# Genus $3$ Mapping Class Groups are not Kähler

Richard Hain  Department of Mathematics
Duke University
Durham, NC 27708-0320<EMAIL_ADDRESS>
###### Abstract.

We prove that finite index subgroups of genus $3$ 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 $\geq 4$; they are quadratic. We also show that groups
commensurable with hyperelliptic mapping class groups and mapping class groups
in genera $\leq 2$ are not Kähler.

###### 1991 Mathematics Subject Classification:

Primary 14H30; Secondary 14H15, 20F34, 32Q15

Supported in part by grant DMS-1005675 from the National Science Foundation

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Presentations of Pronilpotent Lie Algebras
  4. 4 Relative Completion of Discrete Groups
  5. 5 Symplectic Groups and their Representations
  6. 6 Completions of Mapping Class Groups
  7. 7 Presentations of ${\mathfrak{u}}_{g}$ and ${\mathfrak{t}}_{g}$ for $g\geq 4$
  8. 8 Commuting Elements
  9. 9 Presentations of ${\mathfrak{u}}_{3}$ and ${\mathfrak{t}}_{3}$
  10. 10 Genus $3$ Mapping Class Groups are not Kähler Groups
  11. 11 Speculation
  12. 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 $g$ and $n$ are non-negative integers
satisfying $2g-2+n>0$. Let $S$ be a compact oriented surface of genus $g$ and
$P$ a set of $n$ points on $S$. The mapping class group $\Gamma_{g,n}$ is
defined to be the group

$\Gamma_{g,n}:=\pi_{0}\operatorname{Diff}^{+}(S,P),$

of isotopy classes of orientation preserving diffeomorphisms of $S$ that fix
$P$ pointwise. The Torelli group $T_{g,n}$ is the set of elements of
$\Gamma_{g,n}$ that act trivially on $H_{1}(S;{\mathbb{Z}})$. Johnson [18]
proved that $T_{g,n}$ is finitely generated for all $g\geq 3$. The Johnson
group $K_{g,n}$ is the subgroup of $T_{g,n}$ 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 $\leq 2$ are not Kähler.111The genus $2$ case was established by Veliche
[33]. The cases in genera $\leq 1$ 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 $3$ mapping class groups are not Kähler. Suppose $n\geq
0$.

######  Theorem 1.

No finite index subgroup of $\Gamma_{3,n}$ that contains $K_{3,n}$ can be the
fundamental group of a compact Kähler manifold.

Finite index subgroups of $\Gamma_{g,n}$ that contain $K_{g,n}$ when $g\geq 3$
include fundamental groups of moduli spaces of curves with an abelian (i.e.,
standard) level structure; moduli spaces of curves with a Prym-level structure
as defined by Looijenga [23] and Boggi-Pikaart [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, $n$
is a non-negative integer.

######  Theorem 2.

No finite index subgroup of $T_{3,n}$ that contains $K_{3,n}$ 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.222This 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 ${\mathfrak{u}}_{3}$ of the
relative completion of the genus $3$ mapping class group $\Gamma_{3}$. This we
deduce from an explicit presentation of the unipotent completion of the genus
$3$ 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 $\rho:\Gamma_{3}\to{\mathrm{Sp}}_{3}({\mathbb{C}})$ is rigid.
Results of Carlos Simpson [31] then imply that if $\Gamma_{3}$ is the
fundamental group of a compact Kähler manifold $X$, then $\rho$ is the
monodromy representation of a polarized variation of Hodge structure over $X$.
A generalization of the Deligne, Griffiths, Morgan, Sullivan result to
relative completion, proved in [13], implies that if $\Gamma_{3}$ is Kähler,
then ${\mathfrak{u}}_{3}$ would be quadratically presented. Since it is not,
$\Gamma_{3}$ 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 $g\geq 6$ and
partial presentations were derived when $3\leq g<6$. Here we complete the
story by computing explicit presentations when $3\leq g<6$.

Before stating the quadratic presentations we need to fix notation. Denote the
first rational homology $H_{1}(S,{\mathbb{Q}})$ of the reference surface $S$
by $H$. The symplectic group ${\mathrm{Sp}}_{g}$ of rank $g$, also denoted
${\mathrm{Sp}}(H)$, is the group of automorphisms of $H$ that preserve the
intersection pairing. The third fundamental representation of
${\mathrm{Sp}}(H)$ is $\Lambda^{3}_{0}H:=(\Lambda^{3}H)/H$. It plays a
distinguished role as the Johnson homomorphism induces a canonical
${\mathrm{Sp}}_{g}({\mathbb{Z}})$-equivariant isomorphism
$H_{1}(T_{g};{\mathbb{Q}})\cong\Lambda^{3}_{0}H$.

The free Lie algebra generated by the vector space $V$ will be denoted
${\mathbb{L}}(V)$. It is graded; ${\mathbb{L}}_{n}(V)$ will denote its space
of degree $n$ elements. Denote the lower central series of a Lie algebra
${\mathfrak{g}}$ by

${\mathfrak{g}}=L^{1}{\mathfrak{g}}\supseteq L^{2}{\mathfrak{g}}\supseteq
L^{3}{\mathfrak{g}}\supseteq\cdots$

and the associated graded Lie algebra by
$\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{g}}$.

The Lie algebra of the unipotent completion of $T_{g}$ will be denoted by
${\mathfrak{t}}_{g}$ and the Lie algebra of the prounipotent radical of the
completion of the mapping class group $\Gamma_{g}$ with respect to the
homomorphism $\Gamma_{g}\to{\mathrm{Sp}}_{g}$ will be denoted by
${\mathfrak{u}}_{g}$.

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 $\leq 3$; and computations of Sakasai [29], which extend
computations from [14]. The result when $g\geq 6$ was previously proved in
[14].

######  Theorem 3.

If $g\geq 4$, then

${\mathfrak{t}}_{g}\cong\prod_{n\geq
1}\operatorname{Gr}^{n}_{\mathrm{LCS}}{\mathfrak{t}}_{g}.$

and $\operatorname{Gr}^{n}_{\mathrm{LCS}}{\mathfrak{t}}_{g}$ has the quadratic
presentation

$\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{t}}_{g}\cong{\mathbb{L}}(\Lambda^{3}_{0}H)/(R)$

in the category of graded Lie algebras in the category of
${\mathrm{Sp}}_{g}$-modules, where $R$ is the ${\mathrm{Sp}}(H)$ complement of
the unique copy of $V(2\lambda_{2})$ in ${\mathbb{L}}_{2}(\Lambda^{3}_{0}H)$.
Similarly, ${\mathfrak{u}}_{g}$ is isomorphic to the degree completion of the
graded Lie algebra associated to its lower central series. It is has
presentation

$\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{u}}_{g}\cong{\mathbb{L}}(\Lambda^{3}_{0}H)/(R+V(0))$

in the category of ${\mathrm{Sp}}_{g}$-modules, where $V(0)$ is the unique
copy of the trivial representation in ${\mathbb{L}}_{2}(\Lambda^{3}_{0}H)$.

From this one can deduce presentations of all ${\mathfrak{t}}_{g,n}$ and
${\mathfrak{u}}_{g,n}$ — the Lie algebras of the unipotent completion of
$T_{g,n}$ and of the prounipotent radical of the relative completion of
$\Gamma_{g,n}$ — for all $n>0$ when $g\geq 4$. These are also quadratically
presented. This result allows Theorem B of [8] to be extended from $g\geq 6$
to all $g\geq 4$.

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 $3$, the Lie algebras ${\mathfrak{t}}_{3}$ and ${\mathfrak{u}}_{3}$
are isomorphic to the degree completions of the graded Lie algebras associated
to their lower central series:

${\mathfrak{t}}_{3}\cong\prod_{n\geq
1}\operatorname{Gr}^{n}_{\mathrm{LCS}}{\mathfrak{t}}_{3}\text{ and
}{\mathfrak{u}}_{3}\cong\prod_{n\geq
1}\operatorname{Gr}^{n}_{\mathrm{LCS}}{\mathfrak{u}}_{3}$

The graded Lie algebra
$\operatorname{Gr}^{n}_{\mathrm{LCS}}{\mathfrak{t}}_{3}$ has the cubic
presentation

$\operatorname{Gr}^{n}_{\mathrm{LCS}}{\mathfrak{t}}_{3}\cong{\mathbb{L}}(\Lambda^{3}_{0}H)/(R)$

in the category of graded Lie algebras in the category of
${\mathrm{Sp}}_{3}$-modules, where $R$ is the ${\mathrm{Sp}}(H)$ complement of
the unique copy of $V(2\lambda_{1}+\lambda_{3})$ in
${\mathbb{L}}_{3}(\Lambda^{3}_{0}H)$. The Lie algebra
$\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{u}}_{3}$ has
presentation

$\operatorname{Gr}^{n}_{\mathrm{LCS}}{\mathfrak{u}}_{3}\cong{\mathbb{L}}(\Lambda^{3}_{0}H)/(R+V(0))$

in the category of ${\mathrm{Sp}}_{3}$-modules, where $V(0)$ is the unique
copy of the trivial representation in ${\mathbb{L}}_{2}(\Lambda^{3}_{0}H)$.

One consequence of the computations is that the cup product
$\Lambda^{2}H^{1}(T_{3};{\mathbb{Q}})\to H^{2}(T_{3};{\mathbb{Q}})$ vanishes,
which implies that there is a well-defined Massey triple product map

$H^{1}(T_{3};{\mathbb{Q}})^{\otimes 3}\to H^{2}(T_{3};{\mathbb{Q}}).$

The computation of the cubic relations in ${\mathfrak{t}}_{3}$ is dual to the
computation of the Massey triple products.

######  Theorem 5.

The image of the Massey triple product map is an
${\mathrm{Sp}}_{3}({\mathbb{Z}})$ submodule isomorphic to the restriction of
the ${\mathrm{Sp}}_{3}$-module
$V(2\lambda_{2}+\lambda_{3})+V(\lambda_{1}+\lambda_{2})+V(\lambda_{3})$ to
${\mathrm{Sp}}_{3}({\mathbb{Z}})$.

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
${\mathcal{M}}_{g}$ cannot be a finite index subgroup of $\Gamma_{g}$.

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

$0\subseteq\cdots\subseteq F_{m}V\subseteq F_{m+1}V\subseteq\cdots\subseteq V$

of $V$ (a vector space, group, etc) will be denoted with a lower index,
$F_{\bullet}V$. Decreasing filtrations

$U\supseteq\cdots\supseteq F^{m}U\supseteq F^{m+1}U\supseteq\cdots\supseteq 0$

of $U$ (a vector space, a group, etc) will be denoted with an upper index,
$F^{\bullet}U$. The notation for the graded quotients of $F_{\bullet}V$ and
$F^{\bullet}U$ is

$\operatorname{Gr}_{m}^{F}V:=F_{m}V/F_{m-1}V\text{ and
}\operatorname{Gr}^{m}_{F}U:=F^{m}U/F^{m+1}U.$

The lower central series of a Lie algebra (or group) ${\mathfrak{g}}$ will be
denoted by $L^{\bullet}{\mathfrak{g}}$.333When ${\mathfrak{g}}$ is a
topological Lie algebra (or group), $L^{m}{\mathfrak{g}}$ is defined
inductively to be the closure of $[{\mathfrak{g}},L^{m-1}{\mathfrak{g}}]$. It
is indexed so that ${\mathfrak{g}}=L^{1}{\mathfrak{g}}$. With this convention
the bracket preserves $L^{\bullet}$:

$[L^{m}{\mathfrak{g}},L^{n}{\mathfrak{g}}]\subseteq L^{m+n}{\mathfrak{g}}.$

###  2.2. Proalgebraic groups

Suppose that $F$ is a field of characteristic zero. An affine proalgebraic
group $G$ over $F$ is an inverse limit of affine algebraic $F$-groups
$G_{\alpha}$. The coordinate ring ${\mathcal{O}}(G)$ of $G$ the direct limit
of the coordinate rings of the $G_{\alpha}$. The Lie algebra ${\mathfrak{g}}$
of $G$ is the inverse limit of the Lie algebras ${\mathfrak{g}}_{\alpha}$ of
the $G_{\alpha}$. It is a Hausdorff topological Lie algebra. The
neighbourhoods of $0$ are the kernels of the canonical projections
${\mathfrak{g}}\to{\mathfrak{g}}_{\alpha}$.

The continuous cohomology of
${\mathfrak{g}}=\varprojlim{\mathfrak{g}}_{\alpha}$ is defined by

$H^{\bullet}({\mathfrak{g}}):=\varinjlim_{\alpha}H^{\bullet}({\mathfrak{g}}_{\alpha}).$

Its homology is the full dual:

$H_{\bullet}({\mathfrak{g}}):=\operatorname{Hom}_{F}(H^{\bullet}({\mathfrak{g}}),F)\cong\varprojlim
H_{\bullet}({\mathfrak{g}}_{\alpha})$

Each homology group is a Hausdorff topological vector space.

Continuous cohomology can be computed using continuous Chevalley-Eilenberg
cochains:

$C^{\bullet}({\mathfrak{g}}):=\varinjlim_{\alpha}\operatorname{Hom}_{F}(\Lambda^{\bullet}{\mathfrak{g}}_{\alpha},F)$

with the usual differential.

If, instead, ${\mathfrak{g}}=\bigoplus_{m}{\mathfrak{g}}_{m}$ is a graded Lie
algebra, then the homology and cohomology of ${\mathfrak{g}}$ are also graded.
This follows from the fact that the grading of ${\mathfrak{g}}$ induces a
grading of the Chevalley-Eilenberg chains and cochains of ${\mathfrak{g}}$.

###  2.3. Prounipotent groups and pronilpotent Lie algebras

A prounipotent $F$-group is a proalgebraic group that is an inverse limit of
unipotent $F$-groups.

A pronilpotent Lie algebra over a $F$ 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 $F$-groups and the category of pronilpotent Lie algebras over $F$.

###  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

$V_{\mathbb{C}}\cong\bigoplus_{m\in{\mathbb{Z}}}\operatorname{Gr}^{W}_{\bullet}V_{\mathbb{C}}$

of the complex part $V_{\mathbb{C}}$ of a MHS $V$ 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
$\varprojlim{\mathfrak{g}}_{\alpha}$ of Lie algebras in the category of MHS is
an Ind-MHS — that is, an ind-object of the category of MHS.

If the weight graded quotients of ${\mathfrak{g}}$ are finite dimensional,
then the exactness of the functor $\operatorname{Gr}^{W}_{\bullet}$ on the
category of MHS implies that there are natural isomorphisms

$\operatorname{Gr}^{W}_{\bullet}H_{\bullet}({\mathfrak{g}})\cong
H_{\bullet}(\operatorname{Gr}^{W}_{\bullet}{\mathfrak{g}})\text{ and
}\operatorname{Gr}^{W}_{\bullet}H^{\bullet}({\mathfrak{g}})\cong
H^{\bullet}(\operatorname{Gr}^{W}_{\bullet}{\mathfrak{g}}).$

##  3\. Presentations of Pronilpotent Lie Algebras

Here we recall how each pronilpotent and each positively graded Lie algebra
${\mathfrak{n}}$ has a “minimal presentation” with generating set isomorphic
to $H_{1}({\mathfrak{n}})$ and relations isomorphic to
$H_{2}({\mathfrak{n}})$. Denote the free Lie algebra (over $F$) generated by a
vector space $V$ by ${\mathbb{L}}(V)$. This is a graded Lie algebra: degree
$m$ component ${\mathbb{L}}_{m}(V)$ consists of the homogeneous Lie words of
degree $m$. The $m$th term of its lower central series is

$L^{m}{\mathbb{L}}(V)=\bigoplus_{r\geq m}{\mathbb{L}}_{r}(V).$

The lower central series filtration defines a topology on ${\mathbb{L}}(V)$.
Its completion in this topology is the degree completion of ${\mathbb{L}}(V)$:

${\mathbb{L}}(V)^{\wedge}=\varprojlim_{m}{\mathbb{L}}(V)/L^{m}\cong\prod_{m\geq
1}{\mathbb{L}}_{m}(V).$

When $V$ is finite dimensional, ${\mathbb{L}}(V)^{\wedge}$ is pronilpotent.

Suppose that ${\mathfrak{n}}$ is a pronilpotent Lie algebra. For simplicity,
we assume that $H_{1}({\mathfrak{n}})$ is finite dimensional. Each choice of a
continuous splitting of the canonical surjection ${\mathfrak{n}}\to
H_{1}({\mathfrak{n}})$ induces a continuous surjection

${\mathbb{L}}(H_{1}({\mathfrak{n}}))^{\wedge}\to{\mathfrak{n}}.$

The ideal of relations ${\mathfrak{r}}$ is the kernel of this homomorphism.
Note that ${\mathfrak{r}}$ is contained in $L^{2}{\mathbb{L}}(V)^{\wedge}$. By
the Lie algebra analogue of a theorem of Hopf (cf. [14, Prop. 5.6]) there is a
natural isomorphism

(1)
$H_{2}({\mathfrak{n}})\cong{\mathfrak{r}}/[{\mathfrak{r}},{\mathbb{L}}(V)^{\wedge}].$

The image of any continuous section $\psi$ of ${\mathfrak{r}}\to
H_{2}({\mathfrak{n}})$ is a minimal set of relations:

${\mathfrak{n}}\cong{\mathbb{L}}(H_{1}({\mathfrak{n}}))^{\wedge}/(\operatorname{im}\psi).$

Similarly, if ${\mathfrak{n}}=\bigoplus_{m\geq 1}{\mathfrak{n}}_{m}$ is a
graded Lie algebra that is generated by ${\mathfrak{n}}_{1}\cong
H_{1}({\mathfrak{n}})$, it has a presentation of the form

${\mathfrak{n}}\cong{\mathbb{L}}(H_{1}({\mathfrak{n}}))/(\operatorname{im}\psi)$

where $\psi:H_{2}({\mathfrak{n}})\to{\mathbb{L}}(H_{1}({\mathfrak{n}}))$ is an
injective graded linear mapping. Observe that a minimal space of relations of
degree $n$ is the image of

$\psi_{n}:H_{2}({\mathfrak{n}})_{n}\to{\mathbb{L}}_{n}(H_{1}({\mathfrak{n}})).$

###  3.1. Chains

If ${\mathfrak{n}}=\bigoplus_{m\geq 1}{\mathfrak{n}}_{m}$ is a graded Lie
algebra then its Chevalley-Eilenberg complex is graded. The rows of the
diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda^{2}{\mathfrak{n}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\phantom{x},\phantom{x}]}$$\textstyle{{\mathfrak{n}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda^{3}{\mathfrak{n}}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J}$$\textstyle{{\mathfrak{n}}_{1}\otimes{\mathfrak{n}}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[\phantom{x},\phantom{x}]}$$\textstyle{{\mathfrak{n}}_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

are the weight444Here 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 Chevalley-Eilenberg chains of ${\mathfrak{n}}$. Here
$J$ is the “Jacobi identity” map

(2) $J(x\wedge y\wedge z)=x\otimes[y,z]+y\otimes[z,x]+z\otimes[x,y].$

The homology of the $j$th row ($j=1,2$) is the weight $j$ summand
$H_{\bullet}({\mathfrak{n}})_{j}$ of $H_{\bullet}({\mathfrak{n}})$. The dual
complexes compute the weight $j$ summands of $H^{\bullet}({\mathfrak{n}})$.
Observe that the subalgebra

$\bigoplus_{m\geq 0}H^{m}({\mathfrak{n}})_{m}$

is generated by
$H^{1}({\mathfrak{n}})_{1}=\operatorname{Hom}({\mathfrak{n}}_{1},F)$. If
$\Lambda^{2}{\mathfrak{n}}_{1}\to{\mathfrak{n}}_{2}$ is surjective, then
$H^{2}({\mathfrak{n}})_{2}=0$, which implies that
$H^{m}({\mathfrak{n}})_{m}=0$ for all $m\geq 2$.

Note that when ${\mathfrak{n}}$ is generated by ${\mathfrak{n}}_{1}$,
$H_{1}({\mathfrak{n}})={\mathfrak{n}}_{1}$ and the weight $2$ component of
$\psi:H_{2}({\mathfrak{n}})\to{\mathbb{L}}_{2}({\mathfrak{n}}_{1})=\Lambda^{2}{\mathfrak{n}}_{1}$
is dual to the cup product $\Lambda^{2}H^{1}({\mathfrak{n}})\to
H^{2}({\mathfrak{n}})$.

###  3.2. Quadratic presentations

A graded Lie algebra ${\mathfrak{n}}$ is quadratically presented if it has a
graded presentation of the form

${\mathfrak{n}}={\mathbb{L}}(V)/(R)$

where $V$ is in weight 1 and $R$ is a subspace of ${\mathbb{L}}_{2}(V)$. A
pronilpotent Lie algebra ${\mathfrak{n}}$ with $H_{1}({\mathfrak{n}})$ finite
dimensional is quadratically presented if it has a presentation of the form

${\mathfrak{n}}={\mathbb{L}}(V)^{\wedge}/(R)$

where $R$ is a subspace of ${\mathbb{L}}_{2}(V)$. In this case
${\mathfrak{n}}$ is isomorphic to its degree completion
$\prod_{m>0}\operatorname{Gr}^{m}_{\mathrm{LCS}}{\mathfrak{n}}$ as a
pronilpotent Lie algebra. The following result is a straightforward
consequence of the definitions.

######  Proposition 3.1.

Suppose that ${\mathfrak{n}}$ is a pronilpotent Lie algebra with
$H_{1}({\mathfrak{n}})$ finite dimensional. If ${\mathfrak{n}}$ is
quadratically presented, then
$\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{n}}$ is quadratically
presented and ${\mathfrak{n}}$ is isomorphic (as a pronilpotent Lie algebra)
to the degree completion of
$\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{n}}$:

${\mathfrak{n}}\cong\prod_{m\geq
1}\operatorname{Gr}^{m}_{\mathrm{LCS}}{\mathfrak{n}}$

Conversely, if $\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{n}}$ is
quadratically presented and ${\mathfrak{n}}$ is isomorphic to the degree
completion of $\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{n}}$, then
${\mathfrak{n}}$ is quadratically presented.

Quadratically presented graded Lie algebras have a standard characterization.

######  Lemma 3.2.

A graded Lie algebra ${\mathfrak{n}}$ that is generated in weight $1$ is
quadratically presented if and only if $H_{2}({\mathfrak{n}})$ has weight $2$
or, equivalently, the cup product $\Lambda^{2}H^{1}({\mathfrak{n}})\to
H^{2}({\mathfrak{n}})$ is surjective.

###### Proof.

Suppose that ${\mathfrak{n}}={\mathbb{L}}(V)/{\mathfrak{r}}$ is a minimal
presentation of ${\mathfrak{n}}$. This means that
${\mathfrak{r}}\subset[{\mathbb{L}},{\mathbb{L}}]$. Set
${\mathfrak{f}}={\mathbb{L}}(V)$. There is a natural isomorphism

$H_{2}({\mathfrak{n}})\cong{\mathfrak{r}}/[{\mathfrak{f}},{\mathfrak{r}}].$

A minimal graded space of generators of ${\mathfrak{r}}$ is the image of any
section of
${\mathfrak{r}}\to{\mathfrak{r}}/[{\mathfrak{f}},{\mathfrak{r}}]=H_{2}({\mathfrak{n}})$.
The result follows as ${\mathfrak{n}}$ is quadratically presented if and only
if this set of minimal generators of ${\mathfrak{r}}$ has degree $2$. ∎

Since morphisms of graded Lie algebras induce graded morphisms of their
homology (and cohomology), we deduce:

######  Corollary 3.3.

Suppose that ${\mathfrak{g}}\to{\mathfrak{n}}$ is a surjection of graded Lie
algebras, both generated in weight $1$. If ${\mathfrak{g}}$ is quadratically
presented and $H_{2}({\mathfrak{g}})\to H_{2}({\mathfrak{n}})$ is surjective,
then ${\mathfrak{n}}$ is quadratically presented.

##  4\. Relative Completion of Discrete Groups

Suppose that $\Gamma$ is a discrete group and that $R$ is a reductive
algebraic group over a field $F$ of characteristic zero. The completion of
$\Gamma$ relative to a Zariski dense representation $\rho:\Gamma\to R(F)$ is a
proalgebraic $F$-group ${\mathcal{G}}$ which is an extension of $R$ by a
prounipotent group, and a homomorphism
${\hat{\rho}}:\Gamma\to{\mathcal{G}}(F)$ such that the composite

$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\hat{\rho}}}$$\textstyle{{\mathcal{G}}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R(F)}$

is $\rho$. It is universal for such groups: if $G$ is a proalgebraic $F$ group
that is an extension of $R$ by a prounipotent group, and if $\phi:\Gamma\to
G(F)$ is a homomorphism whose composition with $G\to R$ is $\rho$, then there
is a homomorphism ${\hat{\phi}}:{\mathcal{G}}\to G$ of proalgebraic $F$-groups
such that the diagram

$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\hat{\rho}}}$$\scriptstyle{\phi}$$\textstyle{{\mathcal{G}}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\hat{\phi}}}$$\textstyle{G(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R(F)}$

commutes.

When $R$ is trivial, $\rho$ is trivial and ${\mathcal{G}}={\mathcal{U}}$ is
the unipotent completion of $\Gamma$ over $F$.

When discussing the mixed Hodge structure on a relative completion of the
fundamental group of a compact Kähler manifold $X$, we need to be able to
compare the completion of $\pi_{1}(X,x)$ over ${\mathbb{R}}$ (or
${\mathbb{Q}}$) with its completion over ${\mathbb{C}}$. For this reason we
need to discuss the behaviour of relative completion under base change.

If $K$ is an extension field of $F$ and $\rho_{K}:\Gamma\to R(K)$ is Zariski
dense over $K$, then one has the relative completion ${\mathcal{G}}_{K}$ of
$\Gamma$ with respect to $\rho_{K}$. It is an extension of $R\times_{F}K$ by a
prounipotent group. The universal mapping property of ${\mathcal{G}}_{K}$
implies that the homomorphism $\Gamma\to{\mathcal{G}}(K)$ induces a
homomorphism ${\mathcal{G}}_{K}\to{\mathcal{G}}\times_{F}K$ of proalgebraic
$K$-groups. This homomorphism is an isomorphism. This is easily seen when each
irreducible representation $V$ of $R$ remains irreducible after extending
scalars from $F$ to $K$. This follows directly from the cohomological results
below.

###  4.1. Cohomology

We continue with the notation above, where ${\mathcal{G}}$ is the relative
completion of $\Gamma$. When $R$ is reductive, the structure of
${\mathfrak{g}}$ and ${\mathfrak{u}}$ are closely related to the cohomology of
$\Gamma$ with coefficients in rational representations of $R$.

For each rational representation $V$ of $R$ there are natural isomorphisms

$\operatorname{Hom}_{R}(H_{\bullet}({\mathfrak{u}}),V)\cong[H^{\bullet}({\mathfrak{u}})\otimes
V]^{R}\cong H^{\bullet}({\mathfrak{g}},V)^{\pi_{0}(R)}.$

The homomorphism $\Gamma\to{\mathcal{G}}(F)$ induces a homomorphism

(3) $H^{\bullet}({\mathfrak{g}},V)^{\pi_{0}(R)}\to H^{\bullet}(\Gamma,V)$

It is an isomorphism in degrees $\leq 1$ and an injection in degree 2.

Denote the set of isomorphism classes of finite dimensional irreducible
representations of $R$ by $\check{R}$. Fix an $R$-module $V_{\lambda}$ in each
isomorphism class $\lambda\in\check{R}$. If each irreducible representation of
$R$ is absolutely irreducible555This is the case when $R={\mathrm{Sp}}_{g}$
over any field of characteristic zero. and if $H^{j}(\Gamma,V)$ is finite
dimensional for all rational representations $V$ of $R$ when $j\leq 2$, then
there is an isomorphism

$\prod_{\lambda\in\check{R}}[H^{1}(\Gamma,V_{\lambda})]^{\ast}\otimes_{F}V_{\lambda}\cong
H_{1}({\mathfrak{u}})$

of topological modules, and a continuous $R$-invariant surjection

$\prod_{\lambda\in\check{R}}[H^{2}(\Gamma,V_{\lambda})]^{\ast}\otimes_{F}V_{\lambda}\to
H_{2}({\mathfrak{u}}).$

In both cases, the LHS has the product topology.

######  Lemma 4.1.

Suppose that $\rho:\Gamma\to R(F)$ is a Zariski dense representation from a
discrete group to a reductive group. If $\phi:\Gamma^{\prime}\to\Gamma$ is a
homomorphism that such that $\rho\circ\phi$ is Zariski dense and such that,
for all rational representations $V$ of $R$,

$H^{j}(\Gamma,V)\to H^{j}(\Gamma^{\prime},V)$

is an isomorphism when $j=1$ and injective when $j=2$, then the completion
${\mathcal{G}}^{\prime}\to{\mathcal{G}}$ of $\phi$ relative to $\rho$ is an
isomorphism.

###### Proof.

This follows directly from the cohomological results above and that fact that
a homomorphism of pronilpotent Lie algebras
${\mathfrak{u}}^{\prime}\to{\mathfrak{u}}$ is an isomorphism if and only if
$H^{j}({\mathfrak{u}})\to H^{j}({\mathfrak{u}}^{\prime})$ is an isomorphism in
degree 1 and injective in degree 2. ∎

###  4.2. Hodge theory

Suppose that $X$ is the complement of a normal crossings divisor in a compact
Kähler manifold. Suppose that $F={\mathbb{Q}}$ or ${\mathbb{R}}$ and that
${\mathbb{V}}$ is a polarized variation of $F$-Hodge structure (PVHS) over
$X$. Pick a base point $x_{o}\in X$. Denote the fiber over ${\mathbb{V}}$ over
$x_{o}$ by $V_{o}$. The Zariski closure of the image of the monodromy
representation

$\rho:\pi_{1}(X,x_{o})\to\operatorname{Aut}(V_{o})$

is a reductive $F$-group. Denote it by $R$. Then one has the relative
completion ${\mathcal{G}}$ of $\pi_{1}(X,x_{o})$ with respect to
$\rho:\pi_{1}(X,x_{o})\to R(F)$.

######  Theorem 4.2 ([13]).

The coordinate ring ${\mathcal{O}}({\mathcal{G}})$ is a Hopf algebra in the
category of Ind-mixed Hodge structures over $F$. It has the property that
$W_{-1}{\mathcal{O}}({\mathcal{G}})=0$ and
$W_{0}{\mathcal{O}}({\mathcal{G}})={\mathcal{O}}(R)$.

A slightly weaker version of the theorem is stated in terms of Lie algebras.
Denote the prounipotent radical of ${\mathcal{G}}$ by ${\mathcal{U}}$. Denote
their Lie algebras by ${\mathfrak{g}}$ and ${\mathfrak{u}}$, and the Lie
algebra of $R$ by ${\mathfrak{r}}$.

######  Corollary 4.3 ([13]).

The Lie algebra ${\mathfrak{g}}$ is a Lie algebra in the category of pro-mixed
Hodge structures over $F$. It has the property that

${\mathfrak{g}}=W_{0}{\mathfrak{g}},\
{\mathfrak{u}}=W_{-1}{\mathfrak{g}},\text{ and
}\operatorname{Gr}^{W}_{0}{\mathfrak{g}}\cong{\mathfrak{r}}.$

If ${\mathbb{V}}$ is a PVHS over $X$ with fiber $V_{o}$ over the basepoint
$x_{o}$, then the composite

$H^{\bullet}({\mathfrak{g}},V_{o})^{\pi_{0}(R)}\to
H^{\bullet}(\Gamma,V_{o})\to H^{\bullet}(X,{\mathbb{V}})$

of (3) with the canonical homomorphism is a morphism of MHS.

The existence of the mixed Hodge structure on ${\mathfrak{u}}$ in the
unipotent case and when $X$ is not necessarily compact is due to Morgan [24]
and Hain [10].

###  4.3. Presentations

We continue with the notation of Section 4.2. The problem of computing
presentations of ${\mathfrak{g}}$ and ${\mathfrak{u}}$ is simplified as the
exactness of $\operatorname{Gr}^{W}_{\bullet}$ implies that there is a natural
isomorphism

${\mathfrak{g}}\cong\prod_{m\leq 0}\operatorname{Gr}^{W}_{m}{\mathfrak{g}}$

of topological Lie algebras. In order to determine ${\mathfrak{g}}$, it
suffices to compute $\operatorname{Gr}^{W}_{\bullet}{\mathfrak{u}}$ as a
graded Lie algebra in the category of $R$-modules. Exactness of
$\operatorname{Gr}^{W}_{\bullet}$ implies that
$H^{\bullet}(\operatorname{Gr}^{W}_{\bullet}{\mathfrak{u}})\cong\operatorname{Gr}^{W}_{\bullet}H^{\bullet}({\mathfrak{u}})$.

When $H_{1}({\mathfrak{u}})$ is finite dimensional, the Lie algebra
$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{u}}$ has a presentation of the form

$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{u}}\cong{\mathbb{L}}(\operatorname{Gr}^{W}_{\bullet}H_{1}({\mathfrak{u}}))/(\operatorname{im}\psi)$

in the category of Ind $R$-modules for a suitable $R$-module map
$\psi:H_{2}(\operatorname{Gr}^{W}_{\bullet}{\mathfrak{u}})\to
L^{2}{\mathbb{L}}(\operatorname{Gr}^{W}_{\bullet}H_{1}({\mathfrak{u}}))$.

The following result is a formal consequence of results stated above and the
fact, due to Deligne (cf. [35, Thm. 2.9]), that if ${\mathbb{V}}$ is a PVHS
over $X$ of weight $m$, then the smooth forms on $X$ with coefficients in
${\mathbb{V}}$ is a Hodge complex and $H^{j}(X,{\mathbb{V}})$ has a Hodge
structure of weight $m+j$.

######  Theorem 4.4 ([13, Thm. 13.14]).

If $X$ is a compact Kähler manifold, then

  1. (i)

the MHS on $H^{j}({\mathfrak{u}})$ is pure of weight $j$ when $j\leq 2$,

  2. (ii)

the weight filtration of ${\mathfrak{u}}$ is its lower central series
$W_{-m}{\mathfrak{u}}=L^{m}{\mathfrak{u}}$,

  3. (iii)

${\mathfrak{u}}$ 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 $A$ is a commutative ring and that $H$ is a free $A$-module of
rank $2g$ with a unimodular, skew symmetric bilinear pairing
${\langle\phantom{x},\phantom{x}\rangle}$. The symplectic group
${\mathrm{Sp}}(H)$ is defined to be
$\operatorname{Aut}(H,{\langle\phantom{x},\phantom{x}\rangle})$. The choice of
a symplectic basis $a_{1},\dots,a_{g},b_{1},\dots,b_{g}$ of $H$ gives an
isomorphism of ${\mathrm{Sp}}(H)$ with ${\mathrm{Sp}}_{g}(A)$.

When $A$ is a field $F$ of characteristic zero, ${\mathrm{Sp}}(H)$ is a simply
connected simple algebraic $F$-group all of whose irreducible representations
are absolutely irreducible. Denote its Lie algebra by $\mathfrak{sp}(H)$. This
is isomorphic to the Lie algebra $\mathfrak{sp}_{g}(F)$.

Every irreducible representation of ${\mathrm{Sp}}(H)$ is a submodule of some
tensor power $H^{\otimes m}$ of $H$. Fix a symplectic basis of $H$. Set

$\theta:=a_{1}\wedge b_{1}+\dots+a_{g}\wedge b_{g}\in\Lambda^{2}H.$

Every linear automorphism of $H$ extends to a derivation of the tensor algebra
$T(H)$ on $H$. With this convention, $\mathfrak{sp}(H)$ is the subalgebra of
derivations of $T(H)$ that annihilate $\theta$:

$\mathfrak{sp}(H)=\\{\delta:H\to H:\delta(\theta)=0\\}.$

The abelian subalgebra ${\mathfrak{h}}$ of $\mathfrak{sp}(H)$ consisting of
those derivations $t=(t_{1},\dots,t_{g})$ that act on $H$ via

$t\cdot a_{j}=t_{j}a_{j}\text{ and }t\cdot b_{j}=-t_{j}b_{j}.$

is a Cartan subalgebra. The derivations

$S_{i,j}:=a_{i}\partial/\partial a_{j}-b_{j}\partial/\partial b_{i}\
(i<j)\text{ and }F_{i,j}:=a_{j}\partial/\partial b_{i}+a_{i}\partial/\partial
b_{j}(\text{all }i,j)$

of $T(H)$ span a nilpotent subalgebra of $\mathfrak{sp}(H)$ which, together
with ${\mathfrak{h}}$, span a Borel subalgebra of $\mathfrak{sp}(H)$.

A fundamental set of highest weights with respect to this Borel subalgebra is
$\\{\lambda_{j}:1\leq j\leq g\\}$, where $\lambda_{j}:{\mathfrak{h}}\to F$ is
the function defined by

$\lambda_{k}(t)=t_{1}+\dots+t_{k}.$

We will denote the irreducible $\mathfrak{sp}(H)$ module with highest weight
$\lambda=\sum_{j=1}^{g}n_{j}\lambda_{j}$ by $V(\lambda)$.

The exterior algebra $\Lambda^{\bullet}H$ is an algebra in the category of
${\mathrm{Sp}}(H)$-modules. The fundamental representation $V(\lambda_{k})$ is
the degree $k$ part of the quotient of it by the ideal $(\theta)$:

$V(\lambda_{k})=\Lambda^{k}_{0}H:=\Lambda^{k}H/(\theta\wedge\Lambda^{k-2}H).$

Recall that every irreducible $\mathfrak{sp}(H)$-module has a non-degenerate
invariant bilinear pairing. Equivalently, every irreducible
$\mathfrak{sp}(H)$-module is isomorphic to its dual.

###  5.1. Stability in the representation ring of ${\mathrm{Sp}}(H)$

Fix a sequence of inclusions

$\mathfrak{sp}_{1}\subset\mathfrak{sp}_{2}\subset\mathfrak{sp}_{3}\subset\cdots.$

Each of these induces an inclusion
${\mathrm{Rep}}(\mathfrak{sp}_{h})\hookrightarrow{\mathrm{Rep}}(\mathfrak{sp}_{h+1})$
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
$\mathfrak{sp}_{h}$ with highest weight $\lambda$ to the irreducible
representation of $\mathfrak{sp}_{h+1}$ 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 $C$ is a compact Riemann surface of genus $g$ and that $x\in C$.
Denote the Lie algebra of the unipotent completion ${\mathcal{P}}$ of
$\pi_{1}(C,x)$ by ${\mathfrak{p}}$. Set $H=H_{1}(C;{\mathbb{Q}})$. Let
$a_{1},\dots,a_{g},b_{1},\dots,b_{g}$ be a symplectic basis of $H$. Applying
the results of the previous section, ${\mathfrak{p}}$ has a natural MHS whose
associated graded has the presentation:

$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{p}}\cong{\mathbb{L}}(H)/(\scriptstyle{\sum_{j=1}^{g}}[a_{j},b_{j}]).$

This is a Lie algebra in the category of ${\mathrm{Sp}}(H)$-modules.666This
statement is also a consequence of Labute’s Theorem [22].

The $g\geq 3$ case of the following result is [14, Prop. 8.4]. The genus $2$
case is proved similarly.

######  Proposition 5.1.

The highest weight decomposition of the first four weight graded quotients of
${\mathfrak{p}}$ are:

$\operatorname{Gr}^{W}_{-m}{\mathfrak{p}}=\begin{cases}V(\lambda_{1})&m=1,\
g\geq 1,\cr V(\lambda_{2})&m=2,\ g\geq 2,\cr V(\lambda_{1}+\lambda_{2})&m=3,\
g\geq 2,\cr
V(\lambda_{1}+\lambda_{3})+V(2\lambda_{1})+V(2\lambda_{1}+\lambda_{2})&m=4,\
g\geq 3.\end{cases}$

The exactness of $\operatorname{Gr}^{W}_{\bullet}$ implies that there are
natural graded Lie algebra isomorphisms

$\operatorname{Gr}^{W}_{\bullet}\operatorname{Der}{\mathfrak{p}}\cong\operatorname{Der}\operatorname{Gr}^{W}_{\bullet}{\mathfrak{p}}\text{
and
}\operatorname{Gr}^{W}_{\bullet}\operatorname{OutDer}{\mathfrak{p}}\cong\operatorname{OutDer}\operatorname{Gr}^{W}_{\bullet}{\mathfrak{p}}.$

Since $\operatorname{Gr}^{W}_{\bullet}{\mathfrak{p}}$ (and hence
${\mathfrak{p}}$ as well) has trivial center [2], the sequence

$0\to\operatorname{Gr}^{W}_{\bullet}{\mathfrak{p}}\to\operatorname{Der}{\mathfrak{p}}\to\operatorname{OutDer}{\mathfrak{p}}\to
0$

is exact.

######  Corollary 5.2 ([14, Cor. 9.4]).

When $g\geq 3$, we have

$\operatorname{Gr}^{W}_{-m}\operatorname{OutDer}{\mathfrak{p}}=\begin{cases}V(\lambda_{3})&m=1,\cr
V(2\lambda_{2})&m=2,\cr
V(2\lambda_{1}+\lambda_{3})+V(3\lambda_{1})&m=3.\end{cases}$

∎

##  6\. Completions of Mapping Class Groups

In this section $g$ and $n$ are non-negative integers satisfying $2g-2+n>0$.

###  6.1. Relative completion of mapping class groups

Let $S$ be a closed oriented surface of genus $g$. Set
$H=H_{1}(S;{\mathbb{Z}})$. Fix a symplectic basis of $H$. The action of
$\Gamma_{g,n}$ on $S$ induces a surjective homomorphism

$\rho:\Gamma_{g,n}\to\operatorname{Aut}(H,{\langle\phantom{x},\phantom{x}\rangle})\cong{\mathrm{Sp}}_{g}({\mathbb{Z}}).$

Since ${\mathrm{Sp}}_{g}({\mathbb{Z}})$ is Zariski dense in
${\mathrm{Sp}}_{g}({\mathbb{Q}})$, we have the completion
${\mathcal{G}}_{g,n}$ of $\Gamma_{g,n}$ with respect to $\rho$. It is an
extension

$1\to{\mathcal{U}}_{g,n}\to{\mathcal{G}}_{g,n}\to{\mathrm{Sp}}_{g}\to 1$

of the symplectic group by a prounipotent group. Denote the Lie algebras of
${\mathcal{G}}_{g,n}$ and ${\mathcal{U}}_{g,n}$ by ${\mathfrak{g}}_{g,n}$ and
${\mathfrak{u}}_{g,n}$, respectively. When $n=0$, it will be suppressed, so
that ${\mathcal{G}}_{g}={\mathcal{G}}_{g,0}$,
${\mathfrak{g}}_{g}={\mathfrak{g}}_{g,0}$, etc.

When $m>0$, the level $m$ subgroup $\Gamma_{g,n}[m]$ of $\Gamma_{g,n}$ is
defined to be the kernel of the mod $m$ reduction
$\rho_{m}:\Gamma_{g,n}\to{\mathrm{Sp}}_{g}({\mathbb{Z}}/m)$ of $\rho$. When
$g\geq 3$, the homomorphism
$\Gamma_{g,n}[m]\to{\mathcal{G}}_{g,n}({\mathbb{Q}})$ is the completion of
$\Gamma_{g,n}$ relative to
$\rho:\Gamma_{g,n}\to{\mathrm{Sp}}_{g}({\mathbb{Q}})$, [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 $g\geq 3$. When $g\leq 2$ the completion
of $\Gamma_{g,n}[m]$ depends non-trivially on $m$.

The homomorphism $T_{g,n}\to{\mathcal{U}}_{g,n}({\mathbb{Q}})$ induces a
homomorphism ${\mathcal{T}}_{g,n}\to{\mathcal{U}}_{g,n}$ where
${\mathcal{T}}_{g,n}$ denotes the unipotent completion of $T_{g,n}$.

######  Theorem 6.1 ([11],[14, Thm. 3.4]).

If $g\geq 2$, then this homomorphism is surjective. If $g\geq 3$, it has non-
trivial kernel isomorphic to the additive group ${\mathbb{G}_{a}}$. The
sequence

$0\to{\mathbb{G}_{a}}\to{\mathcal{T}}_{g,n}\to{\mathcal{U}}_{g,n}\to 1$

is a non-trivial central extension.

The action of $\Gamma_{g,1}$ on ${\mathcal{P}}$ induces an action
$\Gamma_{g,1}\to\operatorname{Aut}{\mathfrak{p}}$ that preserves the lower
central series filtration. The action of $\Gamma_{g,1}$ on
$\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{p}}$ factors through
$\rho:\Gamma_{g,1}\to{\mathrm{Sp}}(H)$. The universal mapping property of
relative completion, implies that this action induces an action
${\mathcal{G}}_{g,1}\to\operatorname{Aut}{\mathfrak{p}}$. The induced action
${\mathfrak{g}}_{g,1}\to\operatorname{Der}{\mathfrak{p}}$ on Lie algebras is a
tool for understanding ${\mathfrak{u}}_{g,1}$. It induces the outer action
${\mathfrak{g}}_{g}\to\operatorname{OutDer}{\mathfrak{p}}$.

###  6.2. Mapping class groups as fundamental groups of smooth varieties

The moduli space ${\mathcal{M}}_{g,n}$ of smooth complex projective curves of
type $(g,n)$ is the quotient of the Teichmüller space ${\mathcal{X}}_{g,n}$ by
$\Gamma_{g,n}$, which acts properly discontinuously on Teichmüller space.
Although this action is not fixed point free, it is when restricted to
$\Gamma_{g,n}[m]$ when $m\geq 3$. The quotient
${\mathcal{M}}_{g,n}[m]:=\Gamma_{g,n}[m]\backslash{\mathcal{X}}_{g,n}$ is a
smooth quasi projective variety. Consequently, $\Gamma_{g,n}[m]$ is the
fundamental group of a smooth quasi-projective variety for all $m\geq 3$.

One can realize $\Gamma_{g,n}$ as the fundamental group of a smooth quasi-
projective variety $X$ by fixing $m\geq 3$ and taking

$X={\mathcal{M}}_{g,n}[m]\times_{{\mathrm{Sp}}_{g}({\mathbb{Z}}/m)}Y$

where $Y$ is a simply connected projective manifold on which
${\mathrm{Sp}}_{g}({\mathbb{Z}}/m)$ acts fixed point freely.777Such varieties
$Y$ were constructed by Serre. The argument can be found in [30, IX,§4.2].
Alternatively, when $g+n>3$ and $g\geq 2$, one can take $X$ to be the
complement of the locus in ${\mathcal{M}}_{g,n}$ of $n$-pointed curves with a
non-trivial automorphism (cf. [15, Prop. 4.1]).

Define $\widehat{\Gamma}_{g,n}[m]$ to be the orbifold fundamental group of
${\mathcal{C}}_{g}^{n}[m]$, the $n$th power of the universal curve over
${\mathcal{M}}_{g}[m]$. It is an extension

$1\to\pi^{n}\to\widehat{\Gamma}_{g,n}[m]\to\Gamma_{g}[m]\to 1,$

where $\pi$ denotes the fundamental group of a smooth projective curve of
genus $g$. Since ${\mathcal{M}}_{g,n}[m]$ is a Zariski open subset of
${\mathcal{C}}_{g}^{n}[m]$, $\widehat{\Gamma}_{g,n}[m]$ is a quotient of
$\Gamma_{g,n}[m]$.

Since ${\mathcal{C}}_{g}^{n}[m]$ is a smooth quasi-projective variety when
$m\geq 3$, $\widehat{\Gamma}_{g,n}[m]$ is the fundamental group of a smooth
quasi-projective variety for all $m\geq 1$. (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 $\Gamma$ is one of the mapping class groups $\Gamma_{g,n}$ or
$\widehat{\Gamma}_{g,n}$. Then $\Gamma$ is the fundamental group of a pointed
smooth variety $({\mathcal{M}},x)$, where ${\mathcal{M}}$ is one of the smooth
varieties described above. Denote its completion relative to the homomorphism
$\rho:\Gamma\to{\mathrm{Sp}}(H)$ by ${\mathcal{G}}$ and it prounipotent
radical by ${\mathcal{U}}$. One has a family $f:{\mathcal{C}}\to{\mathcal{M}}$
of complete genus $g$ curves. The representation $\rho$ is the monodromy
representation of the local system $R^{1}f_{\ast}{\mathbb{Q}}$, which is a
polarized variation of Hodge structure. The Lie algebras ${\mathfrak{g}}$ of
${\mathcal{G}}$ and ${\mathfrak{u}}$ of ${\mathcal{U}}$ are Lie algebras in
the category of pro-mixed Hodge structures.

The Torelli subgroup $T$ of $\Gamma$ is the kernel of
$\rho:\Gamma\to{\mathrm{Sp}}(H)$. Denote the Lie algebra of its unipotent
completion by ${\mathfrak{t}}$.

The following result summarizes several results from Sections 3 and 4 of [14].

######  Proposition 6.2.

If $g\geq 3$, then

  1. (i)

the MHS on ${\mathfrak{u}}$ can be lifted to a MHS on ${\mathfrak{t}}$ so that
the central extension

(4) $0\to{\mathbb{Q}}(1)\to{\mathfrak{t}}\to{\mathfrak{u}}\to 0$

is a non-trivial central extension of Lie algebras in the category of pro-MHS.
Consequently, the associated graded sequence

$0\to{\mathbb{Q}}(1)\to\operatorname{Gr}^{W}_{\bullet}{\mathfrak{t}}\to\operatorname{Gr}^{W}_{\bullet}{\mathfrak{u}}\to
0$

is an exact sequence of graded Lie algebras in the category of
${\mathrm{Sp}}(H)$ modules.

  2. (ii)

the weight filtrations of ${\mathfrak{u}}$ and ${\mathfrak{t}}$ are their
lower central series:

$W_{-m}{\mathfrak{u}}=L^{m}{\mathfrak{u}}\text{ and
}W_{-m}{\mathfrak{t}}=L^{m}{\mathfrak{t}}.$

In particular,
$H_{1}({\mathfrak{u}})=H_{1}({\mathfrak{t}})=\operatorname{Gr}^{W}_{-1}{\mathfrak{u}}=\operatorname{Gr}^{W}_{-1}{\mathfrak{t}}$.

  3. (iii)

$H_{1}({\mathfrak{u}}_{g,n})\cong V(\lambda_{3})+V(\lambda_{1})^{n}$.

The spectral sequence of the central extension gives a Gysin sequence.

######  Corollary 6.3.

For all $g\geq 3$ there is a Gysin sequence

$0\to{\mathbb{Q}}(-1)\to H^{2}({\mathfrak{u}}_{g})\to
H^{2}({\mathfrak{t}}_{g})\to H^{1}({\mathfrak{u}}_{g})(-1)\cr\to
H^{3}({\mathfrak{u}}_{g})\to H^{3}({\mathfrak{t}}_{g})\to
H^{2}({\mathfrak{u}}_{g})(-1)\to\dots{}$

in the category of mixed Hodge structures. It remains exact after applying
$\operatorname{Gr}^{W}_{m}$ for all $m\in{\mathbb{Z}}$.

###  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, $K_{g,n}$ denotes the subgroup of $T_{g,n}$
generated by Dehn twists on bounding simple closed curves (BSCCs).888Recall
that a bounding pair in a surface $S$ consists of two disjoint non-separating
simple closed curves $B_{+}$ and $B_{-}$ which together divide the surface
into two components. A bounding pair map (BP map) consists of the product
$\smash{t_{B_{+}}t_{B_{-}}^{-1}}$ of a positive Dehn twist about $B_{+}$ and a
negative Dehn twist about $B_{-}$. The following result is a special case of a
result [26, Thm. A] of Putman.

######  Theorem 6.4 (Putman).

Suppose that $g\geq 3$ and that $n\geq 0$. If $S$ is a finite index subgroup
of $T_{g,n}$ that contains $K_{g,n}$, then the inclusion induces an
isomorphism $H_{1}(S,{\mathbb{Q}})\to H_{1}(T_{g,n},{\mathbb{Q}})$.

Denote the image of $K_{g,n}$ in $\widehat{\Gamma}_{g,n}$ by
$\widehat{K}_{g,n}$.

######  Corollary 6.5.

Suppose that $g\geq 3$ and that $n\geq 0$. If $S$ is a finite index subgroup
of $\widehat{T}_{g,n}$ that contains $\widehat{K}_{g,n}$, then the inclusion
induces an isomorphism $H_{1}(S,{\mathbb{Q}})\to
H_{1}(\widehat{T}_{g,n},{\mathbb{Q}})$.

###### Proof.

Observe that $\widehat{T}_{g,n}$ is the fundamental group of the $n$th power
of the universal curve over Torelli space $BT_{g}$ and that $T_{g,n}$ is the
fundamental group of the Zariski open subset obtained by removing the
“diagonals”. It follows that $T_{g,n}\to\widehat{T}_{g,n}$ is surjective.
Consequently, the inverse image $S^{\prime}$ of $S$ in $T_{g,n}$ is a finite
index subgroup that contains $K_{g,n}$. Now consider the diagram:

$\textstyle{S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{g,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{T}_{g,n}}$

Since the left-hand vertical arrow is surjective, it induces a surjection on
$H_{1}$. Putman’s result implies that top map induces an isomorphism on
rational $H_{1}$. The result follows as the right-hand vertical map induces an
isomorphism on rational $H_{1}$, which follows from [12, Prop. 5.2] and an
easy spectral sequence argument for $n$th power of the universal curve over
${\mathcal{T}}_{g}$. ∎

Putman’s result has the following analogue for mapping class groups.

######  Proposition 6.6.

Suppose that $g\geq 3$ and that $n\geq 0$. If $\Gamma$ is a finite index
subgroup of $\Gamma_{g,n}$ (resp. $\widehat{\Gamma}_{g,n}$) that contains
$K_{g,n}$ (resp. $\widehat{K}_{g,n}$), then for all
${\mathrm{Sp}}_{g}$-modules $V$, the map

$H^{\bullet}(\Gamma_{g,n},V)\to H^{\bullet}(\Gamma,V)$

induced by the inclusion $\Gamma\to\Gamma_{g,n}$ is an isomorphism in degrees
$\leq 1$ and an injection in degree $\geq 2$. In particular,

$H^{1}(\Gamma,V(\lambda)\cong\begin{cases}{\mathbb{Q}}&\lambda=\lambda_{3},\cr{\mathbb{Q}}^{n}&\lambda=\lambda_{1},\cr
0&\text{otherwise.}\end{cases}$

As in [12, §5], the vanishing of $H^{1}(\Gamma,{\mathbb{Q}})$ implies that the
Picard group of the corresponding moduli space of curves is finitely
generated.

###### Proof.

We will prove the $\Gamma_{g,n}$ case; the $\widehat{\Gamma}_{g,n}$ case is
similar and left to the reader. Since $V$ is a ${\mathbb{Q}}$-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 $\Gamma$ in ${\mathrm{Sp}}_{g}({\mathbb{Q}})$ has finite index in
${\mathrm{Sp}}_{g}({\mathbb{Z}})$ and is thus Zariski dense.

To prove that the induced map on cohomology is an isomorphism in degree 1,
write $\Gamma$ as an extension

$1\to\Gamma\cap T_{g,n}\to\Gamma\to L\to 1,$

where $L=\operatorname{im}\\{\Gamma\to{\mathrm{Sp}}_{g}({\mathbb{Z}})\\}$. The
kernel is a finite index subgroup $S$ of $T_{g,n}$ that contains $K_{g,n}$ and
so Putman’s theorem implies that
$H^{1}(S,{\mathbb{Q}})=H^{1}(T_{g,n},{\mathbb{Q}})$. Since $g\geq 3$,
Raghunathan’s vanishing theorem [28] implies the vanishing of $H^{1}(L,V)$,
and thus the injectivity of the restriction map

$H^{1}(\Gamma,V)\to H^{0}(L,H^{1}(S)\otimes
V)=\operatorname{Hom}_{L}(H_{1}(S),V).$

On the other hand, there is an isomorphism [12, p.113]

$H^{1}(\Gamma_{g,n},V)\to
H^{0}({\mathrm{Sp}}_{g}({\mathbb{Z}}),H^{1}(T_{g,n})\otimes
V)=\operatorname{Hom}_{{\mathrm{Sp}}_{g}}(H_{1}(T_{g,n}),V)$

Putman’s theorem implies that the restriction mapping
$H^{1}(\Gamma_{g,n},V)\to H^{1}(\Gamma,V)$ is an isomorphism.

The last assertion follows from the computation of
$H^{1}(\Gamma_{g,n},V(\lambda))$ in [12, §5]. ∎

Combining Putman’s theorem and its variants with Lemma 4.1 yields:

######  Corollary 6.7.

Suppose that $g\geq 3$ and $n\geq 0$.

  1. (i)

If $S$ is a finite index subgroup of $T_{g,n}$ that contains $K_{g,n}$, then
the inclusion of $S$ into $T_{g,n}$ induces an isomorphism on unipotent
completions.

  2. (ii)

If $\Gamma$ is a finite index subgroup of $\Gamma_{g,n}$ that contains
$K_{g,n}$, then the inclusion $\Gamma\to\Gamma_{g,n}$ induces an isomorphism
on completions relative to the standard representation $\rho$.

  3. (iii)

If $S$ is a finite index subgroup of $\widehat{T}_{g,n}$ that contains
$\widehat{K}_{g,n}$, then the inclusion of $S$ into $T_{g,n}$ induces an
isomorphism on unipotent completions.

  4. (iv)

If $\Gamma$ is a finite index subgroup of $\widehat{\Gamma}_{g,n}$ that
contains $\widehat{K}_{g,n}$, then the inclusion
$\Gamma\to\widehat{\Gamma}_{g,n}$ induces an isomorphism on completions
relative to the standard representation $\rho$.

There are interesting and natural examples of moduli spaces whose fundamental
groups do not contain the Torelli group, but do contain its Johnson subgroup.

######  Proposition 6.8.

When $g\geq 3$,

  1. (i)

(Boggi [4]) the fundamental group of the “Prym-level” moduli spaces [23, 5]
(these are called the “Prym-level mapping class groups”) of type $(g,n)$
contain $K_{g,n}$,

  2. (ii)

the fundamental group of the moduli space of genus $g$ curves with a
$(2g-2)$nd root of their canonical bundle contains $K_{g}$.

###### Sketch of Proof.

Let $(S;x_{1},\dots,x_{n})$ be an $n$-pointed reference surface of genus $g$,
where $S$ is compact. Set $S^{\prime}=S-\\{x_{1},\dots,x_{n}\\}$. The Prym-
level mapping subgroups of $\Gamma_{g,n}$ are kernels of homomorphisms
$\Gamma_{g,n}\to\operatorname{Out}G$, where $G$ is a finite quotient of
$\pi_{1}(S^{\prime},x)/W_{-3}$ and the weight filtration is pulled back from
that of the unipotent completion of $\pi_{1}(S^{\prime},x)$. Since $K_{g,n}$
acts trivially on $\pi_{1}(S^{\prime},x)/W_{-3}$, Prym-level mapping class
groups of type $(g,n)$ contain $K_{g,n}$. See [5, p. 179] for definitions.

As explained in [12, §13],999There is a typo in the first display of Theorem
13.3: it should be  $0\to
2H_{1}(C,{\mathbb{Z}}/n{\mathbb{Z}})\to{\mathcal{K}}_{n}^{(2)}\overset{\pi}{\to}{\mathrm{Sp}}_{g}({\mathbb{Z}}/n{\mathbb{Z}})\to
1.$ a result of Sipe [32] implies that when $g\geq 3$ the action of the
Torelli group $T_{g}$ on the $(2g-2)$nd roots of the canonical bundle factors
through the Johnson homomorphism
$T_{g}\to\Lambda^{3}H_{\mathbb{Z}}/(\theta\wedge H_{\mathbb{Z}})$. This
implies that $K_{g}$, and hence all $K_{g,n}$, act trivially on roots of the
canonical bundle when $g\geq 3$. ∎

###  6.4. Kabanov purity

Kabanov [21] proved that the natural map from
$H^{2}({\mathcal{M}}_{g},{\mathbb{V}})$ to the degree two intersection
cohomology of the Satake compactification of ${\mathcal{M}}_{g}$ is an
isomorphism when $g\geq 6$. When $3\leq g<6$ he proved a more technical
statement. These statements, via the arguments in [14, §7], imply the
following result, which bounds the weights of the relations in a presentation
of $\operatorname{Gr}^{W}_{\bullet}{\mathfrak{u}}_{g,n}$.

######  Theorem 6.9.

If $g\geq 6$, then $\operatorname{Gr}^{W}_{m}H^{2}({\mathfrak{u}}_{g,n})=0$
when $m\neq 2$. When $3\leq g<6$,
$\operatorname{Gr}^{W}_{m}H^{2}({\mathfrak{u}}_{g,n})=0$ when
$m\notin\\{2,3\\}$. ∎

This implies that the minimal presentations of ${\mathfrak{u}}_{g,n}$ has
quadratic relations when $g\geq 6$ and at worst cubic relations when $3\leq
g<6$.

##  7\. Presentations of ${\mathfrak{u}}_{g}$ and ${\mathfrak{t}}_{g}$ for
$g\geq 4$

In this section we prove that ${\mathfrak{u}}_{g,n}$,
$\hat{{\mathfrak{u}}}_{g,n}$, ${\mathfrak{t}}_{g,n}$ and
$\hat{{\mathfrak{t}}}_{g,n}$ are quadratically presented when $g\geq 4$ and
$n\geq 0$. This was proved in [14] when $g\geq 6$ using Kabanov’s Purity
Theorem. Here we combine Sakasai’s computations [29] with Kabanov purity to
extend this to the cases $g=4,5$. The same tools allow us to bound the
relations in genus $3$.

The starting point is to recall that
$\operatorname{Gr}^{W}_{m}{\mathfrak{t}}_{g}=\operatorname{Gr}^{W}_{m}{\mathfrak{u}}_{g}$
except when $m=-2$, where they differ by a copy of the trivial representation.
To compute $\operatorname{Gr}^{W}_{-3}{\mathfrak{u}}_{g}$, we consider the
homomorphism

(5)
$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{u}}_{g}\to\operatorname{OutDer}\operatorname{Gr}^{W}_{\bullet}{\mathfrak{p}}.$

Johnson’s fundamental work [19] implies that it is an isomorphism in weight
$-1$ when $g\geq 3$, in which case all groups are isomorphic to
$V(\lambda_{3})$.101010We could equally well have used the action
$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{u}}_{g,1}\to\operatorname{Der}\operatorname{Gr}^{W}_{\bullet}{\mathfrak{p}}$.
All results in the sequel have equivalent analogues for it. This is because
${\mathfrak{p}}$ has trivial center [2], which implies that
$\ker\\{{\mathfrak{u}}_{g,1}\to{\mathfrak{u}}_{g}\\}\cong\ker\\{\operatorname{Der}{\mathfrak{p}}\to\operatorname{OutDer}{\mathfrak{p}}\\}$.

######  Proposition 7.1 ([14, Thm. 10.1]).

The homomorphism (5) is an isomorphism in weight $-2$ for all $g\geq 3$. In
this case $\operatorname{Gr}^{W}_{-2}{\mathfrak{u}}_{g}\cong V(2\lambda_{2})$.

In this section we will show that (5) is injective in weight $-3$ for all
$g\geq 4$. In Section 9 we will show that it also injective in weight $-3$
when $g=3$, which will allow us to compute a minimal set of cubic relations in
genus $3$.

It is not known whether the outer action (5) is injective in any genus.
However, we believe (without much evidence) that it is “stably injective”.

######  Question 7.2.

Is there a monotonic unbounded function $w:{\mathbb{N}}\to{\mathbb{N}}$ such
that

$\operatorname{Gr}^{W}_{-m}{\mathfrak{u}}_{g,1}\to\operatorname{Gr}^{W}_{-m}{\mathfrak{p}}$

is injective when $m\leq w(g)$?

This is a weaker version of Question 9.7 in [16].

###  7.1. Quadratic relations

Let ${\mathfrak{n}}$ be ${\mathfrak{u}}_{g}$ or ${\mathfrak{t}}_{g}$ when
$g\geq 3$. Since $H^{1}({\mathfrak{n}})$ has weight $1$,
$H^{1}({\mathfrak{n}})=(\operatorname{Gr}^{W}_{-1}{\mathfrak{n}})^{\ast}$, the
cup product and the quadratic relations in
$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{n}}$ are related by the exact
sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\big{(}\operatorname{Gr}^{W}_{-2}{\mathfrak{n}}\big{)}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{[\phantom{x},\phantom{x}]}}$$\textstyle{\Lambda^{2}H^{1}({\mathfrak{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$cup$\textstyle{\operatorname{Gr}^{W}_{2}H^{2}({\mathfrak{n}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

which is the weight $2$ summand of the Chevalley-Eilenberg cochains of
${\mathfrak{n}}$.

######  Proposition 7.3 ([14, Lem. 10.2]).

As an ${\mathrm{Sp}}_{g}$-module, the second exterior power of
$V:=V(\lambda_{3})$ decomposes:

${\mathbb{L}}_{2}(V)=\Lambda^{2}V=\begin{cases}V(0)+V(2\lambda_{2})&g=3,\cr
V(0)+V(2\lambda_{2})+V(\lambda_{2}+\lambda_{4})+V(\lambda_{2})&g=4,\cr
V(0)+V(2\lambda_{2})+V(\lambda_{2}+\lambda_{4})+V(\lambda_{2})+V(\lambda_{4})&g=5,\cr
V(0)+V(2\lambda_{2})+V(\lambda_{2}+\lambda_{4})+V(\lambda_{2})+V(\lambda_{4})+V(\lambda_{6})&g\geq
6.\end{cases}$

∎

Since the image of the cup product $\Lambda^{2}H^{1}({\mathfrak{t}}_{g})\to
H^{2}({\mathfrak{t}}_{g})$ is dual to the set of quadratic relations in
$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{t}}_{g}$, we have:

######  Corollary 7.4 (Cf. [14, Thm. 10.1]).

The space of quadratic relations $R$ in a minimal presentation of
$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{t}}_{g}$ are dual to (and hence
isomorphic to) the image of the cup product:

$\operatorname{im}\\{\Lambda^{2}H^{1}(T_{g},{\mathbb{Q}})\to
H^{2}(T_{g},{\mathbb{Q}})\\}=\begin{cases}0&g=3,\cr
V(\lambda_{2}+\lambda_{4})+V(\lambda_{2})&g=4,\cr
V(\lambda_{2}+\lambda_{4})+V(\lambda_{2})+V(\lambda_{4})&g=5,\cr
V(\lambda_{2}+\lambda_{4})+V(\lambda_{2})+V(\lambda_{4})+V(\lambda_{6})&g\geq
6.\end{cases}$

In particular, there are no quadratic relations in
$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{t}}_{3}$. The space of quadratic
relations in $\operatorname{Gr}^{W}_{\bullet}{\mathfrak{u}}_{g}$ is $R\oplus
V(0)$.

As explained in [14, §10], a generic vector in the space of quadratic
relations is given by a pair of disjoint bounding pair maps.

###  7.2. Sakasai’s computation

The following assertion is the dual of [29, Prop. 6.2].

######  Proposition 7.5 (Sakasai).

The cokernel of the Jacobi identity map (2)

$J:\Lambda^{3}\operatorname{Gr}^{W}_{-1}{\mathfrak{t}}_{g}\to\operatorname{Gr}^{W}_{-1}{\mathfrak{t}}_{g}\otimes\operatorname{Gr}^{W}_{-2}{\mathfrak{t}}_{g}$

has highest weight decomposition

$\operatorname{coker}J\cong\begin{cases}V(2\lambda_{1}+\lambda_{3})+V(\lambda_{1}+\lambda_{2})+V(2\lambda_{2}+\lambda_{3})+V(\lambda_{3})&g=3,\cr
V(2\lambda_{1}+\lambda_{3})&g\geq 4.\end{cases}$

∎

######  Corollary 7.6.

For all $g\geq 3$, the image of
$\operatorname{Gr}^{W}_{-3}{\mathfrak{u}}_{g}\to\operatorname{Gr}^{W}_{-3}\operatorname{OutDer}{\mathfrak{p}}$
is isomorphic to $V(2\lambda_{1}+\lambda_{3})$. If $g=3$, then

  1. (i)

one has the exact sequence

$0\to\operatorname{Gr}^{W}_{-3}H_{2}({\mathfrak{t}}_{3})\to
V(2\lambda_{1}+\lambda_{3})+V(\lambda_{1}+\lambda_{2})+V(2\lambda_{2}+\lambda_{3})+V(\lambda_{3})\to\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{3}\to
0$

of ${\mathrm{Sp}}(H)$-modules.

  2. (ii)

the sequence

$0\to{\mathbb{Q}}(-1)\to H^{2}({\mathfrak{u}}_{3})\to
H^{2}({\mathfrak{t}}_{3})\to V(-1)\to 0$

is exact. In particular,
$\operatorname{Gr}^{W}_{-m}H^{2}({\mathfrak{t}}_{3})=0$ when $m\neq 3$. Here
$V=V(\lambda_{3})$, which is regarded as having weight $1$.

If $g\geq 4$, $\operatorname{Gr}^{W}_{m}H^{2}({\mathfrak{t}}_{g})$ vanishes
when $m\neq 2$,
$\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{g}=V(2\lambda_{1}+\lambda_{3})$,
and the sequence

$0\to{\mathbb{Q}}(-1)\to H^{2}({\mathfrak{u}}_{g})\to
H^{2}({\mathfrak{t}}_{g})\to 0$

is exact. Consequently, $\operatorname{Gr}^{W}_{m}H^{2}({\mathfrak{u}}_{g})$
vanishes when $m\neq 2$.

Combined with Kabanov purity, this implies that ${\mathfrak{t}}_{3}$ has only
cubic relations and that ${\mathfrak{t}}_{g}$ and ${\mathfrak{u}}_{g}$ are
quadratically presented for all $g\geq 4$ as $H^{2}({\mathfrak{t}}_{g})$ and
$H^{2}({\mathfrak{u}}_{g})$ are pure of weight 2

###### Proof.

Corollary 5.2 provides a lower bound for
$\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{g}$. Since $V(3\lambda_{1})$ does
not occur in ${\mathbb{L}}_{3}(V(\lambda_{3}))$, the image of
$\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{g}$ in
$\operatorname{Gr}^{W}_{-3}\operatorname{Out}\operatorname{Der}{\mathfrak{p}}$
must be $V(2\lambda_{1}+\lambda_{3})$ or trivial. If it were trivial, the
image of ${\mathfrak{t}}_{g}$ in $\operatorname{OutDer}{\mathfrak{p}}$ would
be nilpotent, which is not the case. So the image of
$\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{g}$ must be
$V(2\lambda_{1}+\lambda_{3})$.

The homology of the sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda^{3}\operatorname{Gr}^{W}_{-1}{\mathfrak{t}}_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{J}$$\textstyle{\operatorname{Gr}^{W}_{-1}{\mathfrak{t}}_{g}\otimes\operatorname{Gr}^{W}_{-2}{\mathfrak{t}}_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$bracket$\textstyle{\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{g}\to
0}$

is $\operatorname{Gr}^{W}_{-3}H_{\bullet}({\mathfrak{t}}_{g})$. Here the left-
hand term is placed in degree 3 and the right-hand term in degree 1. Since
$\operatorname{Gr}^{W}_{-3}H_{1}({\mathfrak{t}}_{g})=0$, the right-hand map is
surjective. This implies that the sequence

$0\to\operatorname{Gr}^{W}_{-3}H_{2}({\mathfrak{t}}_{g})\to\operatorname{coker}J\to\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{g}\to
0$

is exact. When $g=3$, this implies assertion (i). When $g\geq 4$, Sakasai’s
computation combined with the first assertion implies that
$\operatorname{Gr}^{W}_{-3}H_{2}({\mathfrak{t}}_{g})$ vanishes and that
$\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{g}=V(2\lambda_{1}+\lambda_{3})$.

Since $H^{1}({\mathfrak{u}}_{g})\to H^{1}({\mathfrak{t}}_{g})$ is an
isomorphism, the Gysin sequence Corollary 6.3 gives an exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{Q}}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}({\mathfrak{u}}_{g})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{H^{2}({\mathfrak{t}}_{g})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{V(-1)}$

of MHS in the category of ${\mathrm{Sp}}(H)$-modules. Kabanov purity implies
$\operatorname{Gr}^{W}_{m}H^{2}({\mathfrak{u}}_{g})=0$ when $m>3$ and $g\geq
3$. Since $V(-1)$ has weight $3$, this implies that
$\operatorname{Gr}^{W}_{m}H^{2}({\mathfrak{t}}_{g})$ also vanishes when $g\geq
3$ and $m>3$.

When $g\geq 4$, the vanishing of
$\operatorname{Gr}^{W}_{3}H^{2}({\mathfrak{t}}_{g})$ implies the exactness of

$0\to{\mathbb{Q}}(-1)\to H^{2}({\mathfrak{u}}_{g})\to
H^{2}({\mathfrak{t}}_{g})\to 0.$

Combined with Kabanov purity, this implies that
$\operatorname{Gr}^{W}_{m}H^{2}({\mathfrak{u}})$ and
$\operatorname{Gr}^{W}_{m}H^{2}({\mathfrak{t}}_{g})$ vanish for all $m\geq 3$.

To complete the proof, we prove assertion (ii). Since ${\mathfrak{t}}_{3}$ has
no quadratic relations, and since the copy of the trivial representation in
$\operatorname{Gr}^{W}_{-2}{\mathfrak{t}}_{3}$ is central, there is a copy of
$V$ in the cubic relations. This relation cannot be a consequence of quadratic
relations as there are none. This implies that the homomorphism
$\psi:H^{2}({\mathfrak{t}}_{3})\to V(-1)$ is surjective. Plugging this into
the Gysin sequence above establishes (ii). ∎

###  7.3. Proof of Theorem 3

By Proposition 3.1 and Lemma 3.2 we only need show that
$H^{2}({\mathfrak{t}}_{g,n})$ and $H^{2}({\mathfrak{u}}_{g,n})$ are pure of
weight 2. When $n=0$, we know this from Corollary 7.6. This yields a quadratic
presentation with the relations computed in Corollary 7.4:

######  Theorem 7.7.

The Lie algebras ${\mathfrak{u}}_{g}$ and ${\mathfrak{t}}_{g}$ are
quadratically presented for all $g\geq 4$. The presentations are

${\mathfrak{t}}_{g}={\mathbb{L}}(\Lambda^{3}_{0}H)^{\wedge}/(R)\text{ and
}{\mathfrak{u}}_{g}={\mathbb{L}}(\Lambda^{3}_{0}H)^{\wedge}/(R+V(0))$

where $R$ is the orthogonal complement of $V(2\lambda_{2})+V(0)$ in
$\Lambda^{2}_{0}\Lambda^{3}_{0}H={\mathbb{L}}_{2}(\Lambda^{3}_{0}H)$ described
explicitly in Corollary 7.4.

####  7.3.1. The case $n>0$

The degree 2 cohomology of each of the Lie algebras ${\mathfrak{u}}_{g,n}$,
$\hat{{\mathfrak{u}}}_{g,n}$, ${\mathfrak{t}}_{g,n}$ and
$\hat{{\mathfrak{t}}}_{g,n}$ is also pure of weight 2 when $n\geq 0$ and
$g\geq 4$. This can be proved using the fact that, in each case,

  1. (i)

the result is true for each series of Lie algebras when $n=0$,

  2. (ii)

$H^{1}$ of each of these Lie algebras is pure of weight $1$, and

  3. (iii)

$H^{2}$ is pure of weight $2$ if and only it is true in the base case, $n=0$.

This can be proved by an easy spectral sequence applied, for example, to the
extension

$0\to{\mathfrak{p}}^{n}\to\hat{{\mathfrak{u}}}_{g,n}\to{\mathfrak{u}}_{g}\to
0.$

Since $H^{2}({\mathfrak{u}}_{g})$ is of weight 2 for all $g\geq 4$, this
establishes the assertion. Details are left to the reader.

As explained above, this result combined with Proposition 3.1 and Lemma 3.2
imply the following result, of which Theorem 3 is a special case.

######  Theorem 7.8.

The Lie algebras ${\mathfrak{u}}_{g,n}$, $\hat{{\mathfrak{u}}}_{g,n}$ and
${\mathfrak{t}}_{g,n}$ and $\hat{{\mathfrak{t}}}_{g,n}$ are quadratically
presented for all $g\geq 4$ and all $n\geq 0$. ∎

###  7.4. Proof of Theorem 2

Recall from Section 6.3 that $K_{g,n}$ is the subgroup of $T_{g,n}$ generated
by Dehn twists on bounding simple closed curves and that $\widehat{K}_{g,n}$
is its image in $\widehat{T}_{g,n}$. A group homomorphism $G\to H$ is
virtually surjective if its image has finite index in $H$.

######  Theorem 7.9.

Suppose that $\Gamma$ is a finitely generated group and that $\Gamma\to T_{3}$
is a virtually surjective homomorphism. If $H_{2}(\Gamma,{\mathbb{Q}})\to
H_{2}(T_{3},{\mathbb{Q}})$ is surjective, then $\Gamma$ is not a Kähler group.
In particular, no finite index subgroup of $T_{3,n}$ that contains $K_{3,n}$
and no finite index subgroup of $\widehat{T}_{g,n}$ that contains
$\widehat{K}_{3,n}$ is a Kähler group.

###### Proof.

From [14, Rem. 10.4] (or Cor. 7.6) we know that
$\operatorname{Gr}^{W}_{3}H^{2}({\mathfrak{t}}_{3})\supseteq V(\lambda_{3})$
and is therefore non-zero. Proposition 3.1 and Lemma 3.2 imply that
${\mathfrak{t}}_{3}$ is not quadratically presented. The remaining cases
follow by applying Corollary 3.3 to the homomorphism
${\mathfrak{g}}\to{\mathfrak{t}}_{3}$ from the Lie algebra of the unipotent
completion of $\Gamma$ to ${\mathfrak{t}}_{3}$. ∎

This completes the proof of Theorem 2. In the proof of Theorem 1, which is
given in Section 10, we will need to know that ${\mathfrak{u}}_{3}$ is not
quadratically presented. This does not follow from the fact that
${\mathfrak{t}}_{3}$ is not quadratically presented. It is conceivable that
${\mathfrak{u}}_{3}$ is quadratically presented even though
${\mathfrak{t}}_{3}$ is not. This would happen if
$\operatorname{Gr}^{W}_{3}H^{2}({\mathfrak{t}}_{3})$ were equal to
$V(\lambda_{3})$. In that case ${\mathfrak{u}}_{3}$ would be isomorphic to
${\mathfrak{p}}_{7}$, the Lie algebra associated to the unipotent fundamental
group of a genus 7 surface $S$, and ${\mathfrak{t}}_{3}$ would be isomorphic
to the Lie algebra ${\mathfrak{p}}_{7,\vec{1}}$ of the unipotent fundamental
group of the unit tangent bundle of $S$. These are related by a central
extension

$0\to{\mathbb{Q}}(1)\to{\mathfrak{p}}_{7,\vec{1}}\to{\mathfrak{p}}_{7}\to 0.$

To prove that ${\mathfrak{u}}_{3}$ is not quadratically presented we will show
that $\operatorname{Gr}^{W}_{3}H^{2}({\mathfrak{t}}_{3})$ is strictly larger
than $V(\lambda_{3})$. To achieve this we compute explicit presentations of
${\mathfrak{t}}_{3}$ and ${\mathfrak{u}}_{3}$ in Section 9.

##  8\. Commuting Elements

In this section, we assume the reader is familiar with the Johnson
homomorphism. Surveys of its construction can be found in [17] and [12]. To
generate cubic relations in ${\mathfrak{t}}_{3}$, we determine the
${\mathrm{Sp}}(H)$-module of cubic relations determined by a certain pair of
commuting elements of $T_{3}$. Bounding pair maps give non-trivial elements of
$\operatorname{Gr}^{W}_{-1}{\mathfrak{t}}_{g}$ via the Johnson homomorphism.
In genus 3, one cannot find distinct commuting bounding pair maps.111111This
provides a heuristic explanation of why there are no quadratic relations in
${\mathfrak{t}}_{3}$. In genus 4 and beyond, commuting BP maps generate all
relations in $\operatorname{Gr}^{W}_{\bullet}{\mathfrak{t}}_{g}$. Cf. [14,
§10]. Because of this, we are forced to consider a commuting pair of elements
of $T_{3}$ consisting of a bounding pair map and a Dehn twist on a bounding
simple closed curve (a BSCC map). For possible future applications, we give a
lower bound on the cubic relations generated by such commuting pairs for all
$g\geq 3$ as the computations are no more difficult than they are in the genus
3 case. In the next section we will show that these are all the cubic
relations, which will allow us to give a presentation of ${\mathfrak{t}}_{3}$.

Let $S$ be a reference surface of genus $g$. Throughout this section, $t_{A}$
will be the BSCC map and $t_{B}:=t_{B_{+}}t_{B_{-}}^{-1}$ will be the BP map
depicted in Figure 1.

Figure 1. The BSCC map $t_{A}$ and the BP map $t_{B}:=t_{B_{+}}t_{B_{-}}^{-1}$

They commute as they have disjoint supports.

If we regard $t_{A}$ and $t_{B}$ as elements of the unipotent completion
${\mathcal{T}}_{g}$ of $T_{g}$, we can take their logarithms $\log t_{A}$ and
$\log t_{B}$. These are commuting elements of ${\mathfrak{t}}_{g}$. Since
$t_{A}$ lies in the kernel of the Johnson homomorphism and since
${\mathfrak{t}}_{g}=W_{-1}{\mathfrak{t}}_{g}$,

$\log t_{A}\in W_{-2}{\mathfrak{t}}_{3}\text{ and }\log t_{B}\in
W_{-1}{\mathfrak{t}}_{3}.$

The image of their tensor product in
$\operatorname{Gr}^{W}_{-2}{\mathfrak{t}}_{3}\otimes\operatorname{Gr}^{W}_{-1}{\mathfrak{t}}_{3}$
thus lies in the kernel of the bracket

(6)
$\operatorname{Gr}^{W}_{-2}{\mathfrak{t}}_{3}\otimes\operatorname{Gr}^{W}_{-1}{\mathfrak{t}}_{3}\to\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{3}.$

The ultimate goal of this section is to prove the following result.

######  Proposition 8.1.

If $g\geq 3$, then the ${\mathrm{Sp}}(H)$-submodule of the kernel of (6)
generated by the image of $\log t_{A}\otimes\log t_{B}$ contains the unique
submodule of
$\operatorname{Gr}^{W}_{-2}{\mathfrak{t}}_{3}\otimes\operatorname{Gr}^{W}_{-1}{\mathfrak{t}}_{3}$
with highest weights $2\lambda_{2}+\lambda_{3}$, $\lambda_{1}+\lambda_{2}$,
and $\lambda_{3}$.

Let $S^{\prime}$ be the genus 1 subsurface of $S$ that lies to the left of the
separating SCC $A$. Choose a symplectic basis
$a_{1},\dots,a_{g},b_{1},\dots,b_{g}$ of $H_{1}(S)$, where $a_{2},b_{2}$ is a
basis of $H_{1}(S^{\prime})$ and where the homology classes of $B_{\pm}$ are
$\pm a_{1}$. The Johnson homomorphism induces an isomorphism

$\operatorname{Gr}^{W}_{-1}{\mathfrak{t}}_{g}\cong\Lambda^{3}_{0}H.$

The image of $\log t_{B}$ in $\Lambda^{3}_{0}H$ under this isomorphism is the
image

(7) $\overline{a_{2}\wedge b_{2}\wedge
a_{1}}\in\Lambda^{3}_{0}H\cong\operatorname{Gr}^{W}_{-1}\operatorname{OutDer}{\mathfrak{p}}$

of $a_{2}\wedge b_{2}\wedge a_{1}\in\Lambda^{3}H$ in $\Lambda^{3}_{0}H$. (Cf.
[14, Prop. 10.3].)

Recall from Corollary 5.2 that
$\operatorname{Gr}^{W}_{-2}\operatorname{OutDer}{\mathfrak{p}}$ is isomorphic
to $V(2\lambda_{2})$. Our first task is to compute the element of
$V(2\lambda_{2})$ that corresponds to $\log t_{A}$. The first step is to
compute the image of a BSCC map under the second Johnson homomorphism that
takes an element of the kernel of the first Johnson homomorphism to
$\operatorname{Gr}^{W}_{-2}\operatorname{OutDer}{\mathfrak{p}}$.

###  8.1. The second Johnson homomorphism

In this section we give an exposition of Morita’s computation [25, p. 308] of
the second Johnson homomorphism. Suppose that $g\geq 2$, that $x$ is a point
in the reference surface $S$ and that $A$ is a bounding simple closed curve in
$S-\\{x\\}$. Denote the Lie algebra of the unipotent completion of
$\pi_{1}(S,x)$ by ${\mathfrak{p}}$. Denote the Dehn twist about $A$ by
$t_{A}$. It is an element of $T_{g,1}$ and has logarithm $\log t_{A}$ in
$W_{-2}{\mathfrak{t}}_{g,1}$. In this section we compute the image of $\log
t_{A}$ in $\operatorname{Gr}^{W}_{-2}\operatorname{Der}{\mathfrak{p}}$.

Figure 2. The BSCC $A$ and the decomposition $S=S^{\prime}\cup
S^{\prime\prime}$

The curve $A$ divides $S$ into two sub-surfaces. Denote the one containing $x$
by $S^{\prime}$ and the other by $S^{\prime\prime}$. Let $g^{\prime}$ and
$g^{\prime\prime}$ be their respective genera. Note that
$g^{\prime}+g^{\prime\prime}=g$. Choose a symplectic basis
$a_{1},\dots,a_{g^{\prime}},b_{1},\dots,b_{g^{\prime}}$ of
$H^{\prime}:=H_{1}(S^{\prime})$ and
$a_{g^{\prime}+1},\dots,a_{g},b_{g^{\prime}+1},\dots,b_{g}$ of
$H^{\prime\prime}:=H_{1}(S^{\prime\prime})$. Set

$\theta^{\prime}=\sum_{j=1}^{g^{\prime}}a_{j}\wedge b_{j},\
\theta^{\prime\prime}=\sum_{j=g^{\prime}+1}^{g}a_{j}\wedge b_{j}\text{ and
}\theta=\theta^{\prime}+\theta^{\prime\prime}.$

Then $H:=H_{1}(S)=H^{\prime}\oplus H^{\prime\prime}$ and $\theta$ corresponds
to the cup product $\Lambda^{2}H^{1}(S)\to{\mathbb{Z}}$.

Define

$\tilde{\varphi}:S^{2}\Lambda^{2}H\to\operatorname{Hom}(H,{\mathbb{L}}_{3}(H))=\operatorname{Gr}^{W}_{-2}\operatorname{Der}{\mathbb{L}}(H)$

by

$\tilde{\varphi}\big{(}(u_{1}\wedge v_{1})(u_{2}\wedge
v_{2})\big{)}:x\mapsto\langle u_{1},x\rangle[v_{1},[u_{2},v_{2}]]-\langle
v_{1},x\rangle[u_{1},[u_{2},v_{2}]]\cr+\langle
u_{2},x\rangle[v_{2},[u_{1},v_{1}]]-\langle
v_{2},x\rangle[u_{2},[u_{1},v_{1}]].$

This homomorphism is easily seen to be ${\mathrm{Sp}}(H)$-equivariant.

######  Lemma 8.2.

For a subset $I$ of $\\{1,\dots,g\\}$, set $\theta_{I}=\sum_{i\in
I}a_{i}\wedge b_{i}$, $H_{I}=\operatorname{span}\\{a_{i},b_{i}:i\in I\\}$, and
$H_{I}^{c}=\operatorname{span}\\{a_{i},b_{i}:i\notin I\\}$. Then

$\tilde{\varphi}(\theta_{I}^{2})(x)=\begin{cases}0&x\in H_{I}^{c},\cr
2[x,\theta_{I}]&x\in H_{I}.\end{cases}$

###### Proof.

Set $\theta_{j}=a_{j}\wedge b_{j}$. The result follows immediately from the
easily verified fact that $\tilde{\varphi}(\theta_{j}\theta_{k})$ takes
$x\in\operatorname{span}\\{a_{i},b_{i}\\}$ to
$\delta_{i,j}[x,\theta_{k}]+\delta_{i,k}[x,\theta_{j}]$. ∎

######  Corollary 8.3.

The homomorphism $\tilde{\varphi}$ induces a well-defined
${\mathrm{Sp}}(H)$-equivariant homomorphism

$\varphi:S^{2}\Lambda^{2}H\to\operatorname{Gr}^{W}_{-2}\operatorname{Der}{\mathfrak{p}}.$

###### Proof.

The lemma implies that
$\tilde{\varphi}(\theta^{2})=-2\operatorname{ad}_{\theta}$. This implies that
$\tilde{\varphi}$ preserves the ideal $(\theta)$ in ${\mathbb{L}}(H)$ and thus
descends to a derivation of
$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{p}}={\mathbb{L}}(H)/(\theta)$ of
weight $-2$. ∎

######  Proposition 8.4 (Morita [25, Prop. 1.1]).

The image of $\log t_{A}$ in
$\operatorname{Gr}^{W}_{-2}\operatorname{Der}{\mathfrak{p}}$ is
$\varphi(\theta^{\prime\prime})/2$.

###### Proof.

Set $\pi=\pi_{1}(S,x)$. First note that there are natural isomorphisms

$\operatorname{Gr}^{W}_{-2}\operatorname{Der}{\mathfrak{p}}\cong\operatorname{Hom}(H,{\mathbb{L}}_{3}(H)/[H,\theta])\cong\operatorname{Hom}(H_{1}(\pi),\operatorname{Gr}^{3}_{\mathrm{LCS}}\pi).$

Regard ${\mathfrak{p}}$ as a subspace of the completion

${\mathbb{Q}}\pi^{\wedge}:=\varprojlim_{m}{\mathbb{Q}}\pi/J^{m}$

of the group algebra of $\pi$ with respect to the powers of its augmentation
ideal $J$, as explained in [27, Appendix A]. The reason for working in the
completed group algebra is that it contains both $\pi$ and ${\mathfrak{p}}$
and is acted on by $\Gamma_{g,1}$.

A standard and elementary fact is that the function $\pi\to J/J^{2}$ that
takes $\gamma$ to $(\gamma-1)+J^{2}$ is a homomorphism that induces an
isomorphism $H=H_{1}(\pi,{\mathbb{Q}})\to J/J^{2}$. Since $T_{g,1}$ acts
trivially on $H=J/J^{2}$, it follows that $\sigma(\gamma)-\gamma\in J^{2}$ for
all $\sigma\in T_{g,1}$ and $\gamma\in\pi$. The induced homomorphism
$T_{g,1}\to\operatorname{Hom}(J/J^{2},J^{2}/J^{3})$ is essentially the Johnson
homomorphism. (Cf. [12, §11].) Since $t_{A}$ lies in the kernel of the Johnson
homomorphism,

$(t_{A}-1)(\gamma)=t_{A}(\gamma)-\gamma\in J^{3}$

for all $\gamma\in\pi$, so that $(t_{A}-1)^{m}:J^{s}\to J^{s+2m}$ for all
$s,m>0$. Consequently

$\displaystyle\log(t_{A})(\log(\gamma))$
$\displaystyle=\big{(}(t_{A}-1)-(t_{A}-1)^{2}/2+\cdots\big{)}\big{(}(\gamma-1)-(\gamma-1)^{2}/2+\cdots\big{)}$
$\displaystyle\equiv t_{A}(\gamma)-\gamma\bmod J^{4}$

We can thus compute the map

(8) $\log t_{A}:J/J^{2}\to J^{3}/J^{4}$

by computing $t_{A}(\gamma)-\gamma$ for a set of loops $\gamma$ that span $H$.

First, if $\gamma$ is the image of a loop in $(S^{\prime},x)$, then
$t_{A}(\gamma)=\gamma$, which implies that the restriction of (8) to
$H^{\prime}$ is trivial. To compute the restriction of (8) to
$H^{\prime\prime}$, we compute $t_{A}$ on loops $\gamma=\rho\mu\rho^{-1}$ that
follow a fixed path $\rho$ in $S^{\prime}$ from $x$ to a point $x^{\prime}\in
A$ and then follow a loop $\mu$ in $S^{\prime\prime}$ and return to $x$ along
$\rho^{-1}$.

Denote by $\alpha$ the loop based at $x^{\prime}$ that traverses $\partial
S^{\prime}$ in the positive direction. Set $\beta=\rho\alpha\rho^{-1}$. Then

$t_{A}(\gamma)=\rho\alpha^{-1}\mu\alpha\rho^{-1}=\beta^{-1}\gamma\beta\equiv\\{\beta^{-1},\gamma\\}+\gamma\bmod
J^{4},$

where $\\{x,y\\}$ denotes the group commutator $xyx^{-1}y^{-1}$. Since

$\beta^{-1}=\prod_{j=1}^{g^{\prime}}\\{\alpha_{j},\beta_{j}\\},$

where $\alpha_{j}$, $\beta_{j}$ is a standard set of generators of
$\pi_{1}(S^{\prime},x)$, it follows that

$\log
t_{A}(\log\gamma)=t_{A}(\gamma)-\gamma\equiv[\theta^{\prime},\log\gamma]=-\operatorname{ad}_{\theta^{\prime\prime}}(\log\gamma).$

This implies that the restriction of $\log t_{A}$ to $H^{\prime\prime}$ is
$-\operatorname{ad}_{\theta^{\prime\prime}}$. ∎

###  8.2. A projection of $S^{2}\Lambda^{2}_{0}H$ onto its $2\lambda_{2}$
isotypical component

The composite

$\textstyle{S^{2}\Lambda^{2}H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\operatorname{Gr}^{W}_{-2}\operatorname{Der}{\mathfrak{p}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Gr}^{W}_{-2}\operatorname{OutDer}{\mathfrak{p}}\cong
V(2\lambda_{2})}$

is an ${\mathrm{Sp}}(H)$-equivariant map onto $V(2\lambda_{2})$. The
representation $V(2\lambda_{2})$ has multiplicity 1 in $S^{2}\Lambda^{2}H$,
and in $S^{2}\Lambda^{2}_{0}H$. In this section, we compute a formula for the
projection of $S^{2}\Lambda^{2}_{0}H$ onto the $2\lambda_{2}$ isotypical
factor.

Recall that $\Lambda^{k}_{0}H$ denotes the ${\mathrm{Sp}}(H)$ module
$\Lambda^{k}H/(\theta\wedge\Lambda^{k-2}H)$. It is isomorphic to
$V(\lambda_{k})$. Note that $\Lambda^{2}_{0}H$ can be identified with the
kernel of the polarization $\Lambda^{2}H\to{\mathbb{Q}}$. The projection of
$u\wedge v\in\Lambda^{2}H$ onto this submodule is $u\wedge v-\langle
u,v\rangle\theta/g$.

######  Lemma 8.5.

If $g\geq 2$ the cup product

$\mathrm{cup}:S^{2}\Lambda^{2}_{0}H\to\Lambda^{4}H$

is a surjective ${\mathrm{Sp}}(H)$-module map. Its kernel is irreducible with
highest weight $2\lambda_{2}$. The ${\mathrm{Sp}}(H)$-module map

$\phi:\Lambda^{4}H\to S^{2}\Lambda^{2}_{0}H$

that takes $x_{1}\wedge x_{2}\wedge x_{3}\wedge x_{4}$ to

$\Big{(}x_{1}\wedge x_{2}-\frac{\langle
x_{1},x_{2}\rangle}{g}\theta\Big{)}\cdot\Big{(}x_{3}\wedge x_{4}-\frac{\langle
x_{3},x_{4}\rangle}{g}\theta\Big{)}\cr-\Big{(}x_{1}\wedge x_{3}-\frac{\langle
x_{1},x_{3}\rangle}{g}\theta\Big{)}\cdot\Big{(}x_{2}\wedge x_{4}-\frac{\langle
x_{2},x_{4}\rangle}{g}\theta\Big{)}\cr+\Big{(}x_{1}\wedge x_{4}-\frac{\langle
x_{1},x_{4}\rangle}{g}\theta\Big{)}\cdot\Big{(}x_{2}\wedge x_{3}-\frac{\langle
x_{2},x_{3}\rangle}{g}\theta\Big{)}$

is injective. The restriction of
$\mathrm{cup}\circ\phi:\Lambda^{4}H\to\Lambda^{4}H$ to the component with
highest weight $\lambda$ is multiplication by
$\begin{cases}3&\lambda=\lambda_{4},\ (g\geq 4\text{
only}),\cr\frac{2g+2}{g}&\lambda=\lambda_{2},\ (g\geq 3\text{
only}),\cr\frac{2g+1}{g}&\lambda=0.\end{cases}$

###### Proof.

Kabanov’s stability theorem [20] implies that the highest weight decomposition
of $S^{2}\Lambda^{2}_{0}H$ is independent of $g$ when $g\geq 4$. One can show
(using LiE or by direct computation) that $S^{2}\Lambda^{2}_{0}H$ is
abstractly isomorphic to $V(2\lambda_{2})\oplus\Lambda^{4}H$ as an
${\mathrm{Sp}}(H)$-module for all $g\geq 2$.

The cup product and $\phi$ are both ${\mathrm{Sp}}(H)$-equivariant. All
assertions follow from the last statement. Schur’s Lemma implies that the
restriction of $\mathrm{cup}\circ\phi$ to each irreducible component of
$\Lambda^{4}H$ is multiplication by a scalar. These scalars can be determined
by computing the composite on a highest weight vector of each irreducible
component of $\Lambda^{4}H$. These are $\theta^{2}$ ($g\geq 2$), $a_{1}\wedge
a_{2}\wedge\theta$ ($g\geq 3$), and $a_{1}\wedge a_{2}\wedge a_{3}\wedge
a_{4}$ ($g\geq 4$). These are easily computed. Details are left to the reader.
∎

######  Corollary 8.6.

The projection of $(a_{j}\wedge b_{j}-\theta/g)^{2}\in S^{2}\Lambda^{2}_{0}H$
onto the weight $2\lambda_{2}$ isotypical component is

$(a_{j}\wedge b_{j}-\theta/g)^{2}+\frac{1}{g+1}\phi\Big{(}a_{j}\wedge
b_{j}\wedge\theta-\frac{1}{(2g+1)}\theta^{2}\Big{)}$

for all $g\geq 2$ and $1\leq j\leq g$.

###### Proof.

The image of $(a_{j}\wedge b_{j}-\theta/g)^{2}$ in $\Lambda^{4}H$ is

$\displaystyle-2\,a_{j}\wedge b_{j}\wedge\theta/g+\theta^{2}/g^{2}$
$\displaystyle=-\frac{2}{g}(a_{j}\wedge
b_{j}\wedge\theta-\theta^{2}/g)-\theta^{2}/g^{2}$
$\displaystyle=-\mathrm{cup}\Big{(}\frac{1}{g+1}\phi(a_{j}\wedge
b_{j}\wedge\theta-\theta^{2}/g)+\frac{1}{g(2g+1)}\phi(\theta^{2})\Big{)}$
$\displaystyle=-\mathrm{cup}\circ\phi\Big{(}\frac{1}{g+1}a_{j}\wedge
b_{j}\wedge\theta-\frac{1}{(g+1)(2g+1)}\theta^{2}\Big{)}$

as the first term of the right-hand side lies in the $V(\lambda_{2})$
component and the second in the trivial component. ∎

Denote the projection of $x\in\Lambda^{3}H$ (resp. $y\in
S^{2}\Lambda^{2}_{0}H$) onto $\Lambda^{3}_{0}H$ (resp. its $2\lambda_{2}$
isotypical component) by $\overline{x}$ (resp. $\overline{y}$). Combining the
previous result with (7), we obtain:

######  Corollary 8.7.

For all $g\geq 3$, the images of $\log t_{A}\in W_{-1}{\mathfrak{t}}_{3}$ and
$\log t_{B}\in W_{-2}{\mathfrak{t}}_{g}$ in
$\operatorname{Gr}^{W}_{\bullet}\operatorname{OutDer}{\mathfrak{p}}$ are

$\overline{\log t_{A}}=\overline{(a_{2}\wedge b_{2}-\theta/g)^{2}}\in
V(2\lambda_{2})\cong\operatorname{Gr}^{W}_{-2}\operatorname{OutDer}{\mathfrak{p}}$

and

$\overline{\log t_{B}}=\overline{a_{2}\wedge b_{2}\wedge
a_{1}}\in\Lambda^{3}_{0}H\cong\operatorname{Gr}^{W}_{-1}\operatorname{OutDer}{\mathfrak{p}}.\qed$

###  8.3. Proof of Proposition 8.1

As explained in the introduction to this section, we want to compute the
${\mathrm{Sp}}(H)$-submodule of

$\operatorname{Gr}^{W}_{-2}{\mathfrak{u}}_{3}\otimes\operatorname{Gr}^{W}_{-1}{\mathfrak{u}}_{3}\cong
V(2\lambda_{2})\otimes V(\lambda_{3})$

generated by the image of $\log t_{A}\otimes\log t_{B}$.

The following computation is equivalent to Sakasai’s computation [29, Lem.
6.1]. It can be proved using ${\sf{LiE}}$ plus Kabanov stability.

######  Lemma 8.8.

The irreducible factors of $V(2\lambda_{2})\otimes V(\lambda_{3})$ have
highest weights

$\begin{cases}2\lambda_{1}+\lambda_{3},\ \lambda_{1}+2\lambda_{2},\
\lambda_{1}+\lambda_{2},\ 2\lambda_{2}+\lambda_{3},\ \lambda_{3}&g=3,\cr
2\lambda_{1}+\lambda_{3},\ \lambda_{1}+2\lambda_{2},\
\lambda_{1}+\lambda_{2}+\lambda_{4},\ \lambda_{1}+\lambda_{2},\
\lambda_{1}+\lambda_{4},\ 2\lambda_{2}+\lambda_{3},\ \lambda_{2}+\lambda_{3},\
\lambda_{3}&g=4,\cr 2\lambda_{1}+\lambda_{3},\ \lambda_{1}+2\lambda_{2},\
\lambda_{1}+\lambda_{2}+\lambda_{4},\ \lambda_{1}+\lambda_{2},\
\lambda_{1}+\lambda_{4},\ 2\lambda_{2}+\lambda_{3},\ \lambda_{2}+\lambda_{3},\
\lambda_{2}+\lambda_{5},\ \lambda_{3}&g\geq 5.\end{cases}$

Each has multiplicity 1. Consequently, there are unique (up to scalar
multiplication) projections of $V(2\lambda_{2})\otimes V(\lambda_{3})$ onto
$V(\lambda_{3})$, $V(\lambda_{1}+\lambda_{2})$ and
$V(2\lambda_{2}+\lambda_{3})$. ∎

Regard $V(2\lambda_{2})$ as the kernel of the cup product
$S^{2}\Lambda^{2}_{0}H\to\Lambda^{4}H$. Regard
$V(\lambda_{3})=\Lambda^{3}_{0}H$ as the kernel of the contraction

$c:\Lambda^{3}H\to H,\quad u\wedge v\wedge w\mapsto\langle u,v\rangle
w+\langle v,w\rangle u+\langle w,u\rangle v$

The projection of $x\in\Lambda^{3}H$ onto $\Lambda^{3}_{0}H$ is
$x-c(x)\wedge\theta/(g-1)$. One also has the injection (the “Jacobi identity
map”)

$\Lambda^{3}H\to\Lambda^{2}H\otimes H,\quad u\wedge v\wedge w\mapsto(u\wedge
v)\otimes w+(v\wedge w)\otimes u+(w\wedge u)\otimes v$

and the standard imbedding $v_{1}v_{1}\mapsto v_{1}\otimes v_{2}+v_{2}\otimes
v_{1}$ of $S^{2}\Lambda^{2}H$ into $(\Lambda^{2}H)^{\otimes 2}$. Finally, one
has the pairing

$(\phantom{x},\phantom{x}):\Lambda^{2}H\otimes\Lambda^{2}S^{2}H\to{\mathbb{Q}}.$
$(u_{1}\wedge v_{1},u_{2}\wedge v_{2})=\left|\begin{matrix}\langle
u_{1},v_{1}\rangle&\langle u_{1},v_{2}\rangle\cr\langle
u_{2},v_{1}\rangle&\langle u_{2},v_{2}\rangle\end{matrix}\right|$

All of these maps are ${\mathrm{Sp}}(H)$-equivariant. They can be assembled
into the ${\mathrm{Sp}}(H)$-equivariant map:

(9) $V(2\lambda_{2})\otimes V(\lambda_{3})\hookrightarrow
S^{2}\Lambda^{2}H\otimes\Lambda^{3}H\hookrightarrow\Lambda^{2}H\otimes(\Lambda^{2}H\otimes\Lambda^{2}H)\otimes
H\overset{1\otimes\det\otimes 1}{\longrightarrow}\Lambda^{2}H\otimes H.$

This can be composed with the multiplication map $\Lambda^{2}H\otimes
H\to\Lambda^{3}H$ to obtain a projection to $V(2\lambda_{2})\otimes
V(\lambda_{3})\to\Lambda^{3}H$.

######  Lemma 8.9.

For all $g\geq 3$, the ${\mathrm{Sp}}_{g}$-submodule $W$ of
$V(\lambda_{2})\otimes V(\lambda_{3})$ generated by

$v:=\overline{\log t_{A}}\otimes\overline{\log t_{B}}=\overline{(a_{2}\wedge
b_{2}-\theta/g)^{2}}\otimes\overline{a_{2}\wedge b_{2}\wedge a_{1}}$

contains the factors with highest weights $2\lambda_{2}+\lambda_{3}$,
$\lambda_{1}+\lambda_{3}$ and $\lambda_{3}$.

###### Proof.

Set $\theta_{j}=a_{j}\wedge b_{j}$, so that $\theta=\sum_{j}\theta_{j}$. Note
that $(\theta_{j},\theta_{k})=\delta_{j,k}$. Since

$F_{2,3}\circ F_{1,2}^{2}(v)=2(a_{1}\wedge a_{2})^{2}\otimes(a_{1}\wedge
a_{2}\wedge a_{3})\in W$

is a highest weight vector of weight $2\lambda_{2}+\lambda_{3}$, $W$ contains
the copy of $V(2\lambda_{2}+\lambda_{3})$.

To show that $W$ contains highest weight $\lambda_{1}+\lambda_{2}$ we first
apply $F_{1,2}$ to $v$ to get

$\Big{(}2(a_{2}\wedge
a_{1})(\theta_{2}-\theta/g)+\frac{1}{g+1}\phi(a_{2}\wedge
a_{1}\wedge\theta)\Big{)}\otimes\big{(}\theta_{2}\wedge a_{1}-\theta\wedge
a_{1}/(g-1)\big{)}\in W.$

This goes to the vector

$\displaystyle\bigg{(}$ $\displaystyle
2\Big{(}(\theta_{2},\theta_{2})-\frac{1}{g}(\theta,\theta_{2})-\frac{1}{g-1}(\theta_{2},\theta)+\frac{1}{g(g-1)}(\theta,\theta-\theta_{1})\Big{)}$
$\displaystyle+\frac{1}{g+1}\sum_{j>2}(\theta_{j}-\theta/g,\theta_{2})-\frac{1}{(g+1)(g-1)}\sum_{j>2}(\theta_{j}-\theta/g,\theta-\theta_{1})\Big{)}(a_{2}\wedge
a_{1})\otimes a_{1}$ $\displaystyle=$
$\displaystyle\bigg{(}2\Big{(}1-\frac{1}{g}-\frac{1}{g-1}+\frac{g-1}{g(g-1)}\Big{)}-\frac{g-2}{g(g+1)}-\frac{g-2}{g(g-1)(g+1)}\bigg{)}(a_{2}\wedge
a_{1})\otimes a_{1}$ $\displaystyle=$
$\displaystyle-\frac{(g-2)(2g+1)}{g^{2}-1}\,(a_{1}\wedge a_{2})\otimes
a_{1}\hskip 202.0pt$

in $\Lambda^{2}H\otimes H$ under the projection (9). This is a non-zero
highest weight vector of weight $\lambda_{1}+\lambda_{2}$ for all $g\geq 3$.

Since we do not need to know the result for $\lambda_{3}$, we only sketch the
argument. Since $\theta\wedge a_{1}\in\Lambda^{3}H$ is a highest weight vector
with highest weight $\lambda_{1}$, and since $V(2\lambda_{2})\otimes
V(\lambda_{1})$ does not contain $V(\lambda_{3})$, it suffices to show that
the ${\mathrm{Sp}}(H)$-module generated by the image of

$w:=\overline{(a_{2}\wedge b_{2}-\theta/g)^{2}}\otimes(a_{2}\wedge b_{2}\wedge
a_{1})\in V(2\lambda_{2})\otimes\Lambda^{3}H$

in $\Lambda^{3}H$ under the projection described above contains
$V(\lambda_{3})$. To do this, first apply $F_{2,3}$ to $w$ to get

$\Big{(}2(a_{2}\wedge
a_{3})(\theta_{2}-\theta/g)+\frac{1}{g+1}\phi(a_{2}\wedge
a_{3}\wedge\theta)\Big{)}\otimes(\theta_{2}\wedge
a_{1})\cr+\bigg{(}(\theta_{2}-\theta/g)^{2}+\frac{1}{g+1}\phi(\theta_{2}\wedge\theta)-\frac{1}{(g+1)(2g+1)}\phi(\theta^{2})\bigg{)}\otimes(a_{1}\wedge
a_{2}\wedge a_{3}).$

This is a vector of weight $\lambda_{3}$ whose image in $\Lambda^{3}H$ is the
highest weight vector

$\bigg{(}\frac{2(g-1)}{g}+\frac{g-1}{g(g+1)}+0+\frac{2}{g+1}-\frac{6}{(g+1)(2g+1)}\bigg{)}\,a_{1}\wedge
a_{2}\wedge a_{3}\cr=(g-1)\frac{4g^{2}+12g+3}{g(g+1)(2g+1)}\,a_{1}\wedge
a_{2}\wedge a_{3}.$

Since this is non-zero for all $g\geq 3$, it follows that $W$ contains a
factor with highest weight $\lambda_{3}$. ∎

##  9\. Presentations of ${\mathfrak{u}}_{3}$ and ${\mathfrak{t}}_{3}$

Recall that ${\mathfrak{t}}_{3}$ is isomorphic to the degree completion of the
graded Lie algebra associated to its weight filtration. So a presentation of
its associated graded Lie algebra also gives a presentation of
${\mathfrak{t}}_{3}$ itself. Similarly, a presentation of
$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{u}}_{3}$ gives a presentation of
${\mathfrak{u}}_{3}$.

######  Theorem 9.1.

A minimal presentation of
$\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{t}}_{3}$ is

$\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{t}}_{3}={\mathbb{L}}(V)/{\mathfrak{r}},$

where $V:=V(\lambda_{3})$ and ${\mathfrak{r}}$ is the graded ideal generated
by the unique ${\mathrm{Sp}}_{3}$-submodule $R$ of ${\mathbb{L}}_{3}(V)$
isomorphic to

$V(2\lambda_{2}+\lambda_{3})+V(\lambda_{1}+\lambda_{2})+V(\lambda_{3}).\qed$

The copy of $V(\lambda_{3})$ in the cubic relations corresponds to the fact
that the generators $\operatorname{Gr}^{W}_{-1}{\mathfrak{t}}_{3}$ commute
with the copy of the trivial representation
${\mathbb{Q}}\psi\in\operatorname{Gr}^{W}_{-2}{\mathfrak{t}}_{3}$. That is,
the $\lambda_{3}$ component of the cubic relations is
$[\psi,\operatorname{Gr}^{W}_{-1}{\mathfrak{t}}_{3}]$. Since $\psi=0$ in
${\mathfrak{u}}_{3}$, we obtain the following presentation of
$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{u}}_{3}$.

######  Corollary 9.2.

A minimal presentation of
$\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{u}}_{3}$ is

$\operatorname{Gr}^{\bullet}_{\mathrm{LCS}}{\mathfrak{u}}_{3}={\mathbb{L}}(V)/{\mathfrak{r}},$

where $V:=V(\lambda_{3})$ and ${\mathfrak{r}}$ is the graded ideal generated
in degrees $2$ and $3$. The space of quadratic relations is the unique copy of
the trivial representation in ${\mathbb{L}}_{2}(V)\cong\Lambda^{2}V$. The
space of cubic relations is the unique ${\mathrm{Sp}}_{3}$-submodule of
${\mathbb{L}}_{3}(V)$ isomorphic to
$V(2\lambda_{2}+\lambda_{3})+V(\lambda_{1}+\lambda_{2})$. ∎

Corollary 7.6 implies that all relations in
$\operatorname{Gr}^{W}_{\bullet}{\mathfrak{t}}_{3}$ are of weight $-3$ — i.e.,
are cubic. The theorem is thus an immediate consequence of the following
computation.

######  Proposition 9.3.

There are ${\mathrm{Sp}}(H)$-module isomorphisms

$\operatorname{Gr}^{W}_{3}H^{2}({\mathfrak{t}}_{3})\cong
V(2\lambda_{2}+\lambda_{3})+V(\lambda_{1}+\lambda_{2})+V(\lambda_{3})$

and
$\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{3}\cong\operatorname{Gr}^{W}_{-3}{\mathfrak{u}}_{3}\cong
V(2\lambda_{1}+\lambda_{3})$. Consequently,

$\operatorname{Gr}^{W}_{3}H^{2}({\mathfrak{u}}_{3})\cong
V(2\lambda_{2}+\lambda_{3})+V(\lambda_{1}+\lambda_{2}).$

###### Proof.

We use the notation of Proposition 7.5. The kernel of
$\operatorname{coker}J\to\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{3}$ is
$\operatorname{Gr}^{W}_{-3}H_{2}({\mathfrak{t}}_{3})$. It contains the
${\mathrm{Sp}}(H)$-submodule generated by $(\log t_{A})\otimes(\log t_{B})$.
We know that the image of $\operatorname{coker}J$ in
$\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{3}$ contains
$V(2\lambda_{1}+\lambda_{3})$. Proposition 8.1 implies that
$\operatorname{Gr}^{W}_{-3}H_{2}({\mathfrak{t}}_{3})$ is generated as an
${\mathrm{Sp}}(H)$-module by the image of $\log t_{A}\otimes\log t_{B}$ and
that $\operatorname{Gr}^{W}_{-3}{\mathfrak{t}}_{3}$ is irreducible with
highest weight $2\lambda_{1}+\lambda_{3}$.

The corresponding statements for ${\mathfrak{u}}_{3}$ are immediate
consequences. ∎

###  9.1. Massey products

The computations above can be stated in terms of Massey triple products.
Suppose that $X$ is a space (such as the classifying space of $T_{3}$). Fix a
coefficient ring $R$ for cohomology. The Massey triple product is the map

$\ker\big{\\{}H^{p}(X)\otimes H^{q}(X)\otimes
H^{r}(X)\to\big{(}H^{p+q}(X)\otimes
H^{r}(X)\big{)}\oplus\big{(}H^{p}(X)\otimes H^{q+r}(X)\big{)}\big{\\}}\cr\to
H^{p+q+r-1}(X)/\big{(}H^{p+q-1}(X)\wedge H^{r}(X)+H^{p}(X)\wedge
H^{q+r-1}(X)\big{)}$

defined by $[w_{1}]\otimes[w_{2}]\otimes[w_{3}]\mapsto[w_{12}\wedge
w_{3}+w_{1}\wedge w_{23}]$ where $w_{1},w_{2},w_{3}$ are cocycles on $X$ of
degrees $a,b,c$, respectively and $w_{12}$ and $w_{23}$ are cochains
satisfying $dw_{12}=w_{1}\wedge w_{2}$ and $dw_{23}=\pm w_{2}\wedge w_{3}$.

In particular, if $p=q=r=1$ and the cup product $H^{1}(X)^{\otimes 2}\to
H^{2}(X)$ vanishes, one has a well defined Massey triple product

$H^{1}(X)^{\otimes 3}\to H^{2}(X).$

When $X$ is the classifying space of $T_{3}$, the cup product
$H^{1}(T_{3};{\mathbb{Q}})^{\otimes 2}\to H^{2}(T_{3};{\mathbb{Q}})$ vanishes
by Corollary 7.4. The Massey triple product map is defined and
${\mathrm{Sp}}_{3}({\mathbb{Z}})$-equivariant.

######  Proposition 9.4.

The image of the Massey triple product map

$H^{1}(T_{3};{\mathbb{Q}})^{\otimes 3}\to H^{2}(T_{3};{\mathbb{Q}})$

is an ${\mathrm{Sp}}_{3}({\mathbb{Z}})$ module isomorphic to the restriction
of $V(2\lambda_{2}+\lambda_{3})+V(\lambda_{1}+\lambda_{2})+V(\lambda_{3})$ to
${\mathrm{Sp}}_{3}({\mathbb{Z}})$. Consequently, $\dim
H_{2}(T_{3},{\mathbb{Q}})\geq 694$.

###### Proof.

The corresponding statement for ${\mathfrak{t}}_{3}$ follows by examining the
weight 3 part of the Chevalley-Eilenberg cochains on ${\mathfrak{t}}_{3}$. It
and the fact that $H^{2}({\mathfrak{t}}_{3})$ is pure of weight 3 imply that
the Massey triple product map $H^{1}({\mathfrak{t}}_{3})^{\otimes 3}\to
H^{3}({\mathfrak{t}}_{3})$ is surjective. The assertion for $T_{3}$ follows as
the map $H^{\bullet}({\mathfrak{t}}_{3})\to H^{\bullet}(T_{3})$ is
${\mathrm{Sp}}_{3}$-equivariant, preserves Massey products and is an
isomorphism in degree 1 and injective in degree 2. ∎

##  10\. Genus $3$ Mapping Class Groups are not Kähler Groups

In this section we derive some general consequences of a mapping class group
of genus $g\geq 3$ being a Kähler group. In particular, we complete the proof
of Theorem 1.

To prove Theorem 1 we need to know that if the genus 3 mapping class group
$\Gamma$ is the fundamental group of a compact Kähler manifold $X$, then the
standard representation $\rho:\Gamma\to{\mathrm{Sp}}(H)$ is the monodromy
representation of a PVHS. This will imply that the completion of $\Gamma$
relative to $\rho$ has a MHS and, by Theorem 4.4, that the Lie algebra of its
prounipotent radical is quadratically presented, a contradiction. To prove
that $\rho$ is the monodromy representation of a PVHS, we appeal to Simpson’s
fundamental work [31].

###  10.1. Rigid representations of Kähler groups

Recall that a representation $\rho:\Gamma\to R$ of a finitely generated group
into an algebraic group $G$ over ${\mathbb{R}}$ is rigid if every
representation $\phi$ that is sufficiently close to $\rho$ in
$\operatorname{Hom}(\Gamma,R)$ is conjugate to $\rho$. It is properly rigid if
it is rigid as a homomorphism to the Zariski closure of its image.

A well-known result of Weil [34] implies that $\rho$ is rigid if
$H^{1}(\Gamma,\operatorname{Ad}(\rho))=0$, where $\operatorname{Ad}(\rho)$ is
the representation

$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{Ad}}$$\textstyle{\operatorname{Aut}({\mathfrak{g}})}$

of $\Gamma$ on the Lie algebra of $G$.

The following result is a special case of a result [31, Thm. 5] of Simpson.

######  Theorem 10.1 (Simpson).

Suppose that $X$ is a connected compact Kähler manifold and that
$\rho:\pi_{1}(X,x)\to{\mathrm{GL}}(V)$ is an irreducible representation on a
rational vector space. If $H^{1}(X,\operatorname{Hom}(V,V))=0$, then

  1. (i)

$\rho$ is rigid and properly rigid,

  2. (ii)

the Zariski closure $R$ of $\rho$ is a reductive group of Hodge type,

  3. (iii)

$\rho:\pi_{1}(X,x)\to R$ is the monodromy representation of a polarized
${\mathbb{Q}}$-VHS over $X$ with fiber $V\oplus V^{\ast}$ over the base point
$x$.

Combining this with the main results of [13], we obtain:

######  Corollary 10.2.

With the same assumptions as in the preceding theorem, the completion of
$\pi_{1}(X,x)$ relative to $\rho$ has a natural ${\mathbb{Q}}$-MHS and the
action of $\pi_{1}(X,x)$ on $H_{1}({\mathfrak{u}}_{X})$ is the monodromy
representation of a polarized ${\mathbb{Q}}$-VHS. ∎

###  10.2. Rigidity for mapping class groups

For the rest of this section, we suppose that $\Gamma$ denotes $\Gamma_{g,n}$
or $\widehat{\Gamma}_{g,n}$ and that $\Gamma^{\prime}$ is a subgroup of
$\Gamma$ that contains its Johnson subgroup. Corollary 6.7 implies that the
completion of $\Gamma^{\prime}$ relative to $\rho$ is isomorphic to the
completion of $\Gamma$ relative to $\rho$. Denote it by ${\mathcal{G}}$.

######  Lemma 10.3.

If $g\geq 3$, then $\rho:\Gamma^{\prime}\to{\mathrm{GL}}(H)$ is rigid and
properly rigid.

###### Proof.

This follows from [34]. Since $\operatorname{Ad}(\rho)\cong H^{\otimes 2}\cong
S^{2}H\oplus\Lambda^{2}_{0}H\oplus{\mathbb{Q}}$, Proposition 6.6 implies that
both $H^{1}(\Gamma^{\prime},\operatorname{Ad}(\mathfrak{sp}_{g}))$ and
$H^{1}(\Gamma^{\prime},\operatorname{Ad}(\operatorname{End}(H)))$ vanish. ∎

Combining Corollary 10.2 with Theorem 4.4 we obtain:

######  Proposition 10.4.

If $\Gamma^{\prime}$ is the fundamental group $\pi_{1}(X,x)$ of a compact
Kähler manifold $X$, then the Lie algebra ${\mathfrak{g}}_{X,x}$ of the
completion ${\mathcal{G}}_{X,x}$ of $\pi_{1}(X,x)$ relative to $\rho$ has a
natural MHS. The weight filtration of the Lie algebra ${\mathfrak{u}}_{X,x}$
of its prounipotent radical is its lower central series:

$W_{-m}{\mathfrak{u}}_{X,x}=L^{m}{\mathfrak{u}}_{X,x}.$

The fact that ${\mathfrak{u}}_{3}$ is not quadratically presented completes
the proof of Theorem 1 via Corollary 3.3.

######  Corollary 10.5.

If $g=3$, then $\Gamma^{\prime}$ cannot be the fundamental group of a compact
Kähler manifold.

##  11\. Speculation

The proof that genus $3$ mapping class groups are not Kähler clearly fails
when $g>3$ as ${\mathfrak{u}}_{g}$ is quadratically presented by Theorem 7.8.
So the question arises as to whether higher genus mapping class groups are
Kähler. My instincts tell me that they are not, but I have no serious evidence
to support this.

One way to attempt to prove that a mapping class group $\Gamma_{g,n}$ (or a
group commensurable with it) is Kähler is to construct a smooth complete
subvariety $X$ of some smooth finite (orbifold) covering ${\mathcal{M}}$ of
${\mathcal{M}}_{g,n}$ such that the inclusion $X\to{\mathcal{M}}$ induces an
isomorphism on fundamental groups. One can attempt to do this using the
generalized Lefschetz Theorem [9, p. 150] as the Satake compactification of
${\mathcal{M}}$ is projective. This implies that the inclusion
$X\hookrightarrow{\mathcal{M}}$ of a generic 2-dimensional linear section $X$
of ${\mathcal{M}}$ induces an isomorphism on fundamental groups. However, such
an $X$ cannot be compact as one of the boundary components of ${\mathcal{M}}$
has codimension 2. The known complete subvarieties of ${\mathcal{M}}$ are far
from generic and have special fundamental groups.

Suppose that $X$ is a subvariety of an orbifold covering ${\mathcal{M}}$ of
${\mathcal{M}}_{g,n}$. Denote the Zariski closure of the image of
$\pi_{1}(X,x)\to{\mathrm{Sp}}(H)$ by $R$ and the completion of $\pi_{1}(X,x)$
relative to $\pi_{1}(X,x)\to R$ by ${\mathcal{G}}_{X,x}$. Denote the
completion of $\pi_{1}({\mathcal{M}},x)\to{\mathrm{Sp}}(H)$ by
${\mathcal{G}}_{{\mathcal{M}},x}$.

######  Conjecture 11.1.

If $X$ is a smooth complete subvariety of a finite orbifold covering
${\mathcal{M}}\to{\mathcal{M}}_{g,n}$ of ${\mathcal{M}}_{g,n}$, then
$\pi_{1}(X,x)\to\pi_{1}({\mathcal{M}},x)$ is not an isomorphism. Even
stronger, the induced homomorphism
${\mathcal{G}}_{X,x}\to{\mathcal{G}}_{{\mathcal{M}},x}$ cannot be an
isomorphism.

Suppose that $X$ is a smooth (quasi-) projective variety with fundamental
group a finite index subgroup $\Gamma$ of $\Gamma_{g,n}$. Let ${\mathcal{M}}$
be the finite orbifold covering of ${\mathcal{M}}_{g,n}$ with fundamental
group $\Gamma$. On a more speculative level, one can ask if there must be a
morphism $X\to{\mathcal{M}}$ that induces an isomorphism on fundamental
groups. Here one may need to insist that $g$ be sufficiently large.

One way to approach this problem is to ask whether there is a family of
principally polarized abelian varieties (PPAVs) $f:A\to X$ whose associated
local system $R^{1}f_{\ast}{\mathbb{Z}}$ is the local system ${\mathbb{H}}$
corresponding the fundamental representation
$\rho:\Gamma\to\Gamma_{g,n}\to{\mathrm{Sp}}_{g}({\mathbb{Z}})$. The PVHS
produced by Simpson’s Theorem (Thm. 10.1) may not have Hodge level
one121212The level of a Hodge structure $V=\oplus V^{p,q}$ is the maximum of
the $p-q$ for which $V^{p,q}\neq 0$. and may not have rank $2g$, so that it
may not arise from a family of PPAVs over $X$. Note that if $X$ is projective,
then the local system whose fiber over $x\in X$ is
$H_{1}({\mathfrak{u}}_{X,x})$ is a PVHS over $X$ which is is isomorphic to the
local system $\Lambda^{3}_{0}{\mathbb{H}}$.

We now enter the realm of the wildly speculative. If such a family $A\to X$ of
PPAVs does exist, one can speculate that it is isogenous to a family of
jacobians. This is not totally unreasonable as the relations in
${\mathfrak{u}}_{g}$ seem to be related to controlling the image of the period
mapping ${\mathcal{M}}_{g}\to{\mathcal{A}}_{g}$. One might call this the
“topological Schottky problem”. If it has a positive solution, we can assume
that $A\to X$ is a family of jacobians of curves of compact type. The final
step would be to argue that one can lift the period map
$X\to{\mathcal{A}}_{g}$ to a morphism $X\to{\mathcal{M}}$.

##  Appendix A Mapping Class Groups in Low Genus are not Kähler

For completeness we prove that no finite index subgroup of a mapping class
group in genus $0$, $1$ or $2$ is Kähler. As remarked in the introduction, the
genus $2$ case is due to Veliche [33] and the genus 0 and 1 cases appear to be
folklore. Since I could not find references for the genus 0 and 1 cases, I am
including complete proofs here. By adapting Veliche’s argument in genus 2, we
will prove the more general statement that groups commensurable with
hyperelliptic mapping class groups are not Kähler.

The hyperelliptic mapping class groups $\Delta_{g,n}$ is the orbifold
fundamental group of the moduli stack ${\mathcal{H}}_{g,n}$ of $n$-pointed
hyperelliptic curves of genus $g\geq 2$. A topological definition will be
given below. Since every curve of genus $2$ is hyperelliptic,
$\Delta_{2,n}=\Gamma_{2,n}$. So the genus 2 case is a special case of the
hyperelliptic case.

######  Theorem A.1.

Suppose that $g$ and $n$ are non-negative integers satisfying $2g-2+n>0$. If
$g\leq 2$, then no finite index subgroup of the mapping class group
$\Gamma_{g,n}$ can be a Kähler group. If $g\geq 2$, then no finite index
subgroup of the hyperelliptic mapping class group $\Delta_{g,n}$ can be a
Kähler group.

The key point in the proof is that hyperelliptic mapping class groups and
mapping class groups in genera $\leq 1$ can be written as an extension of a
non-abelian free group by a finitely generated group. So, since non-abelian
free groups have infinitely many ends, the main technical ingredient of the
proof is the following result of Arapura, Bressler and Ramachandran.

######  Theorem A.2 (Arapura, Bressler, Ramachandran [1]).

If $\Gamma$ is an extension

$1\to K\to\Gamma\to G\to 1$

of a group with infinitely many ends by a finitely generated group, then
$\Gamma$ is not a Kähler group.

Note that a finite index subgroup of such a group $\Gamma$ is also an
extension of a group with infinitely many ends by a finitely generated group.
So no finite index subgroup of such a group $\Gamma$ can be a Kähler group.
Note also that the class of such groups is closed under extension by a
finitely generated group.

Suppose that $n\geq 1$ and that $U$ is a topological space. Denote the
configuration space of $n$ (labelled) points on $U$ by $C_{n}(U)$:

$C_{n}(U)=U^{n}-\text{(the fat diagonal)}$

where the fat diagonal consists of all $(u_{1},\dots,u_{n})\in U^{n}$ where
the $u_{j}$ are not distinct.

###  A.1. Genus $0$

Note that ${\mathcal{M}}_{0,4}$ is isomorphic to ${\mathbb{C}}-\\{0,1\\}$; the
point $t\in{\mathbb{C}}-\\{0,1\\}$ corresponds to the $4$-pointed rational
curve $({\mathbb{P}}^{1};0,1,\infty,t)$. It follows that $\Gamma_{0,4}$ is
free of rank 2 and therefore not Kähler. Suppose that $n\geq 4$. The fiber of
the projection ${\mathcal{M}}_{0,n}\to{\mathcal{M}}_{0,4}$ that forgets all
but the first 4 points is a fibration. Its fiber over the point of
$t\in{\mathbb{C}}-\\{0,1\\}$ is the configuration space
$C_{n-4}({\mathbb{C}}-\\{0,1\\})$. This is a hyperplane complement and has
finitely generated fundamental group. So $\Gamma_{0,n}$ is an extension of a
free group of rank 2 by a finitely generated group. Theorem A.2 implies that
no finite index subgroup of $\Gamma_{0,n}$ can be Kähler.

###  A.2. Genus $1$

The genus 1 mapping class group $\Gamma_{1,1}$ is isomorphic to
${\mathrm{SL}}_{2}({\mathbb{Z}})$. Since ${\mathrm{SL}}_{2}({\mathbb{Z}})$ is
virtually free, none of its finite index subgroups can be Kähler. Let $G$ be a
finite index free subgroup of $\Gamma_{1,1}$. All such $G$ are free of rank
$\geq 2$. Suppose that $n>1$. The inverse image $\Gamma$ of $G$ in
$\Gamma_{1,n}$ is an extension

$1\to K\to\Gamma\to G\to 1$

of $G$ by the fundamental group $K$ of the configuration space
$C_{n-1}(E^{\prime})$ of a once punctured genus 1 surface. Since the group $K$
is finitely generated, this implies that no finite index subgroup of
$\Gamma_{1,n}$ can be Kähler.

###  A.3. Hyperelliptic mapping class groups

We begin by recalling a few facts about hyperelliptic mapping class groups.
Suppose that $g\geq 2$ and that $S$ is a compact oriented surface of genus
$g$. A hyperelliptic involution $\sigma:S\to S$ is an orientation preserving
diffeomorphism of order 2 of $S$ with exactly $2g+2$ fixed points. By the
Riemann-Hurwicz formula, the quotient $S/\langle\sigma\rangle$ is a sphere.
From this it follows that all hyperelliptic involutions are conjugate in
$\operatorname{Diff}^{+}S$.

The hyperelliptic mapping class group $\Delta_{g}$ is defined to be the set of
isotopy classes of orientation preserving diffeomorphisms of $S$ that commute
with $\sigma$:

$\Delta_{g}:=\pi_{0}(\text{centralizer of $\sigma$ in
$\operatorname{Diff}^{+}S$}).$

A result of Birman and Hilden [3] implies that the natural homomorphism
$\Delta_{g}\to\Gamma_{g}$ is injective and that its image is the centralizer
of the isotopy class of $\sigma$ in $\Gamma_{g}$.

Denote the set of fixed points of $\sigma$ by $W$. The natural homomorphism
$\rho_{W}:\Delta_{g}\to\operatorname{Aut}W$ is surjective. There is an obvious
homomorphism $\ker\rho_{W}\to\Gamma_{0,2g+2}$ whose kernel is the subgroup
generated by the hyperelliptic involution. One therefore has an extension:

$1\to\langle\sigma\rangle\to\ker\rho_{W}\to\Gamma_{0,2g+2}\to 1.$

Since $\Gamma_{0,2g+2}$ is an extension of a non-abelian free group by a
finitely generated group, so is $\ker\rho_{W}$. Theorem A.2 implies that no
finite index subgroup of $\ker\rho_{W}$ (and thus of $\Delta_{g}$) can be
Kähler.

The hyperelliptic mapping class group $\Delta_{g,n}$ is defined to be the
restriction of the extension

$1\to\pi_{1}(C_{n}(S),\ast)\to\Gamma_{g,n}\to\Gamma_{g}\to 1$

to $\Delta_{g}$. It is an extension

$1\to\pi_{1}(C_{n}(S),\ast)\to\Delta_{g,n}\to\Delta_{g}\to 1.$

Since $\pi_{1}(C_{n}(S),\ast)$ is finitely generated, the inverse image of
$\ker\rho_{W}$ in $\Delta_{g,n}$ is an extension of a non-abelian free group
by a finitely generated group. Theorem A.2 implies that no finite index
subgroup of $\Delta_{g,n}$ can be Kähler.

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