Topological invariants of groups and Koszul modules
Marian Aprodu, Gavril Farkas, Stefan Papadima, Claudiu Raicu, Jerzy Weyman
aa r X i v : . [ m a t h . G R ] J un TOPOLOGICAL INVARIANTS OF GROUPS AND KOSZUL MODULES
MARIAN APRODU, GAVRIL FARKAS, S¸TEFAN PAPADIMA † , CLAUDIU RAICU,AND JERZY WEYMANA BSTRACT . We provide a uniform vanishing result for the graded components of the fi-nite length Koszul module associated to a subspace K ⊆ V V , as well as a sharp upperbound for its Hilbert function. This purely algebraic statement has interesting appli-cations to the study of a number of invariants associated to finitely generated groups,such as the Alexander invariants, the Chen ranks, or the degree of growth and nilpo-tency class. For instance, we explicitly bound the aforementioned invariants in termsof the first Betti number for the maximal metabelian quotients of (1) the Torelli groupassociated to the moduli space of curves; (2) nilpotent fundamental groups of compactK¨ahler manifolds; (3) the Torelli group of a free group.
1. I
NTRODUCTION
A well-established way of studying the properties of a finitely generated group G isby considering its associated graded Lie algebragr ( G ) := M q ≥ Γ q ( G ) / Γ q +1 ( G ) , defined in terms of its lower central series G = Γ ( G ) ≥ . . . ≥ Γ q ( G ) ≥ Γ q +1 ( G ) ≥ . . . where Γ q +1 ( G ) = [Γ q ( G ) , G ] . The quotients gr q ( G ) = Γ q ( G ) / Γ q +1 ( G ) are finitely generated abelian groups and theirranks φ q ( G ) := rk gr q ( G ) encode fundamental numerical information on G and are ingeneral extremely difficult to compute. For example, Γ ( G ) = G ′ is the commutatorsubgroup of G and gr ( G ) is the abelianization of G , whose rank b ( G ) is the first Bettinumber of G .Following Chen [8], Massey [33], Papadima-Suciu [37] and many others, a more man-ageable task is to pass instead to the metabelian quotient G/G ′′ , where G ′′ = [ G ′ , G ′ ] isthe second commutator subgroup of G , and to consider instead its graded Lie algebragr ( G/G ′′ ) and the corresponding Chen ranks θ q ( G ) := φ q ( G/G ′′ ) = rk gr q ( G/G ′′ ) . One sample application of the results of our work is the following (see Theorem 4.10).
Theorem 1.1.
Let G be the fundamental group of a compact K¨ahler manifold, and suppose itsfirst Betti number is n = b ( G ) ≥ . If G/G ′′ is nilpotent, then θ q ( G ) = 0 for q ≥ n − and θ q ( G ) ≤ (cid:18) n + q − n − (cid:19) · ( n − n − − q ) q , for q = 2 , . . . , n − . S¸tefan Papadima passed away on January 10, 2018.
The first Chen rank always satisfies θ ( G ) = φ ( G ) = b ( G ) , and in general one hasthe inequality φ q ( G ) ≥ θ q ( G ) . For basic properties of these invariants initially consid-ered in knot theory we refer to [37], whereas for applications of Chen ranks to hyper-plane arrangements one can consult [11], [12] and the references therein.It turns out that that in many cases gr ( G/G ′′ ) is an object belonging essentially tocommutative algebra, which can be studied to great effect with homological and algebro-geometric methods. The precise context where these methods are most fruitful is when G is a -formal group in the sense of Sullivan [44]. We recall that for instance fundamen-tal groups of compact K¨ahler manifolds, hyperplane arrangement groups, or the Torelligroup T g of the mapping class group are all known to be -formal. For such groups weprove the following in Section 4.3. Theorem 1.2.
Let G be a finitely generated -formal group, and let M = G/G ′′ denote itsmaximal metabelian quotient. If Γ q ( M ) is finite for q ≫ (in particular if M is nilpotent) andif n = b ( G ) ≥ , then (1) Γ n − ( M ) is finite. (2) M contains a nilpotent finite index subgroup with nilpotency class at most n − . (3) The group M has polynomial growth of degree d ( M ) bounded above by d ( M ) ≤ n + ( n − · (cid:18) n − n − (cid:19) . We recall that the nilpotency class of a nilpotent group H is the largest value of c forwhich Γ c ( H ) = { } , and that a group is virtually nilpotent if it contains a nilpotent finiteindex subgroup. We can then define the virtual nilpotency class (vnc) of a group to bethe minimal nilpotency class of a finite index subgroup, so that the second conclusionof Theorem 1.2 states that the metabelian quotient G/G ′′ has vnc( G/G ′′ ) ≤ n − . Acelebrated theorem of Gromov [22] asserts that virtually nilpotent groups are preciselythe ones that exhibit polynomial growth, and our theorem gives a precise bound forthe degree of growth of M = G/G ′′ solely in terms of the first Betti number of G . Aswe show in Section 4.3, this bound is a consequence of the Bass–Guivarc’h formula andthe estimate on θ q ( G ) as in Theorem 1.1. As explained in Example 4.9 and Remark 4.13,in the absence of -formality (or some appropriate replacement) there can be no suchbound for either vnc( M ) or for d ( M ) .An important special case of Theorem 1.2 concerns the Torelli group T g which mea-sures from a homotopical point of view the difference between the moduli space ofcurves of genus g and that of principally polarized abelian varieties of dimension g . If Mod g denotes the mapping class group, recall that the Torelli group is defined via theexact sequence −→ T g −→ Mod g −→ Sp g ( Z ) −→ . Using Johnson’s fundamental calculation H ( T g , Q ) ∼ = V H Q /H Q , where H is the firstintegral homology of a genus g Riemann surface (and hence rk ( H ) = 2 g ), together withthe recent result by Ershov and He [19, Corollary 1.5] asserting that T g /T ′′ g is nilpo-tent for g ≥ , and the -formality of T g proved in [24], our Theorem 1.2 yields thefollowing consequence (see Section 4.5). OPOLOGICAL INVARIANTS OF GROUPS AND KOSZUL MODULES 3
Theorem 1.3. If g ≥ , the metabelian quotient T g /T ′′ g has virtual nilpotency class at most (cid:18) g (cid:19) − g − . We note that some of the results of [19] have been extended to g ≥ in [9], but thenilpotence of T g /T ′′ g is still open, preventing us from extending the range for g in Theo-rem 1.3.We have similar applications to the Torelli group OA g of the free group F g on g gen-erators, discussed in Section 4.5. We show in Theorem 4.17 that vnc(OA g / OA ′′ g ) ≤ g ( g + 1)( g − − for all g ≥ , which is an analogue of Theorem 1.3. We note that in contrast with the Torelli groups T g ,the groups OA g are not known to be -formal [38, Question 10.6].The proofs of Theorems 1.1 and 1.2 are fundamentally algebraic. In order to make thetransition to commutative algebra and algebraic geometry, for every finitely generatedgroup G we consider its Alexander invariant B ( G ) := H ( G ′ , Z ) = G ′ /G ′′ , viewed as a module over the group ring Z [ G/G ′ ] , with the action being given by con-jugation. Using the augmentation ideal I ⊆ Z [ G/G ′ ] , one then constructs the gradedmodule gr B ( G ) over the ring gr Z [ G/G ′ ] = L q ≥ I q /I q +1 , by settinggr q B ( G ) := I q · B ( G ) /I q +1 · B ( G ) . Massey proves in [33] the existence of canonical isomorphisms Γ q +2 ( G/G ′′ ) ∼ = I q · B ( G ) ,which in particular yield θ q +2 ( G ) = rk gr q B ( G ) . Following [37] and especially [39], it turns out that the graded ring gr B ( G ) can be stud-ied via Koszul cohomology using a purely algebraic construction which we describenext. Koszul modules.
Suppose that V is an n -dimensional complex vector space and fixan m -dimensional subspace K ⊆ V V . We denote by K ⊥ = ( V V /K ) ∨ ⊆ V V ∨ theorthogonal of K . We also denote by S := Sym ( V ) the polynomial algebra over V andconsider the Koszul complex · · · −→ ^ V ⊗ S δ −→ ^ V ⊗ S δ −→ V ⊗ S δ −→ S −→ C −→ . According to [39], the
Koszul module associated to ( V, K ) is the graded S -module W ( V, K ) := Coker n ^ V ⊗ S −→ (cid:16) ^ V /K (cid:17) ⊗ S o , where the map in question is the projection V V ⊗ S → ( V V /K ) ⊗ S composed withthe Koszul differential δ . Our grading conventions are so that W ( V, K ) is generated in M. APRODU, G. FARKAS, S. PAPADIMA, C. RAICU, AND J. WEYMAN degree . Hence, passing to individual graded pieces one has the identification W q ( V, K ) ∼ = ^ V ⊗ Sym q ( V ) / (cid:16) Ker ( δ ,q ) + K ⊗ Sym q ( V ) (cid:17) . If G is a finitely generated group, we define its Koszul module W ( G ) := W (cid:0) H ( G, C ) , K (cid:1) , by setting K ⊥ := Ker n ∪ G : V H ( G, C ) → H ( G, C ) o , or equivalently by letting K bethe dual of Im( ∪ G ) , which embeds naturally into V H ( G, C ) . It is shown in [16] and[37] that the following inequality holds(1) θ q +2 ( G ) = rk gr q B ( G ) ≤ dim W q ( G ) , with equality if the group G is . We prove the following result about W ( G ) . Theorem 1.4.
Let G be a finitely generated group and assume that the cup product map ∪ G : ^ H ( G, C ) → H ( G, C ) does not vanish on any non-zero decomposable elements. If n = b ( G ) ≥ , then (2) W q ( G ) = 0 for all q ≥ n − . If moreover M = G/G ′′ satisfies the condition that Γ q ( M ) be finite for q ≫ , then its virtualnilpotency class and degree of polynomial growth satisfy the inequalities in Theorem 1.2. The vanishing (2) follows from Theorem 4.2, while the inequalities for d ( M ) and vnc( M ) are explained in the proof of Theorem 1.2 (see Remark 4.14). An importantapplication of Theorem 1.4 is when X is a compact K¨ahler manifold and G = π ( X ) , inwhich case H ( G, C ) = H ( X, C ) . In this case, the kernel of the cup-product map ∪ G equals the kernel of the map ∪ X : ^ H ( X, C ) → H ( X, C ) . By a well-known generalization of the Castelnuovo–de Franchis Theorem [6], the con-dition that ∪ X does vanish on decomposable elements amounts to the existence of afibration X → C over a smooth curve of genus g ≥ . Thus, the first part of Theo-rem 1.4 applies to all compact K¨ahler manifolds X with irregularity q ( X ) ≥ whichare not fibred over curves.We obtain the vanishing in Theorem 1.4 as a special instance of a general vanishingresult for Koszul modules, as explained next. We return to the general situation when V is a vector space, ι : K ֒ → V V is a fixed subspace, and K ⊥ := Ker( ι ∨ ) ⊆ V V ∨ isthe space of skew-symmetric bilinear forms on V which vanish identically on K . It isshown in [39, Lemma 2.4] that the support of the Koszul module W ( V, K ) in the affinespace V ∨ coincides (if non-empty) with the resonance variety R ( V, K ) defined as R ( V, K ) := n a ∈ V ∨ | there exists b ∈ V ∨ such that a ∧ b ∈ K ⊥ \ { } o ∪ { } . OPOLOGICAL INVARIANTS OF GROUPS AND KOSZUL MODULES 5
In particular, W ( V, K ) has finite length, that is, W q ( V, K ) = 0 for q ≫ if and only R ( V, K ) = { } . This last condition is equivalent to the fact that the projective sub-space P ( K ⊥ ) ⊆ P ( V V ∨ ) is disjoint from the Grassmann variety Gr ( V ∨ ) in its Pl ¨uckerembedding. Our vanishing result applies uniformly to all subspaces K satisfying thiscondition, as follows (see Theorem 3.1). Main Theorem.
Let V be a complex n -dimensional vector space and let K ⊆ V V be asubspace such that R ( V, K ) = { } . We have that W q ( V, K ) = 0 for all q ≥ n − .In a forthcoming paper, we prove the equivalence between the Main Theorem andGreen’s Vanishing Conjecture for syzygies of generic canonical curves. This leads to analternate approach to Green’s conjecture from the one of Voisin’s [45], [46].The vanishing result in the Main Theorem is responsible for the estimates of thevirtual nilpotency classes that appear in Theorems 1.2, 1.3 and 1.4. In addition to thevanishing result, we also provide a sharp upper bound for the Hilbert function of afinite length Koszul module in Theorem 3.2. In combination with (1) this gives thebound on the Chen ranks in Theorem 1.1, and in conjunction with the Bass–Guivarc’hformula (recalled as formula (29)) it provides the estimate for the degree of polynomialgrowth in Theorem 1.2.To emphasize the robust relationship between algebra and topology in the -formalsetting, we end our Introduction summarizing the connection between some of theinvariants discussed in this setting. Theorem 1.5.
Let G be a finitely generated -formal group, and assume that n = b ( G ) ≥ .The following statements are equivalent: (a) θ q ( G ) = 0 for some (any) q ≫ . (b) θ n − ( G ) = 0 . (c) ∪ G does not vanish on any non-zero decomposable elements. (d) W q ( G ) = 0 for some (any) q ≫ . (e) W n − ( G ) = 0 . The equivalence of (c), (d), (e) comes from the Main Theorem, since all three condi-tions characterize finite dimensional Koszul modules. The equivalence of these state-ments with (a) and (b) follows from the fact that (1) is an equality for -formal groups. Acknowledgment.
Above all, we acknowledge with thanks the contribution of AlexSuciu. This project started with the paper [39], and since then we benefited from nu-merous discussions with him. We also thank A. Beauville, F. Campana, F. Catanese, D.Eisenbud, B. Farb, B. Klingler, P. Pirola, A. Putman and C. Voisin for interesting discus-sions related to this circle of ideas.
Aprodu was partially supported by the Romanian Ministry of Research and Innovation,CNCS - UEFISCDI, grant PN-III-P4-ID-PCE-2016-0030, within PNCDI III. Farkas was supportedby the DFG grant
Syzygien und Moduli . Raicu was supported by the Alfred P. Sloan Foundationand by the NSF Grant No. 1600765. Weyman was partially supported by the Sidney ProfessorialFund and the NSF grant No. 1400740.
M. APRODU, G. FARKAS, S. PAPADIMA, C. RAICU, AND J. WEYMAN
2. K
OSZUL MODULES
Throughout this article, we denote by V a complex vector space of dimension n ≥ ,we write V ∨ = Hom C ( V, C ) for its dual, and S = Sym V for the symmetric algebra of V .We consider the standard grading on S where the elements in V are of degree 1. We fixa subspace K ⊆ V V of dimension m , we denote by ι : K → V V the inclusion andlet K ⊥ := Ker( ι ∨ ) = Coker( ι ) ∨ , where ι ∨ : V V ∨ −→ K ∨ is the dual of ι .2.1. Definitions and basic properties.
We recall from [39] the definition of Koszulmodules. They are defined starting from the classical Koszul differentials δ p : p ^ V ⊗ S → p − ^ V ⊗ S,δ p ( v ∧ · · · ∧ v p ⊗ f ) = p X j =1 ( − j − v ∧ · · · ∧ b v j ∧ · · · ∧ v p ⊗ v j f. Note that we have a decomposition δ p = L q δ p,q , where(3) δ p,q : p ^ V ⊗ Sym q V → p − ^ V ⊗ Sym q +1 V, and that the differentials δ p fit together into the Koszul complex which gives a minimalresolution by free graded S -modules of the residue field C . Definition 2.1 ([39, § . The
Koszul module associated to the pair ( V, K ) is a graded S -module denoted by W ( V, K ) , and defined as(4) W ( V, K ) := Coker n ^ V ⊗ S −→ (cid:16) ^ V /K (cid:17) ⊗ S o where the map V V ⊗ S → ( V V /K ) ⊗ S is the composition of the quotient map V V ⊗ S → ( V V /K ) ⊗ S with the Koszul differential δ . The grading is inherited fromthe symmetric algebra if we make the convention that V V /K is placed in degree and V V is in degree . The degree q component of W ( V, K ) is then given by(5) W q ( V, K ) = Coker n ^ V ⊗ Sym q − V −→ (cid:16) ^ V /K (cid:17) ⊗ Sym q V o . Note that this grading convention guarantees that W ( V, K ) is generated in degree . Inparticular, if W q ( V, K ) = 0 for some q ≥ , then W p ( V, K ) = 0 for all p ≥ q .Since the Koszul complex is exact in homological degree one, we can realize W ( V, K ) as the middle cohomology of the complex(6) K ⊗ S δ | K ⊗ S / / V ⊗ S δ / / S Using the fact that
Ker( δ ,q +1 ) = Im( δ ,q ) , we obtain as in [39, (2.1)](7) W q ( V, K ) = Im( δ ,q ) / Im( δ ,q | K ⊗ S ) . OPOLOGICAL INVARIANTS OF GROUPS AND KOSZUL MODULES 7
This construction is natural in the following sense: for K ⊆ K ′ , the identity map of V V ⊗ S induces a surjective morphism of graded S -modules(8) W ( V, K ) ։ W ( V, K ′ ) . Example 2.2.
In the extremal cases the Koszul modules are easy to compute: • If K = V V then W ( V, K ) = 0 ; • If K = 0 then W q ( V, K ) = H (cid:0) P ( V ∨ ) , Ω P ( V ∨ ) ( q + 2) (cid:1) = 0 for all q ≥ .We will therefore assume when necessary that ≤ m < (cid:0) n (cid:1) . Example 2.3 ([39, § . Any finitely generated group G gives rise to a pair ( V, K ) byletting V := H ( G, C ) ∼ = H ( G, C ) ∨ and K be the the dual of the image of the cup–product map ∪ G : V H ( G, C ) → H ( G, C ) . The corresponding Koszul modules aredenoted W ( G ) . Conversely, it was noted in [39, Proposition 6.2] that every Koszulmodule W ( V, K ) corresponding to a rational subspace K ⊆ V V has the form W ( G ) for some finitely presented group G . A more thorough analysis of the Koszul modulesassociated to groups will be discussed in § Resonance.
Inspired by work of Green and Lazarsfeld in [21], Papadima and Su-ciu defined in [39] resonance varieties in a purely algebraic context as follows.
Definition 2.4.
The resonance variety associated to the pair ( V, K ) is the locus(9) R ( V, K ) := n a ∈ V ∨ | there exists b ∈ V ∨ such that a ∧ b ∈ K ⊥ \ { } o ∪ { } The resonance variety R ( V, K ) is the union of -dimensional subspaces of V ∨ parametrizedby the intersection P ( K ⊥ ) ∩ Gr ( V ∨ ) . More precisely, via the canonical diagram ( P V ∨ × P V ∨ ) \ Diag π / / p (cid:15) (cid:15) Gr ( V ∨ ) P V ∨ R ( V, K ) is the affine cone over p ( π − ( P ( K ⊥ ) ∩ Gr ( V ∨ ))) . It was showed in [39,Lemma 2.4] that resonance and Koszul modules are strongly related: the support ofthe Koszul module in the affine space V ∨ coincides with the resonance variety awayfrom , and hence R ( V, K ) = { } if and only if W ( V, K ) is of finite length. In particular(10) P ( K ⊥ ) ∩ Gr ( V ∨ ) = ∅ ⇐⇒ R ( V, K ) = { } ⇐⇒ dim C W ( V, K ) < ∞ . It was also proved in [39, Proposition 2.10] that there exists a uniform bound q ( n, m ) such that W q ( V, K ) = 0 for all q ≥ q ( n, m ) and for all ( V, K ) satisfying P ( K ⊥ ) ∩ Gr ( V ∨ ) = ∅ . A central goal of our paper is to determine this bound, which we doin Theorem 3.1. For dimension reasons, if P ( K ⊥ ) ∩ Gr ( V ∨ ) = ∅ then m ≥ n − . The borderline case m = 2 n − is therefore of maximal interest, as illustrated for instance in § M. APRODU, G. FARKAS, S. PAPADIMA, C. RAICU, AND J. WEYMAN
3. F
INITE LENGTH K OSZUL MODULES
The goal of this section is to prove the main theorem of our paper, characterizingfinite dimensional Koszul modules (those for which the equivalent conditions in (10)hold) in terms of the vanishing of a fixed graded component. More precisely, we prove:
Theorem 3.1.
Let V be a vector space of dimension n ≥ and let K ⊆ V V be an arbitrarysubspace. We have that (11) R ( V, K ) = { } ⇐⇒ W n − ( V, K ) = 0 . The assumption that n ≥ in the above theorem is made in order to avoid trivialities:when n = 1 we have V V = 0 , while for n = 2 we have that either K = V V or K = 0 , both of which have been discussed in Example 2.2. In addition to the vanishingresult above, in the borderline case m = 2 n − we give an exact description of theHilbert series of the Koszul modules with vanishing resonance, which in turn providesan upper bound for such Hilbert series in the general case m ≥ n − , as follows. Theorem 3.2. If R ( V, K ) = { } , then for q = 0 , · · · , n − , we have that dim W q ( V, K ) ≤ (cid:18) n + q − q (cid:19) ( n − n − q − q + 2 , and equality holds for all ≤ q ≤ n − when dim( K ) = 2 n − . We prove Theorem 3.1 using Bott’s theorem and a hypercohomology spectral se-quence that was featured in the work of Voisin on the Green conjecture [45]. We recallthe special case of Bott’s theorem needed for our argument in Section 3.1, and prove thevanishing theorem in Section 3.2. The proof of Theorem 3.2 is presented in Section 3.3,and is summarized as follows. We reinterpret the characterization (11) in terms of atransversality condition on K with respect to the kernel of the Koszul differential δ ,which allows us to prove the conclusion of Theorem 3.2 in the borderline case. Thegeneral case of Theorem 3.2 reduces to the borderline case via a generic projection ar-gument.3.1. Bott’s Theorem for Grassmannians.
Our main reference for this topic is [47, Ch. 4].Let G = Gr ( V ∨ ) denote the Grassmannian of –dimensional subspaces of V ∨ , whichwe think of by duality as parametrizing -dimensional quotients of V . On G we thenhave the tautological exact sequence(12) −→ U −→ V ⊗ O G −→ Q −→ , where U (respectively Q ) denotes the universal rank ( n − sub–bundle (respectivelyrank quotient bundle) of the trivial bundle V . We write O G (1) for V Q , which is thePl ¨ucker line bundle giving an embedding of G into P ( V V ∨ ) as the projectivization ofthe set of decomposable -forms a ∧ b , with a, b ∈ V ∨ .We write Z r dom for the set of dominant weights in Z r , that is, tuples ν = ( ν , . . . , ν r ) ∈ Z r with ν ≥ ν ≥ . . . ≥ ν r . We write S ν for the Schur functor associated with ν , and recallthat if ν = ( a, , . . . , then S ν = Sym a , and if ν = (1 , , . . . , then S ν = V r . A specialcase that will be of interest to us is when α = ( α , α ) ∈ Z , in which case we have(13) S α Q = Sym α − α Q ⊗ O G ( α ) . OPOLOGICAL INVARIANTS OF GROUPS AND KOSZUL MODULES 9
The cohomology groups of sheaves of the form S α Q ⊗ S β U are completely described byBott’s Theorem [47, Corollary 4.1.9]. We will only need a special case of this theorem,which we record next. Theorem 3.3 (Bott) . Let α ∈ Z and β ∈ Z n − , and let γ = ( α | β ) ∈ Z n denote theirconcatenation. If γ is dominant then (14) H ( G , S α Q ⊗ S β U ) = S γ V, and H j ( G , S α Q ⊗ S β U ) = 0 for all j > . If there exist ≤ x < y ≤ n with γ x − x = γ y − y , then (15) H j ( G , S α Q ⊗ S β U ) = 0 , for all j ≥ . In particular, (1) If α ≥ α ≥ , then H ( G , S α Q ) = S α V and H j ( G , S α Q ) = 0 for all j > . (2) For all a, j ≥ we have that H j ( G , Sym a Q ⊗ U ) = 0 . (3) If − ≥ α ≥ − n , or − ≥ α ≥ − n , then H j ( G , S α Q ) = 0 for all j ≥ . To see how (1) and (3) follow from the general statement of the theorem we specializeto β = (0 , , . . . , , so that γ = ( α , α , , . . . , . In case (1) we have that γ is dominantso we can apply (14), while in case (3) we apply (15) with ( x, y ) = (1 , − α ) when − ≥ α ≥ − n , and with ( x, y ) = (2 , − α ) when − ≥ α ≥ − n . To see how (2)follows from (15) we note that α = ( a, , β = (1 , , . . . , , and take ( x, y ) = (2 , .3.2. Vanishing for Koszul modules.
We begin by giving a geometric construction of
Ker( δ ) , where δ : V ⊗ S −→ S is the first Koszul differential. We let G = Gr ( V ∨ ) asin the previous section, and consider S = Sym O G ( Q ) = O G ⊕ Q ⊕ Sym Q ⊕ · · · as a sheaf of graded O G -algebras on G . Locally on G , the sheaf S can be identified witha polynomial ring in two variables, where Q is the space of linear forms. We can thendefine the Koszul differentials δ Q : Q ⊗ O G S −→ S and δ Q : ^ Q ⊗ O G S −→ Q ⊗ O G S as in Section 2.1. Recalling that V Q = O G (1) , we write V Q ⊗ O G S simply as S (1) . Lemma 3.4.
We have a commutative diagram of graded S –modules (16) V V ⊗ S δ / / f (cid:15) (cid:15) V ⊗ S δ / / f (cid:15) (cid:15) S f (cid:15) (cid:15) / / H ( G , S (1)) H ( G ,δ Q ) / / H ( G , Q ⊗ S ) H ( G ,δ Q ) / / H ( G , S ) where the maps f and f are isomorphisms. Moreover, we have a natural isomorphism between Ker( δ ) and H ( G , S (1)) .Proof. If we identify S ⊗ O G with Sym O G ( V ⊗ O G ) then the tautological quotient map V ⊗ O G ։ Q in (12) yields a natural surjection S ⊗ O G ։ S which can be thought of locally on G as realizing the polynomial ring S in two variablesas a quotient of S by the ideal generated by the linear forms in U . The naturality of theKoszul complex yields a commutative diagram V V ⊗ S ⊗ O G / / (cid:15) (cid:15) V ⊗ S ⊗ O G / / (cid:15) (cid:15) S ⊗ O G (cid:15) (cid:15) / / V Q ⊗ S / / Q ⊗ S / / S where the bottom row is the full Koszul complex since Q has rank . The diagram(16) is then obtained by applying the global sections functor H ( G , • ) . It is sufficientto prove that f and f are isomorphisms, since this implies that f induces an isomor-phism between Ker( δ ) and Ker( H ( G , δ Q )) , and the latter is in turn is isomorphic to H ( G , S (1)) via the map H ( G , δ Q ) .To prove that f is an isomorphism it suffices to prove it is injective, since for each q we have that H ( G , Sym q Q ) = Sym q V by conclusion (1) of Theorem 3.3. If we let Y = Spec G ( S ) denote the total space of the bundle Q ∨ (see [26, Exercise II.5.18]), then Y is a subbundle of the trivial bundle V ∨ over G , which can be described explicitly as Y = n ( c, [ a ∧ b ]) ∈ V ∨ × G : c ∈ Span ( a, b ) o If we write π : Y → V ∨ for the natural projection map, then π is surjective and there-fore the induced pull-back map f : S −→ H ( V ∨ , π ∗ O Y ) = H ( G , S ) is injective, asdesired.To prove that f is an isomorphism, we observe that it factors as the composition of V ⊗ S id V ⊗ f −→ V ⊗ H ( G , S ) = H ( G , V ⊗ S ) , which is an isomorphism, with the map H ( G , V ⊗ S ) −→ H ( G , Q ⊗ S ) induced bythe tautological surjection V ⊗ O G ։ Q . To prove that this map is an isomorphism, itis enough to check that H ( G , U ⊗
Sym q Q ) = H ( G , U ⊗
Sym q Q ) = 0 for all q ≥ , which follows from conclusion (2) of Theorem 3.3. (cid:3) We next recall that W ( V, K ) can be thought of as the middle cohomology of (6), orequivalently as W ( V, K ) = Coker n K ⊗ S δ | K ⊗ S / / Ker( δ ) o ∼ = Coker n K ⊗ S f | K ⊗ S / / H ( G , S (1)) o where the isomorphism above follows from Lemma 3.4. Using again the isomorphism f : S −→ H ( G , S ) and the fact that K ⊗ H ( G , S ) = H ( G , K ⊗ S ) , we can rewrite(17) W ( V, K ) = Coker n H ( G , K ⊗ S ) H ( G ,η ) / / H ( G , S (1)) o OPOLOGICAL INVARIANTS OF GROUPS AND KOSZUL MODULES 11 where(18) η : K ⊗ S −→ S (1) is induced by K ⊗ O G ֒ → ^ V ⊗ O G ։ ^ Q = O G (1) . Equipped with the description (17) of the Koszul modules, we can now give the proofof our main theorem.
Proof of Theorem 3.1. If W n − ( V, K ) = 0 , then dim W ( V, K ) < ∞ , since the graded mod-ule W ( V, K ) is generated in degree . It follows from (10) that R ( V, K ) = { } .Conversely, suppose that R ( V, K ) = { } , which by (10) is equivalent to the fact that P ( K ⊥ ) is disjoint from the Grassmannian G , which in turn is equivalent to the fact thatthe linear system of divisors ( K, O G (1)) is base point free, or to the fact that the map K ⊗ O G −→ O G (1) is surjective. The Koszul complex associated to this map is thereforeexact, and takes the form K • : 0 −→ m ^ K ⊗ O G (1 − m ) −→ · · · −→ ^ K ⊗ O G ( − −→ K ⊗ O G −→ O G (1) −→ where we use a cohomological grading with K − i = V i K ⊗ O G (1 − i ) . Tensoring with Sym n − Q preserves exactness, so the hypercohomology of K • ⊗ Sym n − Q vanishesidentically. The hypercohomology spectral sequence associated with K • ⊗ Sym n − Q then yields that E − i,j = H j (cid:16) G , i ^ K ⊗ O G (1 − i ) ⊗ Sym n − Q (cid:17) = ⇒ . Since
Sym n − Q ⊗ O G (1 − i ) = S ( n − − i, − i ) Q by (13), we can rewrite the terms in thespectral sequence as(19) E − i,j = H j (cid:16) G , i ^ K ⊗ S ( n − − i, − i ) Q (cid:17) = i ^ K ⊗ H j (cid:0) G , S ( n − − i, − i ) Q (cid:1) . We have using (17) the following identification W n − ( V, K ) = Coker (cid:8) E − , −→ E , (cid:9) and we suppose by contradiction that this map is not surjective. Since E , ∞ = 0 , theremust be some non-zero differential E − r,r − r −→ E , r for r ≥ , or E , r −→ E r, − rr for r ≥ . Since E r, − r = 0 for all r ≥ it follows that E r, − rr = 0 as well, so the latter case does notoccur. To prove that the former case does not occur either and obtain a contradiction, itsuffices to check that E − r,r − = 0 for all r ≥ , which follows from (19) if we can provethat(20) H r − (cid:0) G , S ( n − − r, − r ) Q (cid:1) = 0 for r ≥ . If we write α = ( n − − r, − r ) then for ≤ r ≤ n − we have that − ≥ α ≥ − n ,and for n ≤ r ≤ n − we have that − ≥ α ≥ − n , so (20) follows from conclusion(3) of Theorem 3.3. When r > n − we get that r − > n − G ) , so (20) is aconsequence of Grothendieck’s vanishing theorem [26, Theorem III.2.7]. (cid:3) The Hilbert series of Koszul modules.
The goal of this section is to prove Theo-rem 3.2. Recall from equation (7) that W q ( V, K ) is obtained as the cokernel of the map(21) δ ,q : K ⊗ Sym q V −→ Im( δ ,q ) and in particular we have that W q ( V, K ) = 0 when dim(Im( δ ,q )) > dim( K ⊗ Sym q V ) .We say that the divisorial case occurs when the map (21) is given by a square matrix, i.e. dim ( K ⊗ Sym q V ) = dim (Im( δ ,q )) . Proposition 3.5.
Let K ⊆ V V be an arbitrary subspace of dimension m = 2 n − . For any ≤ q ≤ n − we have W q ( V, K ) = 0 , and q = n − is a divisorial case.Proof. Using the fact that dim Sym k V = (cid:0) n + k − k (cid:1) and dim( K ) = 2 n − we obtain dim ( K ⊗ Sym q V ) = (2 n − (cid:18) n + q − q (cid:19) . The exactness of the Koszul complex implies that δ ,q +1 is surjective and that we havethe equality Im( δ ,q ) = Ker( δ ,q +1 ) , so dim(Im( δ ,q )) = dim ( V ⊗ Sym q +1 V ) − dim (Sym q +2 V ) = n · (cid:18) n + qq + 1 (cid:19) − (cid:18) n + q + 1 q + 2 (cid:19) , A direct computation leads for q ≥ to the formula(22) dim(Im( δ ,q )) − dim ( K ⊗ Sym q V ) = (cid:18) n + q − q (cid:19) ( n − n − q − q + 2 , which is positive for ≤ q ≤ n − and vanishes for q = n − , proving that W q ( V, K ) = 0 for ≤ q ≤ n − , and that the case q = n − is divisorial. (cid:3) We next analyze in more detail the divisorial case, where the vanishing of W n − ( V, K ) can be rephrased into a transversality condition as follows. Lemma 3.6.
Let K ⊆ V V be a subspace of dimension n − . The following are equivalent: (a) W n − ( V, K ) = 0 . (b) ( K ⊗ Sym n − V ) ∩ Ker( δ ,n − ) = 0 . (c) ( K ⊗ Sym q V ) ∩ Ker( δ ,q ) = 0 for all q ≤ n − .Proof. It is clear that (c) implies (b), while for the reverse implication it suffices to ob-serve that multiplication by a linear form in V yields for each q ≥ an injective mapfrom ( K ⊗ Sym q V ) ∩ Ker( δ ,q ) to ( K ⊗ Sym q +1 V ) ∩ Ker( δ ,q +1 ) .Using the isomorphism Im( δ ,n − ) = ( V V ⊗ Sym n − V ) / Ker( δ ,n − ) and (7) we get(23) W n − ( V, K ) = (cid:0) ^ V ⊗ Sym n − V (cid:1) / (cid:0) K ⊗ Sym n − V + Ker( δ ,n − ) (cid:1) . Since we are in the divisorial case we have dim(Im( δ ,n − )) = dim( K ⊗ Sym n − V ) , thus(24) dim( ^ V ⊗ Sym n − V ) = dim( K ⊗ Sym n − V ) + dim(Ker( δ ,n − )) . Combining (23) with (24) we conclude that W n − ( V, K ) = 0 if and only if the sum K ⊗ Sym n − V + Ker( δ ,n − ) is direct, proving the equivalence of (a) and (b). (cid:3) OPOLOGICAL INVARIANTS OF GROUPS AND KOSZUL MODULES 13
Corollary 3.7. If K ⊆ V V has dimension n − and R ( V, K ) = { } , then the map (21) isof maximal rank for all q .Proof. Using (22), it suffices to prove that (21) is injective for q ≤ n − and surjectivefor q ≥ n − . Since R ( V, K ) = { } , we know from Theorem 3.1 that W n − ( V, K ) = 0 ,hence the assertion in Lemma 3.6(c) holds, proving the injectivity of (21) in the range q ≤ n − . The surjectivity of (21) is equivalent to the vanishing of W q ( V, K ) , which istrue for q ≥ n − because W n − ( V, K ) = 0 and W ( V, K ) is generated in degree . (cid:3) We now have all the tools necessary to prove Theorem 3.2.
Proof of Theorem 3.2.
We write m = dim( K ) as usual, and suppose first that m = 2 n − .By Corollary 3.7 we know that the maps (21) have maximal rank for all q . Since theircokernels are the graded components of W ( V, K ) , it follows from (22) that dim W q ( V, K ) = (cid:18) n + q − q (cid:19) ( n − n − q − q + 2 for all q = 0 , · · · , n − . We next consider the case m > n − . If we set G := Gr ( V ∨ ) ⊆ P ( V V ∨ ) then (10)implies that P ( K ⊥ ) ∩ G = ∅ . We consider the linear projection away from P ( K ⊥ ) π : P ( ^ V ∨ ) P m − Since dim( G ) = 2 n − it follows that π ( G ) is a subvariety of P m − of codimension atleast m − − (2 n −
4) = m − n + 3 , hence a general linear subspace P m − n +2 ⊂ P m − is disjoint from π ( G ) , i.e. it satisfies π − ( P m − n +2 ) ∩ G = ∅ . Since π − ( P m − n +2 ) isa linear subspace of P ( V V ∨ ) of codimension n − containing P ( K ⊥ ) , we can write π − ( P m − n +2 ) = P ( K ′⊥ ) for some K ′ ⊆ K with dim( K ′ ) = 2 n − . It follows from (8)(with the roles of K and K ′ exchanged) that dim W q ( V, K ) ≤ dim W q ( V, K ′ ) for all q ,which combined with the case m = 2 n − proves the desired inequality and completesour proof. (cid:3) Remark 3.8.
A special case of Theorem 3.2 is obtained by taking V = Sym n − C and K = Sym n − C , in which case W ( V, K ) is the Weyman module W ( n − (see [39,Section 5] and [18, Section 3.I.B]). It follows that Theorem 3.2 computes the Hilbertseries of the said module, answering a problem left open in [39, Section 1.2]. For n ≤ ,Alex Suciu used a computer search to construct subspaces K n ⊆ V C n with m = 2 n − and R ( C n , K n ) = { } . He also computed the Hilbert series of the Koszul modules W ( C n , K n ) . He conjectured the existence of a family of subspaces K n ⊆ V C n with m = 2 n − and R ( C n , K n ) = { } , for all values of n , with Hilbert series given by theformula from Theorem 3.2. Our results prove that the expectations were correct.4. T OPOLOGICAL INVARIANTS OF GROUPS
In this section we explain the topological motivation for investigating Koszul mod-ules, and derive a number of consequences of our results to the study of invariantsof fundamental groups. Some of our main applications concern groups G that are -formal (in the sense of Sullivan [44]), where a direct relationship between certain in-variants of G and those of the corresponding Koszul modules can be derived. The class of -formal groups contains important examples such as the fundamental groups ofcompact K¨ahler manifolds, hyperplane arrangement groups, or the Torelli group of themapping class group of a genus g surface.4.1. The Koszul module and the resonance variety of a group G.
We recall from Ex-ample 2.3 the construction of the
Koszul module W ( G ) associated to a finitely generatedgroup G . The module W ( G ) := W ( V, K ) is obtained by setting V := H ( G, C ) , andletting K ⊥ ⊆ V V ∨ be the kernel of the cup-product map ∪ G : ^ H ( G, C ) → H ( G, C ) . To the Koszul module W ( G ) we can associate its resonance variety R ( G ) as in Defini-tion 2.4. Alternatively, we can define R ( G ) without reference to Koszul modules asfollows. For an element a ∈ H ( G, C ) , we consider the complex of abelian groups ( G, a ) : 0 −→ H ( G, C ) ∪ a −→ H ( G, C ) ∪ a −→ H ( G, C ) ∪ a −→ · · · . The resonance variety R ( G ) is then obtained as R ( G ) = (cid:8) a ∈ H ( G, C ) : H ( G, a ) = 0 (cid:9) . In the -formal context, R ( G ) has yet another interpretation given as follows. Remark 4.1.
For a character ρ ∈ b G := Hom ( G, C ∗ ) , we denote by C ρ the C [ G ] -module C with multiplication given by g · z := ρ ( g ) z . The resonance variety R ( G ) turns out tobe an infinitesimal analogue of the much studied characteristic variety V ( G ) := (cid:8) ρ ∈ b G : H ( G, C ρ ) = 0 (cid:9) , which already appeared in the work of Green–Lazarsfeld [20], Beauville [3], or Libgober[30]. It is shown in [17, Theorems A and C] that when G is -formal, R ( G ) is isomorphicto the tangent cone at of the characteristic variety V ( G ) .Reinterpreted in the context of groups, our Theorems 3.1 and 3.2 imply the following.We write b ( G ) := dim H ( G, C ) for the first Betti number of the group G . Theorem 4.2.
Let G be a finitely generated group and let n = b ( G ) . If n ≥ then we havethe equivalence R ( G ) = { } ⇐⇒ W n − ( G ) = 0 . Moreover, if the equivalent statements above hold then dim W q ( G ) ≤ (cid:18) n + q − q (cid:19) ( n − n − q − q + 2 for ≤ q ≤ n − . This theorem can be used to obtain upper bounds on the Chen ranks of a group G with vanishing resonance, once we establish a relationship between the said ranks andthe Hilbert function of W ( G ) . This is achieved via the study of the Alexander invariantof G in the next section. OPOLOGICAL INVARIANTS OF GROUPS AND KOSZUL MODULES 15
Alexander invariants.
Alexander-type invariants have a long history in topology,starting with the definition of the Alexander polynomial of knots and links [1]. For aconnected CW–complex X with fundamental group G = π ( X ) , each surjective grouphomomorphism α : G → H gives rise to a sequence of Alexander-type invariants of X ,namely the twisted homology groups H i ( X, L ) , where L is the local system C [ H ] . Byconstruction, these invariants are C [ G ] –modules. When i = 1 the construction dependssolely on G , is purely algebraic and can be carried out for any group. The classicalAlexander invariant of a knot Y ⊆ S corresponds to the case when X := S \ Y is theknot complement, i = 1 and α : G → Z is the abelianization map. Definition 4.3. If G is a finitely generated group, its Alexander invariant is defined as B ( G ) := H ( G ′ , Z ) = G ′ /G ′′ , viewed as a module over the group ring Z [ G ab ] , where G ab := G/G ′ , the action being given by conjugation. We write B ( G ) C := B ( G ) ⊗ Z C .Geometrically, if X is a CW–complex with π ( X ) ∼ = G and X ab → X is the maximalabelian cover corresponding to the group H ( X, Z ) , then B ( G ) can be identified with H ( X ′ , Z ) , viewed as a module over G ab ∼ = H ( X, Z ) .The Alexander invariant is equipped with the I –adic filtration, where I := Ker ( ǫ ) is the kernel of the augmentation map ǫ : Z [ G/G ′ ] → Z . A key result of Massey [33]asserts that for each q ≥ , one has an isomorphism(25) gr q B ( G ) := gr Iq B ( G ) ∼ = gr q +2 ( G/G ′′ ) . In particular, the generating series of the Chen ranks of G is given as a Hilbert series ∞ X q =0 θ q +2 ( G ) t q = Hilb (cid:0) gr B ( G ) C , t (cid:1) . Remark 4.4.
The Chen ranks are usually more easily computable than the largely inac-cessible lower central ranks of a group G . An instance of this is the case of hyperplanearrangement groups associated to an arrangement A = { H , . . . , H n } of hyperplanes in C m . Whereas the fundamental group G ( A ) := π (cid:0) C m − ∪ H ∈A H (cid:1) of the arrangementis not determined by the intersection lattice of A , the lower central series and the cor-responding ranks φ q ( G ( A )) and θ q (cid:0) G ( A )) are, and in particular they are combinatorialobjects. An explicit formula for φ q ( G ( A )) remains elusive, but the Chen ranks are sub-ject to a conjecture of Suciu, proven in many cases in [40] and in full generality in [12].It asserts that for q ≫ , one has the formula θ q (cid:0) G ( A ) (cid:1) = X r ≥ h r θ q ( F r +1 ) , where h r is the number of r -dimensional components of the resonance variety R ( G ( A )) and the Chen ranks of the free group are given by (see [8]) θ ( F r +1 ) = r + 1 and θ q ( F r +1 ) = ( q − (cid:18) q + r − q (cid:19) for q ≥ . The Alexander invariants of G exhibit the same type of behaviour as their cousins,the topological jump loci V ( X ) := V ( π ( X )) from Remark 4.1. They are complicated objects encoding useful information, with complexity bounded by invariants depend-ing on the cohomology ring. Work from [16, 33, 37, 38, 43] implies the existence of anatural surjection W ( G ) ։ gr B ( G ) C . In particular, for all q ≥ ,(26) dim gr q B ( G ) C ≤ dim W q ( G ) , with equality when the group G is –formal [39, Corollary 6.1].One should regard the relation between gr B ( G ) C and W ( G ) as a commutative ana-logue of the relationship between the Lie algebra gr ( G ) C and the holonomy Lie algebraof G , whose definition we recall next. Denoting by ∂ G : V H ( G, C ) → H ( G, C ) thedual of the cup product map ∪ G , one defines following Chen [8] the holonomy Lie algebra H ( G ) := L (cid:0) H ( G, C ) (cid:1) / Ideal (cid:0) Im ( ∂ G ) (cid:1) , where L (cid:0) H ( G, C ) (cid:1) is the free complex Lie algebra generated by H ( G, C ) . Work of Sul-livan [44] implies that for a -formal group G one has an isomorphism of Lie algebrasgr ( G ) C ∼ = H ( G ) , that is, the entire lower central series of G is determined by the first cup-product map.In the following applications we relate the I –adic filtrations of the Alexander invari-ant B ( G ) of G to the cohomology ring of G in low degrees in a more precise way. Theorem 4.5.
Let G be a finitely generated group with b ( G ) ≥ . If R ( G ) = { } , then (cid:0) I q · B ( G ) (cid:1) C = (cid:0) I q +1 · B ( G ) (cid:1) C for all q ≥ b ( G ) − .Proof. Combine Theorem 4.2 with (26). (cid:3)
We recall that a module M over a group ring is nilpotent if I q · M = 0 for some q ,where I is the augmentation ideal. Corollary 4.6. If R ( G ) = { } and B ( G ) is nilpotent, then (cid:0) I b ( G ) − · B ( G ) (cid:1) C = 0 . Ifmoreover G is –formal, then dim B ( G ) C = dim W ( G ) < ∞ . In general, the Alexander invariant of a group may not be nilpotent as a module, butthis is the case when the group itself is nilpotent. Recall that a group G is said to benilpotent if there exists c ≥ with Γ c ( G ) = { } . The largest c such that Γ c ( G ) = { } iscalled the nilpotency class of G , and is denoted nc( G ) . A group G is virtually nilpotent ifthere exists a finite index nilpotent subgroup H ≤ G . We define the virtual nilpotencyclass vnc( G ) of G to be the following quantity(27) vnc( G ) := min (cid:8) nc( H ) : H ≤ G is a nilpotent finite index subgroup (cid:9) . Since every virtually nilpotent group contains a torsion free finite index subgroup, andsince the nilpotency class of a torsion free nilpotent group equals that of any of its finiteindex subgroups, it follows that for a virtually nilpotent group G we have(28) vnc( G ) = nc( H ) for any torsion free nilpotent subgroup H ≤ G of finite index . One useful bound for the virtual nilpotency class of G is obtained as follows. OPOLOGICAL INVARIANTS OF GROUPS AND KOSZUL MODULES 17
Lemma 4.7. If G contains a torsion-free finite index subgroup H and Γ c ( G ) is finite, then vnc( G ) = nc( H ) ≤ c − .Proof. Since Γ c ( H ) ≤ Γ c ( G ) and Γ c ( H ) is torsion-free, it follows that Γ c ( H ) = { } andtherefore nc( H ) ≤ c − . The equality vnc( G ) = nc( H ) follows from (28). (cid:3) Corollary 4.8. If G is nilpotent and R ( G ) = { } , then (cid:0) I b ( G ) − · B ( G ) (cid:1) C = 0 . It followsthat vnc( G/G ′′ ) ≤ b ( G ) − .Proof. Since G is nilpotent, the Alexander invariant is also nilpotent so the conclusion (cid:0) I b ( G ) − · B ( G ) (cid:1) C = 0 follows from Corollary 4.6. It follows moreover from (25) and(26) that θ q ( G ) = 0 for q ≥ b ( G ) − . Since G/G ′′ is nilpotent, we must have that θ q ( G ) is finite for q ≥ b ( G ) − , so vnc( G/G ′′ ) ≤ b ( G ) − by Lemma 4.7. (cid:3) The conclusion of Corollary 4.8 does require the vanishing resonance assumption.
Example 4.9.
Indeed, let F n be the free group on n ≥ generators, and considerthe nilpotent group G := F n / Γ k ( F n ) , where k ≥ . As explained for instance in[35, Remark 2.4], the resonance variety of a group depends only on its third nilpo-tent quotient. Since G/ Γ ( G ) ∼ = F n / Γ ( F n ) , it follows that R ( G ) ∼ = R ( F n ) ∼ = C n . Notealso that b ( G ) = n . On the other hand, as already pointed out, the Hilbert series P q ≥ θ q ( F n ) t q was computed by Chen [8] and all of its coefficients are strictly positive.Since θ q ( F n ) = θ q ( G ) for q < k , it follows from (25) that I q · B ( G ) = 0 for q < k − , thusthe nilpotency class cannot be bounded solely in terms of n = b ( G ) .In fact the resonance variety of a nilpotent group of nilpotency class can be arbi-trarily complicated, cf. [35, Remark 2.4]. However, this phenomenon does not occurif we assume -formality. The resonance variety of a finitely generated, nilpotent, -formal group G must vanish, see [5, Lemma 2.4] and the proof of Theorem 4.10 below.Theorem 1.1 in the Introduction is the special case when G is a K¨ahler group of thefollowing. Theorem 4.10.
Let G be a finitely generated -formal group, and suppose its first Betti numberis n = b ( G ) ≥ . If G/G ′′ is nilpotent, then θ q ( G ) = 0 for q ≥ n − and θ q ( G ) ≤ (cid:18) n + q − n − (cid:19) · ( n − n − − q ) q for q = 2 , . . . , n − . Proof.
The condition that
G/G ′′ be nilpotent implies that θ q ( G ) = φ q ( G/G ′′ ) = 0 for q ≫ . Since G is -formal, we have using (25) and (26) that dim W q ( G ) = θ q +2 ( G ) forall q ≥ . In particular W ( G ) is a finite dimensional module, hence R ( G ) = { } andTheorem 4.2 applies to give the desired conclusions about θ q ( G ) = dim W q − ( G ) . (cid:3) We proved in Theorem 3.1 that W n − ( V, K ) = 0 when R ( V, K ) = { } , and therefore W q ( V, K ) = 0 for all q ≥ n − . In the borderline case m = 2 n − , we infer fromProposition 3.5 that this is the best possible vanishing result for Koszul modules. Inthe general case, the following examples coming from nilpotent groups show that thebound q ( n, m ) = n − may be improved, at least in some cases. Example 4.11.
For k ≥ , we denote by H k the fundamental group of the Heisenbergnilmanifold of dimension k + 1 . It is well-known that H k is finitely generated and Γ ( H k ) = { } . We set V := H ( H k , C ) and denote by K ⊥ ⊆ V V ∨ the kernel ofthe cup-product map as before. If k ≥ , we know from [34] that H k is –formal and R ( H k ) = { } . In this case, b ( H k ) = 2 k and K ⊆ V V has codimension . The fact that Γ ( H k ) = { } and (26) together imply that W ( H k ) = 0 , but note that < b ( H k ) − assoon as k > .4.3. A bound on the degree of growth of a group.
The goal of this section is to providea proof of Theorem 1.2. Suppose G is a finitely generated group and let S ⊆ G be a finiteset of generators. For an integer m ≥ , we denote by ℓ G ( m ) = ℓ G,S ( m ) the numberof elements g ∈ G expressible as a product of m elements from S ∪ S − . Bass [2]and Guivarc’h [23] showed that a finitely generated nilpotent group G has polynomialgrowth. More precisely, the inequality ℓ G ( m ) ≤ Cm d ( G ) holds for m ≫ , where d ( G ) is the degree of polynomial growth of the group, which in the case of nilpotent groups isgiven by the formula(29) d ( N ) = X q ≥ q · φ q ( N ) . We also recall a celebrated theorem of Gromov [22] which asserts that conversely, thevirtually nilpotent groups are precisely the ones that have polynomial growth. If G ismerely virtually nilpotent, the Bass-Guivarc’h formula (29) fails as seen for instance bytaking G to be the infinite dihedral group, where d ( G ) = 1 , whereas φ q ( G ) = 0 for all q .For a quantitative version of Gromov’s result, we refer to [31, Theorem 4].Before stating the following result, we recall that the Hirsch index of a finitely gen-erated nilpotent (or more generally polycyclic) group G is defined to be the quantity h ( G ) := P q ≥ φ q ( G ) , see [42, Chapter 1]. It is well-known that for a normal subgroup N E G one has the additivity property h ( G ) = h ( N ) + h ( G/N ) . Lemma 4.12.
Let G be a finitely generated group, and H be a finite index subgroup of G . Forevery k ≥ we have that (30) k X q =1 φ q ( H ) ≥ k X q =1 φ q ( G ) . Moreover, if [ H ∩ Γ k +1 ( G ) : Γ k +1 ( H )] < ∞ , then the above inequality is in fact an equality.Proof. The quotient H := H/ Γ k +1 ( H ) is nilpotent and its Hirsch number is given by h ( H ) = k X q =1 φ q ( H ) . A similar formula holds for G := G/ Γ k +1 ( G ) . Since H ≤ G , we must also have that Γ k +1 ( H ) ≤ Γ k +1 ( G ) , and therefore we obtain an induced group homomorphism ψ : H −→ G. Since [ G : H ] < ∞ , the image of ψ has finite index in G , therefore h (Im ψ ) = h ( G ) . Since Im ψ is a quotient of H , it follows that h ( H ) = h (Ker ψ ) + h (Im ψ ) ≥ h (Im ψ ) , which OPOLOGICAL INVARIANTS OF GROUPS AND KOSZUL MODULES 19 proves that h ( H ) ≥ h ( G ) . If we assume that Γ k +1 ( H ) has finite index in H ∩ Γ k +1 ( G ) then Ker ψ is a finite group and therefore h (Ker ψ ) = 0 , which implies h ( H ) = h ( G ) . (cid:3) In the next proof, recall that M := G/G ′′ denotes the maximal metabelian quotientof a -formal finitely generated group G . Proof of Theorem 1.2.
Since Γ q ( M ) is finite for q ≫ , it follows that θ q ( G ) = φ q ( M ) = 0 for q ≫ . Using (25) and the fact that (26) is an equality when G is -formal, we obtain dim W q ( G ) = φ q +2 ( M ) = 0 for q ≫ . It follows that W ( G ) is finite dimensional, which by (10) and Theorem 3.1 implies that W q ( G ) = 0 for q ≥ n − and thus φ q ( M ) = 0 for q ≥ n − . Combining this with thefact that Γ q ( M ) is finite for q ≫ , we conclude that in fact Γ n − ( M ) is finite provingconclusion (1) of Theorem 1.2.We note that if N is a finite normal subgroup of M = G/G ′′ then the groups M and M/N have the same growth rate. Taking N = Γ q ( M ) where q is such that Γ q ( M ) is finite, we get that M has the same growth rate as the nilpotent group M/N , and inparticular M is virtually nilpotent by Gromov’s theorem. We can then apply Lemma 4.7to deduce part (2) of Theorem 1.2.Consider now H to be a nilpotent finite index subgroup of M . Using the last part ofLemma 4.12 and the fact that Γ n − ( M ) is finite we get that(31) n − X q =1 φ q ( H ) = n − X q =1 φ q ( M ) . Applying the Bass–Guivarc’h Formula 1.2 to H yields d ( M ) = d ( H ) = X q ≥ q · φ q ( H ) = ( n − · n − X q =1 φ q ( H ) − n − X k =1 k X q =1 φ q ( H ) ( ) , ( ) ≤ ( n − · n − X q =1 φ q ( M ) − n − X k =1 k X q =1 φ q ( M ) = n − X q =1 q · φ q ( M )= φ ( M ) + n − X q =0 ( q + 2) · dim W q ( G ) . Using the fact that φ ( M ) = b ( G ) = n and the inequality in Theorem 3.2, it followsthat d ( M ) ≤ n + n − X q =0 (cid:18) n + q − q (cid:19) · ( n − · ( n − q −
3) = n + ( n − · (cid:18) n − n − (cid:19) . (cid:3) Remark 4.13.
Just as in the case of the nilpotency class, in the absence of -formalitythere can be no uniform bound in terms of b ( G ) for the degree of polynomial growth of M = G/G ′′ . Indeed, if we take G = F n / Γ k ( F n ) as in Example 4.9, then M is nilpotent.Combining (29) with the fact that φ q ( M ) = θ q ( G ) = θ q ( F n ) is strictly positive for q < k ,we obtain that d ( M ) can be made arbitrarily large if we let k → ∞ . Remark 4.14.
The only place in the proof of Theorem 1.2 that required -formality isthe assertion that the vanishing of the Chen ranks implies the vanishing of W q ( G ) . Ifwe replace the -formality assumption with the hypothesis that R ( G ) = { } then thevanishing of W q ( G ) follows from Theorem 4.2, and the proof of Theorem 1.2 carriesthrough without any further modifications.4.4. K¨ahler groups.
Our results can be applied to analyze the nilpotency class of fun-damental groups of compact K¨ahler manifolds. Suppose X is a compact K¨ahler man-ifold and G := π ( X ) , hence H ( G, Z ) ∼ = H ( X, Z ) . The first thing we observe is that R ( G ) = 0 if and only if the (2 , -part of the cup product map ψ X = ∪ , X : ^ H ( X, Ω X ) → H ( X, Ω X ) is zero on a degenerate element = ω ∧ ω ∈ V H ( X, Ω X ) . This follows because ∪ , X is the conjugate of ∪ , X , whereas the (1 , -part of the cup-product in the Hodge decom-position ∪ , X : H , ( X ) ⊗ H , ( X ) → H , ( X ) cannot vanish on decomposable tensors.Following Catanese’s generalization of the Castelnuovo-de Franchis Theorem [6], it fol-lows that R ( X ) = 0 if and only if X is fibred onto a curve C of genus at least . In thiscase, we have a surjection π ( X ) ։ π ( C ) , thus π ( X ) is not nilpotent. We say that inthis case X is fibred . Since π ( X ) is -formal, we obtain as a special case of Theorem 1.2the following. Theorem 4.15.
Suppose that X is a non-fibred compact K¨ahler manifold. If π ( X ) /π ( X ) ′′ isnilpotent, then its virtual nilpotency class is at most q ( X ) − . Constructing explicit K¨ahler manifolds with nilpotent fundamental group has provento be a challenging task. Examples of K¨ahler groups with a non-trivial nilpotent filtra-tion (in fact, of nilpotency class ) have been produced first by Campana [4]. On theother hand, a K¨ahler group is either fibred or each of its solvable quotients is virtuallynilpotent [14]. In particular, solvable K¨ahler groups are virtually nilpotent.A recent example is provided by the Schoen surfaces discovered in [41] and studiedfurther in [10]. They are minimal algebraic surfaces X of general type having invariants q ( X ) = 4 , p g ( X ) = 5 , K X = 16 , h , ( X ) = 12 . Remarkably, p g ( X ) = 2 q ( X ) − , that is, we are in the divisorial case when Ker ( ψ X ) maynot contain decomposable elements. That is the case, for Schoen surfaces are not fibred.In fact Ker ( ψ X ) is of dimension one, implying dim Ker ( ∪ X ) ∈ { , } . It is unknownwhether the fundamental group of such an X is nilpotent. Applying Theorem 4.15, weobtain that the virtual nilpotency class of π ( X ) /π ( X ) ′′ is at most b ( X ) − .4.5. The Torelli group of the mapping class group.
A particularly interesting classof applications of our results is provided by the
Torelli groups T g of genus g ≥ . Wefix a closed oriented surface Σ g of genus g and denote by π g := π (Σ g ) its fundamen-tal group and by H := H (Σ g , Z ) its first homology. Let Mod g be the mapping classgroup of isotopy classes of orientation-preserving homeomorphisms of Σ g . If X g de-notes the Teichm ¨uller space of genus g , then Mod g is the orbifold fundamental group ofthe moduli space M g of smooth curves of genus g , which can be realized as the quotient OPOLOGICAL INVARIANTS OF GROUPS AND KOSZUL MODULES 21 M g ∼ = X g / Mod g . Nielsen theory offers an alternative description Mod g ∼ = Out + ( π g ) ofthe mapping class group as the group of orientation preserving outer isomorphismsof π g . The Torelli group consists of those outer automorphisms of π g that act triviallyon H . The Torelli space T g := X g /T g can then be thought of as the moduli space of pairs [ C, α , . . . , α g , β , . . . , β g ] , consistingof a smooth curve C of genus g and a symplectic basis of H ( C, Z ) .Johnson [27] defined the surjective homomorphism τ : T g → ^ H/H, where the Sp g ( Z ) -equivariant injective map H ֒ → V H is given by z z ∧ ω , where ω := α ∧ β + · · · + α g ∧ β g ∈ V H , with ( α , . . . , α g , β , . . . , β g ) being a symplectic basisof H . Then he proved that T g is finitely generated and that the Johnson kernel defined bythe exact sequence −→ K g −→ T g −→ ^ H/H −→ , is precisely the subgroup of Mod g generated by Dehn twists along separated simpleclosed curves in Σ g . Finally, Johnson [28] showed that the abelianization of T g is given(up to -torsion) by the map τ , that is one has an isomorphism τ ∗ : H ( T g , Q ) ∼ = ^ H Q /H Q , which in particular implies that b ( T g ) = (cid:18) g (cid:19) − g. For g ≥ , the Torelli group is -formal [24, 25] and R ( T g ) = { } , see [15, The-orem 4.4]. Furthermore, Dimca, Hain and Papadima [16, Corollary 3.5] proved that B ( T g ) C = H ( K g , C ) is a nilpotent module over C [ T g /T ′ g ] . We deduce from Corollary 4.6the following new estimate of nilpotency of the Alexander invariant of T g . Theorem 4.16. If g ≥ and I ⊆ C [ T g /T ′ g ] denotes the augmentation ideal, then I q · B ( T g ) C = 0 for q ≥ (cid:18) g (cid:19) − g − . More recently, it was shown in [19, Corollary 1.5] that the metabelian quotient T g /T ′′ g is nilpotent for g ≥ , and in particular it follows that Theorem 1.2 applies for theTorelli groups. It follows that vnc( T g /T ′′ g ) ≤ b ( T g ) − , which is the statement of Theo-rem 1.3 in the Introduction. Of course, the groups T g and Mod g are very far from beingnilpotent. In fact, if G ≤ Mod g is a subgroup, either G is virtually abelian, or it containsa non-abelian free subgroup, see [36]. The Torelli group of a free group.
More generally, for an arbitrary fintely gener-ated group G , we define its Torelli group T ( G ) ⊆ Out( G ) to be the subgroup of outer au-tomorphisms of G consisting of automorphisms which induce the identity on H ( G, Z ) .The class of right-angled Artin groups, associated to finite simple graphs, is of consider-able interest in geometric group theory. The groups G in this class interpolate betweenfree abelian groups (corresponding to full graphs) and free groups (corresponding todiscrete graphs). Of particular interest is the case when G = F g is a free group on g generators x , . . . , x g . The corresponding Torelli group T ( F g ) denoted by OA g , is thusdefined by the following exact sequence −→ OA g −→ Out ( F g ) −→ GL g ( Z ) −→ . The Culler-Vogtmann outer space X g of marked metric graphs of rank g can be thoughtof as the classifying space for OA g , see [13]. The moduli space of graphs X g / OA g is (es-sentially) the moduli space M tr g of tropical curves of genus g , see [7] and referencestherein. In analogy with OA g , one defines the subgroup IA g of the group Aut( F g ) ofall automorphisms of the free group which induce the identity on H ( F g , Z ) . Both IA g and OA g are finitely generated for g ≥ by [32], and there exist natural quotient maps IA g ։ OA g .It is known that for g ≥ one has R (OA g ) = { } , see [38, Theorem 9.7]. More-over, the first Betti number of OA g has been computed by Kawazumi [29], see also [38,Theorem 9.1] and the references therein and is given by b (OA g ) = g ( g + 1)( g − . Note [32] that IA g is generated by the following set of Magnus generators of Aut ( F g ) α ij : x i x j x i x − j , x ℓ x ℓ for ℓ = i, and α ijk : x i x i · [ x j , x k ] , x ℓ x ℓ for ℓ = i, where the indices are subject to the conditions ≤ i = j ≤ g in the first case, respec-tively ≤ j < k ≤ g and i = j, k , in the second case. One has g ( g − = b (IA g ) gener-ators, see again [29]. When passing to OA g , one notices the relation α ,ℓ · · · α g,ℓ = 1 for ℓ = 1 , . . . , g and one is left with b (IA g ) − g = g ( g +1)( g − generators, which explains theformula for b (OA g ) .It is shown in [19, Corollary 1.5] that the metabelian quotients of IA g are nilpotent for g ≥ , so the same is true for the metabelian quotients of OA g . Applying Theorem 1.4with G = OA g we obtain the following result. Theorem 4.17. If g ≥ then the metabelian quotient OA g / OA ′′ g has virtual nilpotency classat most g ( g +1)( g − − . Using Corollary 4.6 we also obtain an analogue of Theorem 4.16 for the Alexanderinvariant of OA g . OPOLOGICAL INVARIANTS OF GROUPS AND KOSZUL MODULES 23 R EFERENCES [1] J. W. Alexander,
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ARIAN A PRODU : S
IMION S TOILOW I NSTITUTE OF M ATHEMATICS
P.O. B OX UCHAREST , R
OMANIA , AND F ACULTY OF M ATHEMATICS AND C OMPUTER S CIENCE , U
NIVERSITY OF B UCHAREST , R
OMANIA
E-mail address : [email protected] G AVRIL F ARKAS : I
NSTITUT F ¨ UR M ATHEMATIK , H
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ERLIN , G
ERMANY
E-mail address : [email protected] S¸ TEFAN P APADIMA : S
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EPARTMENT OF M ATHEMATICS , U
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