BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I

Transcription

1 BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I TOBIAS HARTNICK AND ANDREAS OTT Abstract. We present a new technique that employs partial ifferential equations in orer to explicitly construct primitives in the continuous boune cohomology of Lie groups. As an application, we prove a vanishing theorem for the continuous boune cohomology of SL2, R) in egree four, establishing a special case of a conjecture of Mono. Contents 1. Introuction 1 2. Preliminaries on continuous boune cohomology 3 3. Partial ifferential equations 7 4. Construction of primitives Bouneness of primitives 17 Appenix A. Vanishing of o cocycles 22 Appenix B. The Frobenius integrability conition 25 References Introuction Ever since Gromov s seminal paper [25], boune cohomology of iscrete groups has prove a useful tool in geometry, topology an group theory. In recent years the scope of boune cohomology has wiene consierably. An important step was taken by Burger an Mono, who extene the theory to the category of locally compact secon countable groups uner the name of continuous boune cohomology [14, 38]. Not only i this lea to a breakthrough in the unerstaning of boune cohomology of lattices in Lie groups [14], but also triggere a series of iscoveries in rigiity theory e. g. [15, 8, 10, 9, 4, 41, 42, 18, 27, 26]), higher Teichmüller theory e. g. [11, 13, 12, 1]) an symplectic geometry e. g. [43, 20]). At the same time, our unerstaning of the secon boune cohomology has improve. In particular, the approach originally evelope for free groups [24] an hyperbolic groups [21] has been extene to larger classes of groups incluing mapping class groups [2] an acylinrically hyperbolic groups [30, 22]. Moreover, there has been some progress in constructing boune cohomology classes in higher egree [36, 28, 5, 23]. On the other han, our knowlege on vanishing results for boune cohomology in higher egree is still very poor. It was alreay known to Johnson [32] that the boune cohomology of an amenable group vanishes in all positive egrees. Here the primitive of a given cocycle Date: October 9, Mathematics Subject Classification. 20J06, 22E41, 35F35, 35A09, 35A30. 1

2 2 T. HARTNICK AND A. OTT is obtaine by applying an invariant mean. In contrast, for non-amenable groups no general technique for constructing primitives is available so far. In particular, there is not a single nonamenable group whose boune cohomology is known in all egrees. Actually, the situation is even worse. One may efine the boune cohomological imension of a group Γ to be bcγ) := sup { n H n b Γ; R) 0 }, where Hb n Γ; R) is the n-th boune cohomology of Γ with coefficients in the trivial moule R. At present we o not even know whether there exists any group Γ with bcγ) {0, }. The few vanishing results we have for the boune cohomology of non-amenable groups are all base on the vanishing of all cocycles in the respective egree in some resolution. The most farreaching results in this irection were achieve in [40] by choosing efficient resolutions. However, no such resolutions are known for ealing with the continuous boune cohomology Hcb n H; R) of non-amenable connecte Lie groups H. For such groups the most efficient resolution that is presently available is the bounary resolution [31, 14]. In this particular resolution cocycles vanish only in egree at most three; in egree greater than three there will inevitably be nonzero cocycles, an one faces the problem of fining primitives. This explains why the few existing vanishing results such as in [14, 16] o not go beyon egree three. Our goal in this article is to evelop a new approach to the construction of primitives in continuous boune cohomology for real semisimple Lie groups. To emonstrate its effectiveness we settle the following special case of a conjecture ue to Mono [39, Problem A]. Theorem 1.1. Let G be a connecte real Lie group that is locally isomorphic to SL 2 R). Then Hcb 4 G; R) = 0. Actually, for such G Mono conjecture that bcg) = 2, which means that Hcb n G; R) = 0 for all n > 2. In egree n = 3 there are no nonzero cocycles in the bounary resolution at all [16], but this is no longer true for n > 3. In this sense, Theorem 1.1 is the prototype of a vanishing theorem that requires the construction of primitives. We believe that our metho of proof generalizes to arbitrary n > 3, an possibly to other Lie groups. This will be aresse in future work. Mono s conjecture about the boune cohomology of SL 2 R) is a special case of a more general conjecture, which woul allow to compute the continuous boune cohomology of arbitrary connecte Lie groups. In fact, since continuous boune cohomology is invariant uner ivision by the amenable raical [14, 38], it is sufficient to compute the continuous boune cohomology of semsimple Lie groups H without compact factors an with finite center. For such groups it is conjecture [19, 39] that the natural comparison map between the continuous boune cohomology an the continuous cohomology is an isomorphism. This woul imply that bch) coincies with the imension of the associate symmetric space, thereby proviing examples of groups of arbitrary boune cohomological imension. Plenty is known by now about surjectivity of the comparison map [19, 25, 7, 34, 23, 28], while injectivity still remains mysterious in higher egrees. Inee, injectivity has so far been establishe only in egree two for arbitrary H [14], an for some rank one groups in egree three [16, 3, 6]. Theorem 1.1 is the first result in egree greater than three. Incientally, it has an application to the existence of solutions to perturbations of the Spence-Abel functional equation for Rogers ilogarithm, along the lines suggeste in [16]. This will be iscusse in the forthcoming article [29].

3 BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I 3 For the proof of Theorem 1.1 we shall reformulate the problem of constructing boune primitives in terms of a fixe point problem for the action of G on a certain function space. The main iea is then to escribe the fixe points as solutions of a certain system of linear first orer partial ifferential equations. In this way, we obtain primitives by solving the corresponing Cauchy problem. We show that for carefully chosen initial conitions, particular solutions have aitional iscrete symmetries, which we finally use to euce bouneness. We will give a more etaile outline of our strategy of proof in Section 2.6 after introucing some notation. Acknowlegement. We are most inebte to B. Tugemann for generously conucting numerous computer experiments at an early stage of this project, an for many helpful iscussions. We further thank U. Baer, M. Björklun, M. Bucher, M. Burger, A. Derighetti, J. L. Dupont, H. Heenmalm, A. Iozzi, A. Karlsson, H. Knörrer, A. Laptev, N. Mono, A. Nevo, E. Sayag, I. Smith, J. Swoboa, an A. Wienhar for iscussions an suggestions. We are very grateful to the Institut Mittag-Leffler an the organizers of the program Geometric an Analytic Aspects of Group Theory, the Max Planck Institute for Mathematics an the organizers of the program Analysis on Lie Groups, an École Polytechnique Féérale e Lausanne for their hospitality an for proviing us with excellent working conitions. T. H. receive funing from the European Research Council uner the European Union s Seventh Framework Programme FP7/ ), ERC Grant agreement no A. O. wishes to thank the Institut es Hautes Étues Scientifiques, the Isaac Newton Institute for Mathematical Sciences, the Max Planck Institute for Mathematics, the Centre e Recerca Matemàtica Barcelona, the Hausorff Research Institute for Mathematics, an the Department of Pure Mathematics an Mathematical Statistics at Cambrige University for their hospitality an excellent working conitions. He was supporte by a grant from the Klaus Tschira Founation, by grant EP/F005431/1 from the Engineering an Physical Sciences Research Council, an partially supporte by grant ERC-2007-StG from the European Research Council. 2. Preliminaries on continuous boune cohomology 2.1. The bounary action of PSL 2 R). The goal of this subsection is to escribe a moel for the continuous boune cohomology of SL 2 R). Since continuous boune cohomology of connecte Lie groups is invariant uner local isomorphisms [38, Cor ] we have H cb SL 2R); R) = H cb PSL 2R); R) = H cb PU1, 1); R), where the latter isomorphism is inuce by the Cayley transform. We prefer to carry out our computations in the group G := PU1, 1). The group G acts by fractional linear transformations on the Poincaré isc D an thereby ientifies with the group of orientation-preserving isometries of D, which is an inex 2 subgroup in the full isometry group Ĝ. The actions of G an Ĝ exten continuously to the bounary S 1 of D, an the corresponing actions on S 1 will be referre to as the bounary action of G respectively Ĝ. The action of Ĝ on S1 but not on D) may be ientifie with the action of PGU1, 1) by fractional linear transformations. It is well-known that this action is strictly 3-transitive. Explicitly, elements of PGU1, 1) can be represente by matrices of the form ) a b g a,b :=, b ā

4 4 T. HARTNICK AND A. OTT where a, b C with a 2 b 2 {±1}. We enote by [g a,b ] the equivalence class of the matrix g a,b in PGU1, 1) an efine ɛ := [g 0,1 ]. Then Ĝ = G ɛ, where G is given by equivalence classes of matrices of eterminant one, an ɛ acts on G via ɛg a,b ɛ 1 = gā, b. We will use the following notation concerning subgroups of G: If we efine k ξ := [ g e iξ/2, 0], as := [ ] [ ] g cosh s/2), sinh s/2), nt := g 1+ i 2 t, i 2 t, then the maps ξ k ξ, s a s an t n t are one-parameter groups R G whose images we enote by K, A an N, respectively. Our parametrization of these groups is somewhat non-stanar, but is convenient for certain computations in local coorinates, see Lemma 3.2 below. Observe that K = Stab G 0) is a maximal compact subgroup of G, that A normalizes N an that P = AN = Stab G 1) is a parabolic subgroup of G. Since N is the unipotent raical of P we have FixN) = {1}, whereas FixA) = {±1}. Moreover, we obtain an Iwasawa ecomposition G = KAN, an every elliptic hyperbolic, parabolic) element in G is conjugate to an element in K A, N) Cocycles an strict cocycles. We keep the notation introuce in the last subsection an enote by µ K the unique K-invariant measure on S 1. Given n 0, we shall write MS 1 ) n+1 ) for the space of µ n+1) K -measurable real-value functions on S 1 ) n an L S 1 ) n ) for the subspace of boune functions. The quotients of these spaces obtaine by ientifying µ n+1) K -almost everywhere coinciing functions will be enote by MS 1 ) n+1 ) an L S 1 ) n+1 ) respectively. We efine the homogeneous ifferential : MS 1 ) n+1 ) MS 1 ) n+2 ) by n+1 fz 0,..., z n+1 ) = 1) j fz 0,..., ẑ j,..., z n+1 ). This inuces corresponing ifferentials on L S 1 ) n ), MS 1 ) n+1 ) an L S 1 ) n+1 ). Elements in the kernels of these four ifferentials are respectively referre to as strict n-cocycles, strict boune n-cocycles, n-cocycles an boune n-cocycles. The group Ĝ acts iagonally on S1 ) n+1, an this action commutes with the action of the symmetric group S n+1 by permution of the variables. This inuces a Ĝ S n+1-action on each of the spaces MS 1 ) n+1 ), L S 1 ) n+1 ), MS 1 ) n+1 ) an L S 1 ) n+1 ), an these actions are intertwine by the corresponing homogeneous ifferentials, in particular they preserve the corresponing cocycles. Given a subgroup H < Ĝ, an H-invariant strict, boune) cocycle is simply calle a strict, boune) H-cocycle, an it is calle a strict, boune) H-cobounary if it is containe in the image of the H-invariants uner. A major technical inconvenience is cause by the non-surjectivity of the map MS 1 ) n ) G MS 1 ) n ) G, which means that a G-cocycle c may not amit an invariant representative, i.e., a strict G-cocycle which coincies with c almost everywhere. Fortunately, for boune function classes existence of invariant representatives follows from [37, Thm. A]. Lemma 2.1 Invariant representatives). The maps L S 1 ) n+1 ) G L S 1 ) n+1 ) G, n 0, are surjective. In fact, they amit a family of sections compatible with the homogeneous ifferentials.

5 BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I The bounary moel for continuous boune cohomology. A function class f L S 1 ) n+1 ) is calle alternating provie σ.f = 1) σ f for all σ S n+1, an we enote by L alt S1 ) n+1 ) < L S 1 ) n+1 ) the subspace of alternating function classes. Since the actions of Ĝ an S n+1 commute, this subspace is Ĝ-invariant. Proposition 2.2 Bounary moel, [38, Thm ]). Given a close subgroup H < Ĝ, the continuous boune cohomology of H is given by the cohomology of the complex L alt S1 ) +1 ) H, ), i.e., Hcb n H; R) = H n L alt S1 ) +1 ) H, ) for all n Even an o cocycles. Given f L alt S1 ) n+1 ), we enote by f ± the projections of f onto the ±1-eigenspace of ɛ. Explicitly we have f ± = 1 2 f ± ɛ.f) an f = f + + f. We say that f is even if f = f + an o if f = f. Since ɛ normalizes G the projections f f ± preserve the subspace of G-invariants. They also commute with homogeneous ifferentials, since ɛ oes. In particular, every G-invariant alternating strict) cocycle ecomposes uniquely into a sum of a G-invariant alternating strict) even cocycle an a G-invariant alternating strict) o cocycle, an similarly for cobounaries. On the level of cohomology this yiels a ecomposition H cb G; R) = H cb G; R) ev H cb G; R) o. The first summan H n cb G; R) ev can be ientifie with the continuous boune cohomology H n cb Ĝ; R) of the extene group Ĝ. Similarly, if we enote by R ɛ the unique non-trivial oneimensional Ĝ-moule, then H cb G; R) o = H cb Ĝ; R ɛ), whence the above ecomposition can also be written as H cb G; R) = H cb Ĝ; R) H cb Ĝ; R ɛ). In particular, the vanishing of H cb G; R) is equivalent to the vanishing of both H cb Ĝ; R) an H cb Ĝ; R ɛ). It turns out that the vanishing of H cb Ĝ; R ɛ) can be euce using only combinatorial arguments, see Proposition 2.3 below. The vanishing of the first summan is consierably harer an its proof will occupy the rest of this article. Proposition 2.3. Every alternating G-invariant boune 4-cocycle is even. In particular, H 4 cb G; R) o = H 4 cb Ĝ; R ɛ) = 0. The proof of Proposition 2.3 will be given in Appenix A. We remin the reaer that non-zero even alternating G-invariant boune 4-cocycles o exist, see [16] Primitives in the bounary moel. Given a boune G-cocycle c L S 1 ) 5 ) G an any close subgroup H G, we enote by Pc) H := {p M S 1 ) 4) } H p = c the space of H-invariant primitives of c an by P c) H := {p L S 1 ) 4) H } p = c the subspace of boune H-invariant primitives. In view of Proposition 2.2 the proof of Theorem 1.1 amounts to showing that P c) G for any boune alternating G-cocycle c, an

6 6 T. HARTNICK AND A. OTT by Proposition 2.3 we may furthermore assume that c is even. Uner this assumption we will explicitly construct elements in P c) G. For this purpose, we first efine an operator I : L S 1 ) n+1) L S 1 ) n) by ˆ 2.1) Ic)z 1,..., z n ) := cz, z 1..., z n ) µ K z). S 1 It inuces an operator I : L S 1 ) n+1 ) L S 1 ) n ), which by abuse of notation we enote by the same symbol. By integrating the cocycle equation c = 0, we see that Ic)) = c for every cocycle c. From now on we fix a boune G-cocycle c L S 1 ) 5 ) G. For the moment we o not nee to assume that c is either alternating or even. By K-invariance of the measure µ K we see from formula 2.1) that the function Ic) is K-invariant, hence a K-invariant primitive of c. It will, however, in general not be G-invariant. In orer to obtain G-invariant primitives we amen the operator I in the following way. We enote by S 1 ) n) S 1 ) n the subset of n-tuples of pairwise istinct points in S 1. Note in particular that the G-action on S 1 ) 3) is free an has two open orbits given by positively an negatively oriente triples. We write C S 1 ) 3) ) K for the space of K-invariant real-value smooth functions on S 1 ) 3) an consier it as a subspace of MS 1 ) 3 ). We then efine an operator P c : C S 1 ) 3)) K M S 1 ) 4), f Ic) + f. A key observation is that all G-invariant boune primitives of c necessarily lie in the image of the operator P c. This will allow us to express primitives in terms of smooth rather than measurable) solutions to ifferential equations. Proposition 2.4. The image of the operator P c satisfies P c) G P c C S 1 ) 3) ) K) Pc) K. Proof. We have alreay seen above that Ic) P c) K. We conclue that P c f) Pc) K for all f C S 1 ) 3) ) K. It also follows that if p P c) G is any boune primitive then p Ic)) = 0. Hence we must have p = Ic) + f for some measurable function f given by f = Ip Ic)). The nontrivial part of the proof is to show that this function f can be chosen to be smooth. In fact, by Lemma 2.1 we may take invariant representatives p an c an set f := I p I c)) MS 1 ) 3 ) K. We claim that this function is smooth on S 1 ) 3). Inee, for every g G we have ˆ ) fg.z 1, g.z 2, g.z 3 ) = pz, g.z 1, g.z 2, g.z 3 ) I c)z, g.z 1, g.z 2, g.z 3 ) µ K z) S ˆ ˆ 1 g.µ K ) = z) pz, z 1, z 2, z 3 ) g.µ ) K) w) cz, w, z 1, z 2, z 3 ) µ K w) µ K z). S 1 S 1 µ K µ K Smoothness of the Raon-Nikoym erivative [33, Prop. 8.43] hence implies that the function on G given by g fg.z 1, g.z 2, g.z 3 ) is smooth. Since G-orbits are open in S 1 ) 3) we infer that the function f is smooth.

7 BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I Strategy of proof. We briefly outline the strategy for the proof of Theorem 1.1. We shall procee in three steps. 1) In Section 3, we show that for any function f C S 1 ) 3) ) K the primitive P c f) is G- invariant if an only if f satisfies a certain system of linear first orer partial ifferential equations. 2) In Section 4, we explicitly construct solutions f of this system of ifferential equations, showing that Pc) G. 3) In Section 5, we prove that there exist particular solutions f with certain aitional iscrete symmetries. For such functions f we then show bouneness of P c f), establishing that P c) G. While the constructions in 1) an 2) work for arbitrary boune G-cocycles c, the construction in 3) relies on c being alternating an even. 3. Partial ifferential equations 3.1. The bounary action in local coorinates. In orer to escribe the bounary action of G = PU1, 1) explicitly, we introuce coorinates as follows. We consier S 1 as a subset of C an write z S 1 to enote a complex number z of moulus 1. In aition, it will often be convenient to work with the angular coorinate θ [0, 2π) efine by z = e iθ. Corresponingly, on S 1 ) n we will use the two sets of coorinates z 0,..., z n 1 ) an θ 0,..., θ n 1 ). Note that, in angular coorinates on S 1, the measure µ K is given by ˆ fz) µ K z) = fθ) θ := 1 ˆ 2π fθ) θ. S 1 2π 0 Convention 3.1. Throughout, all operations on angular coorinates will implicitly be unerstoo moulo 2π. For example, θ 2 θ 1 enotes the unique point in the interval [0, 2π) that is congruent to θ 2 θ 1 moulo 2π. The G-action on S 1 inuces a G-action on the interval [0, 2π) by the relation g.e iθ = e i g.θ. Note that in particular k ξ.θ = θ + ξ. The next lemma provies formulas for the infinitesimal action of the one-parameter subgroups {a s } an {n t } in angular coorinates. Lemma 3.2. The infinitesimal action of {a s } an {n t } in angular coorinates is given by for η [0, 2π). s a s.η) = sina s.η), t n t.η) = 1 cosn t.η) Proof. To prove the first formula, we compute s a s.φ) = 1 s=0 i e iφ s e i as.φ = 1 cosh s/2) e iφ + sinh s/2) s=0 i e iφ s sinh s/2) e iφ = sinφ). + cosh s/2)) s=0 Since {a s } is a one-parameter group we further infer that s a s.η) = σ a s+σ.η) = σ=0 σ a σ a s.η)) = sina s.η). σ=0

8 8 T. HARTNICK AND A. OTT Likewise, for the secon formula we compute t n t.φ) = 1 t=0 i e iφ t e i nt.φ = 1 t=0 i e iφ an conclue as above. t ) 1 + i 2 t) eiφ i 2 t i 2 t eiφ + 1 i 2 t = 1 cosφ) t= Funamental vector fiels. We enote by L n) K, Ln) A an Ln) N the ifferential operators that appear as funamental vector fiels for the infinitesimal action of the one-parameter groups {k ξ }, {a s } an {n t } on S 1 ) n). By Lemma 3.2, they are given in angular coorinates by L n) K L n) A L n) N n 1 = = n 1 = n 1 θ j sinθ j ) θ j 1 cosθ j )). θ j The next lemma is crucial for applications of the operators L n) K, Ln) A Lemma 3.3. Let L n) enote one of the operators L n) with the homogeneous ifferential in the sense that for every n > 0. K, Ln) A n L n) = L n+1) n Proof. Let λ C [0, 2π)) an consier the ifferential operators L n) n 1 λ := λθ j ) θ j an Ln) N in cohomology. an Ln) N. Then Ln) commutes for every n > 0. For any smooth n 1)-cochain q we compute L n+1) λ n q )) n n θ 0,..., θ n ) = λθ j ) 1) l q θ 0,..., θ θ ) ) l,..., θ n j n = l=0 1) l j l n = = l=0 l=0 λθ j ) q θ j θ0,..., θ l,..., θ n ) 1) l L n) λ q) θ 0,..., θ l,..., θ n ) n L n) λ q)) θ 0,..., θ n ).

9 BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I Infinitesimal invariance of primitives. We are now in a position to characterize G- invariance of primitives P c f) in terms of ifferential equations for the function f. First, let us efine functions c an c by 3.1) c θ 0, θ 1, θ 2 ) := cosϕ) cη, ϕ, θ 0, θ 1, θ 2 ) η ϕ an 3.2) c θ 0, θ 1, θ 2 ) := sinϕ) cη, ϕ, θ 0, θ 1, θ 2 ) η ϕ. An argument as in the proof of Proposition 2.4 using smoothness of the Raon-Nikoym erivative shows that we may choose the functions c an c to be smooth on S 1 ) 3). The next proposition achieves the first step in the agena outline in Section 2.6. Proposition 3.4. Let f C S 1 ) 3) ) K. Then the primitive P c f) Pc) K is G-invariant if an only if there exist functions v, v C S 1 ) 2) ) such that the triple f, v, v ) satisfies the system of partial ifferential equations 3.3) L 3) A f = c + v L 3) N f = c + v. Note that all functions appearing in 3.3) are smooth, so all erivatives can be unerstoo classically. The proof of Proposition 3.4 relies on the following lemma. Lemma 3.5. The function Ic) is smooth along G-orbits an satisfies where c an c are as in 3.1) an 3.2). L 4) A Ic) = c L 4) N Ic) = c, Proof. An argument as in the proof of Proposition 2.4 shows that the function Ic) can be chosen to be smooth along G-orbits. Fix an invariant representative c of c by Lemma 2.1. Using A-invariance of c an Lemma 3.2 we compute L 4) A Ic))θ 0,..., θ 3 ) = s cϕ, a s.θ 0,..., a s.θ 3 ) ϕ s=0 = a s.ϕ) s s=0 ϕ cϕ, θ 0,..., θ 3 ) ϕ a s.ϕ) = ϕ s cϕ, θ 0,..., θ 3 ) ϕ s=0 = ϕ sina s.ϕ) cϕ, θ 0,..., θ 3 ) ϕ s=0 = cosϕ) cϕ, θ 0,..., θ 3 ) ϕ.

10 10 T. HARTNICK AND A. OTT On the other han, the cocycle ientity 0 = cη, ϕ, θ 0,..., θ 3 ) = cϕ, θ 0,..., θ 3 ) cη, θ 0,..., θ 3 ) + integrates against cosϕ) to 3 1) j cη, ϕ, θ 0,..., θ j,..., θ 3 ) 0 = cosϕ) cϕ, θ 0,..., θ 3 ) η ϕ ) j c θ 0,..., θ j,..., θ 3 ) = cosϕ) cϕ, θ 0,..., θ 3 ) ϕ + c θ 0,..., θ 3 ). This establishes the first ientity. Likewise, to prove the secon ientity we compute L 4) N Ic))θ 0,..., θ 3 ) = t cϕ, n t.θ 0,..., n t.θ 3 ) ϕ t=0 n t.ϕ) = ϕ t cϕ, θ 0,..., θ 3 ) ϕ t=0 = ϕ 1 cosnt.ϕ) ) t=0 cϕ, θ 0,..., θ 3 ) ϕ = sinϕ) cϕ, θ 0,..., θ 3 ) ϕ, an then integrate the above cocycle ientity against sinϕ). Proof of Proposition 3.4. First of all, we observe that for f C S 1 ) 3) ) K we may choose the function P c f) = Ic) + f to be smooth along G-orbits. This follows by an argument as in the proof of Proposition 2.4. Hence the primitive P c f) is G-invariant if an only if it is infinitesimally G-invariant. Since the subgroups K, A an N generate the group G, this in turn is equivalent to P c f) satisfying the system of partial ifferential equations L 4) K P cf) = 0 L 4) A P cf) = 0 L 4) N P cf) = 0. The first equation is automatically satisfie since P c f) is K-invariant by Proposition 2.4. Writing out the efinition of P c f) an applying Lemmas 3.3 an 3.5, the remaining two equations are seen to be equivalent to L 3) A f) = c L 3) N f) = c. Applying the operator I as in the proof of Proposition 2.4, we conclue that f C S 1 ) 3) ) K solves this system if an only if there exist measurable functions v an v such that the triple

11 BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I 11 f, v, v ) satisfies system 3.3). However, it is not ifficult to see that v an v can be chosen to be smooth. In fact, by 3.3) the functions v an v are ifferences of smooth functions an hence smooth. Now replace v an v by Iv ) an Iv ) The Frobenius integrability conition. We now turn to the problem of fining solutions of system 3.3). As we shall see in Proposition 3.7 below, it follows from the classical Frobenius theorem that this system amits a smooth solution f, v, v ) if an only if the functions v an v satisfy a certain integrability conition. In orer to state the result we nee to introuce the function 3.4) čφ 1, φ 2 ) := sinη ϕ) cη, ϕ, ψ, φ 1, φ 2 ) η ϕ ψ. Lemma 3.6. The function č is smooth on S 1 ) 2) an K-invariant. Proof. An argument as in the proof of Proposition 2.4 shows that č C S 1 ) 2) ). By K- invariance of c an of the measure, for every ξ [0, 2π] we have čk ξ.φ 1, k ξ.φ 2 ) = sinη ϕ) cη, ϕ, ψ, φ 1 + ξ, φ 2 + ξ) η ϕ ψ = sinη ϕ) cη ξ, ϕ ξ, ψ ξ, φ 1, φ 2 ) η ϕ ψ = sin η + ξ) ϕ + ξ) ) cη, ϕ, ψ, φ 1, φ 2 ) η ϕ ψ = čφ 1, φ 2 ). Proposition 3.7 Integrability conition). The system 3.3) amits a solution f, v, v ) if an only if the pair v, v ) satisfies the system of partial ifferential equations L 2) K v + v ) = 0 3.5) L 2) K v v ) = 0 L 2) K v L 2) N v + L 2) A v č ) = 0. The proof of the proposition relies on the Frobenius theorem an will be eferre to Appenix B. In the sequel, we shall refer to system 3.5) as the Frobenius system. It will be enough for our purposes to fin a function f satisfying system 3.3) for some pair v, v ). We will hence not attempt to fin all solutions of system 3.5). Rather, we will construct a single special solution v, v ). Proposition 3.8. Let r C 0, 2π), C) be a smooth complex-value solution of the orinary ifferential equation 3.6) 1 e iφ ) r = i rφ) č0, φ). φ By Convention 3.1, we may efine a function v C S 1 ) 2), C) by 3.7) vθ 1, θ 2 ) := e iθ 1 rθ 2 θ 1 ). Then the pair v, v ) := Rev), Imv)) is a solution of the Frobenius system 3.5).

12 12 T. HARTNICK AND A. OTT Proof. We remin the reaer that throughout the proof we ahere to Convention 3.1. observe that if v is a solution of the system L 2) K v = i v 3.8) e iθ v 1 + e iθ v 2 = č, θ 1 θ 2 First then v, v ) := Rev), Imv)) is a solution of 3.5). Inee, taking real an imaginary parts of the first equation in 3.8) we obtain L 2) K v = v, L 2) K v = v, while taking the real part of the secon equation yiels Next consier the transformation Then the first equation in 3.8) is equivalent to L 2) K v L 2) N v + L 2) A v = č. uθ 1, θ 2 ) := e iθ 1 vθ 1, θ 2 ). 3.9) L 2) K u = 0, an the secon equation is equivalent to čθ 1, θ 2 ) = e iθ ) 1 e iθ 1 uθ 1, θ 2 ) θ ) = i uθ 1, θ 2 ) + u θ 1 + e iθ 1 θ 2 ) u θ 2 = i u0, θ 2 θ 1 ) + 1 e iθ 1 θ 2 ) ) u θ 1. + e iθ ) 2 e iθ 1 uθ 1, θ 2 ) θ 2 Here we use that θ1 u = θ2 u by 3.9). Let now r be a solution of equation 3.6) an set vθ 1, θ 2 ) := e iθ 1 rθ 2 θ 1 ). Then uθ 1, θ 2 ) = rθ 2 θ 1 ) obviously satisfies 3.9). By K-invariance of č from Lemma 3.6 we have čθ 1, θ 2 ) = č0, θ 2 θ 1 ). It follows that u solves equation 3.10). 4. Construction of primitives 4.1. Solving the Frobenius system. The first step in the construction of the primitive P c f) is to solve the ifferential equation 3.6) for the function r. As we have seen in Propositions 3.8 an 3.7, the function r then gives rise to a special solution v, v ) of the Frobenius system 3.5), which in turn etermines the inhomogeneities in system 3.3) in such a way that this system amits a solution f. The complex orinary ifferential equation 3.6) can be solve by applying the metho of variation of constants. Its general solution r C 0, 2π), C) is given by rφ) = 1 e iφ) C ˆ φ π č0, ζ) 1 cosζ) ζ ),

13 BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I 13 where C 0 is an arbitrary complex constant. Note that ifferent choices of C 0 lea to cohomologous cochains v. We may therefore assume C 0 = 0, obtaining 4.1) rφ) = e iφ ) ˆ φ π č0, ζ) 1 cosζ) ζ. We shall henceforth be working with the function r efine by this formula. A crucial observation is the following lemma. Lemma 4.1 Bouneness of the inhomogeneity). The function r is boune. In particular, the inhomogeneities in system 3.3) are boune. Proof. Observe that č c an 1 cosφ) cotφ/2) = 2 cosφ/2) 2. Hence for all φ 0, 2π) we have an estimate rφ) 1 ˆ 2 č 1 e iφ φ 1 π 1 cosζ) ζ 2 = 2 č 1 cosφ) cotφ/2) c Cauchy initial value problem. Recall that solutions to a first orer linear partial ifferential equation may be constructe explicitly by integration along its characteristic curves, with initial values prescribe on some non-characteristic hypersurface [17, Ch. 3]. We shall now apply this principle in orer to explicitly construct solutions of the system 3.3). Let the function r be given by formula 4.1). By Proposition 3.8 an Lemma 4.1 this etermines a boune solution v, v ) := Rev), Imv)) of the Frobenius system 3.5) by 4.2) vθ 1, θ 2 ) := e iθ 1 rθ 2 θ 1 ). We shall henceforth keep the function v efine that way. By Proposition 3.7 the system 3.3) then amits a solution f C S 1 ) 3) ) satisfying the system of equations 4.3a) L 3) K f = 0 4.3b) L 3) A f = c + v 4.3c) L 3) N f = c + v. We know from Section 3.2 that the characteristic curves for these equations are precisely the orbits for the actions of the one-parameter groups K = {k ξ }, A = {a s } an N = {n t } on S 1 ) 3), respectively. In orer to construct the function f we may therefore procee as follows. 1) We use equation 4.3a) in orer to construct f with initial values f 0 := f H0 prescribe on the hypersurface H 0 := { z 0, z 1, z 2 ) S 1 ) 3) z 0 := 1 }. Of course, equation 4.3a) just says that f is constant along the K-orbits in S 1 ) 3). The hypersurface H 0 is non-characteristic for the equation 4.3a) since it intersects transversally with the K-orbits in S 1 ) 3). Note that in orer for f to be compatible with the remaining equations 4.3b) an 4.3c), the hypersurface H 0 has to be invariant uner

14 14 T. HARTNICK AND A. OTT the actions of A an N. This, however, is inee the case since the point 1 remains fixe uner these two actions. 2) We use equation 4.3c) in orer to construct f 0 with initial values f 1 := f 0 H1 prescribe on the union of A-orbits H 1 := { a s.1, i, i) < s < } { a s.1, i, i) < s < } H 0. Note that H 1 is non-characteristic for the equation 4.3c) since it intersects transversally with the N-orbits in H 0. Moreover, in orer for f 0 to be compatible with the remaining equation 4.3b), the curve H 1 has to be invariant uner the action of A, which is obviously the case. 3) We use equation 4.3b) in orer to construct f 1 with initial values f 2 := f 1 H2 prescribe on the set of base points H 2 := { 1, e 2πi/3, e 4πi/3), 1, e 4πi/3, e 2πi/3)} H 1. Note that equation 4.3b), when restricte to the curve H 1, becomes an orinary ifferential equation which can be solve irectly. In particular, we see that solutions of system 3.3) are uniquely etermine by the initial values of f 2 on the set of base points H 2 an hence form a 2-parameter family. We will work out the etails of 1) in Section 4.3, while the etails of 2) an 3) will be worke out in Section Reuction of variables. In angular coorinates, the hypersurface H 0 introuce in the previous subsection is given by H 0 = { θ 0, θ 1, θ 2 ) [0, 2π) 3 θ 0 = 0, θ 1 θ 2, θ 1 0 θ 2 }. The canonical projection θ 0, θ 1, θ 2 ) θ 1, θ 2 ) ientifies H 0 with the open subset Ω := 0, 2π) 2 \ of the square, where 0, 2π) 2 enotes the iagonal in the open square. The coorinates in Ω will be enote by φ 1, φ 2 ). Moreover, we write Ω ± := {φ 1, φ 2 ) φ 1 φ 2 } for the open subsets corresponing to the two G-orbits in S 1 ) 3) consisting of positively an negatively oriente triples. The restriction f 0 := f H0 is then given by f 0 φ 1, φ 2 ) = f0, φ 1, φ 2 ). Note that by K-invariance the function f is recovere from f 0 by 4.4) fθ 0, θ 1, θ 2 ) = f 0 θ 1 θ 0, θ 2 θ 0 ). Since the hypersurface H 0 is invariant uner the actions of A an N, the system 3.3) restricts to the system L 2) A 4.5) f 0 = c 0 + v ) 0 L 2) N f 0 = c 0 + v ) 0. Here we enote by c 0, c 0, v ) 0 an v ) 0 the respective restrictions of the functions c, c, v an v, i. e., 4.6) c 0 φ 1, φ 2 ) := c 0, φ 1, φ 2 ), c 0φ 1, φ 2 ) := c 0, φ 1, φ 2 ) an 4.7) v ) 0 φ 1, φ 2 ) := v 0, φ 1, φ 2 ), v ) 0 φ 1, φ 2 ) := v 0, φ 1, φ 2 ). Note that these functions are smooth on Ω.

15 BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I 15 2π 2π φ 2 φ 2 π 0 π φ 1 2π 0 φ 1 2π Figure 1. A-orbits left) an N-orbits right) in the omain Ω 4.4. Metho of characteristics. We now construct the function f 0 by integrating equations 4.3b) an 4.3c) along their characteristic curves. Recall that the characteristics for these equations are precisely the orbits for the actions of the one-parameter groups A = {a s } an N = {n t } on Ω see Figure 1). It will be convenient to abbreviate the inhomogeneities appearing on the right-han sie of system 4.5) by 4.8) F c := c 0 + v ) 0 an F c := c 0 + v ) 0. If we prescribe the value of f 0 on a single point in each of the orbits Ω + an Ω, the function f 0 will be uniquely etermine by the relations an More precisely, let us enote by f 0 a S.φ 1, a S.φ 2 ) f 0 φ 1, φ 2 ) = f 0 n T.φ 1, n T.φ 2 ) f 0 φ 1, φ 2 ) = ˆ S 0 ˆ T 0 F c a s.φ 1, a s.φ 2 ) s F c n t.φ 1, n t.φ 2 ) t. op := { φ, 2π φ) φ 0, 2π) \ {π} } Ω the antiiagonal in Ω, which correspons to the hypersurface H 1 introuce in Section 4.2. Note that it has two connecte components. In orer to compute the function f 0 we first introuce new coorinates on Ω that are aapte to the N-orbits. For every point φ 1, φ 2 ) Ω we efine Φφ 1, φ 2 ) 0, π) π, 2π) in such a way that Φφ 1, φ 2 ), 2π Φφ 1, φ 2 )) is the point of intersection of the antiiagonal with the unique N-orbit passing through the point φ 1, φ 2 ). We then efine T φ 1, φ 2 ), ) by the relation φ 1, φ 2 ) = n T φ1,φ 2 ). Φφ 1, φ 2 ), 2π Φφ 1, φ 2 ) ). 1

16 2 16 T. HARTNICK AND A. OTT 2π op φ 2 ω + φ 1,φ 2 ) π Φ, 2π Φ) ω 0 π φ 1 2π Figure 2. A path traveling along A- an N-orbits from the basepoint ω to some point φ 1, φ 2 ) in Ω For later reference we note that 4.9) T φ 1, φ 2 ) = 1 2 cot ) φ1 + cot 2 )) φ2. 2 Here we use that the action of the one-parameter subgroup {n t } is given by the formula n t.φ = 2 cot t + arccotφ/2)). Integrating the secon equation in 4.5) along the N-orbits in Ω, with initial values f 1 prescribe on the antiiagonal op, we then obtain 4.10) f 0 φ 1, φ 2 ) = f 1 Φφ1, φ 2 ), 2π Φφ 1, φ 2 ) ) + ˆ T φ1,φ 2 ) 0 F c nt.φφ 1, φ 2 ), n t.2π Φφ 1, φ 2 )) ) t for every φ 1, φ 2 ) Ω. It remains to compute the function f 1 along the antiiagonal. Let ω + := 2π/3, 4π/3) an ω := 4π/3, 2π/3) be the points in Ω corresponing to the base points in H 2 introuce in Section 4.2. Note that ω + an ω coincie with the barycenters of the triangles enclosing the omains Ω + an Ω see Figure 2). Define a new coorinate Sφ), ) on each component of the antiiagonal op by the relation φ, 2π φ) = a Sφ).ω ±, epening on whether the point φ, 2π φ) lies in Ω + or Ω. Integrating the first equation in 4.5) along the A-orbits in op, with initial values f 2 prescribe on the base points {ω +, ω }, we get 4.11) f 1 φ, 2π φ) = f 2 ω ± ) + for every φ 0, π) π, 2π). ˆ Sφ) 0 F c a s.ω ± ) s

17 BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I Explicit primitives. Combining the results from the previous subsections, we are now in a position to give the following explicit characterization of primitives. Proposition 4.2 Explicit primitives). Let v an v be the real an imaginary parts of the function v efine by formula 4.2), where r is as in 4.1). Then the following hol. i) Let f C S 1 ) 3) ) K. The primitive P c f) Pc) K is G-invariant if an only if the function f solves system 3.3). ii) There is a one-to-one corresponence between solutions f C S 1 ) 3) ) K of system 3.3) an solutions f 0 C Ω) of system 4.5) via the relation fθ 0, θ 1, θ 2 ) = f 0 θ 1 θ 0, θ 2 θ 0 ). iii) Every pair f 0 ω + ), f 0 ω )) R 2 of initial values uniquely etermines a smooth solution f 0 C Ω) of system 4.5) by the formula 4.12) f 0 φ 1, φ 2 ) = f 0 ω ± ) + ˆ SΦφ1,φ 2 )) 0 + F c a s.ω ± ) s ˆ T φ1,φ 2 ) 0 F c nt.φφ 1, φ 2 ), n t.2π Φφ 1, φ 2 )) ) t, where the functions F c an F c are as in 4.8). Conversely, any smooth solution of system 4.5) arises in this way. Proof. Let f C S 1 ) 3) ) K. Assertion i) hols by Proposition 3.4, while ii) was prove in Section 4.3. Finally, our consierations in Section 4.4 show that the function f 0 satisfies 4.5) if an only if it is given by formulas 4.10) an 4.11), in terms of integration along the unique path in Ω starting at ω ± an traveling to φ 1, φ 2 ) along A- an N-orbits via the point Φφ 1, φ 2 ), 2π Φφ 1, φ 2 ) on the antiiagonal see Figure 2), with initial values prescribe at ω ±. This proves iii). The proposition achieves the secon step in the agena outline in Section 2.6. In particular, it shows that solutions f 0 of system 4.5) form a 2-parameter family. 5. Bouneness of primitives 5.1. Symmetries. Our construction of primitives in the previous section was vali for arbitrary G-cocycles c L S 1 ) 5 ). However, for the proof of bouneness of primitives, which we shall iscuss in this section, it turns out to be essential that c be alternating an even. As we have seen in Subsection 2.5, this is not a loss of generality. The following proposition iscusses symmetries of the various functions appearing in the construction of primitives resulting from these aitional assumptions. Proposition 5.1 Symmetries). Assume that the cocycle c is alternating an even. Then the inhomogeneities on the right-han sies of systems 3.3) an 4.5) have the following properties. i) The functions c + v an c + v are alternating. ii) The function F c = c 0 + v ) 0 is antisymmetric about the antiiagonal op in Ω. In particular, it vanishes along the antiiagonal. iii) The function F c = c 0 + v ) 0 is symmetric about the antiiagonal op in Ω.

18 18 T. HARTNICK AND A. OTT Proof. We begin with the following observation. The function č efine in 3.4) is alternating since c is assume to be alternating. By Lemma 3.6, the function č is K-invariant. Hence č0, ζ) = č ζ, 0) = č0, ζ). By Convention 3.1, substituting ζ by 2π ζ we infer from this that 5.1) ˆ φ π č0, ζ) 1 cosζ) ζ = ˆ 2π φ π č0, ζ) 1 cos ζ) ζ = ˆ φ π č0, ζ) 1 cosζ) ζ for every φ 0, 2π). Recall moreover from Section 4.2 that v, v ) := Rev), Imv)), where vθ 1, θ 2 ) = e iθ 1 rθ 2 θ 1 ) an r is as in 4.1). Let us prove i). Since c is alternating, it is immeiate from 3.1) an 3.2) that c an c are alternating. By 4.1) an 5.1) we have 5.2) vθ 1, θ 2 ) = e iθ 1 rθ 2 θ 1 ) = 1 e iθ 1 e iθ ) ˆ θ2 θ π = 1 e iθ 2 e iθ ) ˆ θ1 θ π č0, ζ) 1 cosζ) ζ č0, ζ) 1 cosζ) ζ = vθ 2, θ 1 ). It follows that v, an hence v an v are alternating. This proves i). For the proof of ii) an iii) we have to show that 5.3) F c φ 1, φ 2 ) = F c φ 2, φ 1 ), F c φ 1, φ 2 ) = F c φ 2, φ 1 ). To this en, we first note that by 4.6) an evenness of c we have c 0 φ 2, φ 1 ) = cosϕ) cη, ϕ, 0, φ 2, φ 1 ) η ϕ 5.4) = cosϕ) c η, ϕ, 0, φ 2, φ 1 ) η ϕ = cos ϕ) cη, ϕ, 0, φ 2, φ 1 ) η ϕ = cosϕ) cη, ϕ, 0, φ 1, φ 2 ) η ϕ = c 0 φ 1, φ 2 ), an similarly c 0 φ 2, φ 1 ) = sinϕ) cη, ϕ, 0, φ 2, φ 1 ) η ϕ 5.5) = sin ϕ) cη, ϕ, 0, φ 2, φ 1 ) η ϕ = sinϕ) cη, ϕ, 0, φ 1, φ 2 ) η ϕ = c 0φ 1, φ 2 ).

19 BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I 19 Applying 5.1) as in the proof of i) above, we obtain v θ 1, θ 2 ) = 1 e iθ 1 e iθ ) ˆ θ2 +θ π = 1 e iθ 1 e iθ ) ˆ θ2 θ Combining this with 5.2) we arrive at vθ 1, θ 2 ) = v θ 2, θ 1 ). π č0, ζ) 1 cosζ) ζ č0, ζ) 1 cosζ) ζ = vθ 1, θ 2 ). Consier the function v) 0 φ 1, φ 2 ) := v0, φ 1, φ 2 ). The previous two ientities imply that v) 0 φ 1, φ 2 ) = vφ 1, φ 2 ) v0, φ 2 ) + v0, φ 1 ) = v φ 2, φ 1 ) v0, φ 1 ) + v0, φ 2 ) ) = v) 0 φ 2, φ 1 ). Recall from 4.7) that v ) 0 an v ) 0 are the real an imaginary parts of v) 0. Hence we conclue that 5.6) v ) 0 φ 1, φ 2 ) = v ) 0 φ 2, φ 1 ), v ) 0 φ 1, φ 2 ) = v ) 0 φ 2, φ 1 ). The ientities 5.3) now follow from 5.4), 5.5) an 5.6), which proves ii) an iii). Next we consier symmetries of the solutions of system 4.5). We introuce some notation first. The S 3 -action on S 1 ) 3) commutes with the K-action, whence it escens to an action on Ω. To escribe this action explicitly, we enote by s 1 an s 2 the Coxeter generators of S 3 that act on S 1 ) 3) by swapping coorinates in the pairs θ 0, θ 1 ) an θ 1, θ 2 ), respectively. Then, with respect to the coorinates φ 1, φ 2 ) on Ω, the actions of s 1 an s 2 are given by s 1.φ 1, φ 2 ) = φ 1, φ 2 φ 1 ), s 2.φ 1, φ 2 ) = φ 2, φ 1 ). A function h 0 C Ω) will be calle alternating uner the action of S 3 if s.h 0 = 1) s h 0 for all s S 3. Thus a function h 0 C Ω) is alternating uner the action of S 3 if an only if the function h C S 1 ) 3) ) K efine by is alternating in the usual sense. hθ 0, θ 1, θ 2 ) = h 0 θ 1 θ 0, θ 2 θ 0 ) Proposition 5.2 Alternating solutions). Assume that the cocycle c is alternating an even. A solution f 0 C Ω) of system 4.5) is alternating uner the action of S 3 if an only if 5.7) f 0 ω + ) = f 0 ω ). In this case the primitive P c f) Pc) G, where f is efine by 4.4), is alternating. Proof. First of all, we observe that S 3 acts on the base points {ω +, ω } by 5.8) s 1.ω ± = ω, s 2.ω ± = ω. Hence, if f 0 is alternating uner the action of S 3 it follows that f 0 ω + ) = 1) s 1 f 0 s 1.ω + ) = f 0 ω ).

20 20 T. HARTNICK AND A. OTT Conversely, let f 0 be a solution of system 4.5) that satisfies 5.7). We will prove that f 0 coincies with its antisymmetrization uner the action of S 3. By Proposition 4.2 ii), the function f 0 correspons to a solution f C S 1 ) 3) ) K of system 3.3) via Let now fθ 0, θ 1, θ 2 ) = f 0 θ 1 θ 0, θ 2 θ 0 ). ˆf := 1 6 s S 3 1) s s.f be the antisymmetrization of f. Then ˆf C S 1 ) 3) ) K, an we further claim that ˆf solves system 3.3) as well. To see this, observe that in system 3.3) the operators L 3) A an L3) N are symmetric, while by Proposition 5.1 i) the inhomogeneities c +v an c +v are alternating. Now by K-invariance, the function ˆf gives rise to a function ˆf 0 C Ω) via ˆfθ 0, θ 1, θ 2 ) = ˆf 0 θ 1 θ 0, θ 2 θ 0 ). Then Proposition 4.2 ii) implies that ˆf 0 solves system 4.5). Moreover, we have ˆf 0 = 1 6 s S 3 1) s s.f 0, whence ˆf 0 is alternating uner the action of S 3. It follows from 5.7) an 5.8) that ˆf 0 ω ± ) = f 0 ω ± ). The uniqueness statement in Proposition 4.2 iii) implies that ˆf 0 coincies with f 0. The proposition shows that solutions f 0 of system 4.5) that are alternating uner the action of S 3 form a 1-parameter family Bouneness. In orer to complete the proof of Theorem 1.1 it remains to show that among the G-invariant primitives we constructe in Section 4, there actually exist boune ones. This is the content of the next proposition, which crucially relies on the symmetries unveile in the previous subsection. Proposition 5.3 Bouneness). Assume that the cocycle c is alternating an even. Let f 0 C Ω) be a solution of system 4.5) that is alternating uner the action of S 3. Then the corresponing primitive P c f) Pc) G, where f is efine by 4.4), is boune. The proof of the proposition relies on the following three basic observations. Lemma 5.4. Let the function f 0 be efine by formula 4.12). If f 0 is boune along the line segments 0, 2π/3) 2π/3, 2π) ξ 2π/3, ξ) an 0, 4π/3) 4π/3, 2π) ξ 4π/3, ξ), an f is given by formula 4.4), then the corresponing primitive P c f) Pc) G is boune. Proof. By Proposition 4.2, P c f) = Ic) + f is G-invariant. By 3-transitivity of the G-action on S 1 an since Ic) is boune, we therefore euce that P c f) is boune if an only if the function z f 1, e 2πi/3, e 4πi/3, z )

21 BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I π φ 2 0 2π 4π φ 2π Figure 3. A funamental omain for the S 3 -action on Ω shae), an the images of the line segments ξ 2π/3, ξ) an ξ 4π/3, ξ) therein uner the S 3 -action is boune. Writing z = e iξ, we may express this function as The lemma follows. ξ f2π/3, 4π/3, ξ) f0, 4π/3, ξ) + f0, 2π/3, ξ) f0, 2π/3, 4π/3) = f 0 2π/3, ξ 2π/3) f 0 4π/3, ξ) + f 0 2π/3, ξ) f 0 2π/3, 4π/3). Lemma 5.5. Let C be a compact subset of the open square 0, 2π) 2. If the function f 0 efine by formula 4.12) is boune along the antiiagonal op in Ω, then it is boune on the subset C Ω of Ω. Proof. By Lemma 4.1 the function Fc = c 0 + v ) 0 is boune. Moreover, by assumption we have f 0 op M for some constant M > 0. Hence we obtain from formula 4.12) the estimate f 0 φ 1, φ 2 ) M + Fc T φ 1, φ 2 ) for all φ 1, φ 2 ) Ω. It remains to show that the function T is boune on C Ω. By compactness of C it will be enough to prove that the function T : Ω R extens to a continuous function on the open square 0, 2π) 2. This, however, is immeiate from formula 4.9). Lemma 5.6. Assume that the cocycle c is alternating. Then the function f 0 efine by formula 4.12) is locally constant along the antiiagonal op in Ω. Proof. Since c is alternating, the inhomogeneity Fc vanishes along the antiiagonal op by Proposition 5.1 ii). The lemma now follows from formula 4.12). Example 5.7. Assume that the cocycle c is alternating. Consier the special solution f 0 etermine by the initial values f 0 ω ± ) = 0. It is alternating uner the action of S 3 by Proposition 5.2. Moreover, by Lemma 5.6 it vanishes along the antiiagonal op. Since uner the

22 22 T. HARTNICK AND A. OTT action of S 3 the components of op get ientifie with the meians of the triangles enclosing the omains Ω + an Ω, we further infer that f 0 also vanishes along these meians. Moreover, by Proposition 5.1 iii) the function F c is symmetric about the antiiagonal. Thus we see from formula 4.12) that the special solution f 0 is antisymmetric with respect to the antiiagonal, an hence antisymmetric with respect to all meians. We are now reay to prove Proposition 5.3. Proof of Proposition 5.3. By Proposition 4.2 iii) the function f 0 is given by formula 4.12). Hence by Lemma 5.4 it suffices to show that f 0 is boune along the line segments ξ 2π/3, ξ) an ξ 4π/3, ξ). Since f 0 is alternating uner the action of S 3, it suffices to prove that f 0 is boune along the images of these line segments in any funamental omain for the S 3 - action on Ω. In fact, we may choose the funamental omain in such a way that the image line segments lie insie a compact subset of 0, 2π) 2 see Figure 3). By Lemma 5.6 an Lemma 5.5, the function f 0 is then boune on these line segments. Theorem 1.1 follows by combining Propositions 4.2, 5.2 an 5.3. Appenix A. Vanishing of o cocycles The goal of this appenix is to prove Proposition 2.3, which states that every boune alternating G-invariant 4-cocycle is necessarily even. Our strategy here is inspire by [16] in that we consier Fourier transforms of cocycles an stuy the conitions impose on them by G- invariance. Given n N 0, we enote by cz n+1 ) the space of complex-value sequences inexe by Z n+1 an by l 2 Z n+1 ) the subspace of square-summable sequences. We enote by e 0,..., e n the stanar basis of Z n+1 an use the multi-inex notation n k := k 0,..., k n ) := k j e j. We then write c alt Z n+1 ) an l 2 alt Zn+1 ), respectively, for the corresponing subspaces of alternating sequences an efine two linear operators A n) ± : l2 alt Zn+1 ) c alt Z n+1 ) by A n) n + F )k) := k j + 1) F k + e j ), A n) n F )k) := k j 1) F k e j ). Definition A.1. An element C l 2 alt Zn+1 ) is calle a combinatorial n-cocycle if the following hol: i) Ck 0,..., k n ) = 0 unless k j = 0 for precisely one j {0,..., n}. ii) Ck 0,..., k n ) = 0 unless k k n = 0. iii) C kera n) + ) keran) ). Accoring to [16, Sec. 3.1] the Fourier transform ĉ of a G-invariant alternating boune n- cocycle c is a combinatorial n-cocycle. Observe that c is even if an only if ĉ is even in the sense

23 BOUNDED COHOMOLOGY VIA PARTIAL DIFFERENTIAL EQUATIONS, I 23 that ĉ k) = ĉk). Proposition 2.3 will therefore be a consequence of the following combinatorial result. Proposition A.2. Every combinatorial 4-cocycle is even. For the proof of Proposition A.2 we nee two preparatory lemmas. Given a combinatorial n-cocycle C we enote by suppc) := {k Z n+1 Ck) 0} the support of C. Lemma A.3. Let C be a combinatorial 4-cocycle an k 0, k 1, k 2, k 3, k 4 ) suppc). Then there exists σ S 5 such that k σ0) < k σ1) < k σ2) = 0 < k σ3) < k σ4). Proof. Since C is alternating an vanishes unless precisely one of its entries is 0, it suffices to show that C vanishes on those k Z 5 which satisfy either A.1) k 0 > k 1 = 0 > k 2 > k 3 > k 4 or k 0 < k 1 = 0 < k 2 < k 3 < k 4. We are going to show Ck 0, 0, k 2, k 3, k 4 ) = 0 whenever k satisfies A.1) an leave the secon, analogous case to the reaer. Our proof will be by inuction on k 0. If k 0 5 then the conition k 2 + k 3 + k 4 = k 0 cannot be satisfie for k satisfying A.1), hence Ck) = 0. Otherwise we use C kera 4) + ) to euce that 0 = A 4) + C)k 0 1, 0, k 2, k 3, k 4 ) = k 0 Ck 0, 0, k 2, k 3, k 4 ) k 2 + 1) Ck 0 1, 0, k 2 + 1, k 3, k 4 ) + k 3 + 1) Ck 0 1, 0, k 2, k 3 + 1, k 4 ) + k 2 + 1) Ck 0 1, 0, k 2, k 3, k 4 + 1). The thir summan on the right-han sie vanishes by antisymmetry if k = 0, an by the inuction hypothesis otherwise. Similarly, the last two summans vanish. Now the assumption k 0 0 implies Ck 0, 0, k 2, k 3, k 4 ) = 0. Lemma A.4. Let C, D be combinatorial 4-cocycles such that hols for all n > 0. Then C = D. C n 1, n, 0, n, n + 1) = D n 1, n, 0, n, n + 1) Proof. Since the space of combinatorial cocycles is linear, we may assume D = 0 an hence n > 0 : C n 1, n, 0, n, n + 1) = 0. We have to show that C = 0. Using that C is alternating an Lemma A.3, it suffices to prove that Ck 0, k 1, 0, k 3, k 4 ) = 0 whenever k 0 < k 1 < 0 < k 3 < k 4. Since C kera 4) + ) we have 0 = A 4) + C)k 0, k 1, 0, k 3, k 4 1) = k 0 + 1) Ck 0 + 1, k 1, 0, k 3, k 4 1) + k 1 + 1) Ck 0, k 1 + 1, 0, k 3, k 4 1) + 0 We may rewrite this as A.2) + k 3 + 1) Ck 0, k 1, 0, k 3 + 1, k 4 1) + k 4 Ck 0, k 1, 0, k 3, k 4 ). Ck 0, k 1, 0, k 3, k 4 ) = k k 4 Ck 0 + 1, k 1, 0, k 3, k 4 1) k k 4 Ck 0, k 1 + 1, 0, k 3, k 4 1)