ∪ · · · ∪ Φ ˆ g ( B r ) ⊂ Φ ( B ) ∪ · · · ∪ Φ r ( B r ) . This completes the proof. (cid:3)
Lemma 4.6.
For each ˆ g ∈ ˆ G , the function φ ◦ Φ ˆ g : B → R is an integrable functionon B with respect to ν .Proof. By Lemma 4.5, there exists a compact subset K of ˆ G such that Φ ˆ g ( B ) ⊂ K .Since φ is continuous, φ | K is a bounded function. Hence, since Φ ˆ g ( B ) ⊂ K , thefunction φ ˆ g ◦ Φ : B → R is bounded. Since ν ( B ) = 1 < ∞ , we have that φ ◦ Φ ˆ g isintegrable. (cid:3) Now define a function ˆ φ : ˆ G → R by(4.1) ˆ φ (ˆ g ) = (cid:90) B φ ◦ Φ ˆ g dν = (cid:90) B φ (cid:0) ˆ g · s ( xq (ˆ g )) − s ( x ) (cid:1) dν ( x ) . Then it follows that ˆ φ | G = φ . Indeed, since g ∈ G implies q ( g ) = e Q , we haveˆ φ ( g ) = (cid:90) B φ (cid:0) g · s ( x · e Q ) − s ( x ) (cid:1) dν ( x ) = (cid:90) B φ ( g ) dν ( x ) = φ ( g ) . OMMUTING SYMPLECTOMORPHISMS 17
Hence, for completing the proof of Theorem 1.4, it suffices to show that ˆ φ is aquasimorphism satisfying D ( ˆ φ ) ≤ D ( φ ) + 3 D (cid:48) ( φ ).Before providing the proof of it, we note the following: fix an arbitrary element a ∈ Q . Then since Ba − is relatively compact, Corollary 4.2 implies that thereexist finitely many elements λ , · · · , λ r ∈ Λ such that Ba − ⊂ λ B ∪ · · · ∪ λ r B .Define C , · · · , C r by C i = { x ∈ Ba − | λ − i x ∈ B } . Since Ba − is also a strict fundamental domain of the action Λ (cid:121) Q , we have that Ba − = C (cid:116) · · · (cid:116) C r and B = λ − C (cid:116) · · · (cid:116) λ − r C r . Lemma 4.7.
Assume that f : Q → R is a left Λ -invariant measurable functionwhich is integrable on B with respect to ν . Then for every a ∈ Q , we have (cid:90) B f ( xa ) dν ( x ) = (cid:90) B f ( x ) dν ( x ) . Proof.
Since f and ν are left Λ-invariant, under the setting above, we have (cid:90) B f ( x ) dν ( x ) = r (cid:88) i =1 (cid:90) λ − i C i f ( x ) dν ( x )= r (cid:88) i =1 (cid:90) C i f ( λ i x ) dν ( x )= r (cid:88) i =1 (cid:90) C i f ( x ) dν ( x )= (cid:90) Ba − f ( x ) dν ( x ) . Since ν is right Q -invariant, we have (cid:90) Ba − f ( x ) dν ( x ) = (cid:90) B f ( xa ) dν ( x ) . This completes the proof. (cid:3)
Corollary 4.8.
For each ˆ g ∈ ˆ G and a ∈ Q , we have the following equality: ˆ φ (ˆ g ) = (cid:90) B φ ◦ Φ ˆ g ( x ) dν ( x ) = (cid:90) B φ ◦ Φ ˆ g ( xa ) dν ( x ) . For a, b ∈ R and D ≥
0, we write a ∼ D b to mean | b − a | ≤ D . By Corollary 4.8,we can show ˆ φ (ˆ g ˆ g ) ∼ D ( φ )+3 D (cid:48) ( φ ) ˆ φ (ˆ g ) + ˆ φ (ˆ g ) in the following manner:ˆ φ (ˆ g ˆ g ) = (cid:90) B φ (cid:0) ˆ g ˆ g s ( xq (ˆ g ˆ g )) − s ( x ) (cid:1) dν ( x ) ∼ D (cid:48) (cid:90) B φ (cid:0) s ( x )ˆ g ˆ g s ( xq (ˆ g ˆ g )) − (cid:1) dν ( x )= (cid:90) B φ (cid:0) s ( x )ˆ g s ( xq (ˆ g )) − s ( xq (ˆ g ))ˆ g s ( xq (ˆ g ˆ g )) − (cid:1) dν ( x ) ∼ D (cid:90) B φ (cid:0) s ( x )ˆ g s ( xq (ˆ g )) − (cid:1) dν ( x ) + (cid:90) B φ (cid:0) s ( xq (ˆ g ))ˆ g s ( xq (ˆ g ˆ g )) − (cid:1) dν ( x ) ∼ D (cid:48) (cid:90) B φ (cid:0) ˆ g s ( xq (ˆ g )) − s ( x ) (cid:1) dν ( x ) + (cid:90) B φ (cid:0) ˆ g · s ( x · q (ˆ g ˆ g )) − s ( xq (ˆ g )) (cid:1) dν ( x )= (cid:90) B φ ◦ Φ ˆ g ( x ) dν ( x ) + (cid:90) B φ ◦ Φ ˆ g ( xq (ˆ g )) dν ( x )= ˆ φ (ˆ g ) + ˆ φ (ˆ g ) . Therefore we have | ˆ φ (ˆ g ˆ g ) − ˆ φ (ˆ g ) − ˆ φ (ˆ g ) | ≤ D ( φ ) + 3 D (cid:48) ( φ ) . This completes the proof of Theorem 1.4.
Remark . In the setting of Theorem 1.4, we moreover obtain a homogeneous quasimorphism ( (cid:99) φ h ) h in the following manner. First, we recall that for a quasimor-phism φ : G → R , we can take the homogenization φ h of φ by setting φ h ( g ) := lim n →∞ φ ( g n ) n for each g ∈ G . By Fekete’s lemma, the above limit exists. It is straightforwardto show that φ h is a homogeneous quasimorphism. Indeed, to see the assertion,observe that for every n ∈ N and for every g ∈ G , (cid:12)(cid:12)(cid:12)(cid:12) φ ( g n ) n − φ ( g ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ D ( φ ) . We also note that D ( φ h ) ≤ D ( φ ) and that D (cid:48) ( φ h ) = 0. The resulting φ h mightnot be continuous in general; nevertheless, the Lebesgue dominated convergencetheorem ensures that in the definition (cid:99) φ h : ˆ G → R ; ˆ g (cid:55)→ (cid:90) B φ h (cid:0) ˆ g · s ( x · q (ˆ g )) − s ( x ) (cid:1) dν ( x ) , the integral above makes sense. Then, the above proof of Theorem 1.4 shows that (cid:99) φ h is a quasimorphism with D ( (cid:99) φ h ) ≤ D ( φ h ), and it extends φ h . Finally, by takinghomogenization, we obtain a homogeneous quasimorphism ( (cid:99) φ h ) h : ˆ G → R satisfyingthe following two conditions: ( (cid:99) φ h ) h is an extension of φ h , and D (( (cid:99) φ h ) h ) ≤ D ( φ h ).5. Proof of the principal theorem
The goal in this section is to prove Theorem 1.1. The rough strategy is to applyTheorem 1.4 to µ P . However, the continuity of µ P is a quite subtle problem. Forthis reason, our precise strategy is to employ another quasimorphism µ P − µ B , OMMUTING SYMPLECTOMORPHISMS 19 instead of µ P . Here µ B is Brandenbursky’s Calabi quasimorphism constructed in[Bra15]. Proposition 5.1.
The quasimorphism µ P − µ B : Ham c ( S, ω ) → R is continuouswith respect to the C -topology. To prove Proposition 5.1, we use the following celebrated result by Entov, Polterovich,and Py.
Theorem 5.2 ([EPP12, Theorem 1.7]) . Let S be a closed orientable surface and ω a symplectic form on S . Let µ be a homogeneous quasimorphism on Ham c ( S, ω ) .Assume that there exists a positive number A such that for any disc D ⊂ S of arealess than A , the restriction of µ to Ham c ( D, ω | D ) vanishes. Then, µ is continuouswith respect to the C -topology.Proof of Proposition . . Note that any disc D ⊂ S of area less than (cid:82) S ω isdisplaceable. Hence, since µ P and µ B are Calabi quasimorphisms, µ P − µ B satisfiesthe condition of Theorem 5.2. (cid:3) We also note that µ B is constructed as the restriction of another quasimorphismˆ µ B : Symp c ( S, ω ) → R to Ham c ( S, ω ). Hence the following theorem immediatelyfollows from Theorem 1.3.
Theorem 5.3.
Let S be a closed orientable surface whose genus l is at least two, ω a symplectic form on S , and V a linear subspace of H ( S ; R ) . Assume dim R ( V ) >l . Then, there exists no homogeneous quasimorphism ˆ µ on Flux − ω ( V ) such that ˆ µ | Ham c ( S,ω ) = µ P − µ B .Proof of Theorem . . Let n be a non-negative integer, and A : Z n → Symp c ( S, ω )a group homomorphism. Set f i = A ( e i ) and x i = Flux ω ( f i ), where e , · · · , e n ∈ Z n are the standard basis of Z n . Let V be the linear subspace of H ( S ; R ) generatedby x , · · · , x n .In what follows, we will prove that we can apply Theorem 1.4 applies to µ P − µ B and to the exact sequence1 → Ham c ( S, ω ) → Flux − ω ( V ) → V → . Here, we endow Symp c ( S, ω ) with the C -topology and regard Ham c ( S, ω ) as asubgroup of the topological group Symp c ( S, ω ).Set m = dim R V . Then there exist i , · · · , i m such that x i , · · · , x i m form an( R -)basis of V . Let Λ be the additive subgroup generated by x i , · · · , x i m . ThenΛ is a uniform lattice of V . Since f i , · · · , f i m are commutative, the map s : Λ → Flux − ω ( V ) defined by n x i + · · · n m x i m (cid:55)→ f n i · · · f n m i m is a group homomorphism satisfying Flux ω ◦ s ( x ) = x for every x ∈ Λ.Set B = { t x i + · · · + t m x i m | ≤ t i < } . Then, B is a strict fundamental domain of Λ (cid:121) V , which is a Borel set. For each x ∈ H ( S ; R ), we can easily show that there exists a path ϕ tx : [0 , → Symp c ( S, ω ) suchthat Flux ω ( ϕ tx ) = tx (for example, see the proof of Proposition 3.1.6 of [Ban97]).Thus we have a continuous section s : B → Flux − ω ( V ) defined by s ( t x i + · · · + t m x i m ) = ϕ t x i ◦ · · · ◦ ϕ t m x im . As we have argued above, µ P − µ B is a Symp c ( S, ω )-invariant homogeneous quasi-morphism on Ham c ( S, ω ) and it is continuous with the C -topology. Therefore, wehave verified that all assumptions of Theorem 1.4 are fulfilled in the present setting.Together with Remark 4.9, Theorem 1.4 implies that there exists a homogeneousquasimorphism on Flux − ω ( V ) which extends µ P − µ B . By Theorem 5.3, this forces m = dim V to be strictly smaller than l + 1. This completes the proof of Theoerem1.1. (cid:3) Remark . Let f , · · · , f l ∈ Symp c ( M, ω ) be elements such that Flux ω ( f ) , · · · , Flux ω ( f l )are linearly independent in H ( S ; R ). Let V be the real vector space spanned byFlux ω ( f ) , · · · , Flux ω ( f l ). Exactly the same argument as in the proof above showsthat µ P is extendable to Flux − ω ( V ).In particular, there exists an l -dimensional subspace V of H ( S ; R ) such thatPy’s Calabi quasimorphism is extendable to Flux − ω ( V ). When V = W β , thenthis assertion is easily checked since the restriction Flux − ω ( W β ) → W β of the fluxhomomorphism has a section homomorphism.6. A result towards Conjecture 1.5
Recall that Theorem 1.1 follows from Theorem 1.3. Using Theorem 3.1 insteadof Theorem 1.3, we have the following theorem. Set W β = (cid:104) β , . . . , β l (cid:105) . Theorem 6.1.
Let S be a closed orientable surface whose genus l is at least twoand ω a symplectic form on S . Assume that Flux ω ( h ) ∈ Sp(2 l, Z ) · W β . Then, forevery h , h ∈ Symp c ( S, ω ) with h h = h h , Flux ω ( h ) (cid:94) Flux ω ( h ) = 0 holds true. Here Sp(2 l, Z ) naturally acts on H ( S ; R ), which can be seen as a symplecticvector space; Sp(2 l, Z ) · W β ⊂ H ( S ; R ) is the union of the orbits of elements in W β under this Sp(2 l, Z )-action. Proof.
First we deal with the case where Flux ω ( h ) ∈ W β . Let h , h ∈ Symp c ( S, ω )with h h = h h and Flux ω ( h ) ∈ W β . Suppose Flux ω ( h ) (cid:94) Flux ω ( h ) (cid:54) = 0. Set V = (cid:104) h , h (cid:105) . Since Flux ω ( h ) (cid:94) Flux ω ( h ) (cid:54) = 0, Theorem 3.1 implies that Py’sCalabi quasimorphism µ P is not extendable to Flux − ω ( V ). On the other hand, bythe same argument as the proof of Theorem 1.1 and h h = h h , we can showthat µ P is extendable to Flux − ω ( V ). This is a contradiction, and we conclude thatFlux ω ( h ) (cid:94) Flux ω ( h ) = 0.Finally, we treat the general case. Assume that Flux ω ( h ) ∈ Sp(2 l, Z ) · W β .Then, there exists T ∈ Sp(2 l, Z ) such that Flux ω ( h ) ∈ T · W β . Note that theaction Symp c ( S, ω ) on S naturally induces an action on H ( S ; R ), and that itfactors through the Sp(2 l, Z )-action. In particular, there exists f ∈ Symp c ( S, ω )for which f acts on H ( S ; R ) as T ∈ Sp(2 l, Z ). For this f , replace S with f − ( S );run the argument above for the first case to this f − ( S ), and return to the originalsurface S by applying f . It ends our proof for the general case. (cid:3) Continuity of extended quasimorphisms
Recall that in Theorem 1.4, we do not assume that the homomorphism q : ˆ G → Q is continuous. If we assume that q is continuous, then we can take the resulting OMMUTING SYMPLECTOMORPHISMS 21 extension ˆ φ : ˆ G → R to be continuous. The goal of this section is to show thisassertion. Namely, we show the following theorem. Theorem 7.1.
Let ˆ G be a Hausdorff topological group, G a topological subgroup of ˆ G , Q a locally compact Hausdorff group, and → G → ˆ G q −→ Q → an exact sequence of topological groups. In particular, q is a continuous group ho-momorphism. Let Λ be a discrete subgroup of Q which has a group homomorphism s : Λ → ˆ G satisfying q ◦ s ( x ) = x for every x ∈ Λ . Assume that there exists arelatively compact strict fundamental domain B of the action Λ (cid:121) Q , which sat-isfies condition ( ∗ ) in Theorem . . Then for each ˆ G -quasi-invariant continuousquasimorphism φ : G → R , there exists a continuous quasimorphism ˆ φ : ˆ G → R such that ˆ φ | G = φ and D ( ˆ φ ) ≤ D ( φ ) + 3 D (cid:48) ( φ ) . Here, we note that ˆ G is assumed to be Hausdorff in Theorem 7.1, unlike Theorem1.4. Before giving the proof of Theorem 7.1, we recall the following two well-knownfacts. As in Section 4, let ν be the Haar measure on Q with ν ( B ) = 1. Proposition 7.2 (Tube lemma) . Let K be a compact topological space. Then, forevery topological space X , for every point x of X , and for every neighborhood U of K × { x } in K × X , there exists a neighborhood W of x in X such that K × W ⊂ U . Proposition 7.3 (Strong continuity of the ( L -)right-regular representation) . As-sume that X is a Borel subset of Q with ν ( X ) < ∞ . Then for every positive number ε , there exists a neighborhood W of ˆ g such that ˆ g ∈ W implies ν ( Xq (ˆ g ) (cid:52) Xq (ˆ g )) <ε . Here (cid:52) is the symmetric difference: A (cid:52) B = ( A \ B ) (cid:116) ( B \ A ). Also recall that ν is bi-invariant. Proof of Theorem . . We only prove Theorem 7.1 in the case of p = 1; the prooffor the general case goes along an almost similar way. Assume that the grouphomomorphism q : ˆ G → Q is continuous. Take arbitrary ˆ g ∈ ˆ G . It suffices to showthat the quasimorphism ˆ φ : ˆ G → R defined in (4.1) is continuous at ˆ g . In whatfollows, we take arbitrarily ε > φ is bounded over Φ( { ˆ g } × B ), there exists a positive number C > { ˆ g } × B ) ⊂ ( − C/ , C/ q : ˆ G → Q is continuous and Q is locally compact, there exists an openneighborhood O of ˆ g such that q ( O ) is relatively compact in Q . Since B · q ( O ) isrelatively compact, Corollary 4.2 implies that there exist λ , · · · , λ r ∈ Λ such that B · q ( O ) ⊂ λ B (cid:116) · · · (cid:116) λ r B. For i = 1 , · · · , r , set A i = { (ˆ g, x ) ∈ O × B | xq (ˆ g ) ∈ λ i B } and ˜ B i = ( { ˆ g } × B ) ∩ A i . Here A i means the closure of A i in O × B .Then we have O × B = A (cid:116)· · ·(cid:116) A r and ˜ B i is compact. Now fix i ∈ { , , . . . , r } .Define a continuous function φ i : A i → R by φ i (ˆ g, x ) = φ (cid:0) ˆ g ¯ s λ i ( x · q (ˆ g )) − ¯ s ( x ) (cid:1) . Since { ˆ g } × B is a compact Hausdorff space, it follows from the Tietze extensiontheorem that there exists a continuous function ψ i : { ˆ g }× B → R such that ψ i | ˜ B i = φ i | ˜ B i and ψ i ( { g }× B ) ⊂ ( − C, C ). Set A (cid:48) i = A i ∪ ( { ˆ g }× B ) and define the functionˆ φ i : A (cid:48) i → R by ˆ φ i (ˆ g, x ) = (cid:40) φ i (ˆ g, x ) if (ˆ g, x ) ∈ A i ,ψ i (ˆ g, x ) if (ˆ g, x ) ∈ { ˆ g } × B. Since ψ i | ˜ B i = φ i | ˜ B i , this ˆ φ i is well-defined. Note that since ˆ G is Hausdorff, { ˆ g }× B is a closed subset of A (cid:48) i . Hence ˆ φ i is continuous and ˆ φ i | A i = φ ◦ Φ | A i .Let U i be the subset of A (cid:48) i consisting of the points of A (cid:48) i satisfying both of thefollowing conditions:(1) ˆ φ i (ˆ g, x ) ∈ ( − C, C ) , (2) | ˆ φ i (ˆ g, x ) − ˆ φ i (ˆ g , x ) | < ε/ . Then U i is an open subset of A (cid:48) i containing { ˆ g } × B . Since A (cid:48) i is a closed subsetof O × B , the complement F i = A (cid:48) i \ U i of U i in A (cid:48) i is a closed subset of O × B .Set U (cid:48) = ( O × B ) \ ( F ∪ · · · ∪ F r ). Then U (cid:48) is an open subset of O × B containing { ˆ g }× B . Since B is compact, by Proposition 7.2, there exists an open neighborhood W of ˆ g such that W × B ⊂ U (cid:48) . For each i , define B i to be the subset of Q satisfying˜ B i = { ˆ g } × B i . Apply Proposition 7.3 for B i for each i ∈ { , , . . . , r } , and takethe intersection. Thus, by choosing W to be sufficiently small, we can take theopen neighborhood W above in such a way that for every ˆ g ∈ W and for every i the following inequality holds: ν ( B i q (ˆ g ) (cid:52) B i q (ˆ g )) < ε Cr .
Let ˆ g ∈ W . It suffices to show | ˆ φ (ˆ g ) − ˆ φ (ˆ g ) | < ε for the proof of the continuityof ˆ φ at ˆ g . Set X = { x ∈ B | There is i such that both (ˆ g , x ) and (ˆ g, x ) belong to A i } ,Y = B \ X. Then we have | ˆ φ (ˆ g ) − ˆ φ (ˆ g ) | ≤ (cid:90) X | φ ◦ Φ(ˆ g, x ) − φ ◦ Φ(ˆ g , x ) | dν ( x )+ (cid:90) Y | φ ◦ Φ(ˆ g, x ) − φ ◦ Φ(ˆ g , x ) | dν ( x ) . If x ∈ X , then there exists i such that (ˆ g , x ) ∈ A i and (ˆ g, x ) ∈ A i . Then with theaid of this i , we obtain that | φ ◦ Φ(ˆ g, x ) − φ ◦ Φ(ˆ g , x ) | = | φ i (ˆ g, x ) − φ i (ˆ g , x ) | < ε/ ν ( X ) ≤ ν ( B ) = 1, we have (cid:90) X | φ ◦ Φ(ˆ g, x ) − φ ◦ Φ(ˆ g , x ) | dν ( x ) < ε . Next, we treat the integral above over Y . Since Y ⊂ (cid:83) ri =1 ( B i q (ˆ g ) (cid:52) B i q (ˆ g )), wehave ν ( Y ) ≤ r · ε Cr = ε C .
Since φ ◦ Φ( W × B ) is contained in ( − C, C ), we have (cid:90) Y | φ ◦ Φ(ˆ g, x ) − φ ◦ Φ(ˆ g , x ) | dν ( x ) ≤ C · ν ( Y ) = ε . OMMUTING SYMPLECTOMORPHISMS 23
Therefore, we conclude that | ˆ φ (ˆ g ) − ˆ φ (ˆ g ) | < ε . This proves the continuity of ˆ φ atˆ g . Since ˆ g is arbitrarily taken, we complete the proof of the continuity of ˆ φ . (cid:3) Acknowledgment
The first author, the third author and the fourth author are supported in partby JSPS KAKENHI Grant Number JP18J00765, 19K14536 and 17H04822, respec-tively. We thank Professors Masayuki Asaoka and Kaoru Ono, who have drawn theauthors’ attention to Conjecture 1.5.
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Email address : [email protected] (Mitsuaki Kimura) Graduate School of Mathematical Sciences, The University ofTokyo, Tokyo, 153-8914, Japan
Email address : [email protected] (Takahiro Matsushita) Department of Mathematical Sciences, University of the Ryukyus,Nishihara-cho, Okinawa 903-0213, Japan
Email address : [email protected] (Masato Mimura) Mathematical Institute, Tohoku University, 6-3, Aramaki Aza-Aoba,Aoba-ku, Sendai 9808578, Japan
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