Arithmetic group symmetry and finiteness properties of Torelli groups
aa r X i v : . [ m a t h . G R ] O c t ARITHMETIC GROUP SYMMETRY AND FINITENESSPROPERTIES OF TORELLI GROUPS
ALEXANDRU DIMCA AND STEFAN PAPADIMA Abstract.
We examine groups whose resonance varieties, characteristic varietiesand Sigma-invariants have a natural arithmetic group symmetry, and we exploreimplications on various finiteness properties of subgroups. We compute resonancevarieties, characteristic varieties and Alexander polynomials of Torelli groups, andwe show that all subgroups containing the Johnson kernel have finite first Bettinumber, when the genus is at least 4. We also prove that, in this range, the I -adic completion of the Alexander invariant is finite-dimensional, and the Kahlerproperty for the Torelli group implies the finite generation of the Johnson kernel. Contents
1. Introduction 22. Main general results 53. Associated graded Lie algebra 84. Resonance varieties of Torelli groups 95. Resonance and finiteness properties 126. Characteristic varieties of Torelli groups 177. Sigma-invariants and the Kahler property 22References 25
Mathematics Subject Classification.
Primary 20J05, 57N05; Secondary 11F23, 20G05.
Key words and phrases. resonance variety, characteristic variety, Alexander invariant, Alexan-der polynomial, Sigma-invariant, arithmetic group, semisimple Lie algebra, finiteness properties,Torelli group, Johnson kernel, Kahler group. Partially supported by the French-Romanian Programme LEA Math-Mode and ANR-08-BLAN-0317-02 (SEDIGA). Partially supported by the French-Romanian Programme LEA Math-Mode and PN-II-ID-PCE-2011-3-0288, grant 132/05.10.2011. Introduction
The mapping class group Γ g , consisting of the isotopy classes of orientation–preserving homeomorphisms of a closed, oriented, genus g surface Σ g , is an ubiq-uitous character on the mathematical stage. The Torelli subgroup T g , that is, thekernel of the action on the first Z –homology of Σ g , defined by the exact sequence(1.1) 1 → T g −→ Γ g p −→ Sp g ( Z ) → , brings into play the arithmetic integral symplectic group Sp g ( Z ). (Since T g is trivialfor g ≤
1, we will assume g ≥ p isa surjection. The Johnson kernel K g , analyzed by Dennis Johnson in a remarkableseries of papers [25]–[29], is the subgroup of T g generated by Dehn twists aboutseparating curves on Σ g . Our main result in this note is the following. Theorem A.
For g ≥ , the vector space H ( K g , Q ) is finite–dimensional. The question of whether the first Betti number of K g is finite has been open formany years, since Johnson’s work in the early 80’s, despite the attention of manyexperts. When we started working on this problem, the general consensus seemedto be that the answer is probably no; see for instance Hain’s survey [22] on expectedinfiniteness properties related to Torelli groups. In this direction, Akita’s result [1]guarantees the existence of an infinite Betti number of K g . Biss and Farb tried toprove in [9] that K g is not finitely generated, but a fatal error was found in theirpaper. As we shall see, Theorem A actually holds for all subgroups of T g containing K g , which answers Problem 5.3 from Farb’s list [19].Another important problem concerns the Kahler property: is T g a Kahler group(that is, the fundamental group of a compact Kahler manifold)? The answer isnegative, in low genus: T is not finitely generated, as proved by McCullough andMiller [35], and T violates Dennis Sullivan’s [45] 1-formality property of Kahlergroups (established by Deligne, Griffiths, Morgan and Sullivan in [13]), as shown byHain in [21]. Theorem 4.9 from [19] states that T g is not a Kahler group. Since theproof assumed infinite generation of K g , the Kahler group problem is open as well,for g ≥
4. Our second main result connects the Kahler problem for Torelli groupswith the finite generation question for Johnson kernels.
Theorem B. If T g is a Kahler group, the Johnson kernel K g must be finitely gen-erated, for g ≥ . Finiteness properties of the Johnson filtration.
The group Γ g may beviewed from many angles. By a result of Earle and Eells, the Eilenberg–MacLanespace K (Γ g ,
1) classifies oriented Σ g –bundles. Via Heegaard splittings, Γ g is inti-mately related with the world of 3–manifolds. The group Γ g is the orbifold funda-mental group of the (aspherical) moduli space of genus g compact Riemann surfaces, RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 3 M g : Γ g acts properly discontinuously, with finite isotropy, on the (contractible) Te-ichm¨uller space X g , with quotient X g / Γ g = M g . Furthermore, the so-called (finiteindex, normal) level subgroup Γ g ( ℓ ) is torsion–free, and the quotient X g / Γ g ( ℓ ) isthe (quasi–projective) moduli space of complex curves with level ℓ structure, for ℓ ≥
3. So, Γ g enjoys almost every conceivable finiteness property: it is finitely pre-sentable, it has finite virtual cohomological dimension, and all its homology groupsare finitely generated. However, little is known about finiteness properties of infiniteindex subgroups, such as T g .We are going to adopt a group–theoretical viewpoint on Γ g , that emerges fromNielsen theory: Γ g = Out + ( π g ), the proper outer group of automorphisms of thegroup π g = π (Σ g ). In this way, we may identify both T g and K g with the firstterms in the so-called Johnson filtration of Γ g . This filtration was first introducedand analyzed for Out( F n ), by Andreadakis [2], where F n is the free group on n generators.The construction goes as follows. Given a group π , denote by π s the subgroupgenerated by length s commutators. (Here, π = π , π is the derived subgroup π ′ ,and so on.) Let gr • π = ⊕ s ≥ π s /π s +1 be the associated graded Lie algebra. TheJohnson filtration { J s ( π ) } s ≥ is the normal, descending series defined by(1.2) J s ( π ) = ker { Out( π ) → Out( π/π s +1 ) } (with the convention that J ( π g ) = Out + ( π g )).We infer from (1.1) that J ( π g ) = T g . By Teichm¨uller theory, the complex man-ifold X g /T g = K ( T g ,
1) is the moduli space of curves with a marked symplecticbasis of H (Σ g , Z ). Nevertheless, the (higher) finiteness properties of T g are largelya mystery. A deep theorem of Johnson [26] says that T g is finitely generated for g ≥
3, but finite presentability is open. Another deep result, due to Bestvina, Buxand Margalit [6], is that H g − ( T g , Z ) is infinitely generated, for g ≥ Johnsonhomomorphisms ,(1.3) { τ s : J s ( π ) −→ ODer s (gr • π ) } s ≥ . They are induced by the map that associates to ϕ ∈ J s ( π ) and x ∈ π t the classmodulo π s + t +1 of ϕ ( x ) · x − . They take values in the outer derivations of the Liealgebra gr • π that raise degree by s . The construction works well for a class of groupsthat includes π = π g for g ≥ π = F n for n ≥
2; see [39] for full details.By construction, J s +1 ( π ) = ker( τ s ). Another useful property is related to thenatural symmetry group , J ( π ) /J ( π ), that acts by conjugation on ODer s (gr • π ).When π = π g (respectively π = F n ), this group is Sp g ( Z ) (respectively GL n ( Z )).The presence of this arithmetic group symmetry will be very important later on. Inparticular, it opens the way for the use of classical representation theory. ALEXANDRU DIMCA AND STEFAN PAPADIMA
When π = π g and g ≥ τ is Johnson’s homomorphism constructed in [25], τ : T g → V H (Σ g , Z ) /H (Σ g , Z ). By Johnson’s fundamental result from [29], τ induces an isomorphism, ( T g ) abf ≃ → V H (Σ g , Z ) /H (Σ g , Z ); here, T abf denotes themaximal torsion-free abelian quotient of a group T . Moreover, the natural actionof Sp g ( Z ) on the target of τ extends to a rational, irreducible representation of thelinear algebraic group Sp g ( C ) on the complexification.In his landmark paper [28], Johnson showed that ker( τ ) = J ( π g ) coincides with K g . The Johnson kernel K g has since proved quite important in many studies.For example, it plays a key role in Morita’s work on the Casson invariant [36].Other equivalent definitions of the Johnson homomorphism τ for π g , involving thecohomology rings of mapping tori, and the period map from Teichm¨uller theory,were proposed by Johnson in [27] (see also Hain [20] for complete proofs). Theyled to alternative descriptions of K g , and opened new perspectives in approachingmapping class groups and related moduli spaces. Despite this, the basic finitenessquestions on K g remained unanswered.1.2. Strategy of proof.
Guided by the above results on T g , we are going to definethe Johnson kernel of a group T , denoted K T , to be the kernel of the canonicalprojection, T ։ T abf . Assuming from now on g ≥
3, we recall that K T g = K g . Since T g is finitely generated, work of Dwyer and Fried [17] (as refined in [39]) on finite-ness properties of Betti numbers in abelian covers implies that H ( K g , Q ) is finitedimensional if and only if V ( T g ) is finite. Here, V ( T g ) is a certain algebraic subva-riety of the affine, connected torus T ( T g ) = Hom(( T g ) abf , C ∗ ), called the restrictedcharacteristic variety of T g .Our first step involves the geometry and the symmetry of V ( T g ). The arithmeticgroup Sp g ( Z ) naturally acts on T ( T g ) and preserves V ( T g ). We use a powerfulresult in number theory due to Laurent [31], to show that either V ( T g ) is finite, or V ( T g ) = T ( T g ). This is reminiscent of a foundational result in algebraic geometrydue to Arapura, who determined in [3] the qualitative structure of characteristicvarieties associated to fundamental groups of quasi–Kahler manifolds; this makesthe Kahler problem for T g even more inciting.Another important result on the geometry of characteristic varieties was obtainedby Libgober [32] (and refined in Lemma 6.6). It implies that the tangent cone atthe unit 1 ∈ T ( T g ) of V ( T g ) is contained in the so–called resonance variety R ( T g ).The resonance variety is an algebraic subvariety sitting inside the Lie algebra of thealgebraic group T ( T g ), invariant under the Sp g ( Z )–action.Our second step is the analysis of the geometry and symmetry of R ( T g ). It isknown that resonance varieties are closely connected to associated graded Lie alge-bras. A deep result of Hain [21] gives a presentation for the Lie algebra gr • ( T g ) ⊗ C . RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 5
We combine in Theorem 4.4 Hain’s work with classical methods from the representa-tion theory of Sp g ( C ), to arrive at the following conclusion: R ( T ) = Lie( T ( T )) = { } , and R ( T g ) = { } , for g ≥ g ≥
4, this implies that V ( T g ) = T ( T g ). Hence, V ( T g ) must be finite, and H ( K g , Q ) must be finite dimensional, which proves Theorem A.In Theorem B, we follow a similar approach. The BNSR invariant of the finitelygenerated group T g , denoted Σ( T g ), is a subset of Hom(( T g ) abf , R ) \ { } . A powerfulresult in geometric group theory, due to Bieri, Neumann and Strebel [7], impliesthat K g is finitely generated if and only if Σ( T g ) = Hom(( T g ) abf , R ) \ { } . Again, Sp g ( Z ) naturally acts on Hom(( T g ) abf , R ), preserving Σ( T g ).When T = π ( M ) is the fundamental group of a compact Kahler manifold, akey theorem of Delzant [14] describes Σ( T ) in terms of the geometry of holomorphicpencils on M . We deduce Theorem B from the vanishing of R ( T g ), by using Delzant’sresult, together with the Sp g ( Z )–symmetry of Σ( T g ).The techniques developed in this paper lead to a surprising answer to anotherconjecture, concerning the subgroups of the Johnson filtration associated to auto-morphism groups of free groups. See [39].2. Main general results
As explained in the introduction, we adopt the group–theoretic viewpoint on Γ g , T g and K g . This approach leads us to various other general results, that seem to beof independent interest. In this section, we discuss this more general viewpoint.The natural Aut( T )-symmetry factors through Out( T ), for a good number ofinvariants of a group T . The simplest examples are the abelianization, T ab , andthe quotient of T ab by its torsion, T abf . More well-known examples are provided bythe (complex) cohomology ring H • T := H • ( T, C ) and the (complexified) graded Liealgebra associated to the lower central series, G • ( T ) := gr • T ⊗ C .Our starting point is to consider a group epimorphism, p : Γ ։ D , with finitelygenerated kernel T . This is motivated by the defining exact sequence (1.1) ofTorelli groups. We examine three other known types of invariants with naturalouter symmetry, through the prism of the D -symmetry induced by the canonicalhomomorphism, D → Out( T ). Firstly, we look at the resonance varieties R ik ( T )(i.e., the jump loci for a certain kind of homology, associated to the ring H • T ),sitting inside H T ; they are reviewed in Section 4, and their outer symmetry isdiscussed in Remark 4.1. For our purposes here, the most important resonance va-riety will be R ( T ) := R ( T ), which is Zariski closed in H T . Secondly, we inspectthe characteristic varieties V ik ( T ) (i.e., the jump loci for homology with rank onecomplex local systems), lying inside the character torus T ( T ) := Hom( T ab , C ∗ ), andtheir intersection with the connected component of 1 ∈ T ( T ) of the affine group T ( T ), T ( T ) := Hom( T abf , C ∗ ); their definition is recalled in Section 6, and their ALEXANDRU DIMCA AND STEFAN PAPADIMA outer symmetry is explained in Lemma 6.1. The restricted characteristic variety V ( T ) := V ( T ) ∩ T ( T ) is Zariski closed in T ( T ). Finally, we recollect in Section 7a couple of relevant facts about the Bieri-Neumann-Strebel-Renz (BNSR) invariants,Σ q ( T, Z ) ⊆ H ( T, R ) \ { } , and we point out their outer symmetry in Lemma 7.1.We will be particularly interested in Σ( T ) := Σ ( T, Z ). In particular, the resonanceand characteristic varieties of T g , as well as its BNSR–invariants, acquire a natural Sp g ( Z )-symmetry, for g ≥ K ⊆ T be a subgroup containing the derived group T ′ . Then H K := H ( K, C ) becomes in a natural way a module over the group ring C T ab , with modulestructure induced by T -conjugation. Its I -adic completion is denoted [ H K , where I ⊆ C T ab is the augmentation ideal. When K = T ′ , H K is the classical Alexanderinvariant from link theory (over C ).The technique of I -adic completion was promoted in low-dimensional topology byMassey [34]. A key result related to I -adic completion was obtained by Hain in [21],where he shows that T g is a 1-formal group in the sense of Sullivan [45], for g ≥ T = T g , for g ≥ Theorem C.
Let T be a finitely generated group. (1) Assume T is -formal. Then R ( T ) ⊆ { } if and only if dim C [ H T ′ < ∞ . (2) For any subgroup K ⊆ T containing T ′ , dim C [ H K < ∞ , if R ( T ) ⊆ { } . Aiming at finer finiteness properties, we go on by examining characteristic vari-eties, in Section 6. Here, we start from a basic result of Dwyer and Fried [17], asrefined in [39]. It says that the finiteness of Betti numbers, up to degree q , of normalsubgroups with abelian quotient is detected precisely by the finiteness of the inter-section between characteristic varieties of type V i , for i ≤ q , and the correspondingsubtorus of T .Given the D -symmetry of characteristic varieties, we are thus led to consider thefollowing context, inspired by Torelli groups. Let L be a D -module which is finitelygenerated and free as an abelian group. Assume that D is an arithmetic subgroup ofa simple C -linear algebraic group S defined over Q , with Q − rank( S ) ≥
1. Supposealso that the D -action on L extends to an irreducible, rational S -representation in L ⊗ C . (Note that the above assumptions are satisfied for g ≥ D = Sp g ( Z ) ⊆ Sp g ( C ) = S and L = ( T g ) abf , due to Johnson’s pioneering results on the symplectic RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 7 symmetry of Torelli groups from [25, 29].) The D -representation in L gives rise toa natural D -action on the connected affine torus T ( L ) := Hom( L, C ∗ ). Theorem D.
If the D -module L satisfies the above assumptions, then T ( L ) is ge-ometrically D -irreducible, that is, the only D -invariant, Zariski closed subsets of T ( L ) are either equal to T ( L ) , or finite. We deduce Theorem D from a deep result in diophantine geometry, due to M.Laurent [31], in Section 6. Note that the conclusion of our theorem above is inmarked contrast with the behavior of the induced D -representation in the affinespace L ⊗ C , for which S -invariant, infinite and proper Zariski closed subsets maywell exist. Theorem D enables us to obtain in Section 6 the following consequencesof the triviality of resonance. In the particular context of mapping class groups, werecover from Part (3) below Theorem A, in slightly stronger form. Theorem E.
Let p : Γ ։ D be a group epimorphism with finitely generated kernel T , having the property that R ( T ) ⊆ { } . Assume D ⊆ S is arithmetic, where the C -linear algebraic group S is defined over Q , simple, with Q − rank( S ) ≥ . Sup-pose moreover that the canonical D -representation in T abf extends to an irreducible,rational S -representation in T abf ⊗ C . The following hold. (1) The (restricted) characteristic varieties V k ( T ) ∩ T ( T ) are finite, for k ≥ . (2) If moreover b ( T ) > , the Alexander polynomial ∆ T is a non-zero constant,modulo the units of the group ring Z T abf . (3) For any subgroup N ⊆ T , containing the kernel of the canonical map, T ։ T abf , the first Betti number b ( N ) is finite. Note that the computation of characteristic varieties and Alexander polynomialscan be a very difficult task, in general. What makes life easier in Theorem E,Parts (1) and (2), is the arithmetic symmetry. These two results hold in particularfor T = T g , when g ≥
4. As we pointed out earlier, in this case we may inferfrom Theorem C(2) that dim C [ H T ′ g < ∞ . Note that the finite-dimensionality ofthe (uncompleted) Alexander invariant H ( T ′ g , C ) cannot be deduced from TheoremE(1), since ( T g ) ab contains non-trivial 2-torsion, according to Johnson [29]. To thebest of our knowledge, the finiteness of b ( T ′ g ) is an open question.Theorem E may also be used in the particular case when T = J ( F n ), with n ≥ J ( F n ) is finite; see [39].We investigate in Section 7 the BNSR invariants, which control geometric finite-ness properties of normal subgroups with abelian quotient [7, 8]. In the contextfrom (1.1), we use their arithmetic symmetry to prove Theorem B. ALEXANDRU DIMCA AND STEFAN PAPADIMA Associated graded Lie algebra
The symplectic symmetry is well-known to be an important tool for the studyof Torelli groups. In this section, we recall a basic result of Hain [21], related tothe symplectic symmetry at the level of the associated graded Lie algebra. Hain’sstarting point is Johnson’s pioneering work, which we review first.Let Σ g be a closed, oriented, genus g surface. In the sequel we will assume that g ≥
3. In this range, the group T g is finitely generated. For a group T , we denoteby T abf the quotient of its abelianization T ab by the torsion subgroup. Among otherthings, Johnson [25, 29] gave a very convenient description of ( T g ) abf , in the followingway. Fix a symplectic basis of H := H (Σ g , Z ), { a , . . . , a g , b , . . . , b g } , and denote by ω = P gi =1 a i ∧ b i ∈ V H the symplectic form. Let Sp g ( Z ) be the group of symplecticautomorphisms of H . Note that the Sp g ( Z )-action on H canonically extends toa Sp g ( Z )-action on the exterior algebra V ∗ H , by graded algebra automorphisms.Consider the Sp g ( Z )-equivariant embedding, H ֒ → V H , given by h ∈ H h ∧ ω ∈ V H , and denote by L the Sp g ( Z )-module V H/H . Johnson’s homomorphismconstructed in [25], τ : T g → L , has the following properties. Theorem 3.1 (Johnson) . The group homomorphism τ is Γ g -equivariant, with re-spect to the (left) conjugation action on T g induced by (1.1) , and the restrictionof the Sp g ( Z ) -action on L via p . It induces a Sp g ( Z ) -equivariant isomorphism, τ : ( T g ) abf ≃ −→ L . Note that the arithmetic group Sp g ( Z ) is a Zariski dense subgroup of the semisim-ple algebraic group Sp g ( C ); see e.g. [42, Corollary 5.16]. Setting H ( C ) = H ⊗ C and L ( C ) = L ⊗ C , note also that the canonical representation of Sp g ( Z ), comingfrom (1.1), in ( T g ) ab ⊗ C ≃ L ( C ), extends to a rational representation of Sp g ( C ).This symplectic symmetry propagates to higher degrees, in the following sense.Recall that the associated graded Lie algebra (with respect to the lower centralseries) of a group T , gr • T , is generated as a Lie algebra by gr T = T ab , since theLie bracket is induced by the group commutator. Lemma 3.2.
Given a group extension, (3.1) 1 → T → Γ → D → , assume that T is finitely generated, D is a Zariski dense subgroup of a complex linearalgebraic group S , and the D -action on T ab extends to a rational representation of S in T ab ⊗ C . Then the D -action on gr • T extends to an action of S on gr • T ⊗ C ,in the category of graded rational representations; moreover, every s ∈ S acts on gr • T ⊗ C by a graded Lie algebra automorphism.Proof. Presumably this result is well-known to the experts. Being unable to find areference, we decided to include a proof. Set T • := gr • T ⊗ C , noting that D acts RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 9 on T • by graded Lie algebra automorphisms. For each q ≥
1, denote by K q ⊆ T ⊗ q the kernel of the linear surjection sending t ⊗ · · · ⊗ t q to ad t ◦ · · · ad t q − ( t q ) ∈ T q .By construction, T q = T ⊗ q /K q . Since D ⊆ S is Zariski dense, the linear subspace K q is S -invariant. It remains to show that the rational representations of S in T • constructed in this way, which extend the D -action coming from (3.1), have theproperty that s [ a, b ] = [ sa, sb ], for s ∈ S , a ∈ T q and b ∈ T r . This in turn is easilyproved by induction on q . Induction starts with q = 1, by noting that the iteratedLie bracket, T ⊗ r +11 ։ T r +1 , is S -equivariant by construction. For the inductivestep, write a = [ t, a ′ ], with t ∈ T and a ′ ∈ T q − , and use the Jacobi identity toconclude. (cid:3) By Lemma 3.2 and Theorem 3.1, we have a short exact sequence of rational Sp g ( C )-representations,(3.2) 0 → K → ^ gr T g ⊗ C β → gr T g ⊗ C → , where β denotes the Lie bracket and K is by definition ker( β ). Hain [21] computedthe associated exact sequence of sp g -modules, where sp g is the Lie algebra of Sp g ( C ).To describe his result, we follow the conventions from [21, Section 6]. Our referencesfor algebraic groups (respectively Lie algebras) are [24] (respectively [23]).The Lie algebra of the maximal diagonal torus in Sp g ( C ) is denoted h , and hascoordinates t = ( t , . . . , t g ). Let Φ ⊆ h ∗ be the corresponding root system, with thestandard choice of positive roots, Φ + = { t i − t j , t i + t j | ≤ i < j ≤ g }∪{ t i | ≤ i ≤ g } . Let sp g := s = n − ⊕ h ⊕ n + be the canonical decomposition of the Lie algebra. Wedenote by B the associated Borel subgroup of Sp g ( C ) := S , with unipotent radical U ;the Lie algebra of B is h ⊕ n + , and the Lie algebra of U is n + . We work with the finite-dimensional s -modules associated to rational representations of S . The irreducibleones are of the form V ( λ ), where the dominant weight λ is a positive integral linearcombination of the fundamental weights, { λ j ( t ) = t + · · · + t j | ≤ j ≤ g } .It follows from Theorem 3.1 that gr T g ⊗ C = V ( λ ), as sp g -modules. Accordingto [21, Lemma 10.2], all irreducible submodules of V V ( λ ) occur with multiplicityone, and V V ( λ ) contains V (2 λ ) ⊕ V (0) as a submodule. Theorem 3.3 (Hain) . The sp g -map β from (3.2) is the canonical sp g -equivariantprojection of V V ( λ ) onto the submodule V (2 λ ) ⊕ V (0) . Resonance varieties of Torelli groups
In this section, we use representation theory to compute the resonance varieties(in degree 1 and for depth 1) of the Torelli groups T g , for g ≥
3, over C .We begin by reviewing the resonance varieties , R id ( A • ), associated to a connected,graded-commutative C -algebra A • , for (degree) i ≥ d ≥
1. Given a ∈ A , denote by µ a left-multiplication by a in A • , noting that µ a = 0, due tograded-commutativity. Set(4.1) R id ( A • ) = { a ∈ A | dim C H i ( A • , µ a ) ≥ d } . Remark 4.1.
It seems worth pointing out that, given an arbitrary group extension(3.1), one has the following symmetry property, related to resonance. By standardhomological algebra (see e.g. [11]), the conjugation action of Γ on T induces a (right)action of D on A • = H • ( T, C ), by graded algebra automorphisms. We infer from(4.1) that the D -representation on A leaves R id ( A • ) invariant, for all i and d .In this paper, we need to consider only the resonance varieties R d ( A • ) := R d ( A • ),which plainly depend only on the co-restriction of the multiplication map, ∪ : ∧ A → A , to its image. When dim C A < ∞ , it is easy to see that each R d ( A • )is a Zariski closed, homogeneous subvariety of the affine space A . For a connectedspace M , we denote R id ( H • ( M, C )) by R id ( M ). It is equally easy to check that R d ( M ) = R d ( M (2) ), for all d , where M ( q ) denotes the q –skeleton of a connectedCW-complex M . For a group G , R id ( G ) means R id ( K ( G, R d ( M ) = R d ( π ( M )). We will abbreviate R by R .There is a well-known, useful relation between cup-product in low degrees andgroup commutator, see Sullivan [44], and also Lambe [30] for details. For a group G with finite first Betti number, there is a short exact sequence,(4.2) 0 → (gr G ⊗ C ) ∗ d → H G ∧ H G ∪ → H G , where cohomology is taken with C -coefficients and d is dual to the Lie bracket β from (3.2). (Note that the only finiteness assumption needed in the proof from [30]is b ( G ) < ∞ .)In what follows, we retain the notation from Section 3. Set V = L ( C ) ∗ and R = R ( T g ). We infer from (3.2) and (4.2) that R ⊆ V is Sp g ( C )-invariant. Here isour first step in exploiting the complex symmetry from Theorem 3.3. Lemma 4.2. If R 6 = { } , then R contains a maximal vector of the sp g -module V .Proof. This is an easy consequence of Borel’s fixed point theorem [24, Theorem21.2], which guarantees the existence of a B -invariant line, C · v ⊆ R . Invarianceunder the action of the maximal torus implies that v belongs to a weight space ofthe h -action on V , that is, v ∈ V λ for some λ ∈ h ∗ . Finally, n + · v = 0 follows from U -invariance. (cid:3) Since − sp g , all finite-dimensional representationsof sp g are self-dual [23, Exercise 6 on p.116]. This remark leads to the followingdual reformulation of Theorem 3.3, via (4.2). RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 11
Lemma 4.3.
Set V = V ( λ ) . Then H T g = V , as sp g -modules, and the kernel ofthe cup-product, ∪ : V H T g → H T g , is V (2 λ ) ⊕ V (0) . Our main result in this section is the following.
Theorem 4.4.
For g = 3 , R ( T ) = H T , while R ( T g ) = { } , for g ≥ . The assertion for g = 3 is an immediate consequence of Lemma 4.3, since in thiscase V V = V (2 λ ) ⊕ V (0), cf. Lemma 10.2 from [21]. So, we will assume in thesequel that g ≥ R ( T g ) = { } , and use Lemma 4.2 to derive a contradiction.Set V (0) = C · z . We will need explicit maximal vectors, v for V ( λ ) and u for V (2 λ ). To this end, we recall that V ( λ ) = ( V H/H ) ⊗ C , where the sp g -actionon V • H is by algebra derivations; in particular, s · x ∧ y ∧ z = s · x ∧ y ∧ z + x ∧ s · y ∧ z + x ∧ y ∧ s · z , for s ∈ s and x, y, z ∈ H . Set v ′ = a ∧ a ∧ a , and u ′ = P gk =3 ( a ∧ a ∧ a k ) V ( a ∧ a ∧ b k ). Denote by v the class of v ′ in V = V ( λ ),and let u be the class of u ′ in V V . Using the explicit description of h ⊕ n + from[21, Section 6], it is straightforward to check that v has weight λ , u has weight2 λ , and n + · v = n + · u = 0.To verify that both v and u are non-zero, we will resort to the Sp g ( Z )-equivariantcontraction constructed by Johnson in [25, p.235], C : V H → H , given by(4.3) C ( x ∧ y ∧ z ) = ( x · y ) z + ( y · z ) x + ( z · x ) y , where the dot designates the intersection form on H . Set L ′ = ker( C ), and L ′ ( C ) = L ′ ⊗ C . It follows from [25, pp.238-239] that there is an induced sp g -isomorphism,(4.4) L ′ ( C ) ≃ −→ L ( C ) . Clearly, 0 = v ′ ∈ L ′ and 0 = u ′ ∈ V L ′ . Hence, both v and u are non-zero, asneeded.We infer from uniqueness of maximal vectors [23, Corollary 20.2] that necessarily v ∈ R = R ( T g ), if R 6 = { } ; see Lemma 4.2. Denote by W the s -submodule V (2 λ ) ⊕ V (0) of V V from Lemma 4.3. By definition (4.1) and Lemma 4.3, v ∈ R if and only if im( µ ) ∩ W = 0, where µ : V → V V denotes left-multiplication by v ∈ V in the exterior algebra.Since n + · v = 0, we conclude that µ is n + -equivariant. Since v ∈ V λ , it followsthat(4.5) µ ( V λ ′ ) ⊆ ( ^ V ) λ + λ ′ , for each weight subspace V λ ′ ⊆ V . Lemma 4.5. If R ( T g ) = { } , then im( µ ) ∩ W contains a non-zero vector killed by n + . Proof.
Taking into account the preceding discussion, the assertion will follow fromEngel’s theorem [23, Theorem 3.3], applied to the Lie algebra n + and the n + -moduleim( µ ) ∩ W . It is enough to check that the action of any n ∈ n + on W is nilpotent.This in turn follows from the fact that s α · · · s α r · W λ ′ ⊆ W λ ′′ , where λ ′′ = λ ′ + P ri =1 α i , for any α , . . . , α r ∈ Φ + and any non-trivial weight space W λ ′ . Indeed, W λ ′′ = 0 for r big enough. To check the vanishing claim, one may invoke [23,Theorem 20.2(b)], and then use a height argument. (cid:3) Lemma 4.6.
The following hold. (1)
For g ≥ , the vector v ∧ u ∈ V V is non-zero. (2) The class of ( a ∧ a ∧ a ) V ( b ∧ b ∧ b ) in V V does not belong to V (0) .Proof. (1) By (4.4), it is enough to show that v ′ ∧ u ′ ∈ V ( ∧ H ) is non-zero. Thisis clear, since v ′ ∧ u ′ = P gk =4 ( a ∧ a ∧ a ) V ( a ∧ a ∧ a k ) V ( a ∧ a ∧ b k ).(2) Again by (4.4), it suffices to verify that the element e = ( a ∧ a ∧ a ) V ( b ∧ b ∧ b ) ∈ V ( ∧ H ) ⊗ C is not killed by sp g , since b ∧ b ∧ b ∈ L ′ . Indeed, T · e = ( a ∧ a ∧ a ) V ( a ∧ b ∧ b ) = 0, where T ∈ sp g is described in [21, § (cid:3) We will finish the proof of Theorem 4.4, by showing that Lemmas 4.5 and 4.6lead to a contradiction, assuming g ≥ R ( T g ) = { } .Indeed, in this case there is v ∈ V ( λ ) such that(4.6) 0 = v ∧ v = w + w , with w ∈ V (0), w ∈ V (2 λ ) and n + · w = 0, by Lemma 4.5.If w = 0, then w ∈ C ∗ · u , by uniqueness of maximal vectors. Taking the weightdecomposition of v , it follows from (4.5) that we may suppose that w = 0 in (4.6).Therefore, v ∧ u = 0 ∈ V V , contradicting Lemma 4.6(1).If w = 0, then v ∧ v ∈ C ∗ · z , for some v ∈ V ( λ ) − λ , by the same argumenton weight decomposition as before. Since the weights λ and − λ are conjugateunder the action of the Weyl group, the weight space V ( λ ) − λ is one-dimensional,generated by the class v of b ∧ b ∧ b ∈ L ′ in V ; see [23, Theorems 20.2(c) and21.2], [21, Section 6] and (4.4). Therefore, v ∧ v ∈ V (0), contradicting Lemma4.6(2). The proof of Theorem 4.4 is thus completed.5. Resonance and finiteness properties
We devote this section to a general discussion of finiteness properties related toresonance, with an application to Torelli groups. Unless otherwise mentioned, wework with C -coefficients. For a graded object O • , the notation O means that weforget the grading.We need to review a couple of key notions. We begin with the holonomy Liealgebra associated to a connected CW-complex M with finite 1-skeleton, H • ( M ).This is a quadratic graded Lie algebra, defined as the quotient of the free Lie algebra RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 13 generated by H M , L • ( H M ), graded by bracket length, modulo the ideal generatedby the image of the comultiplication map, ∂ M : H M → V H M ; here we are usingthe standard identification V H M ≡ L ( H M ), given by the Lie bracket. Notethat the dual of ∂ M is the cup-product, ∪ M : V H M → H M .When G is a finitely generated group and M = K ( G, H • ( M ) is denoted H • ( G ).By considering a classifying map, it is easy to see that H • ( M ) = H • ( π ( M )), for aconnected CW-complex M with finite 1-skeleton. The associated graded Lie algebragr • G ⊗ C , denoted G • ( G ), is also finitely generated in degree one, but not necessarilyquadratic. The canonical identification, H G ≡ G ( G ), extends to a graded Liealgebra epimorphism, L • ( H G ) ։ G • ( G ). By (4.2), this factors to a graded Liealgebra surjection,(5.1) H • ( G ) ։ G • ( G ) . Let P • be the polynomial algebra Sym( H G ), endowed with the usual grading.For a Lie algebra H , denote by H ′ := [ H , H ] the derived Lie algebra, and by H ′′ :=[ H ′ , H ′ ] the second derived Lie algebra. The exact sequence of graded Lie algebras(5.2) 0 → H ′• ( G ) / H ′′• ( G ) → H • ( G ) / H ′′• ( G ) → H • ( G ) / H ′• ( G ) → P • -module structure on the infinitesimal Alexander invari-ant , b • ( G ) := H ′• ( G ) / H ′′• ( G ), induced by the adjoint action. A finite P • -presentationof b • ( G ) is described in [37, Theorem 6.2].Let E k − ( b ( G )) ⊆ P be the Fitting ideal generated by the codimension k − P -presentation for b ( G ). The definition of E k − does not depend on the P –presentation; see [18, Corollary 20.4]. Denote by W k ( G ) ⊆ H G the zero setof E k − ( b ( G )). The next lemma explains the relationship between the infinitesimalAlexander invariant and the resonance varieties in degree one. Lemma 5.1.
Let G be a finitely generated group. Then the equality W k ( G ) \ { } = R k ( G ) \ { } holds for all k ≥ .Proof. We know from [37, Theorem 6.2] that b ( G ) = coker( ∇ ), as P -modules, where(5.3) ∇ := δ + id ⊗ ∂ G : P ⊗ (cid:16) ^ H G ⊕ H G (cid:17) → P ⊗ ^ H G , and the P -linear map δ is given by δ ( a ∧ b ∧ c ) = a ⊗ b ∧ c + b ⊗ c ∧ a + c ⊗ a ∧ b , for a, b, c ∈ H G . A polynomial f ∈ P = Sym( H G ) may be evaluated at z ∈ H G . Itwill be convenient to denote by F ( z ) the matrix over C obtained by evaluating at z all entries of the P –matrix F . This may be done for ∇ and all Koszul differentials, δ i : P ⊗ V i H G → P ⊗ V i − H G .Pick any z ∈ H G \ { } . By linear algebra, we infer that z ∈ W k ( G ) if and only ifdim C coker( ∇ ( z )) ≥ k . Consider the exact cochain complex ( V • H G, λ z ), where λ z denotes left multiplication by z , and the dual exact chain complex, ( V • H G, ♯ λ z ).It is straightforward to check that the restriction of ♯ λ z to V i H G equals δ i ( z ), forall i .We obtain from exactness the following isomorphism:coker( ∇ ( z )) ∼ = im( δ ( z )) / im( δ ( z ) ◦ ∂ G ) . By exactness again, dim C im( δ ( z )) −
1, where n := b ( G ). Hence z ∈ W k ( G ) if andonly if rank( δ ( z ) ◦ ∂ G ) ≤ n − − k .The linear map dual to the restriction of ♯ λ z to V H G , δ ( z ), is λ z : H G → V H G . By construction, the dual of ∂ G is the cup–product ∪ G : V H G → H G .Hence, the linear map dual to δ ( z ) ◦ ∂ G is the map µ z : H G → H G from thedefinition of resonance (4.1). Consequently, z ∈ W k ( G ) if and only if rank( µ z ) ≤ n − − k , that is, if and only if z ∈ R k ( G ); see again definition (4.1). (cid:3) We may spell out our first general result relating resonance and finiteness.
Theorem 5.2.
Let G be a finitely generated group. Then R ( G ) ⊆ { } if and onlyif dim C b ( G ) < ∞ .Proof. Lemma 5.1 yields in particular the equality R ( G ) = Zero(ann b ( G )), awayfrom the origin. This is a consequence of the fact that the ideals E ( b ( G )) andann b ( G ) have the same radical; see [18, Proposition 20.7]. Let m ⊆ P be themaximal ideal of 0 ∈ H G .If dim C b ( G ) < ∞ , then m k ⊆ ann b ( G ), for some k , by degree inspection. Takingzero sets, we obtain that R ( G ) ⊆ { } .Conversely, the assumption R ( G ) ⊆ { } implies m ⊆ p ann b ( G ), by Hilbert’sNullstellensatz. Therefore, m k ⊆ ann b ( G ), for some k . Since b ( G ) is finitely gener-ated over P , we infer that dim C b ( G ) < ∞ . (cid:3) Theorem 5.2 has several interesting consequences. To describe them, we firstrecall a couple of notions from rational homotopy theory. A
Malcev Lie algebra (in the sense of Quillen) is a Lie algebra endowed with a decreasing vector spacefiltration satisfying certain axioms; see [41, Appendix A], where Quillen associatesin a functorial way a Malcev Lie algebra, M ( G ), to an arbitrary group G . Theconstruction goes as follows. Start with the group ring C G , and his standard Hopfalgebra structure. The I -adic completion, d C G , becomes a complete Hopf algebra.The Malcev Lie algebra M ( G ) consists of the primitive elements of d C G , with Liebracket given by the algebra commutator in d C G , and filtration induced by thecompletion filtration of d C G .Following Sullivan [45], we say that a finitely generated group G is 1 -formal if M ( G ) is the completion with respect to degree of a quadratic Lie algebra, as afiltered Lie algebra. In Theorem 1.1 from [21], Hain proves that the Torelli group RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 15 T g is 1-formal, for g ≥ T is not 1-formal). Fundamental groups of compactKahler manifolds are 1-formal, as shown by Deligne, Griffiths, Morgan and Sullivanin [13]. Many other interesting examples of 1-formal groups are known; see e.g. [16]and the references therein.Next, we review the Alexander invariant , B ( G ) = G ′ ab ⊗ C , associated to a finitelygenerated group G . The exact sequence of groups(5.4) 1 → G ′ /G ′′ → G/G ′′ → G/G ′ → B ( G ), induced byconjugation, over the Noetherian group ring C G ab . We denote by I ⊆ C G ab theaugmentation ideal, and by b B the I -adic completion of a C G ab -module B .Given a graded vector space V • , b V • means completion with respect to the degreefiltration: b V • = lim ←− q V /F q , where F q = V ≥ q . The canonical filtration of b V • is b F q =ker( π q ), where π q : b V • → V /F q is the q -th projection of the inverse limit. The nextcorollary proves Theorem C(1). Corollary 5.3.
Let G be a finitely generated, -formal group. Then R ( G ) ⊆ { } if and only if dim C [ B ( G ) < ∞ . In particular, R ( G ) ⊆ { } , when dim C B ( G ) < ∞ .Proof. Let m ⊆ P be the maximal ideal of the origin. The 1-formality of G providesa vector space isomorphism between [ B ( G ) and the m -adic completion of b ( G ), cf.Theorem 5.6 from [16]. Since b • ( G ) is generated in degree 0, its m -adic completioncoincides with the degree completion, \ b • ( G ). Clearly, b • ( G ) and \ b • ( G ) are simulta-neously finite-dimensional. Hence, our first claim follows from Theorem 5.2. For thesecond claim, simply note that the completion of a finite-dimensional vector spaceis again finite-dimensional. (cid:3) Remark 5.4.
The finitely generated group G discussed in Example 6.4 from [38] sat-isfies R ( G ) ⊆ { } , yet dim C B ( G ) = ∞ . It can be shown that G is 1-formal. Con-sequently, R ( G ) ⊆ { } is only a necessary condition for the finite-dimensionalityof B ( G ), in general.In the particular case when G is nilpotent, the condition dim C B ( G ) < ∞ isautomatically satisfied, since G ′ is finitely generated. We recover in this way fromCorollary 5.3 the resonance obstruction to 1-formality of finitely generated nilpotentgroups found by Carlson and Toledo in [12, Lemma 2.4]. We refer the reader toMacinic [33], for similar higher-degree obstructions to the formality of a finitelygenerated nilpotent group.As we shall see below, the main point in Corollary 5.3 is the fact that the finite-dimensionality of [ B ( G ) forces R ( G ) ⊆ { } , when G is 1-formal. We point out that1-formality is needed for this implication. Indeed, let G be the finitely generatednilpotent Heisenberg group with b ( G ) = 2. As noted before, dim C [ B ( G ) < ∞ . Yet, the resonance variety R ( G ) = C is non-trivial; see for instance [33, Proposition5.5].Let G be an arbitrary finitely generated group, and K ⊆ G a subgroup containing G ′ . Clearly, K is normal in G , and G -conjugation makes H K a finitely generatedmodule over the Noetherian group ring C [ G/K ]. By restriction via the canonicalepimorphism,
G/G ′ ։ G/K , H K becomes a finitely generated C G ab -module. Wemay now state our next main result from this section, which proves Theorem C(2). Theorem 5.5.
Let G be a finitely generated group, and K ⊆ G a subgroup contain-ing G ′ . If R ( G ) ⊆ { } , the vector space [ H K is finite-dimensional.Proof. We first treat the particular case K = G ′ , where we know from Theorem5.2 that the vector space H • ( G ) / H ′′• ( G ) is finite-dimensional. The canonical gradedLie algebra surjection, G • ( G ) ։ G • ( G/G ′′ ), composed with the epimorphism (5.1),gives a graded Lie algebra surjection, H • ( G ) ։ G • ( G/G ′′ ), that factors through anepimorphism(5.5) H • ( G ) / H ′′• ( G ) ։ G • ( G/G ′′ ) . It follows from (5.5) that G • ( G/G ′′ ) = 0, for • >> I q · H G ′ /I q +1 · H G ′ ≃ G q +2 ( G/G ′′ ) , for q ≥
0; see [34, pp.400–401]. We infer from (5.6) that the I -adic filtration of H G ′ stabilizes, for q >>
0. Therefore, dim C [ H G ′ < ∞ , as asserted.For the general case, consider the extension(5.7) 1 → G ′ → K → A → , where A = K/G ′ is a finitely generated subgroup of G ab , and denote by ( H G ′ ) A theco-invariants of the A -module H G ′ , noting that the canonical projection, H G ′ ։ ( H G ′ ) A , is C G ab -linear.The Hochschild-Serre spectral sequence of (5.7) with trivial C -coefficients (seee.g. [11, p.171]) provides an exact sequence of finitely generated C G ab -modules,(5.8) ( H G ′ ) A → H K → H A → . By standard commutative algebra (see for instance [4, Chapter 10]), the I -adiccompletion of (5.8) is again exact. Since dim C H A < ∞ , our claim about [ H K follows from the finite-dimensionality of [ H G ′ . (cid:3) The next corollary gives an application to Torelli groups. It follows from Theorems4.4 and 5.5.
RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 17
Corollary 5.6.
Let K be a subgroup of T g containing T ′ g . The I -adic completion of H ( K, C ) is finite-dimensional, for g ≥ , where I ⊆ C [( T g ) ab ] is the augmentationideal of the group ring. We will improve Corollary 5.6 in the next section, when K contains the Johnsonkernel K g . 6. Characteristic varieties of Torelli groups
Guided by the interplay between arithmetic and Torelli groups, coming from (1.1),we will examine now groups whose characteristic varieties possess a natural discretesymmetry. We will compute the (restricted) characteristic varieties of T g , in degree1, when g ≥
4, and deduce that b ( K g ) < ∞ .Our setup in this section is the following. Let(6.1) 1 → T → Γ p → D → T is finitely generated and D is an arithmetic subgroupof a complex linear algebraic group S , defined over Q , simple and with Q -rank atleast 1. The motivating examples are the extensions (1.1), for g ≥
3. Under theabove assumptions on D , we recall that any finite index subgroup D ⊆ D is Zariskidense in S , as follows from Borel’s density theorem; see e.g. [42, Corollary 5.16].We continue by reviewing a couple of relevant facts related to character toriand characteristic varieties. Let G be a group, with character group T ( G ) =Hom( G ab , C ∗ ). When G is finitely generated, the character torus T ( G ) is a lin-ear algebraic group, with coordinate ring the group algebra C G ab . The connectedcomponent of 1 ∈ T ( G ) is T ( G ) = T ( G abf ).For the beginning, we need no assumptions on the group extension (6.1). The nat-ural D -representation in T ab (respectively T abf ) induced by conjugation canonicallyextends to Z T ab and C T ab (respectively to Z T abf and C T abf ). The correspondingleft D -action on T ( T ), by group automorphisms, is denoted by d · ρ , for d ∈ D and ρ ∈ T ( T ), and is defined by d · ρ ( u ) = ρ ◦ d − ( u ), for u ∈ T ab . When T is finitelygenerated, the D –action on T ( T ) is algebraic.The characteristic varieties of a group G , V ik ( G ), are defined for (degree) i ≥ k ≥ V ik ( G ) = { ρ ∈ T ( G ) | dim C H i ( G, C ρ ) ≥ k } . Here C ρ denotes the rank one complex G –module given by the change of rings Z G → C , corresponding to ρ . When G is finitely generated, it is easy to checkthat V ik ( G ) is Zariski closed in T ( G ), for i ≤ k ≥
1. We will be mainlyinterested in the (restricted) characteristic variety (in degree i = 1 and depth k = 1) V ( G ) := V ( G ) ∩ T ( G ) associated to a finitely generated group G . By the above discussion, the restricted (degree 1) characteristic variety V k ( G ) ∩ T ( G ) is Zariskiclosed in the affine connected torus T ( G ) = ( C ∗ ) b ( G ) , for all k ≥ Lemma 6.1.
Given an arbitrary group extension (6.1) , the characteristic varieties V ik ( T ) ⊆ T ( T ) are D -invariant subsets, for all i ≥ and k ≥ .Proof. For γ ∈ Γ, denote γ -conjugation by ι γ : T ≃ −→ T . For a ring R , withgroup of units R × , and a group homomorphism, χ : T → R × , the notation R χ means the Z T -module R associated to the change of rings χ : Z T → R . Notethat the pair ( ι γ , id R ) : ( T, R χ ◦ ι γ ) ≃ −→ ( T, R χ ), gives an isomorphism in the categoryof local systems; see e.g. [11, III.8]. Hence, there is an induced isomorphism, H ∗ ( T, R χ ◦ ι γ ) ≃ −→ H ∗ ( T, R χ ). Our claim follows by taking R = C , and inspectingdefinition (6.2). (cid:3) When T is finitely generated, let us consider V k ( T ) ∩ T ( T ). This leads us tolook at a discrete group D acting linearly on a free, finitely generated abelian group L , and examine the D -invariant, Zariski closed subsets W of T ( L ) = Hom( L, C ∗ ).Besides the trivial case W = T ( L ), we find a lot of 0-dimensional examples, bytaking W to be the subgroup of m -torsion elements of T ( L ). This raises a naturalquestion: is there anything else?To present a first answer, we need the following notions. A translated subgroup(torus) is a subset of the character torus T = T ( T ) of the form t · S , where t ∈ T and S ⊆ T is a closed (connected) algebraic subgroup. Note that the direction of the translated subgroup, S , is uniquely determined by t · S . A Zariski closedsubset W ⊆ T is a union of translated tori if each irreducible component of W is atranslated torus; in other words, W is a finite union of translated subgroups. Lemma 6.2.
Let L be a D -module which is finitely generated and free as an abeliangroup. Assume that D is an arithmetic subgroup of a simple C -linear algebraicgroup S defined over Q , with Q − rank( S ) ≥ . Suppose also that the D -actionon L extends to an irreducible, rational S -representation in L ( C ) := L ⊗ C . Let W ⊂ T ( L ) be a D -invariant, Zariski closed, proper subset of T ( L ) . If W is a unionof translated tori, then W is finite.Proof. We know that each irreducible component of W is of the form t · S , as above.We have to show that dim( S ) = 0. To this end, we consider the isotropy group of t · S , denoted D ; it is a finite index subgroup of D . Note that dim( S ) < dim( T ( L )),since W = T ( L ).The D -invariance of t · S forces the direction S to be D -invariant as well. There-fore, the Lie algebra T S is a D -invariant linear subspace of Hom( L, C ) = L ( C ) ∗ .Since D ⊆ S is Zariski dense, it follows that the subspace T S is actually S -invariant. Hence, T S = 0, due to S -irreducibility. (cid:3) RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 19
To improve Lemma 6.2, we need a preliminary result.
Lemma 6.3.
Let L be a D -module which is finitely generated and free as an abeliangroup. Then the subgroup O t of T ( L ) , generated by the D -orbit D · t , is finitelygenerated, for any t ∈ T ( L ) .Proof. Pick a Z -basis { e i } of L ; identify L with Z n , and T ( L ) with ( C ∗ ) n . For t = ( t , . . . , t n ) ∈ T ( L ) and w = P i w i e i ∈ L , set t w := Q i t w i i ∈ C ∗ .The elements of D · t are of the form ( t v , . . . , t v n ), with v i = P j v ij e j ∈ L . Define u ijk = δ ik e j ∈ L , for 1 ≤ i, j, k ≤ n . The equality (cid:16) t v , . . . , t v n (cid:17) = Y ≤ i,j ≤ n (cid:16) t u ij , . . . , t u ijn (cid:17) v ij gives the desired conclusion. (cid:3) The key argument in a stronger version of Lemma 6.2 uses the solution of aconjecture of Lang in diophantine geometry, obtained by Laurent in [31]. We recallthis result. Let W ⊆ T ( L ) be Zariski closed, and let O ⊆ T ( L ) be a subgroup offinite rank. Denote by T the set of translated subgroups of T ( L ) of the form γ · S ,with γ ∈ O and such that γ · S ⊆ W , ordered by inclusion. We extract from [31],Th´eor`eme 2, Lemme 3 and Lemme 4, the following result. Theorem 6.4 (Laurent) . The set T has finitely many maximal elements. Denotethem γ · S , . . . , γ m · S m . Furthermore, W ∩ O = m [ i =1 γ i · ( S i ∩ O ) . Now, we are going to show that the D -irreducibility of L is inherited by the affinetorus T ( L ), in the sense explained below. This proves Theorem D. Theorem 6.5.
Let L be a D -module which is finitely generated and free as an abeliangroup. Assume that D is an arithmetic subgroup of a simple C -linear algebraic group S defined over Q , with Q − rank( S ) ≥ . Suppose also that the D -action on L extendsto an irreducible, rational S -representation in L ( C ) := L ⊗ C . Let W ⊂ T ( L ) be a D -invariant, Zariski closed, proper subset of T ( L ) . Then W is finite.Proof. We first claim that D · t is finite, for every t ∈ W . We will apply Theorem 6.4to the closed subvariety W ⊆ T ( L ) and the subgroup O t ⊆ T ( L ), which is finitelygenerated, by Lemma 6.3. In this way, we obtain the inclusion(6.3) W ∩ O t ⊆ m [ i =1 γ i · S i . Denote by V ⊆ W the right-hand side of (6.3). Since O t is D -invariant by construc-tion, we infer that V is D -invariant as well; this follows from the construction of the D –invariant ordered set T , whose maximal elements are γ · S , . . . , γ m · S m . Hence, V must be finite, by Lemma 6.2. Since clearly D · t ⊆ W ∩ O t , we obtain our claim.Supposing that W is infinite, we may find a smooth point t ∈ W whose tangentspace satisfies 0 = T t W = T t T ( L ). Translating to the origin, we obtain a D t -invariant, proper and non-trivial linear subspace of L ( C ) ∗ , where D t is the isotropygroup of t . By the first step of our proof, D t has finite index in D , which contradictsthe S -irreducibility of L ( C ). (cid:3) We go on by establishing a relation between characteristic and resonance varietiesin degree 1, for finitely generated groups.
Lemma 6.6.
Let G be a finitely generated group. (1) There is a finitely presented group G , together with a group epimorphism, φ : G ։ G , such that φ ab : G ab ≃ −→ G ab is an isomorphism, inducingidentifications, V k ( G ) ≡ V k ( G ) and R k ( G ) ≡ R k ( G ) , for all k ≥ . (2) The tangent cone at of V k ( G ) , T C V k ( G ) , is contained in R k ( G ) , for all k ≥ .Proof. Part (1). Let X be a classifying space for G , having 1-skeleton equal to afinite wedge of circles. Denote by { Z G ab ⊗ C • X D • → Z G ab ⊗ C •− X } the cellular chaincomplex of the universal abelian cover X ab , where { C • X d • → C •− X } is the cellularchain complex of X , over Z . Since the ring Z G ab is noetherian, im( D ) is generatedover Z G ab by the images of finitely many 2-cells, say { e , . . . , e r } .Consider now the comultiplication map of the 2–skeleton, ∂ X (2) : H ( X (2) , C ) →∧ H ( X (2) , C ). Pick finitely many 2-cells, say { e ′ , . . . , e ′ s } , such that the ∂ -imagesof the d • -cycles belonging to Z − span { e ′ , . . . , e ′ s } generate im( ∂ X (2) ) over C .Let Y be the finite subcomplex of X (2) obtained from X (1) by attaching the cells { e i } and { e ′ j } . Set G := π ( Y ). Attach cells of dimension at least 3 to Y , in orderto obtain a classifying space for G , denoted X . Extend the inclusion Y ֒ → X (2) toa map X → X , and consider the induced group epimorphism, φ : G ։ G .By our choice of the e -cells, we infer that φ induces in turn an isomorphism, H ( G, R χ ◦ φ ) ≃ −→ H ( G, R χ ), for any group homomorphism, χ : G → R × , when thering R is commutative. In particular, our claims on φ ab and characteristic varietiesare thus verified. To check the claim on resonance varieties, we may replace G by Y and G by X (2) . From our choice for the e ′ -cells, we deduce that, upon identifying ∧ H ( Y, C ) and ∧ H ( X (2) , C ) via φ , we have the equality im( ∂ Y ) = im( ∂ X (2) ). Byduality, the cup-product maps ∪ Y and ∪ X (2) have isomorphic co-restrictions to theimage, which proves our last claim.Part (2). A result of Libgober from [32] implies that T C V k ( G ) ⊆ R k ( G ), for all k ≥
1, when G is finitely presented. By Part (1), the inclusion still holds for finitelygenerated groups. (cid:3) RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 21
Here is our main result in this section, relating arithmetic symmetry and finitenessproperties, which proves Theorem E. The
Alexander polynomial ∆ T of a finitelygenerated group T appearing in Part (3) below is a classical invariant, with roots inknot theory, defined as follows. Let A ( T ) = Z T abf ⊗ Z T I be the Alexander module.It is a finitely generated Z T abf –module, with Fitting ideal E ( A ( T )) ⊆ Z T abf . Thegreatest common divisor of all elements in E ( A ( T )), defined up to units of Z T abf ,is ∆ T . Corollary 6.7.
Assume in (6.1) that T is finitely generated with R ( T ) ⊆ { } , and D ⊆ S is arithmetic, where the C -linear algebraic group S is defined over Q , simple,with Q − rank( S ) ≥ . Suppose moreover that the canonical D -representation in T abf extends to an irreducible, rational S -representation in T abf ⊗ C . Then the followinghold. (1) The restricted characteristic varieties V k ( T ) ∩ T ( T ) are finite, for all k ≥ . (2) The first Betti number of π − ( A ) is finite, for any subgroup A ⊆ T abf , where π : T ։ T abf is the canonical projection. (3) If moreover b ( T ) > , the Alexander polynomial ∆ T is a non-zero constant c ∈ Z , modulo the units of Z T abf .Proof. We may clearly assume b ( T ) = 0. We want to use Theorem 6.5, appliedto the D -module L = T abf and the closed subvariety W = V k ( T ) ∩ T ( T ). The D -invariance of W follows from Lemma 6.1. The fact that W = T ( L ) is a consequenceof our assumption on R ( T ), due to Lemma 6.6(2). Hence, Part (1) follows fromTheorem 6.5.Set K := ker( π ). A basic result of Dwyer and Fried [17], as refined in [38, Corollary6.2], says that the finiteness of V ( T ) ∩ T ( T ) is equivalent to dim C ( K ab ) ⊗ C < ∞ ,for any finitely generated group T . Consider now the extension 1 → K → π − ( A ) → A →
1. A standard application of the Hochschild-Serre spectral sequence shows thatthe first Betti number of π − ( A ) is finite, since both b ( K ) and b ( A ) are finite. Thiscompletes the proof of Part (2).Part (3) follows from Part (1), via Corollary 3.2 from [15]. (cid:3) Corollary 6.7 leads to the following consequences, for Torelli groups.
Corollary 6.8.
Assume g ≥ . (1) The intersection V k ( T g ) ∩ T ( T g ) is finite, for all k ≥ . (2) The vector space H ( N, C ) is finite-dimensional, for any subgroup N of T g containing the Johnson kernel K g . (3) The Alexander polynomial ∆ T g is a non-zero constant c ∈ Z , modulo units.Proof. The condition on resonance is guaranteed by Theorem 4.4. Since b ( T g ) isan increasing function of g , for g ≥ b ( T g ) ≥
14, when g ≥ (cid:3) Sigma-invariants and the Kahler property
We close by examining groups with natural arithmetic symmetry, at the level ofBieri-Neumann-Strebel-Renz invariants. We first review briefly the definitions andthe main properties; for more details and references, see e.g. [38].We start with the
Novikov-Sikorav completion of a finitely generated group G ,with respect to χ ∈ Hom( G, R ), denoted c Z G − χ . For k ∈ Z , let F k be the abeliansubgroup of Z G generated by the elements g ∈ G with χ ( g ) ≥ k . The completion, c Z G − χ , of Z G with respect to the decreasing filtration { F k } k ∈ Z becomes in a naturalway a ring, containing the group ring Z G .The Sigma-invariants Σ q ( G, Z ) are defined for q ≥ q ( G, Z ) = { = χ ∈ H ( G, R ) | H i ( G, c Z G − χ ) = 0 , ∀ i ≤ q } . When the group G is of type F P k , the above definition coincides with the oneintroduced by Bieri and Renz in [8], for q ≤ k . Note that property F P simplymeans finite generation of G . The Sigma-invariant Σ ( G, Z ) of a finitely generatedgroup G , denoted Σ( G ), coincides with the one defined by Bieri, Neumann andStrebel in [7].It turns out that Σ q ( G, Z ) is an open (possibly empty) conical subset of H ( G, R ),for all q ≤ k , when G is of type F P k . Moreover, in this case we have the followingfundamental property. Given a group epimorphism, ν : G ։ L , onto an abeliangroup, ker( ν ) is of type F P q , with q ≤ k , if and only if(7.2) ν ∗ ( H ( L, R ) \ { } ) ⊆ Σ q ( G, Z ) . Here is the analog of Lemma 6.1 for Sigma-invariants.
Lemma 7.1.
Let p : Γ ։ D be a group epimorphism, with finitely generated kernel T . Then Σ q ( T, Z ) is invariant under the canonical action of D on H ( T, R ) , comingfrom (6.1) , for all q ≥ .Proof. Novikov-Sikorav completion is functorial, in the following sense. Let φ : G → K be a group homomorphism and χ ∈ Hom( K, R ). The induced ring homomor-phism, φ : Z G → Z K , clearly preserves the defining filtrations of c Z G − χ ◦ φ and d Z K − χ .Passing to completions, φ extends to a ring homomorphism, b φ : c Z G − χ ◦ φ → d Z K − χ .The above remark may be applied to γ -conjugation, φ = ι γ : T ≃ −→ T , for any γ ∈ Γ, and an arbitrary additive character χ ∈ H ( T, R ). The pair ( φ, b φ ) givesthen an isomorphism, ( T, c Z T − χ ◦ φ ) ≃ −→ ( T, c Z T − χ ), in the category of local systems.Consequently, there is an induced isomorphism, H ∗ ( T, c Z T − χ ◦ φ ) ≃ −→ H ∗ ( T, c Z T − χ ).We infer from (7.1) that χ ∈ Σ q ( T, Z ) if and only if χ · d ∈ Σ q ( T, Z ), where d = p ( γ ). (cid:3) RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 23
Remark 7.2.
Note that − id ∈ Sp g ( Z ) acts by − id on ∧ H/H . Consequently, − Σ( T g ) = Σ( T g ). This symmetry property of Σ( G ) about the origin does not holdin general.Note also that, when − Σ( G ) = Σ( G ), Σ( G ) = ∅ if and only if there is a finitelygenerated, normal subgroup N of G , with infinite abelian quotient G/N . Moreover,in this statement
G/N may be replaced by Z . Indeed, assuming G/N to be infiniteabelian, we infer that Σ( G ) = ∅ , by resorting to (7.2). Conversely, we know that theimage of Σ( G ) in the quotient sphere, ( H ( G, R ) \ { } ) / R + , is open and nonvoid.The density of rational points on this sphere [7, p.451] implies then that Σ( G )contains an epimorphism, ν : G ։ Z . By antipodal symmetry of Σ( G ) and (7.2)again, ker( ν ) must be finitely generated.Under additional hypotheses, Lemma 7.1 may be used to obtain strong informa-tion on finiteness properties. Proposition 7.3.
Let p : Γ ։ D be a group epimorphism, with finitely generatedkernel T . Assume that D is an arithmetic subgroup of a complex, simple linearalgebraic group S , defined over Q , with Q − rank( S ) ≥ , and the canonical D -action on T abf induced by conjugation extends to a non-trivial, irreducible rationalrepresentation of S in T abf ⊗ C . Suppose moreover that (1) either Σ( T ) is a finite disjoint union of finite intersections of open half-spacesin H ( T, R ) , (2) or Σ( T ) is the complement of a finite union of linear subspaces in H ( T, R ) .Then the kernel K of the natural map, T ։ T abf , is finitely generated if and only if Σ( T ) = ∅ .Proof. According to (7.2), finite generation of K is equivalent to Σ( T ) = H ( T, R ) \{ } . Note that our irreducibility assumptions imply that b ( T ) >
1. We claimthat if Σ( T ) is a proper, non-void subset of H ( T, R ) \ { } , then there is a proper,non-trivial linear subspace E ⊆ H ( T, R ) invariant under the canonical action of afinite index subgroup D ⊆ D . Granting the claim, we may use the fact that D isZariski dense in S to infer that E ⊗ C ⊆ ( T abf ⊗ C ) ∗ is S -invariant, a contradiction.Thus, we only need to verify the above claim, in order to finish the proof.(1) In this case, we know that Σ( T ) = ∪ ri =1 C i , where each C i is a chamber of anon-void, finite hyperplane arrangement A i in H ( T, R ), and the union is disjoint.By Lemma 7.1, there is a finite index subgroup D ⊆ D such that C · d = C , for any d ∈ D . Consider the supporting hyperplanes of C , that is, the set S consisting ofthose hyperplanes E ⊆ H ( T, R ) with the property that the intersection of E withthe boundary of C has non-void interior in E . Standard arguments show that S is a non-void subset of A ; see [10, Chapter V.1]. Clearly, S is D -invariant, so wemay choose D to be the isotropy group of an element E ∈ S . (2) In the second case, the complement of Σ( T ) is the union of a non-void, finitearrangement A of non-trivial, proper linear subspaces of H ( T, R ). Again by Lemma7.1, A is D -invariant. Clearly, the isotropy group D of E ∈ A satisfies the desiredconditions. (cid:3) Property (1) above is verified by 3-manifold groups (that is, fundamental groupsof compact, connected, differentiable 3-manifolds), according to [7]. Property (2)holds for Kahler groups, as shown by Delzant in [14]. For Torelli groups, Proposition7.3(2) may be improved to obtain Theorem B.
Corollary 7.4.
If the Torelli group T g ( g ≥ ) is a Kahler group, then the Johnsonkernel K g is finitely generated.Proof. We have to show that Σ( T g ) = ∅ , for g ≥
4, assuming the Kahler property.According to [14], the complement of Σ( T g ) is a finite union, [ α f ∗ α H ( C α , R ) , coming from irrational pencils on the compact Kahler manifold M , if T g = π ( M ).More precisely, each f α : M → C α is a holomorphic map onto a smooth compactcurve with χ ( C α ) ≤
0, having connected fibers, and dim R f ∗ α H ( C α , R ) = b ( C α ). If b ( C α ) < b ( T g ), for every α , then we are done. Otherwise, χ ( C α ) <
0, for some α , since b ( T g ) >
2. We infer from definition (4.1) that R ( C α ) = H ( C α , C ) and f ∗ α H ( C α , R ) ⊆ R ( T g ) ∩ H ( T g , R ), which contradicts Theorem 4.4. (cid:3) Example 7.5.
We point out that there exist Kahler groups with R ( G ) = 0 andΣ( G ) = ∅ . In particular, (7.2) implies that in this situation the kernel of thecanonical projection, G ։ G abf , is not finitely generated. At the same time, thecondition on R ( G ) implies that there is no group epimorphism, G ։ π (Σ h ), whenthe genus h is at least 2, by the argument in the proof of Corollary 7.4. We givesuch an example, inspired by a construction of Beauville [5, Example 1.8].Let g be a fixed-point free involution of a 1-connected compact Kahler manifold E . The existence of such an object follows for instance from Serre’s result [43],which guarantees the realizability of finite groups as fundamental groups of smooth,projective complex varieties. Let Z act on the Fermat curve F := { x + y + z =0 } ⊆ CP by g ( x : y : z ) = ( y : x : z ). Set M := F × E/ Z , where the quotientis taken with respect to the diagonal action, and C := F/ Z . Note that M is acompact Kahler manifold, and C is an elliptic curve. Set G := π ( M ).The first projection induces a holomorphic surjection, f : M → C , having con-nected fibers, and 4 multiple fibers (of multiplicity 2). By Delzant [14], the subspace f ∗ H ( C, R ) is contained in the complement of Σ( G ). Since f • : H • C → H • M maybe identified with the inclusion of fixed points, ( H • F ) Z ֒ → ( H • F ⊗ H • E ) Z , and RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 25 b ( E ) = 0, we infer that f induces in cohomology an isomorphism in degree 1 and amonomorphism in degree 2. It follows that Σ( G ) = ∅ and R ( G ) = 0, as asserted.By a similar construction, we may exhibit examples of Kahler groups with arbi-trary (non-zero) even first Betti number, having the property that R ( G ) = 0 andΣ( G ) = H ( G, R ) \ { } . As before, these conditions imply that the kernel of thecanonical projection, G ։ G abf , is not finitely generated, and there is no groupepimorphism, G ։ π (Σ h ), for h ≥ Remark 7.6.
In the proof of Corollary 7.4, our strategy involves two steps. Firstly,group surjections T g ։ π (Σ h ) with h ≥ T g ։ π orb1 (Σ ). Example 7.5 shows that the second step is needed forthe proof of Corollary 7.4. Acknowledgments.
We are grateful to Alex Suciu, for useful discussions at anearly stage of this work. Thanks are also due to the referee, whose suggestionshelped us to improve the exposition.
References
1. T. Akita,
Homological infiniteness of Torelli groups , Topology (2001), no. 2, 213–221. 12. S. Andreadakis, On the automorphisms of free groups and free nilpotent groups , Proc. LondonMath. Soc. (1965), no. 15, 239–268. 1.13. D. Arapura, Geometry of cohomology support loci for local systems
I, J. Algebraic Geom. (1997), no. 3, 563–597. 1.24. M. F. Atiyah, I. G. MacDonald, Introduction to commutative algebra , Addison-Wesley, Read-ing, Massachussetts, 1969. 55. A. Beauville,
Annulation du H pour les fibr´es en droites plats , in: Complex algebraic varieties(Bayreuth, 1990) , pp. 1–15, Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992. 7.56. M. Bestvina, K.-U. Bux, D. Margalit,
The dimension of the Torelli group , J. Amer. Math. Soc. (2010), no. 1, 61–105. 1.17. R. Bieri, W. Neumann, R. Strebel, A geometric invariant of discrete groups , Invent. Math. (1987), no. 3, 451–477. 1.2, 2, 7, 7.2, 78. R. Bieri, B. Renz, Valuations on free resolutions and higher geometric invariants of groups ,Comment. Math. Helvetici (1988), no. 3, 464–497. 2, 79. D. Biss, B. Farb, K g is not finitely generated , Invent. Math. (2006), no. 1, 213–226;erratum (2009), no. 1, 229. 110. N. Bourbaki, Groupes et alg`ebres de Lie, Chapitres 4-6 , Hermann, Paris, 1968. 711. K. S. Brown,
Cohomology of groups , Grad. Texts in Math., vol. 87, Springer-Verlag, NewYork-Berlin, 1982. 4.1, 5, 612. J. A. Carlson, D. Toledo,
Quadratic presentations and nilpotent K¨ahler groups , J. Geom. Anal. (1995), no. 3, 359–377; erratum (1997), no. 3, 511–514. 5.413. P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of K¨ahler manifolds ,Invent. Math. (1975), no. 3, 245–274. 1, 5
14. T. Delzant,
L’invariant de Bieri Neumann Strebel des groupes fondamentaux des vari´et´esk¨ahl´eriennes , Math. Annalen (2010), 119–125. 1.2, 7, 7, 7.5, 7.615. A. Dimca, S. Papadima, A. Suciu,
Alexander polynomials: Essential variables and multiplici-ties , Int. Math. Research Notices vol. (2008), no. 3, Art. ID rnm119, 36 pp. 616. A. Dimca, S. Papadima, A. Suciu,
Topology and geometry of cohomology jump loci , Duke Math.Journal (2009), no. 3, 405–457. 5, 517. W. G. Dwyer, D. Fried,
Homology of free abelian covers.
I, Bull. London Math. Soc. (1987),no. 4, 350–352. 1.2, 2, 618. D. Eisenbud, Commutative algebra with a view towards algebraic geometry , Grad. Texts inMath., vol. 150, Springer-Verlag, New York, 1995. 5, 519. B. Farb,
Some problems on mapping class groups and moduli space , in:
Problems on mappingclass groups and related topics , pp. 11–55, Proc. Sympos. Pure Math., vol. 74, Amer. Math.Soc., Providence, RI, 2006. 120. R. Hain,
Torelli groups and geometry of moduli spaces of curves , in:
Current topics in complexalgebraic geometry (Berkeley, 1992/1993) , pp. 97–143, MSRI Publ., vol. 28, 1995. 1.121. R. Hain,
Infinitesimal presentations of the Torelli groups , J. Amer. Math. Soc. (1997),no. 3, 597–651. 1, 1.2, 2, 3, 3, 4, 4, 4, 522. R. Hain, Finiteness and Torelli spaces , in:
Problems on mapping class groups and relatedtopics , pp. 57–70, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI,2006. 123. J. E. Humphreys,
Introduction to Lie algebras and representation theory , Grad. Texts in Math.,vol. 9, Springer-Verlag, New York, 1972. 3, 4, 4, 4, 424. J. E. Humphreys,
Linear algebraic groups , Grad. Texts in Math., vol. 21, Springer-Verlag, NewYork, 1975. 3, 425. D. Johnson,
An abelian quotient of the mapping class group T g , Math. Ann. (1980), 225–242. 1, 1.1, 2, 3, 4, 426. D. Johnson, The structure of the Torelli group I : A finite set of generators for T , Ann. ofMath. (1983), 423–442. 1.127. D. Johnson, A survey of the Torelli group , Contemp. Math. (1983), 165–179. 1.128. D. Johnson, The structure of the Torelli group II : A characterization of the group generatedby twists on bounding curves , Topology (1985), no. 2, 113–126. 1.129. D. Johnson, The structure of the Torelli group
III : The abelianization of T , Topology (1985), no. 2, 127–144. 1, 1.1, 2, 2, 330. L. A. Lambe, Two exact sequences in rational homotopy theory relating cup products andcommutators , Proc. Amer. Math. Soc. (1986), no. 2, 360–364. 4, 431. M. Laurent, Equations diophantiennes exponentielles , Invent. Math. (1984), no. 2, 299–327.1.2, 2, 632. A. Libgober, First order deformations for rank one local systems with a non-vanishing coho-mology , Topology Appl. (2002), no. 1-2, 159–168. 1.2, 633. A. Macinic,
Cohomology rings and formality properties of nilpotent groups , J. Pure Appl.Algebra (2010), 1818–1826. 5.434. W. S. Massey,
Completion of link modules , Duke Math. J. (1980), no. 2, 399–420. 2, 535. D. McCullough, A. Miller, The genus Torelli group is not finitely generated , Topology Appl. (1986), no. 1, 43–49. 136. S. Morita, Casson’s invariant for homology –spheres and characteristic classes of surfacebundles I, Topology (1989), no. 3, 305–323. 1.1 RITHMETIC SYMMETRY AND FINITENESS PROPERTIES 27
37. S. Papadima, A. Suciu,
Chen Lie algebras , Int. Math. Res. Notices , no. 21, 1057–1086.5, 538. S. Papadima, A. Suciu,
Bieri–Neumann–Strebel–Renz invariants and homology jumping loci ,Proc. London Math. Soc. (2010), no. 3, 795–834. 5.4, 6, 739. S. Papadima, A. Suciu,
Homological finiteness in the Johnson filtration of the automorphismgroup of a free group , arXiv:1011.5292 , to appear in J. Topol., doi:10.1112/jtopol/jts023 Cinqui`eme compl´ement `a l’analysis situs , Rend. Circ. Mat. Palermo (1904),45–110. 141. D. Quillen, Rational homotopy theory , Ann. of Math. (1969), 205–295. 542. M. S. Raghunathan, Discrete subgroups of Lie groups , Ergebnisse der Math., vol. 68, Springer-Verlag, Berlin, 1972. 3, 643. J.-P. Serre,
Sur la topologie des vari´et´es alg´ebriques en charact´eristique p , in: Symposiuminternacional de topolog´ıa algebraica (Mexico City, 1958), pp. 24–53. 7.544. D. Sullivan,
On the intersection ring of compact three manifolds , Topology (1975), 275–277.445. D. Sullivan, Infinitesimal computations in topology , Inst. Hautes ´Etudes Sci. Publ. Math. (1977), 269–331. 1, 2, 5 Institut Universitaire de France et Laboratoire J.A. Dieudonn´e, UMR du CNRS7351, Universit´e de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02,France
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