ON the BURNSIDE PROBLEM in Diff

Home , Burnside problem, Finite group, Infinite group

DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 17, Number 2, February 2007 pp. 423–439

ON THE BURNSIDE PROBLEM IN Diﬀ (M)

Julio C. Rebelo Dept. de Matematica, PUC-Rio R. Marques de S. Vicente 225 Rio de Janeiro RJ CEP 22453-900 Brazil Ana L. Silva Dept. de Matematica, CCE UEL Campus Universitario, Caixa Postal 6001 Londrina PR CEP 86051-990 Brazil

Dedicated to C. Gutierrez and to M. A. Teixeira on the occasion of their 60th birthday.

Abstract. In this paper we obtain some non-linear analogues of Schur’s theorem asserting that a finitely generated subgroup of a linear group all of whose elements have finite order is, in fact, finite. The main result concerns groups of symplectomorphisms of certain manifolds of dimension 4 including the torus T4.

1. Introduction. The main result of this paper reads as follows: Theorem A: Let M be a compact 4-dimensional symplectic manifold and denote by Symp (M) the group of symplectomorphisms of M (say of class C2). Suppose that the fundamental class in H4(M, Z) is a product of classes in H1(M, Z). Then any finitely generated subgroup G ⊂ Symp (M) having only elements of finite order is, in fact, finite. In the proof of Theorem A, an important role is played by Theorem B below. Consider now a compact (oriented) manifold M and let Diff2(M) be the group of orientation-preserving C2-diffeomorphisms of M. If µ is a probability measure on 2 2 M, denote by Diffµ the subgroup of Diff (M) consisting of those diffeomorphisms preserving µ, then one has:

2 Theorem B: Let M, µ and Diff µ(M) be as above. Suppose that the fundamental class in Hn(M, Z) is a product of elements in H1(M, Z). Suppose in addition that the mapping class group of M is finite. Then any finitely generated subgroup G of 2 Diffµ(M) all of whose elements have finite order is, in fact, finite. Remark: Naturally we can dispense with the condition regarding the mapping class group of M if we assume directly that G consists of diffeomorphisms isotopic to the identity. Some comments on the above statements are necessary. First the assumption concerning the finiteness of the mapping class group of M made in Theorem B is definitely strong, especially because mapping class groups are notoriously hard to

2000 Mathematics Subject Classiﬁcation. 53D25, 37C50 . Key words and phrases. Burnside problem .

423 424 JULIO C. REBELO AND ANA L. SILVA be computed. The reader who feels uncomfortable with this condition may simply regard Theorem B as a technical statement used in the proof of Theorem A (further motivation for Theorem A will be discussed in the sequel). Still we would like to point out that in dimension 3 there is a good number of examples of manifolds having finite mapping class group. These include Haken manifolds or, more generally, manifolds carrying genuine laminations, hyperbolic manifolds and some Seifert manifolds (for terminology and proofs the reader may check [7] and references therein). It should also be emphasized that our statements aim at manifolds whose Euler characteristic e(M) vanishes. Indeed, when e(M) 6= 0, these statements can easily be greatly generalized as explained in Theorem (3.1). The corresponding argument is similar in “spirit” to those employed in [8] and [6]. On the other hand little seems to be known about manifolds verifying e(M) = 0 except for dimensions 1 and 2. Let us also point out that throughout this paper we work with C2 diffeomorphisms. This is naturally required by the Hodge theoretic arguments (cf. Section 2) and we have not searched for the best condition in terms of regularity. Let us now situate the above theorems with respect to other works. In the present days, linear representation theory of groups is a classical and well-developed subject that appears in different aspects of mathematics. However, there are also many relevant cases where a group acts through automorphisms on some manifold and this action is not “linear” i.e. it does not embed in a finite dimensional Lie group acting on M. This suggests us to consider a “non-linear” theory of representations where the target group is no longer a matrix group (or a finite dimensional Lie group) but the whole group of diffeomorphisms of some compact manifold M. Here the amount of regularity of the diffeomorphisms in question is not specified so that we may consider real analytic ones if necessary. This problem is equally interesting if restricted to the groups of diffeomorphisms of M that preserve a volume form or a symplectic structure. Indeed our attention is focused on groups of symplectomorphisms due to two main reasons. The first one is that these groups are really huge and “essentially” non-linear which is necessary to be coherent with the above proposal. On the other hand, there is evidence that they share deep properties with linear groups, at least on the connected component of the identity. This is particu- larly clear from the work of Polterovich (cf. [18]) who has intensively investigated their structure. The point of view of (cf. [18]), however, is rather different from ours. The possibility of studying homomorphisms from abstract groups to a group of the form Diff (M) was discussed for example in [2]. This paper seems to have led E. Ghys to suggest to the first author that it would be interesting to consider actions of “torsion groups”. More precisely, a finitely generated, infinite, group all of whose elements have finite order is going to be called a Burnside group. The idea of E. Ghys was to discuss possible Burnside group actions on compact manifolds. As mentioned, his motivation apparently stemmed from [2] where the authors have suggested that a “generic” group may not act on a compact manifold. According to the works of Champetier and Ol’shanskii, Burnside groups are “generic” in the topological sense, i.e. they form a Gδ-dense set for a suitable topology on the space of finitely generated groups (cf. [10]). Nonetheless they are not “generic” in the probabilistic sense (see [10] for different notions of genericity used in Group Theory). Yet one first answer to the question of [2] might be achieved by showing ON THE BURNSIDE PROBLEM 425 that a Burnside group does not act on any compact manifold. Our theorems then can be viewed as a contribution to this problem in higher dimensions. Remark: After a classical result of Hölder stating that any non-Abelian group acting on the circle has points with non-trivial stabilizers, it immediately follows that a Burnside group cannot act faithfully on the circle. In fact, a diffeomorphism of finite order of the circle possessing fixed points must coincide with the identity. On the other hand, our main motivation to approach the question concerning the existence of an effective Burnside group action on a compact manifold comes, in an indirect way, from the Zimmer program as explained below. Note that years before [2], Zimmer had already proposed the program of studying homomorphisms from lattices of semi-simple Lie groups to Diff (M) with a view to obtaining “non-linear” versions of Margulis’s superrigidity theorems. Due to the difficulties involved and, to some extent, to the lack of suitable techniques to deal with them, definite progress in this program has been accomplished only in dimension 1 (cf. [9], [16], [25]). In view of the difficulties in carrying out the programs mentioned above, in this paper we turn to a question that, at least in principle, should be much simpler. Besides restricting attention to the symplectic category, we also propose to study infinite groups having “many” torsion elements. Of course, the prototype of the groups we have in mind are Burnside groups again. As a matter of fact, the quest for the properties of Burnside groups, in particular their very existence, is closely related to the development of Combinatorial Group Theory. Historically they have preceded the deep results on the structure of arithmetic subgroups and more general lattices in semi-simple Lie groups. Hence, it seemed to us that might be more reasonable to begin the study of “non-linear” representations of groups by considering possible embeddings of Burnside groups in the group of symplectomorphisms of a compact manifold. In this way, we are trying to obtain the non-linear analogue of a classical result due to Schur, cf. [23], rather than searching for the non-linear analogues of the very hard theorems of Margulis (the Zimmer program). This added to the independent interest of this study raised by Ghys observation quoted above. A consequence of our point of view is that, in some sense, this paper lies in the borderline between Dynamics and Topology. In fact, actions of infinite groups are associated to Dynamics since their orbits are not compact. On the other hand, actions of finite groups is a classical subject of Topology. Since our infinite groups have elements of finite order their possible actions have a significative topological component. To close this Introduction let us present a short description of the contents of the paper. The first ingredient needed is a criterion to guarantee that a given diffeomorphism coincides with the identity. We choose here one due to S. Weinberger ([20] and [24]) which is described in Section 2 along with examples of Burnside group actions on open manifolds. The second ingredient is the Flux homomorphism which is defined on the connected component of the identity of Diffµ(M) (for a given probability µ). The key point for Theorem B is a natural relation between this homomorphism and Wein- berger’s construction discussed in Section 2 (Proposition (3.2)). This proposition will be established in the Appendix where these objects will also be related to more general and dynamical constructions involving Rotation vectors due to Pollicott and Schwartzmann. Some of these interpretations are not strictly necessary for the proof of Proposition (3.2) but they might indicate directions to generalize the material presented in Section 2. 426 JULIO C. REBELO AND ANA L. SILVA

Theorem A is then proved in Section 3. The main difficulty now is that the mapping class group of the manifold in question is not necessarily finite. Therefore we are going to need a very fine control of the “iterations” of the elements in the group. To handle this situation we shall combine Theorem B with the fundamental result of Donaldson asserting the existence of a Lefschetz pencil on the symplectic manifold M.

2. Preliminaries. We shall not discuss the combinatorial structure of Burnside groups, instead we refer the reader to the survey [14]. Some examples of these groups can be obtained as follows. Given positive integers m,n ≥ 2, let Fm (resp. n Fm) be the free group on m generators (resp. the subgroup of Fm generated by th n n powers). The quotient B(m,n)= Fm/Fm is called the free Burnside group on m generators and exponent n. If n is larger than 248 then B(m,n) is infinite (and therefore a Burnside group in our sense). Also it should be pointed out that, if n is odd, then every finite subgroup of B(m,n) is cyclic. Therefore the usual theory of compact transformation groups does not provide much information concerning a possible effective action of B(m,n) on a chosen compact manifold. It is easy to construct examples of effective actions of Burnside groups on open manifolds. This goes as follows. First we recall that any finitely presented group Γ can be realized as the fundamental group of a compact 4-dimensional manifold (which can also be made symplectic thanks to a result of Gompf [11]). Hence Γ acts effectively on an open 4-dimensional manifold, namely the universal covering M of M. On the other hand, according to a theorem due to Higman [12], a finitely gen- f erated group embeds in some finitely presented group Γ if and only if the group in question is recursively presented. It follows that every finitely generated recursively presented group admits a faithful action on a compact open manifold of dimension 4. The class formed by these groups is however very large, in particular, it encompasses the Burnside groups B(m,n). This suggests that stronger constraints on the structure of a group G as above exist only when M is compact. The rest of this section is devoted to explaining a criterion to ensure that a diffeomorphism of finite order must be the identity. The material is borrowed from [20], [24]. Let M be a compact (oriented) n-dimensional manifold equipped with a Riemannian metric g. This metric determines a volume form det(gij )dx1 ∧··· dxn and, with the help of this volume form, we define the star operap tor ∗ : Ωk(M) → Ωn−k(M) (where Ωk(M) stands for the sheaf of differential k-forms on M). The operator δ : Ωk(M) → Ωk−1(M) is now defined by setting δ = (−1)n(k+1)+1 ∗ d∗ where d denotes the exterior differentiation. Finally the Laplacian ∆ (or Laplace- Beltrami operator) from Ωk(M) to Ωk(M) is obtained by letting ∆ = dδ + δd. A form ω ∈ Ωk(M) is said to be harmonic if and only if ∆(ω) = 0. The classical Hodge Theorem asserts that, fixed a Riemannian metric g on M, each cohomology k class c ∈ H (M, Z) admits a unique harmonic representative ηc, k =1,...,n. Now consider the dimension d of the first cohomology group H1(M, Z). The torus of dimension d can be thought of as the space Hom(H1(M, Z) , R/Z) consisting of the homomorphisms from H1(M, Z) to R/Z. Consider a given point p ∈ M and a Riemannian metric g on M. The Jacobian map Jac : M → Td = Hom (H1(M, Z) , R/Z) is defined by saying that Jac (x) ∈ Hom (H1(M, Z) , R/Z) is characterized by x Jac(x)(c)= ηc mod Z Zp ON THE BURNSIDE PROBLEM 427

1 where ηc stands for the harmonic representative of c ∈ H (M, Z). Note that the map Jac is well-defined since a closed form η represents an element in H1(M, Z) if and only if it has integral periods. Furthermore Jac induces an isomorphism between H1(M, Z) and H1(Td, Z). Theorem 2.1. ([20]) Let M be a smooth manifold and suppose that Γ is a finite group acting faithfully on M. Then there is an affine Γ-action on the torus Td = Hom (H1(M, Z) , R/Z) which is equivariant with respect to the Jacobian map Jac : M → Td = Hom (H1(M, Z) , R/Z) arising from a certain Riemannian metric g on M. Sketch of Proof : Since Γ is finite, we can choose a metric g so that Γ acts on M by isometries of g. Setting Td = Hom (H1(M, Z) , R/Z) we define an affine Γ action on Td as follows. Let γ ∈ Γ be identified with the corresponding automorphism of M. Given a homomorphism α : H1(M, Z) → R/Z we define a new homomorphism γ . α by the formula γ(p) ∗ (γ . α)(c)= α(γ (c)) + ηc mod Z . Zp Here c is a class of integral cohomology i.e. an element of H1(M, Z), γ∗(c) stands for the class of integral cohomology given as the pull-back of c by γ with γ viewed as an automorphism of M. Finally ηc is the unique harmonic representative of c. The fact that an automorphism of M induces a homomorphism in cohomology ∗ ∗ (i.e. γ (c1 + c2)= γ (c1)+ γ∗(c2) combines with the linearity of the Laplacian (i.e. 1 ∆(η1+η2) = ∆(η1)+∆(η2)) to show that γ . α is a homomorphism from H (M, Z) to R/Z, that is, a point in T d = Hom (H1(M, Z) R/Z). To check that above equation d gives rise to an action of Γ on T , we have to check that (γ1.γ2) . (α)= γ1 . (γ2 . α). This is straightforward and left to the reader. Finally it remains to check that the original Γ-action on M and the above con- structed Γ-action on Td are equivariant with respect to the Jacobian map Jac : M → Td defined by means of g. This amounts to showing that Jac (γ.x) = γ.(Jac (x)) where the left hand side refer to the action on M and the right hand side to the action on Td. To verify the last equation we evaluate both sides on a cohomology class c ∈ H1(M, Z). The verification is again straightforward, we only need to recall that the pull-back of a harmonic form by an isometry is again harmonic. Consider the previously defined Jacobian map Jac : M → Td. Note that an 1 d element c of Hom (H (M, Z) , R/Z) ≃ T is determined by its values on η1,...,ηd, 1 where η1,...,ηd stands for a basis of H (M, Z) consisting of harmonic forms. This observation allows us to identify Td = Hom (H1(M, Z) , R/Z) with R/Z×···×R/Z (d factors). Furthermore, under this identification, the Jacobian map Jac : M → Td is given by x x Jac(x) = η1,..., ηd . (1) Zp Zp

Also we assume from now on that b1(M) = d ≥ 1 so that the Jacobian map is not constant. Let Im[Jac(M)] ⊆ Td denote the image of M by Jac. To abridge notations, the largest integer m for which the image of the induced homomorphism m d m Jac♯ : H (T , Z) → H (M, Z) is not trivial is going to be referred to as the coho- mological dimension of Im [Jac (M)] ⊆ Td (notation: c-dimension). Naturally here Jac♯ stands for the homomorphism in cohomology induced by Jac. The following lemma clariﬁes the nature of the c-dimension. 428 JULIO C. REBELO AND ANA L. SILVA

Lemma 2.2. The c-dimension of Im[Jac(M)] is equal to the maximum number of 1 classes η1,...,ηm in H (M, Z) for which the cup-product η1 ∧···∧ ηm is not trivial in Hm(M, Z). 1 d d Proof : Let e1,..., ed be a basis for H (T , Z) where T is thought of as R/Z ×···× 1 R/Z (d factors) and each e1 is identified with the generator of H ((R/Z)i, Z) (with th d (R/Z)i denotinge thee ı -component of T ). Since, by construction, Jac induces an 1 1 d isomorphism Jac♯ betweeneH (M, Z) and H (T , Z), we conclude that ηi = Jac♯(ei), i =1,...,d, form a basis for H1(M, Z). However, the Kunneth Formula shows that m Td Z e H ( , ) is generated by the products ei1 ∧···∧ eim . Hence the image of Jac♯ m Td Z m Z from H ( , ) to H (M, ) is generated by the products ηi1 ∧···∧ ηim . The statement results at once. e e In particular, when the fundamental class in Hn(M, Z) is a product of classes in H1(M, Z), the c-dimension of Im [Jac (M)] is precisely n. Now we consider a compact manifold M together with a C1-diffeomorphism f : M → M having finite order. Applying Theorem (2.1), we obtain a Jacobian map d d d Jac : M → T which is equivariant w.r.t. an affine diffeomorphism hf : T → T . d Furthermore, by construction, hf leaves Im [Jac (M)] ⊆ T invariant. Next let Fix (f) ⊆ M denote the set of fixed points of f. Proposition (2.3) below is an immediate extension of the arguments presented in [24].

Proposition 2.3. ([24]) Let M, f, hf and Jac be as above. Suppose that f acts 1 trivially on H (M, Z). Then hf is a translation. Now suppose in addition that the fundamental class of M in Hn(M, Z) is a product of classes in H1(M, Z). Then f coincides with the identity provided that so does hf .

Sketch of Proof : To check that hf is a translation we proceed as follows. Suppose for a contradiction that this is not the case. Since hf is aﬃne, it follows that the induced 1 d action of hf on H (T , Z) is not trivial. Hence there is a closed 1-form η such that ∗ ∗ η and hf η are not cohomologous. Set η = Jac η and notice that the equivariance of ∗ ∗ ∗ ∗ Jac yields f η = Jac (hf η). However f η is cohomologous to η since the action of e ∗ ∗ ∗ f on H1(M, Z) is trivial. Therefore Jac η is cohomologous to Jac (h η). Because e e e f Jac induces an isomorphism at the level of H1(M, Z), H1(Td, Z) the last statement ∗ gives a contradiction with the fact that η and hf η are not cohomologous. It remains to show the second part of the statement which relies on equivariant cohomology and, more precisely, on the corresponding “Localization Principle”. A i Z standard reference for what follows is the Chapter 3 of [1]. Let Hf (M, ) (resp. i Td Z th Td Hhf ( , )) denote the i -equivariant cohomology group of M (resp. ) with respect to the ﬁnite group consisting of the powers of f (resp. hf ). Thanks to its i Td Z i Z equivariant character, Jac induces a homomorphism from Hhf ( , ) to Hf (M, ). Next we assume that hf is the identity so that the equivariant cohomology group n Td Z n Td Z Hhf ( , ) is naturally isomorphic to the (ordinary) cohomology group H ( , ). Because the fundamental class of Hn(M, Z) is a product of classes in H1(M, Z), n Z Lemma (2.2) ensures that Hf (M, ) is not trivial (again the equivariance of Jac pulls every class in Hn(Td, Z) back to a class in Hn(M, Z)) which is automatically f-invariant). Finally, if the order of f is a prime, then the Localization Theorem provides n Z n Z H (Fix (f), ) ≃ Hf (M, ) (cf. [1], pg 120). Since the right hand side is not trivial, it can be concluded that Fix (f) possesses non-trivial cohomology in dimension n (equal to the dimension of ON THE BURNSIDE PROBLEM 429

M). Thus Fix (f) must be the whole of M. The general case where the order of f is not a prime can trivially be reduced to the preceding one by inducting on the number of primes dividing this order. Here is a by-product that will be used later on.

Corollary 2.4. Assume that the fundamental class in Hn(M, Z) is a product of classes in H1(M, Z). Let f : M → M be a diffeomorphism of finite order acting trivially on H1(M, Z). Then either f coincides with the identity or it has no fixed points.

d d Proof : If the translation hf : T → T is trivial, then hf coincides with the identity after the preceding lemma. On the other hand, if hf is not trivial then hf does not have ﬁxed points. The equivariance of Jac implies that f does not have ﬁxed points either.

3. Proofs for the main results. Let us begin this section by stating and proving Theorem (3.1) already mentioned in the Introduction. As observed there, this theorem shows that most problems discussed in the beginning of this paper become much easier if the ambient manifold has Euler characteristic diﬀerent from zero. The proof relies on the Lefschetz formula, on Schur theorem [23] and on Corollary (2.4). It is somehow similar to results in [6] and [8].

Theorem 3.1. Assume that M is a n-dimensional compact manifold whose fundamental class in Hn(M, Z) is a product of classes in H1(M, Z). Assume also that the Euler characteristic of M is different from zero. Then any finitely generated subgroup G ⊂ Diff2(M) all of whose elements have finite order is, in fact, finite.

Proof : Suppose for a contradiction that G is a Burnside group acting effectively i i Z on M. For every i = 1,...,n, let G♯ be the group of automorphisms of H (M, ) i R N induced by G. Each G♯ is naturally a subgroup of GL (Ni, ) for a suitable Ni ∈ . Now note that Schur’s negative solution to the Burnside problem in GL (Ni, R) i implies that each G♯ is finite, i = 1,...,n. In other words, modulo passing to a finite index subgroup, we can suppose without loss of generality that G acts trivially on the cohomology ring of M. Given f as above, the Lefschetz number of f is nothing but the Euler characteristic of M. Since by assumption this number is not zero, Lefschetz theorem guarantees the existence of fixed points for f. According to Corollary (2.4) such f has to coincide with the identity. The resulting contradiction establishes the theorem. The object of this section is actually the proof of Theorem A. However, since this proof will require Theorem B, we are going to prove this theorem first. The proof of Theorem B is now an easy consequence of Proposition (3.2) below whose proof, as already mentioned, will be deferred to the Appendix. Note that the homomorphism H mentioned in the statement of this proposition is precisely the well-known Flux homomorphism.

2 Td Proposition 3.2. There is a natural homomorphism H : Diff0,µ(M) → where 2 2 Diff0,µ(M) stands for the connected component of the identity of the group of C - 2 diffeomorphisms preserving µ. Furthermore, if f ∈ Diff0,µ(M) has finite order and H(f) vanishes, then the translation hf used in Proposition (2.3) is trivial. 430 JULIO C. REBELO AND ANA L. SILVA

Proof of Theorem B : If M is acted upon by a Burnside group in a faithful way, then there is also a Burnside group acting faithfully on M and having all its elements contained in the connected component of the identity. Indeed this is an easy consequence of the finiteness of the mapping class group of M. Let then G be a finitely generated group contained in the connected component of the identity and acting faithfully on M. To prove the statement it suffices to show that G is Abelian provided that all its elements are of torsion. In fact, a finitely generated Abelian group all of whose elements have finite order is clearly finite. To check that G is Abelian let f be an element in the first derived group D1G of G. This means that f is a product of commutators of elements in G. Therefore H(f) = 0 since H gives a homomorphism from G to an Abelian group. On the other hand f acts trivially on the cohomology ring of M since it is isotopic to the identity. Besides, Proposition (3.2) tells us that the translation associated to f through the construction described in Section 2 is trivial. Finally Proposition (2.3) implies that f is the identity. The theorem is proved. The remainder of this section is devoted to the proof of Theorem A. Therefore we fix a symplectic manifold (M,ω) of dimension 4 whose fundamental class is a product of classes in H1(M, Z). Also let Symp (M) denote the group of C2- symplectomorphisms of (M,ω). The argument to establish Theorem A will be carried out by contradiction. Thus, in what follows, we suppose that G ⊂ Symp (M) is a Burnside group which acts effectively on M. Besides, by virtue of Schur theorem, we can assume without loss of generality that G acts trivially on the cohomology ring of M. Let f be an element of G ⊂ Symp (M), f 6= id. Corollary (2.4) shows that f has no fixed points in M. For most purposes, we can also assume that f is not isotopic to the identity through homeomorphisms of M. Denoting by Homeo (M) the group of (orientation-preserving) homeomorphisms of M, the last assertion is a consequence of the lemma below:

Lemma 3.3. The intersection of G with the connected component Homeo0(M) of the identity in Homeo (M) is finite. Proof : Recall that the elements of Symp (M) possess a common invariant probability measure, namely the measure associated to the volume form ω ∧ ω. Let G0 = G ∩ Homeo0(M) and note that G0 is a subgroup of G. Now the arguments employed in the proof of Theorem B imply that G0 is an Abelian group. Therefore it must be finite. Summarizing, there are infinitely many elements f ∈ G satisfying all the conditions below: • f acts trivially on the cohomology ring of M. • f is not isotopic to the identity through homeomorphisms of M. • f does not have fixed points. The fundamental ingredient still needed for the proof of Theorem A is Don- aldson’s theorem concerning the existence of Lefschetz pencils on symplectic 4- manifolds (see [3], [4]). For the basics of Symplectic Geometry the reader is referred to [15]. Let (M,ω) be a symplectic manifold of dimension 4. A topological Lefschetz pencil on M consists of a compact Riemann surface S, a smooth function P : M \ B → S and an almost complex structure J on M satisfying the following conditions: ON THE BURNSIDE PROBLEM 431

1. B is a finite set. The map P has finitely many critical values x1,...,xl ∈ S and, k −1 for every j ∈{1,...,l}, there are finitely many critical points pj ∈P (xj ) which, furthermore, are non-degenerate for P. k 2. The projection P is holomorphic in a small neighborhood of each pj . Besides the fibers of P are connected. 3. The symplectic form ω is non-degenerate on the vertical spaces Ker DpP for every p ∈ M \{p1,...,pl}. Remark 3.4. Usually the Riemann surface S is taken to be CP(1). We have decided to consider also higher genus Riemann surfaces to be able to state a simple “equivariant” version of the mentioned theorem. Nonetheless it is an additional issue of Donaldson’s results mentioned above that the base S can be taken to be CP(1) (keeping the fibers connected). Also Donaldson’s construction provides a −1 unique critical point pj on each of the fibers P (xj ). The set B is said to be the set of indeterminacy points of the pencil P. When B is empty then P actually defines a topological Lefschetz fibration. In general, a pencil P can be turned into a fibration by performing a finite number of blow-ups centered at points in B. The existence of Lefschetz pencils on (M,ω) will be exploited in our context through the following simple remark: Lemma 3.5. Let f ∈ G be different from the identity. Then there exists a Lefschetz pencil P : M \ B → S on M which is preserved by f. Proof : Since f does not have fixed points, it acts properly discontinuously on M. Hence the quotient M/(f) is itself a 4-dimensional symplectic manifold. Next we equip M/(f) with a pencil P1 whose basis is CP(1). The original manifold M is then equipped with the lift P = P1 ◦ π of P1 (where π stands for the covering map π : M → M/(f)). However the fibers of P are no longer necessarily connected. e To remedy the situation, we consider an appropriate ramified covering S of CP(1) e whose degree is the number of connected components of the generic fiber of P. Then P naturally induces a pencil P on M taking values on S and having connected e fibers. e ¿From now on we consider a pencil P : M \ B → S as above and an element f ∈ G, f 6= id, preserving P. In particular, f induces a diffeomorphism f of S viewed as the space of fibers of P. Besides one has: Lemma 3.6. The diffeomorphism f acts trivially on H1(S, Z). Proof : Suppose for a contradiction that the statement is false and let η be a closed ∗ 1-form on S such that η and f η do not belong to the same cohomology class. Then ∗ there exists a closed curve c ⊂ S over which the integrals of η and f η are different from each other. Besides we clearly can choose c lying entirely in the complement ∗ of the singular values of P. Next we consider the 1-form ηP = P η defined on M. ∗ ∗ ∗ Since f preserves the pencil, it follows that the 1-form f ηP is nothing but P (f η). 1 ∗ Because f acts trivially on H (M, Z), we conclude that the 1−forms ηP = P η and ∗ P∗(f η) are cohomologous. On the other hand, since c does not contain singular values of P, it admits a lift c ⊂ M which is still a closed path (take the natural lift of c with respect to any connection on the regular part of P and, if this lift is not closed, we make it closede by joining its extremities through a path entirely 432 JULIO C. REBELO AND ANA L. SILVA contained in a regular fiber of P). This construction then gives us

∗ ∗ P∗η = η 6= f η = P∗(f η) . Zce Zc Zc Zce ∗ This however contradicts the fact that P∗η and P∗(f η) are cohomologous. Now we have one last lemma. Lemma 3.7. With the preceding notations S is, in fact, isomorphic to CP(1). Proof : Recall that S is naturally identified with the space of the fibers of P in M. Let us first show that the Euler characteristic of S cannot be negative. In fact, if it were negative, the Lefschetz Fixed Point Formula together with the fact that f acts trivially on the cohomology of S (Lemma 3.6) would yield a fixed point for f. Since f has itself finite order and the fundamental class of S is a product of elements in H1(S, Z), it would follows from Corollary (2.4) that f is the identity. However the quotient of S by f is nothing but the space of fibers of Donaldson’s pencil used in the proof of Lemma (3.5). Indeed, if the fibers of P are disconnected, then f has to be different from the identity by construction. However the latter space is the e 2-dimensional sphere by construction whereas the former is a surface S of negative Euler characteristic. The resulting contradiction establishes the claim. It remains to check that the Euler characteristic of S cannot be zero either. For this we note that the preceding argument implies in the present case that f has no fixed points. Therefore f is conjugate to a translation. We can conclude that the quotient S/(f) is a torus rather than the sphere. The lemma is proved.

Proof of Theorem A : Let f 6= id be an element of G acting trivially on the cohomology ring of M. The strategy of the proof will consist of showing that f is isotopic to the identity through homeomorphisms of M. As already seen, this fact will imply the statement. To begin with, let us consider a pencil P : M \ B → CP(1) which is preserved by f (Lemma 3.5). Again we denote by f the diffeomorphism of the sphere CP(1) induced by f and P. Clearly f has a fixed point p so that the fiber P−1(p) is invariant under f. We also note that f : CP(1) → CP(1) is isotopic to the identity as a particular case of a well-known theorem due to Smale [22]. Besides we can suppose that the isotopy in question leaves the point p fixed. Now we remind the reader that M possesses a natural connection with respect to the symplectic pencil P. Namely, given a fiber P−1(q) and a regular point x ∈ P−1(q), Condition 3 in the definition of symplectic pencil states that the −1 restriction of ω to TxP (q) is non-degenerate. Thus we can define a “horizontal” 2- −1 plane in TxM simply by taking the symplectic orthogonal complement of TxP (q). The resulting (singular) connection is going to be referred to as the symplectic connection. Next let H : [0, 1] × CP(1) → CP(1) be an isotopy between H(0, . ) = f and H(1, . )= id which keeps p ∈ CP(1) fixed. Using the symplectic connection, we can lift H to an isotopy of M that goes from f to some diffeomorphism fP : M → M having the two properties below:

• fP leaves every ﬁber of P invariant. −1 • fP coincides with f on P (p). The proof of the theorem is now essentially reduced to the following claim: ON THE BURNSIDE PROBLEM 433

Claim : The regular fibers of P are tori. Furthermore the restriction of fP to each such fiber is isotopic to the identity on the fiber in question. Proof of the Claim : Recall that f has no fixed point. Since fP coincides with −1 −1 f on P (p), it follows that fP restricted to P (p) does not have fixed points either. Note however that P−1(p) may be a singular fiber. This possibility will make our discussion slightly longer. Let us consider a point q ∈ CP(1) different from p. Whether or not P−1(p) is singular, P−1(q) is regular provided that q is sufficiently close to p. Now the compactness of the fibers and the continuity of P −1 and fP imply that fP restricted to P (q) does not have any fixed point. In other −1 −1 −1 words, fP : P (q) → P (q) has no fixed point. In particular, P (q) is not the (2-dimensional) sphere. On the other hand, f is by assumption of finite order equal to m ∈ N (i.e. m m −1 m −1 −1 f = id). It follows that fP restricted to P (p), fP : P (p) → P (p), is the m −1 −1 identity. Again by continuity, we conclude that fP : P (q) → P (q) can be supposed arbitrarily close to the identity modulo choosing q very close to p. If q m −1 −1 is close enough to p, we conclude that fP : P (q) →P (q) is homotopic to the −1 m −1 identity on P (q). In particular, fP acts trivially on the cohomology of P (q). 1 −1 It follows that the eigenvalues of the action of (the original) fP on H (P (q), Z) are all equal to 1. The trace of this action must however be equal 2 since fP : P−1(q) → P−1(q) has no fixed point (Lefschetz Formula). We then conclude that P−1(q) is a torus. Finally the mapping class group of the 2-dimensional torus is well-known to be isomorphic to SL (2, Z). Therefore, in the present case, the fact −1 −1 that fP : P (q) →P (q) has no fixed points provides the extra information that fP must be isotopic to the identity on this torus. To complete the proof of the theorem, we just need to show that fP is isotopic to the identity on M. This can be done by performing isotopies along the regular fibers of P. Simple arguments of compactness and continuity guarantees that this procedure is well-defined. Theorem A is proved.

• Final Remarks: We would like to close this article with some questions. It seems clear from the discussion above that the techniques involving Donaldson’s Lefschetz pencils might be suited to discuss Burnside group actions on symplectic manifolds beyond the class of those manifolds whose fundamental class can be rep- resented by a product of classes in H1(M, Z). In fact, we believe that a Burnside group cannot embed in the group of symplectomorphisms of a compact symplectic manifold. It seems also interesting to determine whether or not a Burnside group can be realized inside the mapping class group of a symplectic manifold (or maybe inside the “symplectic mapping class group”). Some more subtle phenomena are likely to arise in this question. Finally considering the case of general compact manifolds and their diffeomorphisms, the existence of an effective action of a Burnside group appears unlikely. Nonetheless we do not have any definite feeling with regard to this last problem.

4. Appendix: Rotation Vectors. In this appendix we shall prove Proposi- tion (3.2). We also interpret the translation appearing in Section 2 in a more intrinsic way, namely as a rotation vector. This interpretation may also be useful to obtain specific results on Burnside group actions when we know some non-trivial relations in the group. Let us begin by recalling the notion of rotation set introduced by Pollicott in [17]. Let C0(M, S1) be the set consisting of continuous functions from M to S1 viewed as 434 JULIO C. REBELO AND ANA L. SILVA the unit circle in C. Then given two functions f, g ∈ C0(M, S1) the multiplication fg is a function in C0(M, S1) as well. The Bruschlinsky group C0(M, S1)/ ∼ is defined as the group of homotopy classes of functions in C0(M, S1) (to refer to 0 1 homotopic functions h1, h2 in C (M, S ), we use the notation h1 ∼ h2). It is well- known that C0(M, S1)/ ∼ is naturally isomorphic to the first cohomology group with integral coefficients H1(M, Z) of M. Now consider a diffeomorphism f of M and assume that f is isotopic to the identity. The Mapping Torus of f is the manifold V obtained as the quotient of M×[0, 1] by the equivalence relation identifying the points (x, 1) and (f(x), 0). Since f is isotopic to the identity, it follows that V is diffeomorphic to the product M ×S1. In particular we have the following natural identifications: H1(V, Z)= H1(M, Z)⊕Z and H1(V, R)= H1(M, R) ⊕ R. The suspension of f (or the suspended flow relative to f) is a flow f˜ on V whose diffeomorphism f˜T : V → V induced at time T ∈ R satisfies f˜T (x, u) = (x, u + T ) n with the identification f˜T (x, u) = (f (x),u+T −n) provided that n ≤ u+T ≤ n+1. Having chosen v ∈ V and T > 0, we define a linear functional on C0(V, S1) (i.e. an 0 1 ∗ element of C (V, S ) )Λv,T by letting 1 T d Λv,T (k)= arg[k](f˜t(v))dt , T Z0 dt where k : V → S1 is a continuous function having argument arg[k] (i.e. k(v) = ei arg[k](v)). Strictly speaking, the above formula holds for smooth functions k as above since it involves derivatives. The resulting linear functional is then extended to all continuous functions by continuity. Also we point out that, for the purposes of the above integral, the argument of k is chosen so as to be continuous along the orbit of v by the suspended flow f˜. According to Schwartzmann one has:

Proposition 4.1. ([21]) For each v ∈ V the family of linear functionals {Λv,T }T>0 contained in C0(V, S1)∗ is equicontinuous with respect to the weak* topology. Fur- 0 1 ∗ thermore, for a ﬁxed v ∈ V , the accumulation points Fv ⊂ C (V, S ) of the family {Λv,T }T>0 are constant on equivalence classes of homotopy. In other words, if Λv,∞ is an element of Fv, then

Λv,∞(k1)=Λv,∞(k2) , 1 provided that k1, k2 : V → S are homotopic.

In particular, we see that an accumulation point Λv,∞ as above defines an element in the dual of C0(V, S1)/ ∼ and hence in (H1(V, Z))∗ = Hom (H1(M, Z) , R). 1 Since the dual of H (V, Z) is nothing but the homology group H1(V, R), we conclude that the functional Λv,∞ defines an element in the first homology group with real coefficients of V which, in turn, is isomorphic to H1(M, R) ⊕ R. Because of the explicit construction of the functionals Λv,T , it easily follows that the second component with respect to the decomposition H1(M, R) ⊕ R is always 1 i.e. it is independent on all the previously made choices. It is also obvious that, given v = (x, u) the first component of Λv,T depends solely on x. Thus we can identify the accumulation points Λv,∞ with a set of elements in H1(M, R) denoted by Fx. For a given point x ∈ M, the set Fx ⊆ H1(M, R) is said to be the rotation set of x ∈ M (relative to the homeomorphism f). The rotation set of f will be denoted by ρ(f) and defined as the union of the rotation sets of each point x ∈ M, i.e. R ρ(f)= x∈M Fx ⊆ H1(M, ). S ON THE BURNSIDE PROBLEM 435

Remark 4.2. Whenever necessary we can also think of the rotation set as being contained in Hom (H1(M, Z) , R) (i.e. we do not need to apply PoincaréDuality in the preceding discussion). In fact, the rotation set should be interpreted as defined modulo integral cohomology and therefore should be contained in Hom (H1(M, Z) , R/Z) (formally we can consider its reduction mod Z in the above definition). Here is a simple lemma. Lemma 4.3. Let f be a homeomorphism of M which is isotopic to the identity and has finite order (i.e. f m = id for some m ∈ N∗). Then the rotation set ρ(f) of f consists of a single element Λ∞. Proof : Fix a point v = (x, u) in V . First we are going to prove that the limit limT →∞(Λv,T ) with respect to the weak* topology always exists. In order to do this, note that 1 T d 1 Λv,T (k)= arg[k](f˜t(v))dt = (arg[k](f˜T (v)) − arg[k](v)) T Z0 dt T since f˜0(v)= v. Now let T0 be the period of f i.e. the smallest positive integer such T0 that f (x)= x for every x ∈ M. Given T ∈ R very large, we have T = mT0 + r(T ) N ˜ where m ∈ and 0 ≤ r(t)

Proposition 4.4. With the preceding notations one has hf = ρ (f). 436 JULIO C. REBELO AND ANA L. SILVA

Proof : Consider a continuous function k : M × S1 = V → S1 representing some cohomology class (c, 0) ∈ H1(V, Z)= H1(M, Z) ⊕ H1(S1). Without loss of generality, we can suppose that k is constant in the second coordinate, i.e. k is a function defined on M. Also fix a base point p ∈ M ≃ (p, 0) ∈ V . Since M is a Riemannian manifold, we consider the corresponding space of harmonic forms and denote by ηc the harmonic form representing c. We need to prove that A f(p) = ηc mod Z (2) T0 Zp where T 0 = m is the smallest positive integer for which f T0 (x) = f m(x) = x for every x ∈ M and A is the winding number of the argument arg[k] of k over the orbit Op of (p, 0) by the suspended flow f˜ from 0 to T0. ∗ ∗ We first claim that f ηc = ηc. In fact, note that f ηc and ηc belong to the same class of cohomology since f is isotopic to the identity. Besides f is also an isometry ∗ of the Riemannian metric fixed at the beginning so that f ηc is itself a harmonic form. The claim then follows from the uniqueness part of Hodge Theorem. Now consider the orbit Op. Since the Mapping Torus V , obtained from M and 1 f, is diffeomorphic to M × S , Op can be projected on the fiber (M, 0) to yield a path c in M joining p and f(p). The preceding claim yields 2 f(p) f (p) p=f m(p) ηc = ηc = · · · = ηc , (3) Zp Zf(p) Zf m−1(p) where the integral from p to f(p) should be understood as the integral over c 2 (similarly the integral from f(p) to f (p) is over f(c) and so on). Denote by Cf the path obtained by concatenation of the paths c, f(c),f 2(c),...,f m−1(c). The Z integral of ηc over Cf is an integer ΠCf ∈ which is going to be called the period of 0 ηc over Cf . Recalling that T = m, Equation (2) follows at once from Equation (3) combined to Lemma (4.5) below. The proof of the proposition is over.

Lemma 4.5. With the above notations, one has ΠCf = A. Proof : Recall that the explicit isomorphism B between the Bruschlinsky group C0(M, S1)/ ∼ and H1(M, Z) is given as follows. Fix an element γ ∈ C0(M, S1)/∼ and choose a function f : M → S1 representing γ. The function f induces a homomorphism f ∗ : H1(S1, Z) → H1(M, Z) which depends only on γ and not on the chosen f. We then set B(γ) = f ∗(σ) where σ stands for the generator of the cyclic group H1(S1, Z) (see [13]). Since k : M × S1 = V → S1 represents the class (c, 0) ∈ H1(V, Z)= H1(M, Z) ⊕ 1 1 ∗ 1 1 H (S ), it follows that k (σ) and ηc are the same element of H (M, S ). Therefore their evaluations (pairing) at the class of Cf ∈ H1(M, Z) coincide with each other.

In De Rham Theory, the evaluation of ηc is exactly ΠCf . In Bruschlinsky theory, the evaluation is precisely the winding number of k over the orbit from p through p = f n(p) which is A by assumption. The lemma is proved. Let f be as above. The rotation vector also admits an alternative description similar to the description of the Jacobian map given after Theorem (2.1). Namely 0 1 1 let k1,...,kd be a basis for C (M, S )/∼ (isomorphic to H (M, Z)). With respect to this basis, we again obtain an identiﬁcation of Hom (H1(M, Z) , R/Z) with Td (cf. Remark 4.2). Now the rotation vector is simply (as follows for example from Proposition (4.4)) ρ (f)=(Λ∞(k1),..., Λ∞(kd)) . ON THE BURNSIDE PROBLEM 437

In the sequel we fix the above mentioned identification of Hom (H1(M, Z) , R/Z) and Td. Rotation vectors have a natural relation with invariant measures which will finally lead to the proof of Proposition (3.2). The subsequent discussion is, indeed, a particular case of [21] that takes a nicer form in the present setting since ρ (f) is reduced to a single vector. First observe that there is an obvious correspondence between measures µ invariant by f and measures ν invariant by the suspended flow f˜. Such correspondence takes the local form ν = µ × L where L stands for the one-dimensional Lebesgue measure. 1 Proof of Proposition (3.2) : Consider one of the functions ki : M → R/Z ≃ S as above, i =1,...,d. Given a diffeomorphism f : M → M isotopic to the identity, we consider the corresponding Mapping Torus V ≃ M × S1 and the obvious identifications involving V and M (µ and ν etc). Let h be a function from M to R/Z ≃ S1 and consider arg[h ◦ f˜t − h] that can also be thought of as the difference h ◦ f˜t − h where the horizontal bar denotes a lift to R of h ◦ f˜t − h satisfying h ◦ f˜0 − h = 0 (here h is identified with a function on V in the obvious way). We define H (f) by

H (f)= arg[(k1 ◦ f˜1 − k1)]dν,..., arg[(kd ◦ f˜1 − kd)]dν ZM ZM where ν = µ×L. Note also that f˜1 induces f : M → M. Hence to check that H is a homomorphism it is enough to verify that the assignment f 7→ M arg[(ki ◦f −ki)]dµ 1 is so. However, if f,g are two elements of Diﬀ0,µ(M), one hasR

arg[(ki ◦(g◦f)−ki)]dµ = arg[((ki ◦g)◦f −(ki◦g))]dµ+ arg[(ki ◦g−ki)]dµ . ZM ZM ZM

Since g preserves ν, it results that M arg[((ki ◦ g) ◦ f − (ki ◦ g))]dµ = M arg[(ki ◦ f − ki)]dµ thus proving that H is aR homomorphism. R 1 For the second part of the statement, fix an element f of Diff 0,µ(M) having finite order. Let us begin with two general and simple observations. First we note that ˜ ˜ ˜ ˜ arg[ki ◦ ft1+t2 − ki] = arg[ki ◦ ft2 − ki] ◦ ft1 + arg[ki ◦ ft1 − ki] or ˜ ˜ ˜ ˜ arg[ki ◦ ft2 − ki] ◦ ft1 = arg[ki ◦ ft1+t2 − ki] − arg[ki ◦ ft1 − ki] . (4)

The second observation concerns the existence of a uniform constant Const such that ˜ ˜ arg[ki ◦ ft1+t2 − ki](x) − arg[ki ◦ ft1 − ki](x) < Const (5)

for every t1 ∈ R, x ∈ M and 0 ≤ t2 ≤ 1. Estimate (5) is nothing but an immediate consequence of the compactness of M, V . For a point x ∈ M we now consider

1 T d 1 Λ∞(ki) = lim arg[ki](f˜t(x))dt = lim (arg[ki](f˜T (x)) − arg[ki](x)) . →∞ →∞ T T Z0 dt T T According to Lemma (4.3) the above limits always exist and do not depend on x (being precisely Λ∞(ki)). Next we set

T B(x) = lim arg[ki ◦ f˜1 − ki] ◦ f˜t(x)dt . →∞ T Z0 438 JULIO C. REBELO AND ANA L. SILVA

Again Lemma (4.3) guarantees that this limit exists and is constant for every x ∈ M. Combining with Birkhoﬀ Ergodic Theorem we also obtain that

B(x)= B(x)= arg[ki ◦ f˜1 − ki]dµ . (6) ZM ZM On the other hand Equation (4) yields T B(x) = lim arg[ki ◦ f˜1 − ki] ◦ f˜t(x)dµ →∞ T Z0 T = lim (arg[ki ◦ f˜t+1 − ki](x) − arg[ki ◦ f˜t − ki](x))dt →∞ T Z0 T +1 1 = lim arg[ki ◦ f˜t − ki](x)dt − lim arg[ki ◦ f˜t − ki](x)dt . →∞ →∞ T ZT T Z0 −1 T +1 ˜ Hence B(x) = limT →∞ T T arg[ki ◦ ft − ki](x)dt. Nonetheless Estimate (5) easily implies that the precedingR integral coincides with limT →∞ arg[ki ◦ f˜T − ki](x) which, in turn, is nothing but Λ∞(ki). Thus B(x)=Λ∞(ki) and the proposition results from Equation (6).

Acknowledgements: This work was carried out during a visit of both authors to the Institute for Mathematical Sciences (IMS) at Stony Brook. We would like to thank our colleagues from the IMS for their kind hospitality. We are also grateful to E. Ghys for having drawn our attention to Burnside groups and their generic character and to the referees for valuable suggestions.

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