Genus Mapping Class Groups are not Kähler
Abstract.
We prove that finite index subgroups of genus mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kähler. These results are deduced from explicit presentations of the unipotent (aka, Malcev) completion of genus 3 Torelli groups and of the relative completions of genus 3 mapping class groups. The main results follow from the fact that these presentations are not quadratic. To complete the picture, we compute presentations of completed Torelli and mapping class in genera ; they are quadratic. We also show that groups commensurable with hyperelliptic mapping class groups and mapping class groups in genera are not Kähler.
1991 Mathematics Subject Classification:
Primary 14H30; Secondary 14H15, 20F34, 32Q15Contents
 1 Introduction
 2 Preliminaries
 3 Presentations of Pronilpotent Lie Algebras
 4 Relative Completion of Discrete Groups
 5 Symplectic Groups and their Representations
 6 Completions of Mapping Class Groups
 7 Presentations of and for
 8 Commuting Elements
 9 Presentations of and
 10 Genus Mapping Class Groups are not Kähler Groups
 11 Speculation
 A Mapping Class Groups in Low Genus are not Kähler
1. Introduction
Mapping class groups are groups of topological symmetries of a compact surface. More precisely, suppose that and are nonnegative integers satisfying . Let be a compact oriented surface of genus and a set of points on . The mapping class group is defined to be the group
of isotopy classes of orientation preserving diffeomorphisms of that fix pointwise. The Torelli group is the set of elements of that act trivially on . Johnson [18] proved that is finitely generated for all . The Johnson group is the subgroup of generated by Dehn twists on bounding simple closed curves (BSCC maps, for short).
A Kähler group is a group that can be realized as the fundamental group of a compact Kähler manifold. Groups commensurable with mapping class groups in genus are not Kähler.^{1}^{1}1The genus case was established by Veliche [33]. The cases in genera seem to be folklore. We give complete proofs in the appendix and also prove that groups commensurable with hyperelliptic mapping class groups of any genus cannot be Kähler. Our first result is that genus mapping class groups are not Kähler. Suppose .
Theorem 1.
No finite index subgroup of that contains can be the fundamental group of a compact Kähler manifold.
Finite index subgroups of that contain when include fundamental groups of moduli spaces of curves with an abelian (i.e., standard) level structure; moduli spaces of curves with a Prymlevel structure as defined by Looijenga [23] and BoggiPikaart [5]; and fundamental groups of moduli spaces of curves with an arbitrary root of their canonical bundle.
An easier result is that genus 3 Torelli groups are not Kähler. As above, is a nonnegative integer.
Theorem 2.
No finite index subgroup of that contains can the be fundamental group of a compact Kähler manifold.
To prove the second result, we prove that the unipotent (aka Malcev) completion of each genus 3 Torelli group is not quadratically presented.^{2}^{2}2This result was stated informally in [14, Rem. 10.4]. One then concludes that these groups are not Kähler by applying the well known result of Deligne, Griffiths, Morgan and Sullivan [7]. This computation is extended to finite index subgroups of genus 3 Torelli groups that contain the Johnson subgroups using recent work of Andrew Putman [26].
The proof of Theorem 1 is more involved. The first step is to compute an explicit presentation of the unipotent radical of the relative completion of the genus mapping class group . This we deduce from an explicit presentation of the unipotent completion of the genus Torelli group in Section 9 using computations from Section 8 and the work of Sakasai [29]. Dennis Johnson’s work [19] implies that the standard representation is rigid. Results of Carlos Simpson [31] then imply that if is the fundamental group of a compact Kähler manifold , then is the monodromy representation of a polarized variation of Hodge structure over . A generalization of the Deligne, Griffiths, Morgan, Sullivan result to relative completion, proved in [13], implies that if is Kähler, then would be quadratically presented. Since it is not, is not Kähler.
This paper may be regarded as a sequel to [14] where explicit presentations of the unipotent completion of Torelli groups and the unipotent radical of relative completions of mapping class groups were given for all and partial presentations were derived when . Here we complete the story by computing explicit presentations when .
Before stating the quadratic presentations we need to fix notation. Denote the first rational homology of the reference surface by . The symplectic group of rank , also denoted , is the group of automorphisms of that preserve the intersection pairing. The third fundamental representation of is . It plays a distinguished role as the Johnson homomorphism induces a canonical equivariant isomorphism .
The free Lie algebra generated by the vector space will be denoted . It is graded; will denote its space of degree elements. Denote the lower central series of a Lie algebra by
and the associated graded Lie algebra by .
The Lie algebra of the unipotent completion of will be denoted by and the Lie algebra of the prounipotent radical of the completion of the mapping class group with respect to the homomorphism will be denoted by .
The following Theorem is proved in Section 7.1. The main ingredients in the proof are Hodge theory, which implies that each of these Lie algebras is isomorphic to the degree completion of the associated graded Lie algebra of its lower central series; Kabanov purity [21] (see also Section 6.4), which implies that the degrees of relations in a minimal presentation of these Lie algebras is ; and computations of Sakasai [29], which extend computations from [14]. The result when was previously proved in [14].
Theorem 3.
If , then
and has the quadratic presentation
in the category of graded Lie algebras in the category of modules, where is the complement of the unique copy of in . Similarly, is isomorphic to the degree completion of the graded Lie algebra associated to its lower central series. It is has presentation
in the category of modules, where is the unique copy of the trivial representation in .
From this one can deduce presentations of all and — the Lie algebras of the unipotent completion of and of the prounipotent radical of the relative completion of — for all when . These are also quadratically presented. This result allows Theorem B of [8] to be extended from to all .
The derivation of the genus 3 presentations is more difficult and involves the computation of the cubic relations generated by a commuting pair consisting of a Dehn twist on a bounding simple closed curve and a bounding pair map. These computations are carried out in Section 8 and also use Sakasai’s computations [29]. As in higher genus case, the proof uses Hodge theory and Kabanov purity.
Theorem 4.
In genus , the Lie algebras and are isomorphic to the degree completions of the graded Lie algebras associated to their lower central series:
The graded Lie algebra has the cubic presentation
in the category of graded Lie algebras in the category of modules, where is the complement of the unique copy of in . The Lie algebra has presentation
in the category of modules, where is the unique copy of the trivial representation in .
One consequence of the computations is that the cup product vanishes, which implies that there is a welldefined Massey triple product map
The computation of the cubic relations in is dual to the computation of the Massey triple products.
Theorem 5.
The image of the Massey triple product map is an submodule isomorphic to the restriction of the module to .
A word about exposition: I could have written a shorter paper. However, in the interests of making the proofs of the main results comprehensible and the paper reasonably coherent, I have included expository material, especially where I felt I could improve on earlier expositions. In the final section I speculate on how the problem of deciding whether higher genus mapping class groups are Kähler might be related to the problem of understanding the topology of complete subvarieties of moduli spaces of curves. In particular, I conjecture that the fundamental group of a smooth complete subvariety of cannot be a finite index subgroup of .
Acknowledgments: I am grateful to Carlos Simpson for helpful discussions related to his work. I am also indebted to the referee for his/her careful reading of the manuscript and for pointing out numerous typos in the original manuscript.
2. Preliminaries
This section is included for clarity. In it we introduce some notation and conventions, and recall several definitions. Readers should skip this section and refer to it as needed.
2.1. Filtrations
Increasing filtrations
of (a vector space, group, etc) will be denoted with a lower index, . Decreasing filtrations
of (a vector space, a group, etc) will be denoted with an upper index, . The notation for the graded quotients of and is
The lower central series of a Lie algebra (or group) will be denoted by .^{3}^{3}3When is a topological Lie algebra (or group), is defined inductively to be the closure of . It is indexed so that . With this convention the bracket preserves :
2.2. Proalgebraic groups
Suppose that is a field of characteristic zero. An affine proalgebraic group over is an inverse limit of affine algebraic groups . The coordinate ring of the direct limit of the coordinate rings of the . The Lie algebra of is the inverse limit of the Lie algebras of the . It is a Hausdorff topological Lie algebra. The neighbourhoods of are the kernels of the canonical projections .
The continuous cohomology of is defined by
Its homology is the full dual:
Each homology group is a Hausdorff topological vector space.
Continuous cohomology can be computed using continuous ChevalleyEilenberg cochains:
with the usual differential.
If, instead, is a graded Lie algebra, then the homology and cohomology of are also graded. This follows from the fact that the grading of induces a grading of the ChevalleyEilenberg chains and cochains of .
2.3. Prounipotent groups and pronilpotent Lie algebras
A prounipotent group is a proalgebraic group that is an inverse limit of unipotent groups.
A pronilpotent Lie algebra over a is an inverse limit of finite dimensional nilpotent Lie algebras. The Lie algebra of a prounipotent groups is a pronilpotent Lie algebra. The functor that takes a prounipotent group to its Lie algebra is an equivalence of categories between the category of unipotent groups and the category of pronilpotent Lie algebras over .
2.4. Hodge theory
The reader is assumed to be familiar with the basic properties of mixed Hodge structures (MHSs), which can be found in [6]. One property that we will exploit is that there is a natural (though not canonical) isomorphism
of the complex part of a MHS with the direct sum its weight graded quotients. These isomorphisms are compatible with tensor products and duals.
A Lie algebra in the category of (rational or real) MHS is a finite dimensional Lie algebra endowed with a MHS with the property that the bracket is a morphism of MHS. The cohomology of an inverse limit of Lie algebras in the category of MHS is an IndMHS — that is, an indobject of the category of MHS.
If the weight graded quotients of are finite dimensional, then the exactness of the functor on the category of MHS implies that there are natural isomorphisms
3. Presentations of Pronilpotent Lie Algebras
Here we recall how each pronilpotent and each positively graded Lie algebra has a “minimal presentation” with generating set isomorphic to and relations isomorphic to . Denote the free Lie algebra (over ) generated by a vector space by . This is a graded Lie algebra: degree component consists of the homogeneous Lie words of degree . The th term of its lower central series is
The lower central series filtration defines a topology on . Its completion in this topology is the degree completion of :
When is finite dimensional, is pronilpotent.
Suppose that is a pronilpotent Lie algebra. For simplicity, we assume that is finite dimensional. Each choice of a continuous splitting of the canonical surjection induces a continuous surjection
The ideal of relations is the kernel of this homomorphism. Note that is contained in . By the Lie algebra analogue of a theorem of Hopf (cf. [14, Prop. 5.6]) there is a natural isomorphism
(1) 
The image of any continuous section of is a minimal set of relations:
Similarly, if is a graded Lie algebra that is generated by , it has a presentation of the form
where is an injective graded linear mapping. Observe that a minimal space of relations of degree is the image of
3.1. Chains
If is a graded Lie algebra then its ChevalleyEilenberg complex is graded. The rows of the diagram
are the weight^{4}^{4}4Here we use the word “weight” to refer the grading index. Later this index will be the Hodge theoretic weight, which will be negative. 2 and 3 components of the ChevalleyEilenberg chains of . Here is the “Jacobi identity” map
(2) 
The homology of the th row () is the weight summand of . The dual complexes compute the weight summands of . Observe that the subalgebra
is generated by . If is surjective, then , which implies that for all .
Note that when is generated by , and the weight component of is dual to the cup product .
3.2. Quadratic presentations
A graded Lie algebra is quadratically presented if it has a graded presentation of the form
where is in weight 1 and is a subspace of . A pronilpotent Lie algebra with finite dimensional is quadratically presented if it has a presentation of the form
where is a subspace of . In this case is isomorphic to its degree completion as a pronilpotent Lie algebra. The following result is a straightforward consequence of the definitions.
Proposition 3.1.
Suppose that is a pronilpotent Lie algebra with finite dimensional. If is quadratically presented, then is quadratically presented and is isomorphic (as a pronilpotent Lie algebra) to the degree completion of :
Conversely, if is quadratically presented and is isomorphic to the degree completion of , then is quadratically presented.
Quadratically presented graded Lie algebras have a standard characterization.
Lemma 3.2.
A graded Lie algebra that is generated in weight is quadratically presented if and only if has weight or, equivalently, the cup product is surjective.
Proof.
Suppose that is a minimal presentation of . This means that . Set . There is a natural isomorphism
A minimal graded space of generators of is the image of any section of . The result follows as is quadratically presented if and only if this set of minimal generators of has degree . ∎
Since morphisms of graded Lie algebras induce graded morphisms of their homology (and cohomology), we deduce:
Corollary 3.3.
Suppose that is a surjection of graded Lie algebras, both generated in weight . If is quadratically presented and is surjective, then is quadratically presented.
4. Relative Completion of Discrete Groups
Suppose that is a discrete group and that is a reductive algebraic group over a field of characteristic zero. The completion of relative to a Zariski dense representation is a proalgebraic group which is an extension of by a prounipotent group, and a homomorphism such that the composite
is . It is universal for such groups: if is a proalgebraic group that is an extension of by a prounipotent group, and if is a homomorphism whose composition with is , then there is a homomorphism of proalgebraic groups such that the diagram
commutes.
When is trivial, is trivial and is the unipotent completion of over .
When discussing the mixed Hodge structure on a relative completion of the fundamental group of a compact Kähler manifold , we need to be able to compare the completion of over (or ) with its completion over . For this reason we need to discuss the behaviour of relative completion under base change.
If is an extension field of and is Zariski dense over , then one has the relative completion of with respect to . It is an extension of by a prounipotent group. The universal mapping property of implies that the homomorphism induces a homomorphism of proalgebraic groups. This homomorphism is an isomorphism. This is easily seen when each irreducible representation of remains irreducible after extending scalars from to . This follows directly from the cohomological results below.
4.1. Cohomology
We continue with the notation above, where is the relative completion of . When is reductive, the structure of and are closely related to the cohomology of with coefficients in rational representations of .
For each rational representation of there are natural isomorphisms
The homomorphism induces a homomorphism
(3) 
It is an isomorphism in degrees and an injection in degree 2.
Denote the set of isomorphism classes of finite dimensional irreducible representations of by . Fix an module in each isomorphism class . If each irreducible representation of is absolutely irreducible^{5}^{5}5This is the case when over any field of characteristic zero. and if is finite dimensional for all rational representations of when , then there is an isomorphism
of topological modules, and a continuous invariant surjection
In both cases, the LHS has the product topology.
Lemma 4.1.
Suppose that is a Zariski dense representation from a discrete group to a reductive group. If is a homomorphism that such that is Zariski dense and such that, for all rational representations of ,
is an isomorphism when and injective when , then the completion of relative to is an isomorphism.
Proof.
This follows directly from the cohomological results above and that fact that a homomorphism of pronilpotent Lie algebras is an isomorphism if and only if is an isomorphism in degree 1 and injective in degree 2. ∎
4.2. Hodge theory
Suppose that is the complement of a normal crossings divisor in a compact Kähler manifold. Suppose that or and that is a polarized variation of Hodge structure (PVHS) over . Pick a base point . Denote the fiber over over by . The Zariski closure of the image of the monodromy representation
is a reductive group. Denote it by . Then one has the relative completion of with respect to .
Theorem 4.2 ([13]).
The coordinate ring is a Hopf algebra in the category of Indmixed Hodge structures over . It has the property that and .
A slightly weaker version of the theorem is stated in terms of Lie algebras. Denote the prounipotent radical of by . Denote their Lie algebras by and , and the Lie algebra of by .
4.3. Presentations
We continue with the notation of Section 4.2. The problem of computing presentations of and is simplified as the exactness of implies that there is a natural isomorphism
of topological Lie algebras. In order to determine , it suffices to compute as a graded Lie algebra in the category of modules. Exactness of implies that .
When is finite dimensional, the Lie algebra has a presentation of the form
in the category of Ind modules for a suitable module map .
The following result is a formal consequence of results stated above and the fact, due to Deligne (cf. [35, Thm. 2.9]), that if is a PVHS over of weight , then the smooth forms on with coefficients in is a Hodge complex and has a Hodge structure of weight .
Theorem 4.4 ([13, Thm. 13.14]).
If is a compact Kähler manifold, then

the MHS on is pure of weight when ,

the weight filtration of is its lower central series ,

is quadratically presented.
In the case of unipotent completion, this was first proved by Deligne, Griffiths, Morgan and Sullivan in [7].
5. Symplectic Groups and their Representations
Suppose that is a commutative ring and that is a free module of rank with a unimodular, skew symmetric bilinear pairing . The symplectic group is defined to be . The choice of a symplectic basis of gives an isomorphism of with .
When is a field of characteristic zero, is a simply connected simple algebraic group all of whose irreducible representations are absolutely irreducible. Denote its Lie algebra by . This is isomorphic to the Lie algebra .
Every irreducible representation of is a submodule of some tensor power of . Fix a symplectic basis of . Set
Every linear automorphism of extends to a derivation of the tensor algebra on . With this convention, is the subalgebra of derivations of that annihilate :
The abelian subalgebra of consisting of those derivations that act on via
is a Cartan subalgebra. The derivations
of span a nilpotent subalgebra of which, together with , span a Borel subalgebra of .
A fundamental set of highest weights with respect to this Borel subalgebra is , where is the function defined by
We will denote the irreducible module with highest weight by .
The exterior algebra is an algebra in the category of modules. The fundamental representation is the degree part of the quotient of it by the ideal :
Recall that every irreducible module has a nondegenerate invariant bilinear pairing. Equivalently, every irreducible module is isomorphic to its dual.
5.1. Stability in the representation ring of
Fix a sequence of inclusions
Each of these induces an inclusion of representation rings. One can describe this in terms of symmetric polynomials (i.e., characters), dominant integral weights or partitions. For example, the inclusion takes the irreducible representation of with highest weight to the irreducible representation of with the corresponding weight. See [20] and [14, §6] for details. In [20], Kabanov proves that the decomposition of Schur functors and tensor products of representations stabilize in this sense. Stability allows one to compute stable decompositions of tensor products and Schur functors, by computing the decomposition in one case in the stable range. See [14, §6] for a more detailed discussion.
5.2. Unipotent completion of surface groups
Suppose that is a compact Riemann surface of genus and that . Denote the Lie algebra of the unipotent completion of by . Set . Let be a symplectic basis of . Applying the results of the previous section, has a natural MHS whose associated graded has the presentation:
This is a Lie algebra in the category of modules.^{6}^{6}6This statement is also a consequence of Labute’s Theorem [22].
The case of the following result is [14, Prop. 8.4]. The genus case is proved similarly.
Proposition 5.1.
The highest weight decomposition of the first four weight graded quotients of are:
The exactness of implies that there are natural graded Lie algebra isomorphisms
Since (and hence as well) has trivial center [2], the sequence
is exact.
Corollary 5.2 ([14, Cor. 9.4]).
When , we have
∎
6. Completions of Mapping Class Groups
In this section and are nonnegative integers satisfying .
6.1. Relative completion of mapping class groups
Let be a closed oriented surface of genus . Set . Fix a symplectic basis of . The action of on induces a surjective homomorphism
Since is Zariski dense in , we have the completion of with respect to . It is an extension
of the symplectic group by a prounipotent group. Denote the Lie algebras of and by and , respectively. When , it will be suppressed, so that , , etc.
When , the level subgroup of is defined to be the kernel of the mod reduction of . When , the homomorphism is the completion of relative to , [14, Prop. 3.3]. In Section 6.3 we will see that a recent result of Putman [26] implies that this result extends to finite index subgroups of mapping class groups that contain its Johnson subgroup, provided that . When the completion of depends nontrivially on .
The homomorphism induces a homomorphism where denotes the unipotent completion of .
Theorem 6.1 ([11],[14, Thm. 3.4]).
If , then this homomorphism is surjective. If , it has nontrivial kernel isomorphic to the additive group . The sequence
is a nontrivial central extension.
The action of on induces an action that preserves the lower central series filtration. The action of on factors through . The universal mapping property of relative completion, implies that this action induces an action . The induced action on Lie algebras is a tool for understanding . It induces the outer action .
6.2. Mapping class groups as fundamental groups of smooth varieties
The moduli space of smooth complex projective curves of type is the quotient of the Teichmüller space by , which acts properly discontinuously on Teichmüller space. Although this action is not fixed point free, it is when restricted to when . The quotient is a smooth quasi projective variety. Consequently, is the fundamental group of a smooth quasiprojective variety for all .
One can realize as the fundamental group of a smooth quasiprojective variety by fixing and taking
where is a simply connected projective manifold on which acts fixed point freely.^{7}^{7}7Such varieties were constructed by Serre. The argument can be found in [30, IX,§4.2]. Alternatively, when and , one can take to be the complement of the locus in of pointed curves with a nontrivial automorphism (cf. [15, Prop. 4.1]).
Define to be the orbifold fundamental group of , the th power of the universal curve over . It is an extension
where denotes the fundamental group of a smooth projective curve of genus . Since is a Zariski open subset of , is a quotient of .
Since is a smooth quasiprojective variety when , is the fundamental group of a smooth quasiprojective variety for all . (Use the trick above.)
6.2.1. Associated Hodge theory
The fact that mapping class groups are fundamental groups of smooth varieties allows us to study their relative completions via Hodge theory.
Suppose that is one of the mapping class groups or . Then is the fundamental group of a pointed smooth variety , where is one of the smooth varieties described above. Denote its completion relative to the homomorphism by and it prounipotent radical by . One has a family of complete genus curves. The representation is the monodromy representation of the local system , which is a polarized variation of Hodge structure. The Lie algebras of and of are Lie algebras in the category of promixed Hodge structures.
The Torelli subgroup of is the kernel of . Denote the Lie algebra of its unipotent completion by .
The following result summarizes several results from Sections 3 and 4 of [14].
Proposition 6.2.
If , then

the MHS on can be lifted to a MHS on so that the central extension
(4) is a nontrivial central extension of Lie algebras in the category of proMHS. Consequently, the associated graded sequence
is an exact sequence of graded Lie algebras in the category of modules.

the weight filtrations of and are their lower central series:
In particular, .

.
The spectral sequence of the central extension gives a Gysin sequence.
Corollary 6.3.
For all there is a Gysin sequence
in the category of mixed Hodge structures. It remains exact after applying for all .
6.3. Variation on a theme of Putman
Here we recall a result of Putman and extract some useful consequences from it. As in the introduction, denotes the subgroup of generated by Dehn twists on bounding simple closed curves (BSCCs).^{8}^{8}8Recall that a bounding pair in a surface consists of two disjoint nonseparating simple closed curves and which together divide the surface into two components. A bounding pair map (BP map) consists of the product of a positive Dehn twist about and a negative Dehn twist about . The following result is a special case of a result [26, Thm. A] of Putman.
Theorem 6.4 (Putman).
Suppose that and that . If is a finite index subgroup of that contains , then the inclusion induces an isomorphism .
Denote the image of in by .
Corollary 6.5.
Suppose that and that . If is a finite index subgroup of that contains , then the inclusion induces an isomorphism .
Proof.
Observe that is the fundamental group of the th power of the universal curve over Torelli space and that is the fundamental group of the Zariski open subset obtained by removing the “diagonals”. It follows that is surjective. Consequently, the inverse image of in is a finite index subgroup that contains . Now consider the diagram:
Since the lefthand vertical arrow is surjective, it induces a surjection on . Putman’s result implies that top map induces an isomorphism on rational . The result follows as the righthand vertical map induces an isomorphism on rational , which follows from [12, Prop. 5.2] and an easy spectral sequence argument for th power of the universal curve over . ∎
Putman’s result has the following analogue for mapping class groups.
Proposition 6.6.
Suppose that and that . If is a finite index subgroup of (resp. ) that contains (resp. ), then for all modules , the map
induced by the inclusion is an isomorphism in degrees and an injection in degree . In particular,
As in [12, §5], the vanishing of implies that the Picard group of the corresponding moduli space of curves is finitely generated.
Proof.
We will prove the case; the case is similar and left to the reader. Since is a module, a standard trace argument implies that the induced map on cohomology is injective in all degrees. Surjectivity in degree 0 follows from the fact that the image of in has finite index in and is thus Zariski dense.
To prove that the induced map on cohomology is an isomorphism in degree 1, write as an extension
where . The kernel is a finite index subgroup of that contains and so Putman’s theorem implies that . Since , Raghunathan’s vanishing theorem [28] implies the vanishing of