# Orbifold Kähler Groups related to Mapping Class groups

Philippe Eyssidieux  Institut Fourier, Laboratoire de Mathématiques UMR 5582,
Université Grenoble Alpes, CS 40700, 38058 Grenoble, France
<EMAIL_ADDRESS>http://www-fourier.univ-grenoble-
alpes.fr/$\sim$eyssi/ and  Louis Funar  Institut Fourier, Laboratoire de
Mathématiques UMR 5582, Université Grenoble Alpes, CS 40700, 38058 Grenoble,
France<EMAIL_ADDRESS>http://www-fourier.univ-grenoble-
alpes.fr/$\sim$funar/

(Date: December, 13, 2021)

###### Abstract.

We construct certain orbifold compactifications of the moduli stack of pointed
stable curves over $\mathbb{C}$ and study their fundamental groups by means of
their quantum representations. This enables to construct interesting Kähler
groups and to settle most of the candidates for a counter-example to the
Shafarevich conjecture on holomorphic convexity proposed in 1998 by Bogomolov
and Katzarkov, using TQFT representations of the mapping class groups.

2010 MSC Classification: 14 H 10, 32 Q 30 (Primary) 32 J 25, 32 G 15, 14 D 23,
57 M 07, 20 F 38, 20 G 20, 22 E 40 (Secondary).

Keywords: Kähler Orbifolds, Kähler Groups, Mapping class group, Moduli space
of Curves, Deligne-Mumford stacks, Topological Quantum Field Theory.

The research of P. E. was partially supported by the ANR project Hodgefun
ANR-16-CE40-0011-01.

###### Contents

  1. 1 Introduction and statements
  2. 2 The Kähler Orbifold ${\overline{\mathcal{M}_{g}^{n\;an}}}[p]$ and its fundamental group
  3. 3 Quantum mapping class group representations
  4. 4 TQFT representations on isotropy groups of stable curves
  5. 5 Bogomolov-Katzarkov Surfaces
  6. 6 Using TQFT representations for the Shafarevich Conjecture

##  1\. Introduction and statements

We denote by ${\rm{Mod}}(\Sigma_{g})$ the mapping class group of a closed
oriented surface $\Sigma_{g}$ of genus $g$. Further
${\rm{Mod}}(\Sigma_{g}^{1})$ denotes the mapping class group of the pair
$(\Sigma_{g},\ast)$, namely the group of isotopy classes of orientation
preserving homeomorphisms of $\Sigma_{g}$ fixing $\ast$. It is well-known that
${\rm{Mod}}(\Sigma_{g}^{1})$ is isomorphic to the mapping class group of the
punctured surface $(\Sigma_{g},\ast)$. By forgetting the marked point one
obtains a surjective homomorphism
${\rm{Mod}}(\Sigma_{g}^{1})\to{\rm{Mod}}(\Sigma_{g})$ which is part of the
Birman exact sequence:

$1\to\pi_{1}(\Sigma_{g})\to{\rm{Mod}}(\Sigma_{g}^{1})\to{\rm{Mod}}(\Sigma_{g})\to
1$

The Dehn-Nielsen-Baer theorem states that the map associating to
$\varphi\in{\rm{Mod}}(\Sigma_{g}^{1})$ the automorphism
$\varphi_{*}:\pi_{1}(\Sigma_{g},\ast)\to\pi_{1}(\Sigma_{g},\ast)$ provides an
isomorphism between ${\rm{Mod}}(\Sigma_{g}^{1})$ and ${\rm{Aut}}^{+}(\pi_{g})$
and induces an isomorphism ${\rm{Mod}}(\Sigma_{g})\to{\rm{Out}}^{+}(\pi_{g})$.
Furthermore, the diagram below is commutative:

$\begin{array}[]{ccccccccc}1&\to&\pi_{1}(\Sigma_{g})&\to&{\rm{Mod}}(\Sigma_{g}^{1})&\to&{\rm{Mod}}(\Sigma_{g})&\to&1\\\
&&\downarrow&&\downarrow&&\downarrow&&\\\
1&\to&\pi_{1}(\Sigma_{g})&\to&{\rm{Aut}}^{+}(\pi_{g})&\to&{\rm{Out}}^{+}(\pi_{g})&\to&1\\\
\end{array}$

where the top horizontal line is the Birman exact sequence.

The groups ${\rm{Mod}}(\Sigma_{g})$ and ${\rm{Mod}}(\Sigma_{g}^{1})$ occur in
complex algebraic geometry as the fundamental groups of the analytifications
of the moduli stacks of curves $\mathcal{M}_{g}^{an}$ and $\mathcal{M}_{g}^{1\
an}$ respectively. More generally, let ${\rm{Mod}}(\Sigma_{g}^{n})$ be the
mapping class group of the genus $g$ orientable surface $\Sigma_{g}^{n}$ with
$n$ punctures (or marked points) and ${\rm{PMod}}(\Sigma_{g}^{n})$ denote the
pure mapping class group of those classes which fix pointwise the punctures.
Then ${\rm{PMod}}(\Sigma_{g}^{n})$ and ${\rm{Mod}}(\Sigma_{g}^{n})$ also occur
as the fundamental groups of the analytification $\mathcal{M}^{n\ an}_{g}$ of
the moduli stacks of curves and $[S_{n}\backslash{\mathcal{M}^{n\ an}_{g}}]$,
respectively, where $S_{n}$ acts on $\mathcal{M}^{n\ an}_{g}$ permuting the
markings. These stacks are separated, smooth, Deligne-Mumford and their moduli
spaces are quasiprojective, but non proper except in the trivial case
$g=0,n=3$.

Constructing stacky compactifications of such objects can be used an
intermediate step to construct interesting Kähler groups [18, 19] and in the
present case, to recast some of the basic ideas of Teichmüller level
structures [7, 9, 42].

For every integer $p\geq 2$ we introduce the subgroup
${\rm{Mod}}(\Sigma_{g}^{n})[p]\subset{\rm{Mod}}(\Sigma_{g})$ generated by the
$p$-th powers of Dehn twists along simple closed curves in $\Sigma_{g}^{n}$.
Note that ${\rm{Mod}}(\Sigma_{g}^{n})[p]\subset{\rm{PMod}}(\Sigma_{g}^{n})$ is
a normal subgroup of ${\rm{Mod}}(\Sigma_{g})$ and in fact a characteristic
one. More generally, there are only finitely many, say $N_{g,n}$ conjugacy
classes of Dehn twists, or equivalently, distinct orbits of essential simple
closed curves on $\Sigma_{g}^{n}$ under the mapping class group action. For
instance $N_{g}=\lfloor\frac{g}{2}\rfloor+1$, $N_{g,1}=g$. Simple closed
curves orbits are determined by the homeomorphism type of the complementary
subsurface, which might be either connected for non-separating curves or
disconnected and hence determined by the set/pair of genera of its two
components, for separating curves. Fix an enumeration of these homeomorphism
types starting with the non-separating one. For each vector
$\mathbf{k}=(k_{0},k_{1},k_{2},\ldots,k_{N_{g,n}-1})$ we define
${\rm{Mod}}(\Sigma_{g}^{n})[\mathbf{k}]$ as the (normal) subgroup generated by
$k_{0}$-th powers of Dehn twist along non-separating simple closed curves and
$k_{j}$-th powers of Dehn twists along simple closed curves of type $j$. As a
shortcut we use ${\rm{Mod}}(\Sigma_{g}^{n})[k;m]$ for
$\mathbf{k}=(k,m,m,m\ldots,m)$ and ${\rm{Mod}}(\Sigma_{g}^{n})[k;-]$ for
$\mathbf{k}=(k;)$, where $k_{i}$ are absent for $i>0$.

The groups ${\rm{Mod}}(\Sigma_{g})/{\rm{Mod}}(\Sigma_{g})[p]$ were introduced
and studied in [23] and
${\rm{Mod}}(\Sigma_{g}^{1})/{\rm{Mod}}(\Sigma_{g}^{1})[p]$ in [25]. In [2] the
authors proved that they are virtually Kähler groups by considering the
Deligne-Mumford compactifications of the moduli spaces of curves for some
level structures for which they are smooth projective varieties (see e.g.
[6]). To show that these groups are actually Kähler we study the images of the
isotropy groups under quantum representations of mapping class groups in order
to be able to use the uniformizability criterion of [17]. Note that quantum
representations of these groups cannot be injective unless their images are
non-arithmetic subgroups of higher rank Lie groups (see [26]).

Our first main result is:

######  Theorem 1.1.

Let $g,n\in\mathbb{N}$ such that $2-2g-n<0$, $(g,n)\not=(1,1),(2,0)$.

For every ${\mathbf{k}}\in\mathbb{N}_{>0}^{N_{g,n}}$ the quotient
${\rm{PMod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[\mathbf{k}]$ is the
fundamental group of a compact Kähler orbifold ${\overline{\mathcal{M}_{g}^{n\
an}}}[\mathbf{k}]$ compactifying $\mathcal{M}_{g}^{n\ an}$ whose moduli space
is the moduli space ${\overline{{M}_{g}^{n\ an}}}$ of stable $n$-punctured
curves of genus $g$.

If $p\geq 5$ is odd, ${\overline{\mathcal{M}_{g}^{n\ an}}}[p]$ is
uniformizable, hence
${\rm{PMod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[p]$ is a Kähler group,
i.e. it occurs as the fundamental group of a compact Kähler manifold.

If $p\geq 10$ is even and $(g,n,p)\neq(2,0,12)$,
$\overline{{\mathcal{M}}_{g}^{n\;an}}[2p,p/{\rm g.c.d.}(p,4)]$ is
uniformizable and
${\rm{Mod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[2p,p/{\rm
g.c.d.}(p,4)]$ is a Kähler group.

It is an important open question whether one can construct a complex
projective surface whose fundamental group is infinite and has no complex
linear finite dimensional representation with infinite image. For most $p$,
one knows that ${\rm{Mod}}(\Sigma_{g})/{\rm{Mod}}(\Sigma_{g})[p]$ is infinite
but the proof uses a linear representation [23]. This representation can be
lifted to a representation of
${\rm{PMod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[p]$ with infinite
image. Hence our construction gives infinite Kähler groups but will not solve
that question. The fact that the representation comes from a Reshetikhin-
Turaev TQFT is however striking from an algebro-geometric perspective since it
is not known to the authors if these representations are motivic or rigid.

However, our construction was motivated by its relation to certain complex
projective surfaces proposed by Bogomolov and Katzarkov [8] as potential
counterexamples to the Shafarevich conjecture on holomorphic convexity that
remained unsettled up to now.

These surfaces are obtained in the following way. Let $\pi:S\to C$ be a smooth
projective surface fibered over a curve such that the regular fibers have
genus $g\geq 2$ and the singular fibers are reduced and have only nodes. Let
$N\in\mathbb{N}^{*}$ be a positive integer. Perform a base change along a
covering $C^{\prime}\to C$ with uniform ramification index $N$ at the images
of the singular fibers. Then, for most $N$, and if $\pi$ is stable, the
surface $S\times_{C}C^{\prime}$, which has a finite number of singular points
of type $A_{N-1}$ lying over the nodes in the singular fibers, has a quasi-
étale cover $S(N)$ which is a smooth projective surface inheriting from $\pi$
a fibration over some curve. The image $I_{N}$ of the fundamental group of a
general fiber $F$ of this fibration in $\pi_{1}(S(N))$ is what they call a
Burnside type quotient of $\pi_{g}$, and is an infinite group if $N$ is large
enough. The only ‘obvious’ linear representations of $\pi_{1}(S(N))$ with
infinite image have finite image when restricted to $I_{N}$. If $\pi$ is not
stable, or $N$ is even, then $S(N)$ may be only a non developable orbisurface.

However, we shall see that quantum representations give rise to
representations of $\pi_{1}(S(N))$, the reason being that $S(N)$ maps non
trivially to $\overline{\mathcal{M}^{1\ an}_{g}}[N]$. The recent result [36]
translates into the fact that most of them have an infinite image on its
general fiber. A remarkable property of TQFT representations, their
functoriality when passing to subsurfaces, actually enables to settle the
question raised by [8] in most cases:

######  Theorem 1.2.

If $N\not\in\\{1,2,3,4,6,8,12,16,20,24\\}$ the universal covering space of the
Bogomolov-Katzarkov surface ${S}(N)$ is holomorphically convex. Moreover, if
$\pi:S\to C$ is a stable curve over $C$, then it is Stein.

If $S(N)$ is not developpable, its universal covering space is a complex
analytic Deligne-Mumford stack, and we say it is Stein, resp. holomorphically
convex, if such is its moduli space (a normal complex analytic space with only
quotient singularities).

We believe the examples constructed here to be extremely interesting from the
point of view of uniformization in several complex variables. It does not seem
to be trivial task to settle the Shafarevich and the Toledo conjectures for
${\overline{\mathcal{M}_{g}}}[p]$. We also conjecture Kazhdan’s property (T)
for its fundamental group since it would follow from Kazhdan’s property (T)
for the mapping class group - in particular we would have
$q^{+}({\overline{\mathcal{M}_{g}}}[p])=0$, which follows from the Ivanov
conjecture. We hope to come back on these questions in future work.

In the next section we define the Mumford-Deligne stack
${\overline{\mathcal{M}_{g}^{n\;an}}}[\mathbf{k}]$ and show that it is an
orbifold with the fundamental group
${\rm{PMod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[\mathbf{k}]$. In
section 3, we introduce quantum representations of mapping class groups,
compute the orders of Dehn twists and prove they factor through
${\rm{Mod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[p]$ if $p$ is odd and
${\rm{Mod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[2p,p/g.c.d(4,p)]$ if
$p$ even. We prove that these quantum representations have infinite images in
most cases. These results are further used in section 4 to show that the
orbifolds ${\overline{\mathcal{M}_{g}^{n\;an}}}[p]$ are uniformizable, thereby
proving Theorem 1.1. In section 5, we explain the construction of the
Bogomolov-Katzarkov surfaces by starting with a non isotrivial fibration of a
smooth complex projective surface, namely the pull-back of the universal curve
${\overline{\mathcal{M}_{g}^{1\;an}}}$ along some holomorphic maps from a
smooth algebraic curve into ${\overline{\mathcal{M}_{g}^{an}}}$ and then
taking the cartesian product with ${\overline{\mathcal{M}_{g}^{1\;an}}}[p]$.
We show that the Bogomolov-Katzarkov surfaces obtained in that manner are
uniformizable Kähler orbifolds, when the orbifolds
${\overline{\mathcal{M}_{g}^{1\;an}}}[p]$ and
${\overline{\mathcal{M}_{g}^{1\;an}}}[2p,p/{\rm g.c.d}(4,p)]$ for even $p$
respectively, are uniformizable. Section 6 contains a proof of Theorem 1.2
answering in the affirmative the Shafarevich conjecture on holomorphic
convexity for these surfaces in most cases, by analyzing the images of their
fundamental groups under the quantum representations above and using [20] (we
actually only need the reductive surface case proved in [33]).

All algebraic varieties and stacks will be over $\mathbb{C}$ and we will
mainly think of them through their analytification as complex-analytic
objects.

Acknowledgements. The authors are indebted to J. Aramayona, M. Boggi, L.
Katzarkov, B. Klingler, J. Marché and C. Simpson for helpful discussions.

##  2\. The Kähler Orbifold ${\overline{\mathcal{M}_{g}^{n\;an}}}[p]$ and its
fundamental group

###  2.1. Conventions

As advocated by [41], we follow the conventions of [19] and view an orbifold
as a smooth Deligne-Mumford stack with trivial generic isotropy groups
relative to the category of complex analytic spaces with Hausdorff topology.
There is an analytification $2$-functor from algebraic Deligne-Mumford stacks
over $\mathbb{C}$ to complex analytic Deligne-Mumford stacks and an underlying
topological stack functor from complex analytic Deligne-Mumford stacks to
topological Deligne-Mumford stacks [45, 46, 3]. We will only consider
separated quasi-compact algebraic stacks locally of finite type and separated
analytical/topological stacks with proper diagonal.

On a complex or topological stack we use the Hausdorff topology, see [51] for
the relation with fancier topologies in the algebraic case. In the complex
analytic and topological case, we can always refine a covering of a Deligne-
Mumford stack by a covering by open substacks that are quotients of a proper
action of a finite group. If $\mathcal{X}$ is an algebraic Deligne-Mumford
stack $\mathcal{X}$ there is a structural sheaf $\mathcal{O}_{\mathcal{X}}$,
an abelian category of coherent algebraic sheaves and similarly in the complex
analytic case, see [39, 12]. When we represent a stack by an étale groupoid a
sheaf comes by descent from an equivariant sheaf on the groupoid.

An effective Cartier divisor of an algebraic/analytic is a closed substack
which is defined by an invertible ideal sheaf, locally generated by a nonzero
divisor. It is actually better to slightly enlarge the notion and in our
terminology a Cartier divisor on $\mathcal{X}$ will be a pair
$(s,\mathcal{L})$ of an invertible sheaf $\mathcal{L}$ and a section
$s:\mathcal{O}_{\mathcal{X}}\to\mathcal{L}$ up to equivalence (multiplication
by an invertible function). The resulting groupoid of Cartier divisors is
naturally equivalent to the groupoid ${\rm
Hom}(\mathcal{X},[\mathbb{A}^{1}/\mathbb{G}_{m}])$ (see [47], p.778) where
$[\mathbb{A}^{1}/\mathbb{G}_{m}])$ is the quotient stack of the usual action
of $\mathbb{G}_{m}$ on $\mathbb{A}^{1}$, and is an algebraic stack which is
not Deligne-Mumford.

The multiplication map
$m:[\mathbb{A}^{1}/\mathbb{G}_{m}]\times[\mathbb{A}^{1}/\mathbb{G}_{m}]\to[\mathbb{A}^{1}/\mathbb{G}_{m}]$
enables one to define the sum of two divisors by the composition
$\mathcal{X}\to[\mathbb{A}^{1}/\mathbb{G}_{m}]\times[\mathbb{A}^{1}/\mathbb{G}_{m}]\buildrel
m\over{\longrightarrow}[\mathbb{A}^{1}/\mathbb{G}_{m}]$ and the pull back
$f^{*}=\circ f:{\rm Hom}(\mathcal{X},[\mathbb{A}^{1}/\mathbb{G}_{m}])\to{\rm
Hom}(\mathcal{Y},[\mathbb{A}^{1}/\mathbb{G}_{m}])$, for every $f\in Ob({\rm
Hom}(\mathcal{Y},\mathcal{X}))$.

### 2.2.

Recall (see [46, 41] that an orbifold is a smooth Deligne-Mumford stack with
trivial generic isotropy groups relative to the category of complex analytic
spaces with the classical topology. An orbifold $\mathcal{X}$ is called
developable if its universal covering stack is a smooth manifold, namely if
every local inertia morphism
$\pi_{1}(\mathcal{X},x)_{loc}\to\pi_{1}(\mathcal{X},x)$ is injective, for $x$
an orbifold point of $\mathcal{X}$. The orbifold $\mathcal{X}$ is
uniformizable whenever the profinite completion of a local inertia morphism,
namely $\pi_{1}(\mathcal{X},x)_{loc}\to\widehat{\pi_{1}(\mathcal{X},x)}$ is
still injective. In particular developable orbifolds with residually finite
fundamental groups are uniformizable.

### 2.3.

For $2g-2+n>0$ the stack $\overline{\mathcal{M}_{g}^{n}}$ of $n$-punctured
genus $g$ stable curves is a smooth proper Deligne-Mumford stack, whose
analytification $\overline{\mathcal{M}_{g}^{n\;an}}$ is a smooth proper
complex analytic Deligne-Mumford stack too (see [34], Thm. 2.7), actually an
orbifold if $(g,n)\neq(2,0)$. Moreover, $\overline{\mathcal{M}_{g}^{n\;an}}$
carries a Kähler metric in the sense of [21], from Knudsen-Mumford’s theorem
that the moduli space $\overline{M_{g}^{n}}$ is projective (see [35],
Thm.6.1). Observe that the forgetful map
$f_{1}:\overline{\mathcal{M}^{1}_{g}}\to\overline{\mathcal{M}_{g}}$ is proper
and representable ([15]), more precisely it is schematic (we use the
convention of [39] for representability).

### 2.4.

The stack of smooth curves is realized as an open substack
$j:\mathcal{M}_{g}\hookrightarrow\overline{\mathcal{M}_{g}}$ and
$\overline{\mathcal{M}_{g}}\setminus\mathcal{M}_{g}$ is a union of irreducible
Cartier divisors
$\mathcal{D}_{0},\mathcal{D}_{1},\ldots,\mathcal{D}_{\lfloor\frac{g}{2}\rfloor}$.

The generic point of $\mathcal{D}_{0}$ is a genus $g-1$ curve with one node
and the generic point of $\mathcal{D}_{k}$ has two irreducible smooth curves
as components which are meeting at a node and whose respective genera are $k$
and $g-k$. The total divisor

$\mathcal{D}=\sum_{i=0}^{\lfloor\frac{g}{2}\rfloor}\mathcal{D}_{i},$

is a normal crossing divisor (see [15]).

### 2.5.

The map $j^{*}f_{1}$ is the forgetful map
$\mathcal{M}_{g}^{1}\to\mathcal{M}_{g}$ and is representable, smooth and
proper. Hence
$\mathcal{D}^{1}:=\overline{\mathcal{M}^{1}_{g}}\setminus\mathcal{M}_{g}^{1}$
is a closed substack with a representable proper map
$f_{1}|_{\mathcal{D}^{1}}:\mathcal{D}^{1}\to\mathcal{D}$. Now,

$\mathcal{D}^{1}=\sum_{i=0}^{g-1}\mathcal{D}^{1}_{i}$

where the generic point of $\mathcal{D}^{1}_{i}$ for $i>0$ represents a nodal
curve with a component of genus $i$ carrying the one marked point and a
component of genus $g-i$, both meeting at a node which is distinct from the
marked point. On the other hand the generic point of $\mathcal{D}^{1}_{0}$
represents an irreducible nodal curve of genus $g-1$ with one marked point. It
follows that
$f_{1}^{*}(\mathcal{D}_{i})=\mathcal{D}_{1,i}+\mathcal{D}_{1,g-i}$, is a
divisor whose two components meet transversally in codimension $1$, when
$0<i<\frac{g}{2}$ whereas $f^{*}_{1}(\mathcal{D}_{0})=\mathcal{D}^{1}_{0}$. If
$g\equiv 0\;({\rm mod}\;2)$, then
$f^{*}_{1}(\mathcal{D}_{\frac{g}{2}})=\mathcal{D}^{1}_{\frac{g}{2}}$. The
divisors $\mathcal{D}^{1}_{i}$ are irreducible, the total divisor
$\mathcal{D}^{1}$ is a normal crossing divisor (see [34], Thm.2.7) but not a
simple normal crossing one. For instance $\mathcal{D}^{1}_{0}$ is singular at
a singular point of the generic fiber of
$\mathcal{D}^{1}_{0}\to\mathcal{D}_{0}$.

More generally, the stack ${\mathcal{M}}_{g}^{n}$ of smooth $n$-pointed smooth
curves of genus $g$ is an open substack
$j:{\mathcal{M}}_{g}^{n}\to\overline{{\mathcal{M}}_{g}^{n}}$ of the Deligne-
Mumford stack of stable $n$-pointed curves of genus $g$, when $2g-2+n>0$. Then
$\mathcal{D}^{n}=\overline{{\mathcal{M}}_{g}^{n}}\setminus{\mathcal{M}}_{g}^{n}$
is a union of irreducible Cartier divisors $\mathcal{D}_{i}$,
$i=0,\ldots,N_{g,n}-1$. Knudsen proved in [34] that
$\overline{{\mathcal{M}}_{g}^{n+1}}$ is equivalent to the stack of $n$-pointed
stable curves endowed with an extra section, namely the universal $n$-pointed
stable curve. Moreover,
$\mathcal{D}^{n}=\overline{{\mathcal{M}}_{g}^{n}}\setminus{\mathcal{M}}_{g}^{n}$
is a union of irreducible Cartier divisors $\mathcal{D}_{i}$,
$i=0,\ldots,N_{g,n}-1$ and a normal crossings divisor. More specifically the
projection map
$f_{n+1}:\overline{{\mathcal{M}}_{g}^{n+1}}\to\overline{{\mathcal{M}}_{g}^{n}}$
has the property that

$D^{n+1}=f_{n+1}^{-1}(D^{n})+\sum_{i=1}^{n}S^{i,n+1},$

where a point of $S_{n+1}^{i}$ corresponds to a stable curve with a rational
component having two marked points labeled $i$ and $n+1$, namely $S^{i,n+1}$
is the image of the canonical $i$-th section
$\overline{{\mathcal{M}}_{g}^{n}}\to\overline{{\mathcal{M}}_{g}^{n+1}}$.

######  Lemma 2.1.

We have $f_{n+1}^{*}\mathcal{D}^{n}=\mathcal{D}^{n+1}-\sum_{i=1}^{n}S^{i,n+1}$
and in particular, $f_{1}^{*}\mathcal{D}=\mathcal{D}^{1}$.

###### Proof.

See ([34], Proof of Thm.2.7). ∎

###  2.6. The canonical stack of the root stack of a normal crossing divisor
that may not be simple

The first ingredient of our construction is the root stack
$\mathcal{S}\left[\sqrt[p]{\mathcal{E}}\right]$ associated to a Cartier
divisor $\mathcal{E}$ on a stack $\mathcal{S}$, which was introduced
independently in [1, 10] and used in [18]. Specifically, it is defined as:

$\mathcal{S}\left[\sqrt[p]{\mathcal{E}}\right]:=\mathcal{S}\times_{\phi_{\mathcal{E}},[\mathbb{A}^{1}/\mathbb{G}_{m}],\\_^{p}}\left[\mathbb{A}^{1}/\mathbb{G}_{m}\right],$

where
$\phi_{\mathcal{E}}:\mathcal{S}\to\left[\mathbb{A}^{1}/\mathbb{G}_{m}\right]$
is the natural map and $\\_^{p}$ is the $p$-th power map,
$p\in\mathbb{N}^{*}$, the case $p=1$ being trivial. We use $2$-fibered
products above as one should (see [39]).

Note that the morphism
$\mathcal{S}\left[\sqrt[p]{\mathcal{E}}\right]\to{\mathcal{S}}$ is universal
among morphisms for which $\mathcal{E}$ pulls back to $p$ times a Cartier
divisor.

The same construction can be performed with respect to the topological and the
analytic categories after application of $\\_^{an},\\_^{top}$ since these lax
2-functors preserve finite homotopy limits.

The second ingredient is the construction of the canonical stack due to
Vistoli (see [53]). Recall that a Deligne-Mumford stack $\mathcal{X}$ over a
field of characteristic zero has quotient singularities if it has a (locally
étale in the algebraic case) cover which is a scheme with quotient
singularities, namely locally the quotient of a smooth variety by a finite
group. If $\mathcal{X}$ is a separated analytic Deligne-Mumford stack of
finite type with generic trivial isotropy having only quotient singularities
then there is a smooth canonical stack $\mathcal{X}^{can}\to\mathcal{X}$ which
is an equivalence in codimension 1 and which is unique up to equivalence.

Let $\mathcal{S}^{an}$ be an analytic smooth Deligne-Mumford stack and
$\mathcal{E}$ be a normal crossings divisor. Set $\Delta^{n}$ for the open
disk around the origin in $\mathbb{C}^{n}$ and
$E_{j}=\\{z_{j}=0\\}\subset\Delta^{n}$ be the coordinate hyperplanes
associated to a system of complex coordinates $(z_{j})$. An étale map
$\eta:\Delta^{n}\to\mathcal{S}^{an}$ is said to be a chart adapted to
$\mathcal{E}$ if

$\eta^{*}\mathcal{E}=\left\\{(z_{i})_{i=1}^{n};\
\prod_{c=1}^{k}z_{i_{c}}=0\right\\}=\sum_{c=1}^{k}{E}^{\eta}_{i_{c}},$

where the map $c\mapsto i_{c}$ is injective. In order to be precise, we shall
at some point display the dependence of $k$ in $\eta$ and use the notation
$k=k(\eta)$.

######  Proposition 2.2.

If $\mathcal{S}^{an}$ is an analytic smooth Deligne-Mumford stack with trivial
generic isotropy and $\mathcal{E}$ is a normal crossing divisor on
$\mathcal{S}^{an}$, then there exists a smooth Deligne-Mumford stack
$e_{p}:\mathcal{S}^{an}[p,\mathcal{E}]\to\mathcal{S}^{an}$ unique up to
equivalence with the same moduli space as $\mathcal{S}^{an}$ such that for
every adapted chart $\eta$ we have an equivalence:

$\Delta^{n}\times_{\mathcal{S}^{an},\eta,e_{p}}\mathcal{S}^{an}\left[p,\mathcal{E}\right]\simeq\Delta^{n}\left[\sqrt[p]{E_{i_{1}}^{\eta}}\right]\times_{\Delta^{n}}\times\ldots\times_{\Delta^{n}}\Delta^{n}\left[\sqrt[p]{E_{i_{k}}^{\eta}}\right].$

###### Proof.

We first claim that
$\mathcal{S}^{an}\left[\sqrt[p]{\mathcal{E}_{1}}\right]\times_{\mathcal{S}}\mathcal{S}\left[\sqrt[p]{\mathcal{E}_{2}}\right]\times_{\mathcal{S}}\cdots\times_{\mathcal{S}}\mathcal{S}\left[\sqrt[p]{\mathcal{E}_{k}}\right]$
has quotient singularities.

Indeed, if one removes a codimension $\geq 2$ closed substack $\mathcal{Z}$
from the smooth stack

$\mathcal{V}=\Delta^{n}\left[\sqrt[p]{E_{i_{1}}^{\eta}}\right]\times_{\Delta^{n}}\cdots\times_{\Delta^{n}}\Delta^{n}\left[\sqrt[p]{E_{i_{k}}^{\eta}}\right]$

the fundamental group will not change. Such a $\mathcal{Z}$ is actually the
substack over $Z\subset\Delta^{n}$ where $Z$ is a closed analytic subspace.
Then the universal covering stack of $\mathcal{V}$ is $\Delta^{n}$ and
$\mathcal{V}=[\mu_{p}^{k}\backslash\Delta^{n}]$. Here $\mu_{p}$ is the group
of $p$-th roots of unity and the $s$-th factor acts by multiplication of the
$s$-th coordinate.

Consider now the stack
$\mathcal{U}=\Delta^{n}\left[\sqrt[p]{E_{I_{1}}}\right]\times_{\Delta^{n}}\cdots\times_{\Delta^{n}}\Delta^{n}\left[\sqrt[p]{E_{I_{m}}}\right]$,
where we denote $E_{I}=\sum_{j\in I}E_{j}^{\eta}$, and $I_{j}$, for $1\leq
l\leq m$ form a partition of the set $\\{i_{1},\ldots,i_{k}\\}$. The inclusion
of the residual gerbe at the origin in $\mathcal{U}$ is a deformation retract.
Then the local uniformizability of analytic Deligne-Mumford stacks implies
that $\mathcal{U}\simeq[G\backslash U]$, where $U$ is an analytic space, $G$
has a fixed point over the origin and the natural map
$m_{\mathcal{U}}:U\to\Delta^{n}$ is $G$-equivariant. Then
$\mathcal{U}\setminus m^{-1}(Z)$ is a connected uniformizable stack étale over
$\Delta^{n}\setminus Z$, in particular $[\mathcal{U}\setminus
m^{-1}(Z)]\simeq[G^{\prime}\backslash(\Delta^{n}\setminus
m_{\mathcal{V}}^{-1}(Z))]$, for some finite group $G^{\prime}$. As the root
stack construction behaves well under pull back, there is an isomorphism
${U}=G^{\prime}\backslash\Delta^{n}$, where
$G^{\prime}=\ker(\mu_{p}^{k}\to\mu_{p}^{m})$ is the kernel of the homomorphism
$(\zeta_{l})\mapsto(\prod_{l\in I_{1}}\zeta_{l},\ldots,\prod_{l\in
I_{m}}\zeta_{l})$. This proves the claim.

Furthermore, we can define, using Vistoli’s canonical stack which applies to a
Deligne-Mumford stack $\mathcal{X}$ with quotient singularities and yields a
smooth Deligne-Mumford stack together with a map
$\psi:\mathcal{X}^{can}\to\mathcal{X}$ which is an equivalence in codimension
one, see [27]:

$\mathcal{S}^{an}[p,{\mathcal{E}}]=\left(\mathcal{S}^{an}\left[\sqrt[p]{\mathcal{E}_{1}}\right]\times_{\mathcal{S}}\mathcal{S}\left[\sqrt[p]{\mathcal{E}_{2}}\right]\times_{\mathcal{S}}\cdots\times_{\mathcal{S}}\mathcal{S}\left[\sqrt[p]{\mathcal{E}_{k}}\right]\right)^{can}.$

The uniqueness statement is the main theorem in [27] at least in the algebraic
case. The proof given there can be adapted the (easier) analytic case.

∎

Obviously, the construction does not require the ramification indices to be
equal and the proof above shows that:

######  Proposition 2.3.

Let $\mathcal{E}_{1},\ldots,\mathcal{E}_{r}$, $1\leq r\leq\infty$, be the
irreducible components of $\mathcal{E}$ and fix
$\mathbf{p}:=(p_{1},\ldots,p_{r})\in\mathbb{N}^{*r}$. There is a unique (up to
equivalence) smooth Deligne-Mumford stack
$\mathcal{S}^{an}\left[\mathbf{p},\mathcal{E}\right]\to\mathcal{S}^{an}$ which
is an equivalence outside $\mathcal{E}$ and ramifies with index $p_{i}$ on
$\mathcal{E}_{i}$.

This is probably valid in the algebraic category over an algebraically closed
field assuming tame ramification.

######  Remark 2.4.

(See [21] for a proof.) If $\mathcal{S}^{an}$ carries a Kähler metric, so does
$\mathcal{S}^{an}\left[\mathbf{p},\mathcal{E}\right]$.

###  2.7. The analytic stacks $\overline{\mathcal{M}^{an}_{g}}[\mathbf{p}]$

######  Definition 2.5.

Given $\mathbf{p}\in(\mathbb{N}^{*})^{N_{g,n}}$ we denote by
${\overline{\mathcal{M}_{g}^{n\ an}}}[\mathbf{p}]$ the smooth proper Deligne
Mumford stack ${\overline{\mathcal{M}_{g}^{n\ an}}}[\mathbf{p},\mathcal{D}]$.
If $p,q\geq 2$ we shall use the specific notation
${\overline{\mathcal{M}_{g}^{n\ an}}}[p,q]$ for the smooth proper Deligne
Mumford stack ${\overline{\mathcal{M}_{g}^{n\
an}}}[(p,q,q,\ldots,q),\mathcal{D}]$ and will drop $q$ from the notation, when
$q=p$.

######  Theorem 2.6.

Let $g,n\in\mathbb{N}$ such that $2g-2+n>0$, $(g,n)\not\in\\{(1,1),(2,0)\\}$.
The quotient
${\rm{PMod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[\mathbf{p}]$ is the
fundamental group of the compact Kähler orbifold
${\overline{\mathcal{M}_{g}^{n\ an}}}[\mathbf{p}]$.

###### Proof of Theorem 2.6.

There is an isomorphism $\pi_{1}(\mathcal{M}_{g}^{n\
an})\simeq{\rm{Mod}}(\Sigma_{g}^{n})$ which carries the conjugate of the
meridian loops around the various components of $\mathcal{D}^{n}$ to the Dehn
twists along simple closed curves. By Proposition 2.2 the analytic stack
${\overline{\mathcal{M}_{g}^{n\ an}}}[\mathbf{p}]$ is a proper smooth Deligne
Mumford orbifold endowed with a map ${\overline{\mathcal{M}_{g}^{n\
an}}}[\mathbf{p}]\to{\overline{\mathcal{M}_{g}^{n\ an}}}$ inducing an
isomorphism on the moduli space. As ${\overline{\mathcal{M}_{g}^{n\
an}}}[\mathbf{p}]$ is smooth, we can delete the codimension $2$ singular
substack of $\mathcal{D}^{n}$ without changing the orbifold fundamental group.
The stack-theoretic Van Kampen theorem from [58] implies that the fundamental
group of the orbifold ${\overline{\mathcal{M}_{g}^{n\ an}}}[\mathbf{p}]$ is
${\rm{PMod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[\mathbf{p}]$.

Eventually, the orbifold ${\overline{\mathcal{M}_{g}^{n\ an}}}[\mathbf{p}]$
has projective moduli space and hence is a Kähler orbifold by [21]. ∎

For the easy description of the isotropy groups of these orbifolds, see
section 4.2.

######  Example 2.7.

The group ${\rm{PMod}}(\Sigma_{0}^{5})/{\rm{Mod}}(\Sigma_{0}^{5})[5]$ is a
uniform complex hyperbolic group constructed by Hirzebruch: it corresponds to
the quotient of the surface of degree $5^{5}$ over $\mathbb{P}^{2}$ which
ramifies with multiplicity $5$ on $CEVA(2)$ by the (abelian) group of this
Galois covering. For general $p$ see [18].

A similar statement works for
${\rm{Mod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[p]$ taking the quotient
of ${\overline{\mathcal{M}_{g}^{n\ an}}}[p]$ by the group of permutations of
$n$ punctures, see [50] for group actions on stacks.

######  Remark 2.8.

If we delete the singular locus $\mathcal{D}^{n,sing}$ of $\mathcal{D}$, the
divisor of the pair

$({\overline{\mathcal{M}_{g}^{n\
an}}}\setminus\mathcal{D}^{n,sing},\mathcal{D}^{n,reg})$

is actually an effective smooth divisor. The root stack

${\overline{\mathcal{M}_{g}^{n\ an\
0}}}[\mathbf{p}]:=({\overline{\mathcal{M}_{g}^{n\
an}}}\setminus\mathcal{D}^{n,sing})[\sqrt[\mathbf{p}]{\mathcal{D}^{n,reg}}]$

is the complement of an closed substack of codimension at least $2$ in
${\overline{\mathcal{M}_{g}^{n\ an}}}[\mathbf{p}]$ hence the natural map
${\overline{\mathcal{M}_{g}^{n\ an\
0}}}[\mathbf{p}]\to{\overline{\mathcal{M}_{g}^{n\ an}}}[\mathbf{p}]$ induces
an isomorphism of the fundamental groups.

Uniformizability of ${\overline{\mathcal{M}_{g}^{n\ an}}}[\mathbf{p}]$ can
then be formulated only in terms of the moduli map

$\chi:{\overline{\mathcal{M}_{g}^{n\ an\
0}}}[\mathbf{p}]\to{\overline{{M}_{g}^{n\ an}}}.$

Indeed the isotropy morphism at (a lift of) a point
$x\in{\overline{{M}_{g}^{n\ an}}}$ identifies with the morphism
$\pi_{1}(\chi^{-1}(U))\to{\overline{\mathcal{M}_{g}^{n\ an\ 0}}}[\mathbf{p}]$
where $U\subset{\overline{{M}_{g}^{n\ an}}}$ is a small neighborhood of $x$.
If it holds we can construct an étale Galois covering scheme
$Z\to{\overline{\mathcal{M}_{g}^{n\ 0}}}[\mathbf{p}]$, smooth and
quasiprojective, such that, if $G$ is the Galois group,
$[Z/G]={\overline{\mathcal{M}_{g}^{n\ 0}}}[\mathbf{p}]$, the normalization
$\overline{Z}$ of ${\overline{{M}_{g}^{n}}}$ in the field of rational
functions of $Z$ is smooth, the $G$ action extends to $\overline{Z}$ and
$[\overline{Z}/G]\cong{\overline{\mathcal{M}_{g}^{n}}}[\mathbf{p}]$

###  2.8. Orbifold compactifications and Teichmüller level structures

A Teichmüller level structure $\lambda$ is a finite index normal subgroup
$H_{\lambda}\triangleleft{\rm{PMod}}(\Sigma_{g}^{n})$ (see [9]). The natural
finite étale
$G_{\lambda}:=H_{\lambda}\backslash{\rm{PMod}}(\Sigma_{g}^{n})$-covering
$\mathcal{M}_{g}^{n,\lambda}\to\mathcal{M}_{g}^{n}$ can be compactified
uniquely to a finite representable $G_{\lambda}$-ramified covering
$\overline{\mathcal{M}_{g}^{n,\lambda}}\to\overline{\mathcal{M}_{g}^{n}}$ (see
[7]).

Denote by $\overline{\mathcal{M}_{g}^{n}}[\lambda]$ the stack quotient
$[G_{\lambda}\backslash\overline{\mathcal{M}_{g}^{n,\lambda}}]$ (in the sense
of [50]). Then $\overline{\mathcal{M}_{g}^{n}}[\lambda]$ is a Deligne-Mumford
stack compactifying ${\mathcal{M}_{g}^{n}}$ and dominating
$\overline{\mathcal{M}_{g}^{n}}$. In particular, its moduli space is
$\overline{M_{g}^{n}}$.

The scheme $\overline{\mathcal{M}_{g}^{n,\lambda}}$ or equivalently the stack
$\overline{\mathcal{M}_{g}^{n}}[\lambda]$ may or may not be smooth. The
abelian level structure correspond to root stacks of $\mathcal{D}_{0}$ that
are not smooth at the double locus of $\mathcal{D}_{0}$. For some level
structures $\lambda$, $\overline{\mathcal{M}_{g}^{n}}[\lambda]$ is however
smooth (which is equivalent to $\overline{\mathcal{M}_{g}^{n,\lambda}}$ being
smooth) and uniformizable (see [42] for $n=0$, and [6] in general), hence
$\overline{M^{n}_{g}}$ is a quotient of a smooth variety by a finite group.

Let
$\mathcal{D}_{g}^{n}:=\overline{\mathcal{M}_{g}^{n}}\setminus{\mathcal{M}_{g}^{n}}$.
Then according to Knudsen (see [34]) this is a divisor with normal crossings
with an irreducible decomposition

$\mathcal{D}_{g}^{n}=\sum_{i=1}^{N_{g,n}}\mathcal{D}_{g,i}^{n}.$

Given a Teichmüller level structure $\lambda$, we define
$k_{i}(\lambda)\in\mathbb{N}^{*}$ to be the order of the image of a Dehn twist
corresponding to a loop encircling once $D_{g,i}^{n}$ in $G_{\lambda}$ and
denote by $\mathbf{k}(\lambda)$ the vector
$(k_{i})_{i}=(k_{i}(\lambda))_{i}\in\mathbb{N}_{\geq 1}^{N_{g,n}}$.

######  Proposition 2.9.

The Deligne-Mumford stack $\overline{\mathcal{M}_{g}^{n}}[\lambda]$ lies in a
diagram of Deligne-Mumford stacks whose moduli space is $\overline{M_{g}^{n}}$
in which all maps are étale in codimension 1:

$\overline{\mathcal{M}_{g}^{n}}[\mathbf{k}(\lambda),\mathcal{D}_{g}^{n}]\to\overline{\mathcal{M}_{g}^{n}}[\lambda]\to{\overline{\mathcal{M}_{g}^{n}}}\left[\sqrt[k_{1}]{\mathcal{D}_{g,1}^{n}}\right]\times_{\overline{\mathcal{M}_{g}^{n}}}{\overline{\mathcal{M}_{g}^{n}}}\left[\sqrt[k_{2}]{\mathcal{D}_{g,2}^{n}}\right]\times_{\overline{\mathcal{M}_{g}^{n}}}\cdots$

Furthermore, $\overline{\mathcal{M}_{g}^{n}}[\lambda]$ is smooth if and only
if the first map is an equivalence and
$\overline{\mathcal{M}_{g}^{n}}[\lambda]\simeq\overline{\mathcal{M}_{g}^{n}}[\mathbf{k}(\lambda),\mathcal{D}_{g}^{n}]$

###### Proof.

The first statement is clear, the second one is a consequence of the purity of
the branch locus. ∎

It seems to be a rather delicate problem to understand the
$\mathbf{k}(\lambda)$ that occur and give rise to a smooth uniformizable
compactification.

##  3\. Quantum mapping class group representations

###  3.1. The setting of the skein TQFT

A 3d TQFT is, loosely speaking111See [52] for a correct definition and a
discussion of TQFT with anomalies., a functor from the category of surfaces
into the category of finite dimensional vector spaces. Specifically, the
objects of the first category are closed oriented surfaces endowed with
colored banded points and morphisms between two objects are oriented
cobordisms decorated by colored uni-trivalent ribbon graphs compatible with
the banded points. A banded point on a surface is a point with a tangent
vector at that point, or equivalently a germ of an oriented interval embedded
in the surface. There is a corresponding surface with colored boundary
obtained by deleting a small neighborhood of the banded points and letting the
boundary circles inherit the colors of the respective points.

We will use the TQFT functor $\mathcal{V}_{p}$, for $p\geq 3$ and a primitive
root of unity $A$ of order $2p$, as defined in [5]. The vector space
associated by the functor $\mathcal{V}_{p}$ to a surface is called the space
of conformal blocks. Let $\Sigma_{g}$ denote the genus $g$ closed orientable
surface, $H_{g}$ be a genus $g$ handlebody with $\partial H_{g}=\Sigma_{g}$.
Assume given a finite set $\mathcal{Y}$ of banded points on $\Sigma_{g}$. Let
$G$ be a uni-trivalent ribbon graph embedded in $H_{g}$ in such a way that
$H_{g}$ retracts onto $G$, its univalent vertices are the banded points
$\mathcal{Y}$ and it has no other intersections with $\Sigma_{g}$.

The natural number $p\geq 3$ is called the level of the TQFT. We define the
set of colors in level $p$ to be ${\mathcal{C}}_{p}=\\{0,2,4,\ldots,p-3\\}$,
if $p$ is odd and ${\mathcal{C}}_{p}=\\{0,1,2,\ldots,\frac{p-4}{2}\\}$, if $p$
is even, respectively.

An edge coloring of $G$ is called $p$-admissible if the following conditions
are satisfied:

  1. (1)

The triangle inequality is satisfied at each trivalent vertex of $G$. Namely,
if $i,j,k$ are the colors of the incoming edges, one has $|j-k|\leq i\leq
j+k$.

  2. (2)

The sum of of the three colors around a vertex is even, this condition being
empty for odd $p$.

  3. (3)

The sum of the three colors around a vertex is at most $q=2p-4$, if $p$ is odd
and at most $q=p-4$, if $p$ is even, respectively. We call it the $q$-bound
inequality.

Fix a coloring of the banded points $\mathcal{Y}$. Then there exists a basis
of the space of conformal blocks associated to the surface
$(\Sigma_{g},\mathcal{Y})$ with the colored banded points (or the
corresponding surface with colored boundary) which is indexed by the set of
all $p$-admissible colorings of $G$ extending the boundary coloring. We denote
by $W_{g,p,(i_{1},i_{2},\ldots,i_{r})}$ (or, dropping $p$ from the notation,
by $W_{g,(i_{1},i_{2},\ldots,i_{r})}$, if $p$ is uniquely determined by the
context) the vector space associated to the closed surface $\Sigma_{g}$ with
$r$ banded points colored by $i_{1},i_{2},\ldots,i_{r}\in\mathcal{C}_{p}$.
Note that banded points colored by $0$ do not contribute.

Observe that an admissible $p$-coloring of $G$ provides an element of the
skein module [49] $S_{A}(H_{g},\mathcal{Y},(i_{1},i_{2},\ldots,i_{r}))$ 222If
the context allows, we will use the shorthand notations
$S_{A}(H_{g},\mathcal{Y},(i_{1},i_{2},\ldots,i_{r}))=S_{A}(H_{g},\mathcal{Y})=S_{A}(H_{g})$.
of the handlebody with the banded boundary points $\mathcal{Y}$ colored by
$(i_{1},i_{2},\ldots,i_{r})$, evaluated at the primitive $2p$-th root of unity
$A$. This skein element is obtained by cabling the edges of $G$ by the Jones-
Wenzl idempotents prescribed by the coloring and having banded points colors
fixed. Jones-Wenzl idempotents were discovered by V. Jones [31] and their
inductive construction is due to Wenzl [54].

We suppose that $H_{g}$ is embedded in a standard way into the $3$-sphere
$S^{3}$, so that the closure of its complement is also a genus $g$ handlebody
$\overline{H}_{g}$. There is then a sesquilinear form:

$\langle\;,\;\rangle=\langle\;,\;\rangle_{p}:S_{A}(H_{g},\mathcal{Y},(i_{1},i_{2},\ldots,i_{r}))\times
S_{A}(\overline{H}_{g},-\mathcal{Y},(i_{1},i_{2},\ldots,i_{r}))\to\mathbb{C}$

defined by

$\langle x,y\rangle=\langle x\sqcup y\rangle.$

Here $x\sqcup y$ is the element of $S_{A}(S^{3})$ is the union of $x$ and $y$
along $\mathcal{Y}$ in $H_{g}\cup\overline{H}_{g}=S^{3}$, and
$\langle\;\rangle:S_{A}(S^{3})\to\mathbb{C}$ is the Kauffman bracket invariant
[32].

The space of conformal blocks $W_{g,(i_{1},i_{2},\ldots,i_{r})}$ is the
quotient $S_{A}(H_{g})/\ker\langle\;,\;\rangle$ by the left kernel of the
sesquilinear form above. It follows that $W_{g,(i_{1},i_{2},\ldots,i_{r})}$ is
endowed with an induced Hermitian form $H_{A}$.

The projections of skein elements associated to the $p$-admissible colorings
of a trivalent graph $G$ as above form an orthogonal basis of
$W_{g,(i_{1},i_{2},\ldots,i_{r})}$ with respect to $H_{A}$. It is known ([5])
that $H_{A}$ only depends on the $p$-th root of unity $\zeta_{p}=A^{2}$ and
that in this orthogonal basis the diagonal entries belong to the totally real
maximal subfield $\mathbb{Q}(\zeta_{p}+\overline{\zeta_{p}})$ (after
rescaling). If $\sigma$ is a $p$-admissible coloring of $G$ extending the
boundary coloring, then the diagonal term of $H_{A}$ associated to this basis
vector is

(1) $\eta^{g+1}\frac{\prod_{v\;{\rm vertex}}\langle v\rangle}{\prod_{e\;{\rm
edge}}\langle e\rangle},$

where products are taken over the set of all trivalent vertices $v$ and all
edges $e$ of the graph $G$, and

$\eta=\frac{1}{2p}(A\kappa)^{3}(A^{2}-A^{-2})\sum_{j=1}^{2p}(-1)^{j}A^{j^{2}},$

where

$\kappa^{6}=A^{-6-\frac{p(p+1)}{2}}$

is a fixed parameter of the theory. Note that changing $\kappa$ to $-\kappa$
will change the signature of $H_{A}$ into its opposite. For an edge $e$
labeled by the color $a$ we put

$\langle e\rangle=(-1)^{a}[a+1],$

and for a trivalent vertex whose adjacent edges are labeled in
counterclockwise order by the colors $a,b,c$ we put

(2) $\langle
v\rangle=(-1)^{i+j+k+1}\frac{[i+j+k+1]![i]![j]![k]!}{[a]![b]![c]!},$

where

(3) $i=\frac{a+b-c}{2},\;j=\frac{b+c-a}{2},\;k=\frac{c+a-b}{2},$ (4)
$[m]=\frac{A^{2m}-A^{-2m}}{A^{2}-A^{-2}},\;[m]!=[1][2]\cdots[m-1][m].$

Let $G^{\prime}\subset G$ be a uni-trivalent subgraph whose degree one
vertices are colored, corresponding to a sub-surface $\Sigma^{\prime}$ of
$\Sigma_{g}$ with colored boundary. The projections in
$W_{g,(i_{1},i_{2},\ldots,i_{r})}$ of skein elements associated to the
$p$-admissible colorings of $G^{\prime}$ form an orthogonal basis of the space
of conformal blocks associated to the surface $\Sigma^{\prime}$ with colored
boundary components.

There is a linear geometric action of the mapping class groups of the
handlebodies with marked banded boundary $(H_{g},\mathcal{Y})$ and
$(\overline{H}_{g},-\mathcal{Y})$ respectively on their skein modules and
hence on the space of conformal blocks. Moreover, these actions extend to a
projective action $\rho_{g,p,(i_{1},\ldots,i_{r}),A}$ of
${\rm{Mod}}(\Sigma_{g}^{r})$ on $W_{g,(i_{1},i_{2},\ldots,i_{r})}$ respecting
the Hermitian form $H_{\zeta_{p}}=H_{A}$. When referring to
$\rho_{g,p,(i_{1},\ldots,i_{r}),A}$ the subscript specifying the genus $g$
will most often be dropped when its value will be clear from the context.
Notice that the mapping class group of an essential (i.e. without annuli or
disks complements) sub-surface $\Sigma^{\prime}\subset\Sigma_{g}$ is a
subgroup of ${\rm{Mod}}(\Sigma_{g})$ which preserves the subspace of conformal
blocs associated to $\Sigma^{\prime}$ with colored boundary. It is worthy to
note that $\rho_{p,(i_{1},\ldots,i_{r}),A}$ only depends on $\zeta_{p}=A^{2}$,
so we can unambigously shift the notation for this representation to
$\rho_{p,(i_{1},\ldots,i_{r}),\zeta_{p}}$. We will often drop the reference to
the root of unity $\zeta_{p}$, as changing it amounts to use a Galois
conjugacy of the real cyclotomic field.

There is a central extension $\widetilde{{\rm{Mod}}(\Sigma_{g})}$ of
${\rm{Mod}}(\Sigma_{g})$ by $\mathbb{Z}$ and a linear representation
${\widetilde{\rho}}_{p,\zeta_{p}}$ on $W_{g}$ which resolves the projective
ambiguity of $\rho_{p,\zeta_{p}}$. The largest such central extension has
class $12$ times the Euler class (see [30, 44]), but the central extension
considered in this paper is an index $12$ subgroup of it, called
$\widetilde{\Gamma}_{1}$ in [44].

We denote by $\Sigma_{g,n}^{r}$ the compact orientable surface of genus $g$
with $n$ boundary components and $r$ marked points. Then
${\rm{Mod}}(\Sigma_{g,n}^{r})$ denotes the pure mapping class group of
$\Sigma_{g,n}^{r}$ which fixes pointwise boundary components and marked
points.

We consider a subsurface $\Sigma_{g,r}\subset\Sigma_{g+r}$ whose complement
consists of $r$ copies of $\Sigma_{1,1}$. Let $\widetilde{Mod(\Sigma_{g,r})}$
be the pull-back of the central extension $\widetilde{{\rm{Mod}}(\Sigma_{g})}$
to the subgroup ${\rm{Mod}}(\Sigma_{g,r})\subset{\rm{Mod}}(\Sigma_{g+r})$.
Then $\widetilde{{\rm{Mod}}(\Sigma_{g,r})}$ is also a central extension, which
we denote $\widetilde{{\rm{Mod}}(\Sigma_{g}^{r})}$, of
${\rm{Mod}}(\Sigma_{g}^{r})$ by $\mathbb{Z}^{r+1}$.

######  Definition 3.1.

Let $p\geq 4$ and $\zeta_{p}$ a primitive $p$-th root of unity. We denote by
${\widetilde{\rho}}_{p,\zeta_{p},(i_{1},i_{2},\ldots,i_{r})}$ the linear
representation of the central extension
$\widetilde{{\rm{Mod}}(\Sigma_{g}^{r})}$ which acts on the vector space
$W_{g,p,(i_{1},i_{2},\ldots,i_{r})}$ associated by the TQFT to the surface
with the corresponding colored banded points (see [30, 44]).

The functor $\mathcal{V}_{p}$ associates to a handlebody $H_{g}$ the
projection of the skein element corresponding to the trivial coloring of the
trivalent graph $G$ by $0$. The invariant associated to a closed 3-manifold is
given by pairing the two vectors associated to handlebodies in a Heegaard
decomposition of some genus $g$ and taking into account the twisting by the
gluing mapping class action on $W_{g}$.

One should notice that the skein TQFT $\mathcal{V}_{p}$ is unitary, in the
sense that $H_{\zeta_{p}}$ is a positive definite Hermitian form when
$\zeta_{p}=(-1)^{p}\exp\left(\frac{2\pi i}{p}\right)$, corresponding to
$A_{p}=(-1)^{\frac{p-1}{2}}\exp\left(\frac{(p+1)\pi i}{2p}\right)$. For the
sake of notational simplicity, from now we will drop the subscript $p$ in
$\zeta_{p}$, when the order of the root of unity will be clear from the
context and the precise choice of the root of given order won’t matter. Note
that for a general primitive $p$-th root of unity, the isometries of
$H_{\zeta}$ form a pseudo-unitary group.

Now, the image $\rho_{p,\zeta}(T_{\gamma})$ of a right hand Dehn twist
$T_{\gamma}$ in a convenient basis given by a trivalent graph is easy to
describe. Assume that the simple curve $\gamma$ is the boundary of a small
disk intersecting once transversely an edge $e$ of the graph $G$. Consider
$v\in W_{g}$ be a vector of the basis given by colorings of the graph $G$ and
assume that edge $e$ is labeled by the color $c(e)\in\mathcal{C}_{p}$. Then
the action of the (canonical) lift $\widetilde{T_{\gamma}}$ of the Dehn twist
$T_{\gamma}$ in $\widetilde{{\rm{Mod}}(\Sigma_{g})}$ is given by (see [5],
5.8) :

${\widetilde{\rho}}_{p,\zeta}(\widetilde{T_{\gamma}})v=(-1)^{c(e)}A^{c(e)(c(e)+2)}v.$

Actually the central extension
$1\to\mathbb{Z}\to\widetilde{{\rm{Mod}}(\Sigma_{g})}\to{\rm{Mod}}(\Sigma_{g})\to
1$ is trivial over ${\rm{Mod}}(H_{g})<{\rm{Mod}}(\Sigma_{g})$ which enables to
define ${\widetilde{\rho}}_{p,\zeta}(\widetilde{\sigma})$ for every mapping
class $\sigma\in{\rm{Mod}}(H_{g})$ as the geometric action of $\sigma$ on the
reduced skein module of $H_{g}$. The condition on $\gamma$ implies that
$T_{\gamma}\in{\rm{Mod}}(H_{g})$.

###  3.2. Spaces of conformal blocks in genus zero

In the rest of the article, $p$ will be a positive integer, $n$ a non negative
integer, $i_{1},\ldots,i_{n}$ a choice of $n$ colors in $\mathcal{C}_{p}$ and
$\rho_{p,(i_{1},\ldots,i_{n})}$ will denote the homomorphism of
${\rm{Mod}}(\Sigma_{g}^{n})$ in the projective general linear group of the
space of conformal blocks deduced from $\tilde{\rho}_{p}$. We use the
convention that the color $0$ be dropped from the boundary colors, as the
corresponding representations are the same.

A simple closed curve on a surface is essential if it is neither null-
homotopic nor homotopic to a boundary component. Essential simple closed
curves are either called non-separating, when their complement is connected,
or separating, otherwise. The genus of a separating simple closed curve is the
minimum of the genera of the two connected subsurfaces which it bounds.

######  Definition 3.2.

Define $\delta_{p}(\mathbf{i})$, where
$\mathbf{i}=(i_{1},i_{2},\ldots,i_{n})$,
$i_{1},i_{2},\ldots,i_{n}\in\mathcal{C}_{p}$, as the number of those
$j\in\mathcal{C}_{p}$ such that $\dim W_{0,p,(i_{1},i_{2},\ldots,i_{n},j)}\neq
0$. The $n$-tuple $\mathbf{i}=(i_{1},i_{2},\ldots,i_{n})$ is said to be
generic when $\delta_{p}(\mathbf{i})\geq 3$.

The aim of this section is to give explicit formulas for
$\delta_{p}(i_{1},i_{2},\ldots,i_{n})$.

######  Lemma 3.3.

Let $p\geq 5$ be odd. The space of conformal blocks
$W_{0,p,(i_{1},i_{2},\ldots,i_{m})}$, $m\geq 3$, the colors
$i_{1},i_{2},\ldots,i_{m}$ not all equal to $0$, is non-zero if and only if
the condition $P(k,m)$ below is satisfied:

(5) $2\sum_{\alpha=1}^{2k+1}i_{s_{\alpha}}\leq 2k(p-2)+\sum_{t=1}^{m}i_{t},$

for all $0\leq k\leq\frac{m-1}{2}$ and all subsets
$\\{s_{1},s_{2},\ldots,s_{2k+1}\\}\subset\\{1,2,\ldots,m\\}$.

###### Proof.

These conditions for $k=0$ (triangle inequalities) and $k=1$ (the $q$-bound
inequality) are equivalent to the $p$-admissibility, in the first non-trivial
case $m=3$.

Consider now a pants decomposition of the holed sphere, every pair of pants
having one boundary curve from $\partial\Sigma_{0,m}$ except two of them each
of which has precisely two boundary curves from $\partial\Sigma_{0,m}$. We
choose a subset of $k$ vertices of the dual graph such that their stars are
disjoint. We sum the $q$-bound inequalities corresponding to these $k$
vertices with the triangle inequalities of the remaining vertices, which
concern the lower bounds for the internal edges, and obtain the inequality
$P(k,m)$.

Conversely, we claim that the $P(k,m)$ conditions are sufficient to guarantee
the existence of a $p$-admissible coloring of the dual graph. We proceed by
induction on $m$. Suppose the claim is true for all spheres with at most $m$
boundary components and consider $(i_{1},i_{2},\ldots,i_{m+1})$ satisfying the
conditions $P(k,m+1)$. We want to find $j_{m}\in\mathcal{C}_{p}$ such that the
triple $(j_{m},i_{m},i_{m+1})$ satisfies the $p$-admissibility condition and
$(i_{1},i_{2},\ldots,i_{m-1},j_{m})$ satisfies the conditions $P(k,m)$. This
amounts to the following two sets of inequalities:

(6) $|i_{m+1}-i_{m}|\leq
j_{m}\leq\min(i_{m}+i_{m+1},2(p-2)-(i_{m}+i_{m+1}),p-3),$ (7)
$2\sum_{\alpha=1}^{2k+1}i_{s_{\alpha}}-2k(p-2)-\sum_{t=1}^{m-1}i_{t}\leq
j_{m}\leq 2k(p-2)+\sum_{t=1}^{m-1}i_{t}-2\sum_{\alpha=1}^{2k}i_{s_{\alpha}}.$

These inequalities have solutions $j_{m}\in\mathcal{C}_{p}$ if and only if the
intervals defined above have non-empty intersection, namely if

(8) $|i_{m+1}-i_{m}|\leq
2k(p-2)+\sum_{t=1}^{m-1}i_{t}-2\sum_{\alpha=1}^{2k}i_{s_{\alpha}}$

and

(9)
$2\sum_{\alpha=1}^{2k+1}i_{s_{\alpha}}-2k(p-2)-\sum_{t=1}^{m-1}i_{t}\leq\min(i_{m}+i_{m+1},2(p-2)-(i_{m}+i_{m+1}),p-3).$

Inequality (8) is equivalent to $P(k,m+1)$. Inequality (9) consists of three
inequalities, which are respectively equivalent to $P(k,m+1)$, $P(k+1,m+1)$
and

(10) $2\sum_{\alpha=1}^{2k+1}i_{s_{\alpha}}\leq
2k(p-2)+(p-3)+\sum_{t=1}^{m-1}i_{t}.$

This is a consequence of the two former inequalities and the fact that all
$i_{j}$ are even. ∎

######  Lemma 3.4.

Let $p\geq 5$ be odd. Then for every $n\geq 2$,
$\mathbf{i}=(i_{1},i_{2},\ldots,i_{n})$,
$i_{1},i_{2},\ldots,i_{n}\in\mathcal{C}_{p}$ such that $i_{1}\leq
i_{2}\leq\cdots\leq i_{n}$ 333There is no loss of generality in assuming
$i_{1}\leq i_{2}\leq\cdots\leq i_{m}$., we have:

$\delta_{p}(\mathbf{i})=1+\frac{1}{2}\left(J_{p,\max}(\mathbf{i})-J_{p,\min}(\mathbf{i})\right),$

where

(11)
$J_{p,\max}(\mathbf{i})=\min_{0\leq\ell\leq\frac{n}{2}}\min\left(p-3,\sum_{t=1}^{n-2\ell}i_{t}-\sum_{s=n-2\ell+1}^{n}i_{s}+2\ell(p-2)\right),$
(12) $J_{p,\min}(\mathbf{i})=\max_{0\leq
k\leq\frac{n}{2}}\max\left(0,\sum_{s=n-2k}^{n}i_{s}-\sum_{t=1}^{n-2k-1}i_{t}-2k(p-2)\right).$

In particular, $\delta_{p}(\mathbf{i})\geq 1$.

###### Proof.

From Lemma 3.3 it suffices to analyze the solutions $j\in\mathcal{C}_{p}$ of
the system of inequalities $P(k,n+1)$ and prove they are all even colors $j$
such that $J_{p,\min}\leq j\leq J_{p,\max}$ observing that $J_{p,\min}\equiv
0\;({\rm mod}\;2),\ J_{p,\max}\equiv 0\;({\rm mod}\;2)$.

This is equivalent to finding $j\in\mathcal{C}_{p}$ such that:

(13) $\sum_{s=n-2k}^{n}i_{s}-\sum_{t=1}^{n-2k-1}i_{t}-2k(p-2)\leq
j\leq\sum_{t=1}^{n-2\ell}i_{t}-\sum_{s=n-2\ell+1}^{n}i_{s}+2\ell(p-2),$

for all $0\leq k\leq\frac{n}{2},0\leq l\leq\frac{n}{2}$. This system has
solutions if and only if for every $k,\ell$ we have

(14)
$\max\left(0,\sum_{s=n-2k}^{n}i_{s}-\sum_{t=1}^{n-2k-1}i_{t}-2k(p-2)\right)\leq\min\left(p-3,\sum_{t=1}^{n-2\ell}i_{t}-\sum_{s=n-2\ell+1}^{n}i_{s}+2\ell(p-2)\right).$

Consider one of the inequalities involved in (14), say:

$\sum_{s=n-2k}^{n}i_{s}-\sum_{t=1}^{n-2k-1}i_{t}-2k(p-2)\leq\sum_{t=1}^{n-2\ell}i_{t}-\sum_{s=n-2\ell+1}^{n}i_{s}+2\ell(p-2),$

which reads:

$\sum_{s=n-2k}^{n}i_{s}+\sum_{s=n-2\ell+1}^{n}i_{s}\leq\sum_{t=1}^{n-2\ell}i_{t}+\sum_{t=1}^{n-2k-1}i_{t}+2(k+\ell)(p-2).$

For $k+1\leq\ell$ this is equivalent to:

$2\sum_{s=n-2k}^{n}i_{s}\leq 2\sum_{t=1}^{n-2\ell}i_{t}+2(k+\ell)(p-2).$

This inequality follows from:

$2\sum_{s=n-2k}^{n}i_{s}<2(2k+1)(p-2)\leq
2\sum_{t=1}^{n-2\ell}i_{t}+2(k+\ell)(p-2)$

and equality cannot occur. Further, if $k\geq\ell$ this amounts to

$2\sum_{s=n-2\ell+1}^{n}i_{s}\leq 2\sum_{t=1}^{n-2k-1}i_{t}+2(k+\ell)(p-2),$

which follows again from

$2\sum_{s=n-2\ell+1}^{n}i_{s}\leq 4\ell(p-2)\leq
2\sum_{t=1}^{n-2k-1}i_{t}+2(k+\ell)(p-2)$

with equality only if $k=\ell=0$ and

$i_{1}=i_{2}=\cdots i_{n-1}=0.$

Similar arguments show that all inequalities involved in (14) are valid for
all $i_{s}\in\mathcal{C}_{p}$ and hence
$\delta_{p}(i_{1},i_{2},\ldots,i_{n})\geq 1$. ∎

We will need later also the following related function:

######  Definition 3.5.

We let $\delta^{(1)}_{p}(\mathbf{i})$,
$\mathbf{i}=(i_{1},i_{2},\ldots,i_{n})$,
$i_{1},i_{2},\ldots,i_{n}\in\mathcal{C}_{p}$, be the number of those
$j\in\mathcal{C}_{p}$ with the property that $\dim
W_{0,p,(i_{1},i_{2},\ldots,i_{n},j,j)}\neq 0$.

######  Lemma 3.6.

Let $p\geq 5$ be odd. Then for every $n\geq 1$,
$\mathbf{i}=(i_{1},i_{2},\ldots,i_{n})$, $i_{s}\in\mathcal{C}_{p}$ such that
$i_{1}\leq i_{2}\leq\cdots\leq i_{n}$, we have

$\delta^{(1)}_{p}(\mathbf{i})=1+\left\lfloor\frac{J^{(1)}_{p,\max}(\mathbf{i})}{4}\right\rfloor-\left\lceil\frac{J_{p,\min}(\mathbf{i})}{4}\right\rceil,$

where

(15)
$J^{(1)}_{p,\max}(\mathbf{i})=\min_{1\leq\ell\leq\frac{n+1}{2}}\min\left(2(p-3),\sum_{t=1}^{n-2\ell+1}i_{t}-\sum_{s=n-2\ell+2}^{n}i_{s}+2\ell(p-2)\right),$
(16) $J^{(1)}_{p,\min}(\mathbf{i})=\max_{0\leq
k\leq\frac{n-1}{2}}\max\left(0,\sum_{s=n-2k}^{n}i_{s}-\sum_{t=1}^{n-2k-1}i_{t}-2k(p-2)\right).$

###### Proof.

The claim follows by direct computation from Lemma 3.4, reducing us to count
the number of even solutions $j\in\mathcal{C}_{p}$ of:

$J^{(1)}_{p,\min}(\mathbf{i})\leq 2j\leq J^{(1)}_{p,\max}(\mathbf{i}).$

∎

######  Remark 3.7.

Observe we always have $\delta^{(1)}_{p}(\mathbf{i})\geq 1$ if
$\mathbf{i}\not=(0,\ldots,0)$. This follows from
$\delta_{p}(i_{1},\ldots,i_{n},i_{n})\geq 1$. It is not completely obvious
from the formulas given above that

$\left\lfloor\frac{J^{(1)}_{p,\max}(\mathbf{i})}{4}\right\rfloor\geq\left\lceil\frac{J_{p,\min}(\mathbf{i})}{4}\right\rceil.$

######  Lemma 3.8.

Let $p\geq 6$ be even. The space of conformal blocks
$W_{0,p,(i_{1},i_{2},\ldots,i_{m})}$, $m\geq 3$, the colors
$i_{1},i_{2},\ldots,i_{m}$ not all equal to $0$, is non-zero if and only if,
first the following parity obstruction

(17) $\sum_{t=1}^{m}i_{t}\equiv 0\;({\rm mod}\;2)$

holds and second the condition $P(k,m)$ below is satisfied:

(18) $2\sum_{\alpha=1}^{2k+1}i_{s_{\alpha}}\leq k(p-4)+\sum_{t=1}^{m}i_{t}$

for all $0\leq k\leq\frac{m-1}{2}$ and all subsets
$\\{s_{1},s_{2},\ldots,s_{2k+1}\\}\subset\\{1,2,\ldots,m\\}$.

###### Proof.

The proof follows the pattern of the corresponding result for odd $p$. The
parity obstruction, and the $P(k,m)$ condition for $k=0$ (triangle
inequalities) and $k=1$ (the $q$-bound inequality) are equivalent to the
$p$-admissibility, in the first non-trivial case $m=3$.

Consider now a pants decomposition of the holed sphere, every pair of pants
having one boundary curve from $\partial\Sigma_{0,m}$ except two of them each
of which has precisely two boundary curves from $\partial\Sigma_{0,m}$.
Summing up the parity obstruction for vertices we obtain the parity
obstruction for arbitrary $m$. The $P(k,m)$ condition is proved as above.

Conversely, we claim that the $P(k,m)$ conditions are sufficient to guarantee
the existence of a $p$-admissible coloring of the dual graph. We proceed by
induction on $m$. Suppose the claim is true for all spheres with at most $m$
boundary components and consider $(i_{1},i_{2},\ldots,i_{m+1})$ satisfying the
conditions $P(k,m+1)$. We want to find $j_{m}\in\mathcal{C}_{p}$ such that the
triple $(j_{m},i_{m},i_{m+1})$ satisfies the $p$-admissibility condition and
$(i_{1},i_{2},\ldots,i_{m-1},j_{m})$ satisfies the conditions $P(k,m)$. This
amounts to one parity obstruction

(19) $i_{m}+i_{m+1}+j_{m}\equiv 0\;({\rm mod}\;2),$

along with the following two sets of inequalities:

(20) $|i_{m+1}-i_{m}|\leq
j_{m}\leq\min\left(i_{m}+i_{m+1},p-4-(i_{m}+i_{m+1}),\frac{p-4}{2}\right),$
(21) $2\sum_{\alpha=1}^{2k+1}i_{s_{\alpha}}-k(p-4)-\sum_{t=1}^{m-1}i_{t}\leq
j_{m}\leq k(p-4)+\sum_{t=1}^{m-1}i_{t}-2\sum_{\alpha=1}^{2k}i_{s_{\alpha}}.$

These inequalities have solutions $j_{m}\in\mathcal{C}_{p}$ if and only if

(22) $|i_{m+1}-i_{m}|\leq
k(p-4)+\sum_{t=1}^{m-1}i_{t}-2\sum_{\alpha=1}^{2k}i_{s_{\alpha}}$

and

(23)
$2\sum_{\alpha=1}^{2k+1}i_{s_{\alpha}}-k(p-4)-\sum_{t=1}^{m-1}i_{t}\leq\min\left(i_{m}+i_{m+1},p-4-(i_{m}+i_{m+1}),\frac{p-4}{2}\right).$

Inequality (22) is equivalent to $P(k,m+1)$. Inequality (23) consists of three
inequalities, which respectively equivalent to $P(k,m+1)$, $P(k+1,m+1)$ and

(24)
$2\sum_{\alpha=1}^{2k+1}i_{s_{\alpha}}\leq\left(k+\frac{1}{2}\right)(p-4)+\sum_{t=1}^{m-1}i_{t}.$

This is a consequence of the former two inequalities.

It remains to observe that there exist solutions satisfying the parity
condition above. In fact both endpoints of the interval specified by
inequality (20) are compatible with the parity obstruction (19), while the
endpoints of the interval given by (21) are both congruent to
$i_{m}+i_{m+1}\;({\rm mod}\;2)$. It follows that there exists a solution
$j_{m}$ satisfying the parity obstruction. ∎

######  Lemma 3.9.

Let $p\geq 6$ be even. Then for every $n\geq 2$,
$\mathbf{i}=(i_{1},i_{2},\ldots,i_{n})$,
$i_{1},i_{2},\ldots,i_{n}\in\mathcal{C}_{p}$ such that $i_{1}\leq
i_{2}\leq\cdots\leq i_{n}$, we have

$\delta_{p}(\mathbf{i})=1+\frac{1}{2}\left(J_{p,\max}(\mathbf{i})-J_{p,\min}(\mathbf{i})\right),$

where

(25)
$J_{p,\max}(\mathbf{i})=\min_{0\leq\ell\leq\frac{n}{2}}\min\left(\frac{p-4}{2}-\varepsilon_{\mathbf{i}}(p),\sum_{t=1}^{n-2\ell}i_{t}-\sum_{s=n-2\ell+1}^{n}i_{s}+\ell(p-4)\right),$
(26) $J_{p,\min}(\mathbf{i})=\max_{0\leq
k\leq\frac{n}{2}}\max\left(\varepsilon_{\mathbf{i}},\sum_{s=n-2k}^{m}i_{s}-\sum_{t=1}^{n-2k-1}i_{t}-k(p-4)\right),$
(27) $\varepsilon_{\mathbf{i}}\in\\{0,1\\}\;{\rm
such\;that}\;\varepsilon_{\mathbf{i}}\equiv\sum_{s=1}^{n}i_{s}\;({\rm
mod}\;2),$ (28)
$\varepsilon_{\mathbf{i}}(p)=\left\\{\begin{array}[]{ll}\varepsilon_{\mathbf{i}},&\;{\rm
if}\;p\equiv 0\;({\rm mod}\;4);\\\ 1-\varepsilon_{\mathbf{i}},&\;{\rm
if}\;p\equiv 2\;({\rm mod}\;4).\end{array}\right.$

In particular, $\delta_{p}(\mathbf{i})\geq 1$.

###### Proof.

From Lemma 3.8 it suffices to analyze the solutions $j\in\mathcal{C}_{p}$ of
the system of inequalities $P(k,n+1)$ with the additional parity constraint
(17). There is no loss of generality in assuming that $i_{1}\leq
i_{2}\leq\cdots\leq i_{n}$. Therefore our system is equivalent to finding
$j\in\mathcal{C}_{p}$ such that:

(29) $\sum_{s=m-2k}^{n}i_{s}-\sum_{t=1}^{n-2k-1}i_{t}-k(p-4)\leq
j\leq\sum_{t=1}^{n-2\ell}i_{t}-\sum_{s=n-2\ell+1}^{m}i_{s}+\ell(p-4).$

This system has solutions if and only if for every $k,\ell$ we have

(30)
$\max\left(0,\sum_{s=n-2k}^{m}i_{s}-\sum_{t=1}^{n-2k-1}i_{t}-k(p-4)\right)\leq\min\left(\frac{p-4}{2},\sum_{t=1}^{n-2\ell}i_{t}-\sum_{s=n-2\ell+1}^{n}i_{s}+\ell(p-4)\right).$

Moreover, we have solutions $j$ which also satisfy the parity condition (17)
if and only if

$J_{p,\min}(\mathbf{i})\leq J_{p,\max}(\mathbf{i})$

and then $\delta_{p}(i_{1},i_{2},\ldots,i_{n})$ is the number of integers in
this interval congruent to $\varepsilon_{\mathbf{i}}$ modulo $2$, as claimed.

Consider one of the inequalities involved in (30), say:

$\sum_{s=n-2k}^{n}i_{s}-\sum_{t=1}^{n-2k-1}i_{t}-k(p-4)\leq\sum_{t=1}^{n-2\ell}i_{t}-\sum_{s=n-2\ell+1}^{n}i_{s}+\ell(p-4),$

which reads:

$\sum_{s=n-2k}^{m}i_{s}+\sum_{s=n-2\ell+1}^{n}i_{s}\leq\sum_{t=1}^{n-2\ell}i_{t}+\sum_{t=1}^{n-2k-1}i_{t}+(k+\ell)(p-4).$

For $k+1\leq\ell$ this is equivalent to:

$2\sum_{s=n-2k}^{n}i_{s}\leq 2\sum_{t=1}^{n-2\ell}i_{t}+(k+\ell)(p-4).$

This inequality follows from:

$2\sum_{s=n-2k}^{n}i_{s}\leq(2k+1)(p-4)\leq
2\sum_{t=1}^{n-2\ell}i_{t}+2(k+\ell)(p-4).$

Further, if $k\geq\ell$ this amounts to

$2\sum_{s=n-2\ell+1}^{n}i_{s}\leq 2\sum_{t=1}^{n-2k-1}i_{t}+(k+\ell)(p-4),$

which follows again from

$2\sum_{s=n-2\ell+1}^{n}i_{s}\leq 2\ell(p-4)\leq
2\sum_{t=1}^{n-2k-1}i_{t}+(k+\ell)(p-4)$

with equality only if $k=\ell=0$ and

$i_{1}=i_{2}=\cdots i_{n-1}=0.$

Similar arguments show that all inequalities involved in (30) are valid for
all $i_{s}\in\mathcal{C}_{p}$ and hence $\delta_{p}(\mathbf{i})\geq 1$. ∎

######  Lemma 3.10.

Let $p\geq 6$ be even. Then for every $n\geq 1$,
$\mathbf{i}=(i_{1},i_{2},\ldots,i_{n})$,
$i_{1},i_{2},\ldots,i_{n}\in\mathcal{C}_{p}$ such that $i_{1}\leq
i_{2}\leq\cdots\leq i_{n}$, we have:

(31) $\delta^{(1)}_{p}(\mathbf{i})=\left\\{\begin{array}[]{ll}0,&\;{\rm
if}\;\varepsilon_{(i_{1},i_{2},\ldots,i_{n})}=1;\\\
1+\frac{J^{(1)}_{p,\max}(\mathbf{i})-J^{(1)}_{p,\min}(\mathbf{i})}{2},&\;{\rm
if}\;\varepsilon_{(i_{1},i_{2},\ldots,i_{n})}=0,\\\ \end{array}\right.$

where

(32)
$J^{(1)}_{p,\max}(\mathbf{i})=\min_{1\leq\ell\leq\frac{n+1}{2}}\min\left(p-4,\sum_{t=1}^{n-2\ell}i_{t}-\sum_{s=n-2\ell+1}^{n}i_{s}+\ell(p-4)\right),$
(33) $J^{(1)}_{p,\min}(\mathbf{i})=\max_{0\leq
k\leq\frac{n+1}{2}}\max\left(0,\sum_{s=n-2k}^{n}i_{s}-\sum_{t=1}^{n-2k-1}i_{t}-k(p-4)\right).$

###### Proof.

Observe first from Lemma 3.8 that $\varepsilon_{(i_{1},i_{2},\ldots,i_{n})}=0$
is a necessary condition for the existence of such $j$. Further, from the
proof of Lemma 3.9 we have to count the number of solutions
$j\in\mathcal{C}_{p}$ of the double inequalities:

$J^{(1)}_{p,\min}(\mathbf{i})\leq 2j\leq J^{(1)}_{p,\max}(\mathbf{i})$

The claim follows by direct computation. ∎

###  3.3. The order of Dehn twists in the projective quantum representations

The explicit description from the previous section leads to the following
computation:

######  Lemma 3.11.

Let $p\geq 5$ be odd, $c$ be an essential simple closed curve on
$\Sigma_{g,n}$, $g\geq 1$ and $i_{1},i_{2},\ldots,i_{n}$ be non-zero colors
from $\mathcal{C}_{p}$. Then $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})^{p}=1$.
Furthermore,

  1. (1)

Assume that one of the following conditions is satisfied:

    1. (a)

$c$ is separating of strictly positive genus;

    2. (b)

$c$ is non-separating and either $g\geq 2$ or $g=1$ and $n=0$;

    3. (c)

$c$ bounds a holed sphere with other boundary curves colored by the non-zero
colors $i_{1},i_{2},\ldots,i_{m}$, for some $2\leq m\leq n$, where

      1. (i)

$\delta_{p}(i_{1},i_{2},\ldots,i_{m})\geq 3$, or

      2. (ii)

$\delta_{p}(i_{1},i_{2},\ldots,i_{m})=2$ and
$J_{p,\min}(i_{1},i_{2},\ldots,i_{m})=0$.

    4. (d)

$c$ is non-separating, $g=1$ and

      1. (i)

$n\geq 1$, $\delta^{(1)}_{p}(i_{1},i_{2},\ldots,i_{n})\geq 3$, or

      2. (ii)

$n\geq 1$, $\delta^{(1)}_{p}(i_{1},i_{2},\ldots,i_{n})=2$ and
$J_{p,\min}(i_{1},i_{2},\ldots,i_{n})=0$.

Then $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order $p$.

  2. (2)

If $c$ bounds a holed sphere with other boundary curves colored by the non-
zero colors $i_{1},i_{2},\ldots,i_{m}$, for some $2\leq m\leq n$, where
$\delta_{p}(i_{1},i_{2},\ldots,i_{m})=2$. Then
$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order

$p/{\rm g.c.d.}\left(J_{p,\max}(i_{1},i_{2},\ldots,i_{m}),p\right).$

  3. (3)

$c$ is non-separating, $g=1$ and $n\geq 1$,
$\delta^{(1)}_{p}(i_{1},i_{2},\ldots,i_{n})=2$ and
$J_{p,min}(i_{1},i_{2},\ldots,i_{n})>0$. Then
$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order

$p/{\rm
g.c.d.}\left(\left\lfloor\frac{J^{(1)}_{p,\max}(i_{1},i_{2},\ldots,i_{n})}{4}\right\rfloor,p\right).$

  4. (4)

Suppose that:

    1. (a)

either $c$ bounds a holed sphere with other boundary curves colored by the
non-zero colors $i_{1},i_{2},\ldots,i_{m}$, for some $2\leq m\leq n$, where
$\delta_{p}(i_{1},i_{2},\ldots,i_{m})=1$, or

    2. (b)

$c$ is non-separating and $g=1$ $n\geq 1$ and
$\delta^{(1)}_{p}(i_{1},i_{2},\ldots,i_{n})=1$.

Then $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order $1$.

Proof: Choose a pants decomposition of the surface $\Sigma_{g,n}$ obtained by
deleting disjoint disks centered at the $n$ punctures, which contains the
simple closed curve $c$. Note that boundary circles have to be part of the
decomposition and are already colored as $(i_{1},i_{2},\ldots,i_{r})$. If we
cut open $\Sigma_{g,n}$ along $c$ we either obtain $\Sigma_{g-1,n+2}$ – when
$c$ is non-separating – or else a disjoint union
$\sigma_{h,s}\sqcup\Sigma_{g-h,n+1-s}$. We color the two new boundary circles
of the resulting possibly disconnected surface by some color $j$ and let
$W(\Sigma_{g,n}\setminus\\{c\\};j)$ denote the space of conformal blocks
associated to it. Observe also that its identification with a subsurface of
$\Sigma_{g,n}$ induces an injection of $W(\Sigma_{g,n}\setminus\\{c\\};j)$
into $W_{g,(i_{1},i_{2},\ldots,i_{n})}$, and we have the following
decomposition:

$W_{g,(i_{1},i_{2},\ldots,i_{n})}=\oplus_{j\in\mathcal{C}_{p}}W(\Sigma_{g,n}\setminus\\{c\\};j)$

Now $\rho_{p,(i_{1},i_{2},\ldots,i_{n})}(T_{c})$ acts as a scalar on every
subspace $W(\Sigma_{g,n}\setminus\\{c\\};j)$ and with respect to the direct
sum decomposition above we have:

$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})=\oplus_{j\in\mathcal{C}_{p}}(-1)^{j}A^{j(j+2)}{\mathbf{1}}_{W(\Sigma_{g,n}\setminus\\{c\\};j)}$

where ${\mathbf{1}}_{W}$ is the identity operator on the subspace $W$.

On one hand eigenvalues of $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ are powers
of $A^{8}$ and hence

$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})^{p}=1$

In order to compute the order of $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ it
suffices to find which colors $j$ appear effectively in the direct sum above,
namely for which $j\in\mathcal{C}_{p}$ the vector space
$W(\Sigma_{g,n}\setminus\\{c\\};j)$ is non-zero.

According to ([25], Lemma 3.3) we have

$\dim W_{1,p,(i,j)}=\frac{(p-1-\max(i,j))(\min(i,j)+1)}{2}\neq 0$

Thus, if $c$ bounds subsurfaces of positive genera on both sides, then all
possible colors could be realized effectively.

If $c$ is non-separating and $g\geq 2$, the colors $j=0$ and $j=2$ do appear
effectively and, in particular, $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has
among its eigenvalues $1$ and $A^{8}$. Since the order of $A^{8}$ is $p$, the
order of $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ is actually $p$. This proves
the first two items of our Lemma.

If $c$ is like in (1.c.(i)), then there at least three colors $j$ which appear
effectively. The proof of Lemma 3.4 shows that these colors can be chosen to
be consecutive even numbers $2k,2k+2,2k+4$. Since
$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ is the image of a diagonal matrix with
$1,A^{8(k+1)},A^{8(2k+3)}$ as coefficients, $A^{8}$ is primitive $p-th$ root
of $1$, and ${\rm g.c.d.}(p,k+1,2k+3)=1$, the order of
$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ is $p$.

If $c$ is like in (1.c.(ii)), then there are two colors which appear
effectively, hence they are $0$ and $2$, which implies that the order
$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ is the order of $A^{8}$, namely $p$.

When $g=1$ and $c$ is separating, then two copies of $c$ bound a holed sphere.
Observe that $j,j,i_{1},\ldots,i_{m}$ satisfies the conditions $P(k,m+2)$ for
$j\in\\{0,2\\}$, if $i_{1},\ldots,i_{m}$ satisfies the conditions $P(k,m)$.
Then Lemma 3.3 permits to conclude the proof of the last item (1.d).

If $c$ bounds a holed sphere and $\delta_{p}(i_{1},i_{2},\ldots,i_{m})=2$,
then those $j$ which appear effectively are the consecutive even integers
$J_{p,\min}(i_{1},i_{2},\ldots,i_{m})$ and
$J_{p,\max}(i_{1},i_{2},\ldots,i_{m})$. Then the order of
$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ is the order of
$A^{4J_{p,\max}(i_{1},i_{2},\ldots,i_{m})}$, which is $p/{\rm
g.c.d.}(J_{p,\max}(i_{1},i_{2},\ldots,i_{m}),p)$.

If $c$ is non-separating, $g=1$ and
$\delta^{(1)}_{p}(i_{1},i_{2},\ldots,i_{m})=2$, then the $j$ which appear
effectively are the consecutive even integers
$2\left\lceil\frac{J_{p,\min}(i_{1},i_{2},\ldots,i_{m})}{4}\right\rceil$ and
$2\left\lfloor\frac{J^{(1)}_{p,\max}(i_{1},i_{2},\ldots,i_{m})}{4}\right\rfloor$.
Then the order of $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ is the order of
$A^{8\left\lfloor\frac{J^{(1)}_{p,\max}(i_{1},i_{2},\ldots,i_{m})}{4}\right\rfloor}$,
which is $p/{\rm
g.c.d.}\left(2\left\lfloor\frac{J^{(1)}_{p,\max}(i_{1},i_{2},\ldots,i_{m})}{4}\right\rfloor,p\right)$.

Eventually, if $\delta_{p}(i_{1},i_{2},\ldots,i_{m})=1$, or
$\delta^{(1)}_{p}(i_{1},i_{2},\ldots,i_{n})=1$ then there is only one color
$j$ which could appear effectively as label of the curve $c$ and hence
$\rho_{p,(i_{1},i_{2},\ldots,i_{n})}(T_{c})$ is a scalar, namely a trivial
element in the projective unitary group. $\Box$

######  Lemma 3.12.

Let $p\geq 6$ be even, $c$ an essential simple closed curve on $\Sigma_{g,n}$,
$g\geq 1$ and $(i_{1},i_{2},\ldots,i_{n})$ be non-zero colors from
$\mathcal{C}_{p}$. We assume that

(34) $\sum_{s=1}^{n}i_{s}\equiv 0\;({\rm mod}\;2),$

as otherwise $W_{p,g,(i_{1},i_{2},\ldots,i_{n})}$ is a null vector space. Then
$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})^{2p}=1.$ Furthermore:

  1. (1)

If $c$ is separating of strictly positive genus, assume that the other
boundary components of one connected complementary subsurface are colored
$i_{1},i_{2},\ldots,i_{m}$, with $0\leq m\leq n$.

    1. (a)

If

$\sum_{s=1}^{m}i_{s}\equiv 0\;({\rm mod}\;2)$

then $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order

$\left\\{\begin{array}[]{ll}1,&{\rm if}\;p=6;\\\ \frac{p}{4},&{\rm
if}\;p\equiv 0\;({\rm mod}\;4),p\geq 8;\\\ \frac{p}{2},&{\rm if}\;p\equiv
2\;({\rm mod}\;4),p\geq 10.\\\ \end{array}\right.$

    2. (b)

If

$\sum_{s=1}^{m}i_{s}\equiv 1\;({\rm mod}\;2)$

then $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order

$\left\\{\begin{array}[]{ll}1,&{\rm if}\;p\in\\{6,8\\};\\\ 5,&{\rm
if}\;p=10;\\\ 2,&{\rm if}\;p=12;\\\ \frac{p}{2},&{\rm if}\;p\geq 14.\\\
\end{array}\right.$

  2. (2)

If $c$ is non-separating and $g\geq 2$, or $g=1$ and $n=0$, then
$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order

$\left\\{\begin{array}[]{ll}4,&{\rm if}\;p=6;\\\ 2p,&{\rm if}\;p\geq 8.\\\
\end{array}\right.$

  3. (3)

Suppose that $c$ bounds a holed sphere with other boundary curves colored by
the non-zero colors $i_{1},i_{2},\ldots,i_{m}$, for some $2\leq m\leq n$,
where $\delta_{p}(i_{1},i_{2},\ldots,i_{m})\geq 3$.

    1. (a)

If

$\sum_{s=1}^{m}i_{s}\equiv 0\;({\rm mod}\;2),$

then $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order

$\left\\{\begin{array}[]{ll}1,&{\rm if}\;p=6;\\\ \frac{p}{4},&{\rm
if}\;p\equiv 0\;({\rm mod}\;4),p\geq 8\\\ \frac{p}{2},&{\rm if}\;p\equiv
2\;({\rm mod}\;4),p\geq 8\\\ \end{array}\right.$

    2. (b)

If

$\sum_{s=1}^{m}i_{s}\equiv 1\;({\rm mod}\;2),$

then $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order

$\left\\{\begin{array}[]{ll}1,&{\rm if}\;p\in\\{6,8\\};\\\ 5,&{\rm
if}\;p=10;\\\ 2,&{\rm if}\;p=12;\\\ \frac{p}{2},&{\rm if}\;p\geq 14.\\\
\end{array}\right.$

  4. (4)

If $c$ is non-separating, $g=1$ and
$\delta^{(1)}_{p}(i_{1},i_{2},\ldots,i_{n})\geq 3$, so that $p\geq 8$, then
$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order $2p$.

  5. (5)

Let $c$ bound a holed sphere with other boundary curves colored by the non-
zero colors $i_{1},i_{2},\ldots,i_{m}$, for some $2\leq m\leq n$, where
$\delta_{p}(i_{1},i_{2},\ldots,i_{m})=2$. Then
$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order

$p/2\,{\rm g.c.d.}\left(J_{p,\max}(i_{1},i_{2},\ldots,i_{m}),p/2\right).$

  6. (6)

Let $c$ be non-separating, $g=1$, $n\geq 1$ ,
$\delta^{(1)}_{p}(i_{1},i_{2},\ldots,i_{n})=2$. Then
$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order

$2p/{\rm g.c.d.}\left(2+J^{(1)}_{p,\max}(i_{1},i_{2},\ldots,i_{n}),p\right).$

  7. (7)

Suppose that:

    1. (a)

either $c$ bounds a holed sphere with other boundary curves colored by the
non-zero colors $i_{1},i_{2},\ldots,i_{m}$, for some $2\leq m\leq n$, where
$\delta_{p}(i_{1},i_{2},\ldots,i_{m})=1$, or

    2. (b)

$c$ is non-separating and $g=1$, $n\geq 1$ and
$\delta^{(1)}_{p}(i_{1},i_{2},\ldots,i_{n})=1$.

Then $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ has order $1$.

Proof: First note that the sum of the parity conditions over all vertices
yields the global parity condition (34). Further, we follow the arguments from
Lemma 3.11 above and use the same notation. We then have:

$\widetilde{\rho}_{p,(i_{1},\ldots,i_{n})}(\widetilde{T_{c}})=\oplus_{j\in\mathcal{C}_{p}}(-1)^{j}A^{j(j+2)}{\mathbf{1}}_{W(\Sigma_{g,n}\setminus\\{c\\};j)}$

where ${\mathbf{1}}_{W}$ is the identity operator on the subspace $W$. The
eigenvalues of $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ are powers of $A$ and
hence

$\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})^{2p}=1$

In order to compute the order of $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$ it
suffices to find which colors $j$ appear effectively in the direct sum above,
namely for which $j\in\mathcal{C}_{p}$ the vector space
$W(\Sigma_{g,n}\setminus\\{c\\};j)$ is non-zero. Let us first give some
information about spaces of conformal blocks in positive genus:

######  Lemma 3.13.

(35) $\dim
W_{1,p,(i,j)}=\left\\{\begin{array}[]{ll}\left(\frac{p-2}{2}-\max(i,j)\right)\left(1+\min(i,j)\right),&{\rm
if}\;i\equiv j\;({\rm mod}\;2);\\\ 0,&\;{\rm if}\;i\not\equiv j\;({\rm
mod}\;2).\end{array}\right.$

Hence $\dim W_{1,p,(i,j)}=0$ if only if $i\not\equiv j\;({\rm mod}\;2)$.

Proof: Direct computation. $\Box$

######  Corollary 3.14.

If $g\geq 2$, then $\dim W_{g,p,(i,j)}=0$ if only if $i\not\equiv j\;({\rm
mod}\;2)$.

Proof: Decompose a 2-holed surface of genus $g$ as
$\Sigma_{g,2}=\Sigma_{1,2}\cup_{S^{1}}\Sigma_{1,2}\cup_{S^{1}}\ldots\cup_{S^{1}}\Sigma_{1,2}$
and apply the splitting principle coloring each glued boundary circle with the
color $i$. $\Box$

######  Corollary 3.15.

If $g\geq 1$, $m\geq 2$ $\dim W_{g,p,(i_{1},\ldots i_{m})}\not=0$ if only if

$\sum_{t=1}^{m}i_{t}\equiv 0\;({\rm mod}\;2).$

Proof: The case $m=2$ follows from the previous statements. So we may assume
$m\geq 3$. Decompose a 2-holed surface of genus $g$ as
$\Sigma_{g,m}=\Sigma_{1,2}\cup_{S^{1}}\Sigma_{1,2}\cup_{S^{1}}\ldots\cup_{S^{1}}\Sigma_{1,2}\cup_{S^{1}}\Sigma_{0,m}$
assuming $i_{1}$ colors the left boundary curve of the first $\Sigma_{1,2}$
and apply the splitting principle coloring each glued boundary circle with a
color $j_{1}$ such that $W_{0,p,(j_{1},i_{2},\ldots,i_{m})}\not=0$ which
exists by Lemma 3.9 and has the same parity as $i_{1}$. $\Box$

Suppose now that $c$ bounds two subsurfaces of positive genera on both sides.
As above, the colors $j$ that could be realized effectively on $c$ are
precisely those satisfying the parity condition:

$j\equiv\sum_{s=1}^{m}i_{s}\;({\rm mod}\;2)$

and we call the splitting of the surface odd/even according to the parity of
the sum of colors on either side.

Let $c$ provide an even splitting. If $p=6$ there is only one even color,
namely $0$ and hence

$\rho_{g,6,(i_{1},i_{2},\ldots,i_{n})}(T_{c})=1$

For even $p\geq 8$, there exist at least two even colors $0$ and $2$ which
appear effectively thanks to Corollary 3.15. All other even labels yield
eigenvalues which are powers of $A^{8}$. Since $A^{8}$ has order
$\frac{p}{4}$, when $p\equiv 0\;({\rm mod}\;4)$ and $\frac{p}{2}$, otherwise,
the same holds for $\rho_{g,p,(i_{1},i_{2},\ldots,i_{n})}(T_{c})$.

Let $c$ provide an odd splitting. If $p=6$ or $p=8$ there is only one odd
color in $\mathcal{C}_{p}$, namely $1$. The matrix so obtained is then scalar
and hence its image in the projective unitary group is trivial:

$\rho_{g,6,(i_{1},i_{2},\ldots,i_{n})}(T_{c})=1,\;\rho_{g,8,(i_{1},i_{2},\ldots,i_{n})}(T_{c})=1$

If $p=10$ or $p=12$, there are exactly two odd colors $1$ and $3$. They
effectively appear by Corollary 3.15. The associated eigenvalues are $-A^{3}$
and $-A^{15}$. The smallest exponent for which the powers of these two
eigenvalues coincide are $2$, for $p=12$ and $5$ for $p=10$. For $p\geq 14$ ,
there are at least three odd colors $1,3$ and $5$, with associated eigenvalues
$-A^{3},-A^{15}$ and $-A^{35}$. The smallest $k$ for which $k$-th powers are
equal should verify $A^{12k}=A^{20k}=1$ and hence $A^{4k}=1$, so that
$k=\frac{p}{2}$. Higher odd colors lead to eigenvalues of the form powers of
$A^{4}$ times $-A^{3}$. Thus the order of
$\rho_{g,p,(i_{1},i_{2},\ldots,i_{n})}(T_{c})$ is $\frac{p}{2}$. This proves
the first item of our Lemma.

Assume that $c$ is nonseparating and $g\geq 2$. Choose a pants decomposition
such that $c$ is a meridian of a one holed torus bounded by the simple closed
curve $d$. The parity obstruction for the complementary subsurface shows that
$d$ must be even in any $p$-admissible coloring. By Corollary 3.15, any even
color for $d$ appears effectively. Therefore any color for $c$ appears
effectively, in particular the colors $j=0,1$ and also $j=2$, when $p\geq 8$.
Now $-A^{3}$ had order $4$, for $p=6$. If $p\geq 8$, the l.c.m. of orders of
$-A^{3}$ and $A^{8}$ is $2p$ and hence $\rho_{p,(i_{1},\ldots,i_{n})}(T_{c})$
has order $2p$. The proof is similar for $g=1$, $n=0$. This proves the second
item of the Lemma.

Assume $c$ is separating a holed sphere, providing an even splitting. If
$p=6$, there is only one even color $0$, and hence
$\rho_{6,(i_{1},i_{2},\ldots,i_{n})}=1$. If
$\delta_{p}(i_{1},i_{2},\ldots,i_{m})\geq 3$, there are at least three
consecutive even colors $j$ which appear effectively for $c$. Then the order
of $\rho_{p,(i_{1},i_{2},\ldots,i_{n})}$ is the order of $A^{8}$, which is as
stated, by the same argument as in the odd case. If
$\delta_{p}(i_{1},i_{2},\ldots,i_{m})=2$ and
$J_{p,\min}(i_{1},i_{2},\ldots,i_{m})=0$, then the colors $0$ and $2$ appear
effectively and hence the order of $\rho_{p,(i_{1},i_{2},\ldots,i_{n})}$ is
the order of $A^{8}$ again.

Assume $c$ is separating a holed sphere, providing an odd splitting. If $p=6$
or $p=8$, there is only one odd color $1$, and hence
$\rho_{p,(i_{1},i_{2},\ldots,i_{n})}=1$. If
$\delta_{p}(i_{1},i_{2},\ldots,i_{m})\geq 3$, there are at least three
consecutive odd colors $j$ which appear effectively for $c$ and the order
$\rho_{p,(i_{1},i_{2},\ldots,i_{n})}$ is the order of $A^{4}$, which is as
stated, except for small values of $p\leq 12$. This concludes the proof of the
third item.

When $g=1$ and $c$ is non-separating and
$\delta^{(1)}_{p}(i_{1},i_{2},\ldots,i_{m})\geq 3$ then there are at least
three consecutive colors $j,j+1,j+2$ which appear effectively. The
corresponding eigenvalues have a common $N$-th power iff
$(-A)^{(2j+3)N}=(-A)^{(2j+5)N}=1$, and hence $A^{N}=1$, so that $N$ is
divisible by $2p$. This proves the fourth item.

If $c$ bounds a holed sphere and $\delta_{p}(i_{1},i_{2},\ldots,i_{m})=2$,
then $j$ must be even and there are exactly two even colors $j$ and $j+2$
which appear effectively and the order of
$\rho_{p,(i_{1},i_{2},\ldots,i_{n})}$ is the order of $A^{4(j+2)}$, which is
as claimed.

If $c$ is non-separating, $g=1$,
$\delta^{(1)}_{p}(i_{1},i_{2},\ldots,i_{m})=2$, then there are exactly two
colors $j$ and $j+1$ which appear effectively and the order of
$\rho_{p,(i_{1},i_{2},\ldots,i_{n})}$ is the order of $A^{2j+3}$ which is as
claimed.

Eventually, recall that if $\delta_{p}(i_{1},i_{2},\ldots,i_{m})=1$ or
$\delta^{(1)}_{p}(i_{1},i_{2},\ldots,i_{m})=1$, then
$\rho_{p,(i_{1},i_{2},\ldots,i_{n})}$ is a scalar and hence of order $1$.
$\Box$

######  Remark 3.16.

Assume now that all $i_{s}$ are equal to $\frac{p-4}{2}$. We claim that the
space $W_{0,p,(i_{1},\ldots,i_{n})}$ could only be non-trivial when $n\geq 4$
is even, in which case it is of dimension $1$. This follows either by using
Lemma 3.8, or else directly by analyzing the dual tree of the pants
decomposition used in its proof. The internal edges form a chain; their labels
are forced by the $p$-admissibility condition to be alternatively
$0,\frac{p-4}{2},0,\frac{p-4}{2},\ldots$. This implies that the only color
which could appear effectively on $c$ is $0$, if $m$ is odd and
$\frac{p-4}{2}$, if $m$ is even. In both cases
$\rho_{p,(i_{1},i_{2},\ldots,i_{n})}$ is a scalar and hence it is of order
$1$.

###  3.4. Infiniteness results

Recall from the introduction that ${\rm{Mod}}(\Sigma_{g}^{n}[\mathbf{k}])$
denotes the normal subgroup generated by $k_{0}$-th powers of Dehn twist along
non-separating simple closed curves and $k_{j}$-th powers of Dehn twists along
simple closed curves of type $j$, where
$\mathbf{k}=(k_{0};k_{1},k_{2},\ldots,k_{N_{g,n}})$. As a shortcut we use
${\rm{Mod}}(\Sigma_{g}^{n}[k;m])$ for $\mathbf{k}=(k,m,m,m\ldots,m)$ and
${\rm{Mod}}(\Sigma_{g}^{n}[k;-])$ for $\mathbf{k}=(k;)$, where $k_{i}$ are
absent for $i>0$.

######  Proposition 3.17.

Assume $g\geq 1$, $n\geq 0$. Let $(i_{1},\ldots,i_{n})\in\mathcal{C}_{p}^{n}$
such that

$\sum_{t=1}^{n}i_{t}\equiv 0\;({\rm mod}\;2).$

  1. (1)

For odd $p\geq 5$, $\rho_{p,(i_{1},i_{2},\ldots,i_{n})}$ factors through
${\rm{Mod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[p]$.

  2. (2)

For even $p\geq 10$, $\rho_{p,(i_{1},i_{2},\ldots,i_{n})}$ factors through
${\rm{Mod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[2p;p/{\rm
g.c.d.}(4,p)]$, if all colors are even, and through out
${\rm{Mod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[2p;p/2]$, in general.

###### Proof.

This is immediate from Lemmas 3.11 and 3.12. ∎

######  Proposition 3.18.

Assume that $g\geq 2$ and $p\geq 5$, if $p$ is odd, or $p\geq 10$ and
$(g,p)\neq(2,12)$, if $p$ is even. Then $\rho_{p}({\rm{Mod}}(\Sigma_{g,n}))$
is infinite.

###### Proof.

For all $p\geq 5$ except $p\in\\{6,8,12\\}$, $(g,p)=(2,20)$ this is proved in
[23]; Masbaum found explicit elements of infinite order in [43]. The case
$p=20$ is settled in ([16], Appendix A).

It remains to prove the claim for $g\geq 3$ and $p=12$. Note that Wright
proved that the image is finite when $p=12$ and $g=2$ (see [57]). We follow
the approach developed by Coxeter (see [13, 14]) who used it to prove that
quotients of braid groups by powers of braid generators is infinite. This was
used in [38] to study the case of punctured torus.

######  Lemma 3.19 ([13],p.116, or [14],p.121).

If a group acts irreducibly on a finite dimensional complex vector space
keeping invariant a non-degenerate indefinite Hermitian form, then this group
must be infinite.

We need also the following:

######  Lemma 3.20 ([37]).

Let $p$ be even. Then the action of
$\widetilde{\rho}_{p,(1,i_{2},\ldots,i_{n})}(\widetilde{{\rm{Mod}}(\Sigma_{g,n})})$
on the space $W_{g,p,(1,i_{2},\ldots,i_{n})}$ is irreducible for all $g\geq 1$
and all colors $(1,i_{2},\ldots,i_{n})$ for which this vector space is non-
zero.

Consider now a torus $\Sigma_{1,2}$ with two boundary components colored
$(1,1)$. We need the following result whose proof will be postponed a few
lines later:

######  Lemma 3.21.

The Hermitian form on the space $W_{1,p,(1,1)}$ is indefinite, if $p\geq 10$
is even or $p\geq 5$ is odd.

The three lemmas above imply that the image of
$\rho_{12,(1,1)}({\rm{Mod}}(\Sigma_{1,2}))$ is infinite. Further observe that
for every $g\geq 3$ the embedding $\Sigma_{1,2}\subset\Sigma_{g}$ obtained by
gluing $\Sigma_{g-2,2}$ to $\Sigma_{1,2}$ along the boundary components
induces an injection at the mapping class group level:
${\rm{Mod}}(\Sigma_{1,2})\to{\rm{Mod}}(\Sigma_{g})$. It follows that
$\rho_{12}({\rm{Mod}}(\Sigma_{g}))$ contains the subgroup
$\rho_{12,(1,1)}({\rm{Mod}}(\Sigma_{1,2}))$ and it is therefore infinite. This
proves Proposition 3.18. ∎

###### Proof of Lemma 3.21.

The graph $G$ consists of a loop with two pending edges, labeled $1$. Let $p$
be even. The two remaining edges, whose union form the loop, are colored by
$(i,j)$; then by the admissibility conditions at level $p$ we need that $i+j$
be odd, smaller than or equal to $p-5$ and such that $(1,i,j)$ satisfy the
triangle inequalities, so that $|i-j|=1$. Thus the admissible colorings are of
the form $(k,k+1)$ and $(k+1,k)$, with
$k=\left\\{0,1,2,\frac{p-4}{2}\right\\}$, so that $\dim W_{1,p,(1,1)}=p-4$.
Moreover, by the formulas above the diagonal term of the Hermitian form
corresponding to the coloring $(i,j)$ is $\eta^{2}\frac{1}{[i]![j]!}$. The
quotient of two diagonal terms associated to the colorings $(k,k+1)$ and
$(k+1,k+2)$ is therefore given by:

$[k+1][k+2].$

Since $A$ was a primitive $2p$-th root of unity,
$A=\exp\left(\frac{i\pi\ell}{p}\right)$, with ${\rm g.c.d.}(\ell,2p)=1$. Now
$[k]=\frac{\sin\left(\frac{2i\pi\ell
k}{p}\right)}{\sin\left(\frac{2i\pi\ell}{p}\right)}$. If $p\geq 10$, there
exists always some $\ell$ such that $[1][2]<0$, namely the two diagonal terms
have opposite sign and thus the Hermitian form is indefinite. The case $p$ odd
is similar and we skip the details. ∎

The remaining cases, not covered by the Proposition 3.18 will be discussed in
the next section.

Choose a basepoint on the first boundary circle of $\Sigma_{g,n}$. If we
adjoin a disk capping off the first hole the embedding
$\Sigma_{g,n}\to\Sigma_{g,n-1}$ induces an exact sequence:

$1\to\pi_{1}(U\Sigma_{g,n-1})\to{\rm{Mod}}(\Sigma_{g,n})\to{\rm{Mod}}(\Sigma_{g,n-1})\to
1.$

Here $\pi_{1}(U\Sigma_{g,n-1})$ is the fundamental group of the unit tangent
bundle at $\Sigma_{g,n-1}$, namely a central extension of
$\pi_{1}(\Sigma_{g,n-1})$. Then the projective representation
$\rho_{p,(i_{1},i_{2},\ldots,i_{n})}$ of ${\rm{Mod}}(\Sigma_{g,n})$ induces a
representation for which we keep the same notation:

$\rho_{p,(i_{1},i_{2},\ldots,i_{n})}:\pi_{1}(\Sigma_{g,n-1})\to
PU(W_{g,p,(i_{1},i_{2},\ldots,i_{n})}).$

Recall that a subsurface $\Sigma_{g^{\prime},n^{\prime}}$ of $\Sigma_{g,n}$ is
essential if the morphism
$\pi_{1}(\Sigma_{g^{\prime},n^{\prime}})\to\pi_{1}(\Sigma_{g,n})$ induced by
the inclusion is injective. It is well-known that a subsurface is essential if
its complement does not contain disk components.

######  Proposition 3.22.

Assume $n\geq 1$. Let $\Sigma\cong\Sigma_{0,3}$ or $\Sigma_{1,1}$ be an
essential subsurface of $\Sigma_{g}^{n-1}$ equipped with an orientation
preserving homeomorphism onto a general fiber of the map
$f^{1}_{n}:{{\mathcal{M}}_{g}^{n}}\to{{\mathcal{M}}_{g}^{n-1}}$ which forgets
the first puncture.

  1. (1)

Assume that $g\geq 2$ and $p\geq 5$ is odd. Then
$\rho_{p,(i_{1},i_{2},\ldots,i_{n})}(\pi_{1}(\Sigma))$ contains a free
nonabelian group, if $i_{1}\neq 0$.

  2. (2)

Assume that $g\geq 2$ and $p\geq 10$ is even, $(g,p)\neq(2,12)$, $n\geq 2$ and
$1+i_{2}+\cdots+i_{n}\equiv 0\;({\rm mod}\;2)$. Then
$\rho_{p,(1,i_{2},\ldots,i_{n})}(\pi_{1}(\Sigma))$ contains a free nonabelian
group.

  3. (3)

Assume that $g\geq 2$ and $p\geq 10$ is even, $(g,p)\neq(2,12)$, $n\geq 1$ and
$i_{2}+\cdots+i_{n}\equiv 0\;({\rm mod}\;2)$. Then
$\rho_{p,(2,i_{2},\ldots,i_{n})}(\pi_{1}(\Sigma))$ contains a free nonabelian
group.

###### Proof.

When $i_{1}=2$ this is the main result of ([36], Thm.4.1) noting that their
proof works for all $p\geq 5$ not only for large enough $p$. For other values
of $i_{1}\neq 0$ this is contained in the proof of ([25], Prop. 3.2, see also
[24]).

Let $c$ be a curve separating the subsurface $\Sigma_{1,2}$ of $\Sigma_{g,n}$
with boundary labeled $1$. As $g\geq 2$, the color $j=1$ appears effectively
for the curve $c$. Then Lemma 3.21 implies that the Hermitian form on the
subspace $W_{1,p,(1,1)}$ is indefinite, and hence the Hermitian form on
$W_{g,p,(1,i_{2},\ldots,i_{n})}$ is indefinite (when the space is nonzero).
Then the claim follows from Lemmas 3.19 and 3.20.

The last item follows by observing that the proof given in [36] for $i_{1}=2$,
can be adapted without any change to even $p$ (see [25]). ∎

###  3.5. Finiteness for small values of $p$

We start by recalling the following well-known results of Humphreis ([29]):

######  Lemma 3.23 ([29]).

  1. (1)

The group ${\rm{Mod}}(\Sigma_{g,n})/{\rm{Mod}}(\Sigma_{g,n})[2;-]$ is finite
for every $g\geq 1$, $n\leq 1$, and, in particular,
${\rm{Mod}}(\Sigma_{g,n})/{\rm{Mod}}(\Sigma_{g,n})[2]$ is finite. Moreover,

${\rm{Mod}}(\Sigma_{g,n})[2;-]={\rm{Mod}}(\Sigma_{g,n})[2;1]=\ker({\rm{Mod}}(\Sigma_{g,n})\to
Sp(2g,\mathbb{Z}/2\mathbb{Z})).$

  2. (2)

The group ${\rm{Mod}}(\Sigma_{g,n})/{\rm{Mod}}(\Sigma_{g,n})[3;-]$ is finite
for $(g,n)\in\\{(2,0),(2,1),(3,0)\\}$, and, in particular,
${\rm{Mod}}(\Sigma_{g,n})/{\rm{Mod}}(\Sigma_{g,n})[3]$ is finite. Moreover,

${\rm{Mod}}(\Sigma_{2})[3;-]={\rm{Mod}}(\Sigma_{2})[3;1]\supset\ker({\rm{Mod}}(\Sigma_{2})\to
Sp(2g,\mathbb{Z})),$
${\rm{Mod}}(\Sigma_{3})[3;-]={\rm{Mod}}(\Sigma_{3})[3;1].$

  3. (3)

For $n\geq 0$, $p\geq 4$, the group
${\rm{Mod}}(\Sigma_{2,n})/{\rm{Mod}}(\Sigma_{2,n})[p;-]$ is infinite.

Further $\rho_{6}$ factors through
${\rm{Mod}}(\Sigma_{g})/{\rm{Mod}}(\Sigma_{g})[4;1]$, which is finite.

Wright proved in [56] and [57] the following finiteness results:

######  Lemma 3.24 ([56]).

The image $\rho_{8}({\rm{Mod}}(\Sigma_{g}))$ is finite for every $g\geq 1$.

######  Lemma 3.25 ([57]).

The image $\rho_{12}({\rm{Mod}}(\Sigma_{2}))$ is finite for $g=2$.

Note that $\rho_{12}({\rm{Mod}}(\Sigma_{g}))$ is a quotient of
${\rm{Mod}}(\Sigma_{g})/{\rm{Mod}}(\Sigma_{g})[24;3]$.

This has to be compared with the finiteness question about
${\rm{Mod}}(\Sigma_{g})/{\rm{Mod}}(\Sigma_{g})[3]$, which was also answered
affirmatively only for $g\in\\{2,3\\}$, by Humphries (see [29]).

##  4\. TQFT representations on isotropy groups of stable curves

###  4.1. Stabilizers of pants decompositions

The aim of this section is to describe the image of the stabilizer of a pants
decomposition by quantum representations. If $P$ is a pants decomposition of
some surface $\Sigma_{g,n}$ we keep the same notation for the isotopy class of
the multicurve consisting of all loops from $P$. Note that the order of the
curves is irrelevant. Set further ${\rm{Mod}}(\Sigma_{g,n},P)$ for the pure
mapping classes of orientation-preserving homeomorphisms which preserve the
isotopy class of the multicurve $P$ and preserve pointwise the boundary
components.

Let $G$ be a uni-trivalent graph embedded in the handlebody $H_{g}$ in such a
way that the endpoints of $G$ sit on the boundary and $H_{g}$ retracts onto
$G$. Let $\Sigma_{g,n}$ be the result of drilling small holes around the
endpoints of $G$ within the boundary surface. Then $\Sigma_{g,n}$ inherits a
pants decomposition $P=P(G)$ which consists of the set of simple loops
$\gamma_{e}$, where $\gamma_{e}$ bounds a small disk intersecting once $e$,
for every edge $e$ of $G$.

Let ${\rm{Aut}}(G)$ denote the group of automorphisms of $G$ which preserve
pointwise the leaves.

The graph $G$ has two types of internal trivalent vertices, as follows. A
generic vertex has 3 distinct incoming edges and provides an essential
$\Sigma_{0,3}\subset\Sigma_{g,n}$ and the tadpole has only 2 distinct incoming
edges and gives raise to an essential $\Sigma_{1,1}\subset\Sigma_{g,n}$. Fix
an orientation of the tadpole loops and define
${\rm{Aut}}^{+}(G)\subset{\rm{Aut}}(G)$ to be the subgroup preserving the
orientation of the tadpoles.

If $P\subset\Sigma_{g,n}$ is a pants decomposition of $\Sigma_{g,n}$ we denote
by $T(P)$ the free abelian group generated by the Dehn twists along the non-
peripheral curves in $P$ and the $n$ boundary components. Thus $T(P)$ is
isomorphic to $\mathbb{Z}^{3g-3+n}\times\mathbb{Z}^{n}$.

######  Lemma 4.1.

The group ${\rm{Mod}}(\Sigma_{g,n},P)$ is an extension

(36) $1\to T(P)\to{\rm{Mod}}(\Sigma_{g,n},P)\to{\rm{Aut}}(G)\to 1.$

###### Proof.

Any mapping class preserving $P$ induces an automorphism of $G$. On tadpole
loops, the orientation is preserved if the mapping class preserves the two
connected components of a deleted annulus neighborhood of the simple curve in
the one holed torus corresponding to the loop and reversed if these two
components are exchanged. If this automorphism is trivial then each pair of
pants in the decomposition $P$ is fixed. It is well-known that the mapping
class group ${\rm{Mod}}(\Sigma_{0,3})$ of a pair of pants is the abelian group
generated by the boundary Dehn twists.

This establishes the exact sequence above, except for the surjectivity. First
of all, observe
${\rm{Aut}}(G)=(\mathbb{Z}/2\mathbb{Z})^{t}\rtimes{\rm{Aut}}^{+}(G)$ where $t$
is the number of tadpoles. There is a mapping class
$h\in{\rm{Mod}}(\Sigma_{1,1})$ which lifts the elliptic involution
$-\mathbf{1}\in SL_{2}(\mathbb{Z})={\rm{Mod}}(\Sigma_{1}^{1})$. Its square is
the Dehn twist along the boundary curve. Given $(\epsilon_{i})_{1\leq i\leq
t}\in(\mathbb{Z}/2\mathbb{Z})^{t}$, we can glue $h^{\epsilon_{k}}$ on the
$k$-th tadpole and the identity on the rest of the surface to get a mapping
class that projects to $(\epsilon_{i})_{1\leq i\leq t}$ in the first factor.

If $\phi\in{\rm{Aut}}^{+}(G)$ and $\Pi$ is the pair of pants corresponding to
a generic vertex $v$, we construct a homeomorphism from $\Pi$ to
$\Pi^{\prime}$ the pair of pants corresponding to $\phi(v)$ mapping in an
orientation preserving fashion the three boundary curves represented by the
incident edges $\\{e,e^{\prime},e^{\prime\prime}\\}$ to the three boundary
curves represented by $\\{\phi(e),\phi(e^{\prime}),\phi(e^{\prime\prime})\\}$.
These homeomorphisms glue into a homeomorphism of
$\Sigma_{g-t,n+t}=\Sigma_{g,n}\setminus\cup_{1\leq k\leq t}\Sigma_{1,1}$
preserving $P\cap\Sigma_{g-t,n+t}$. Then, we can extend $h$ to a homeomorphism
$\bar{h}$ of $\Sigma_{g,n}$ preserving $P$ and the orientation on tadpoles.
Obviously $\bar{h}$ maps to $\phi$.

∎

The following lemma will not be used here, but we include it nevertheless.

######  Lemma 4.2.

The exact sequence (36) splits over ${\rm{Aut}}^{+}(G)$.

###### Proof.

Each pair of pants decomposes into the union of two hexagons glued along three
disjoint segments in their boundary. On each pair of pants we color one
hexagon in black, the other one in white. This expresses $\Sigma_{g,n}$ as an
union of $6g-6+2n$ hexagons, $3g-3+n$ of them being black and $3g-3+n$ being
white.

At each boundary component of a pant we may adjust locally the hexagons so
that the parts of their boundaries that are internal to the pant match and
glue into a union of simple closed curves and that the black hexagons of two
adjacent pants (including self-adjacent ones) match at the boundary of the
pant.

This induces a partition of $\Sigma_{g,n}$ into two (non necessarily
connected) subsurfaces, one being white, the other one black. We consider the
group $A$ of mapping classes of orientation preserving homeomorphisms of
$\Sigma_{g,n}$ that permute the maximal set of $3g-3+n$ non-peripheral simple
curves and preserve the black/white partition.

There is natural morphism $A\to{\rm{Aut}}^{+}(G)$, where ${\rm{Aut}}^{+}(G)$
fixes an orientation of the tadpole loops. Now an element of
${\rm{Aut}}^{+}(G)$ defines a permutation of the set of pairs of black and
white hexagons in the same pant respecting boundary hence lifts to an element
of $A$. The kernel of this morphism is trivial since the mapping class group
of an hexagon (or a disk) modulo its boundary is trivial, by Alexander’s
lemma. ∎

Let $\mathbf{k}$ be a integral vector indexed by the set of
${\rm{Aut}}(G)$-orbits of non-peripheral simple closed curves and a component
$1$ attached to the boundary components on $(\Sigma_{g,n},P)$. We denote by
$T(P)[\mathbf{k}]$ the free abelian subgroup of $T(P)$ generated by $k_{j}$-th
powers of Dehn twists along curves in $P$ of type $j$. Then $T(P)[\mathbf{k}]$
is a normal subgroup of ${\rm{Mod}}(\Sigma_{g,n},P)$ and we have an extension

(37)
$1\to\frac{T(P)}{T(P)[\mathbf{k}]}\to\frac{{\rm{Mod}}(\Sigma_{g,n},P)}{T(P)[\mathbf{k}]}\to{\rm{Aut}}(G)\to
1.$

We use the shorthand notation $[q,r]$ to denote the integral vector whose
components are $q$ on the ${\rm{Aut}}(G))$-orbits of non-separating curves and
$r$ on the orbits of the separating ones.

Our main result here is then the following:

######  Proposition 4.3.

Let $\mathbf{i}=(2,2,\ldots,2)$, $g>0$, $n\geq 0$ and $2g-2+n>0$ and
$(g,n)\not=(1,0),(1,1),(2,0)$.

  1. (1)

If $p\geq 5$ is odd, then $\rho_{p,(\mathbf{i})}({\rm{Mod}}(\Sigma_{g,n},P))$
is isomorphic to $\frac{{\rm{Mod}}(\Sigma_{g,n},P)}{T(P)[p]}$.

  2. (2)

If $p\geq 8$ is even then $\rho_{p,(\mathbf{i})}({\rm{Mod}}(\Sigma_{g,n},P))$
is isomorphic to $\frac{{\rm{Mod}}(\Sigma_{g,n},P)}{T(P)[2p,p/{\rm
g.c.d.}(p,4)]}$.

###### Proof.

Let us first investigate $\ker\rho_{p,(\mathbf{i})}\cap T(P)$. The description
for $\rho_{p}$ shows that $\rho_{p}|_{T(P)}$ maps into a maximal torus of the
projective group. In the standard basis attached to $P$, all matrices
associated to the Dehn twists along curves in $P$ are simultaneously diagonal.

We did not look at the order of Dehn twists in genus $0$. However, for this
particular vector of colors $\mathbf{i}=(2,\ldots,2)$, in any genus, we can
always color the internal edges except one with the color $2$ and the
remaining one with the colors $0$ or $2$. Hence we are reduced to the same
calculation as in the proof of Lemma 3.11 .(1).(c) resp. Lemma 3.12 .(3).(a)
for separating curves. So the order is always $p$ if $p$ is odd and is $p/{\rm
g.c.d.}(p,4)$ if $p$ even. In higher genus, nothing changes for separating
curves. Since the non-separating edges can be colored with the colors $0,2$ if
$p$ is odd, with the colors $0,1,2$ if $p$ is even, the non-separating order
is $p$ when $p$ is odd, $2p$ when $p$ is even.

It follows that $T(P)[p]\subseteq\ker\rho_{p,(\mathbf{i})}\cap T(P)$, if $p$
is odd and $T(P)[2p,p/{\rm g.c.d.}(p,4)]\subseteq\ker\rho_{p,(\mathbf{i})}\cap
T(P)$, if $p$ is even, respectively.

We prove by induction the genus that these inclusions are equalities.

First if $g=0$ the case $n=3$ is trivial, the cases $n=4,5$ follow by
inspection. Assume now that the induction hypothesis holds for all $k\leq n$
and consider $(\Sigma_{0},_{n+1},P)$ a pants decomposition. Let us fix an
enumeration $(e_{i})_{1\leq i\leq 3g-3+n+1}$ of the internal edges of $G$
which is a tree. Suppose that we have a relation of the form

$\prod_{i=1}^{3g-3+n+1}\rho_{p,(\mathbf{i})}(T_{c_{i}})^{m_{i}}=1.$

We view this as an identity in $PGL(W(\Sigma_{0,n+1})_{(\mathbf{i})})$. Let us
cut along the internal edge $e_{1}$ and get two holed spheres with $m$, resp.
$m^{\prime}$ number of holes $3\leq m,m^{\prime}\leq n$. Color the respective
boundary components corresponding to $e_{1}$ with the color $2$. This gives an
inclusion of vector spaces

$W((\Sigma_{0,m})_{(\mathbf{i})})\oplus
W((\Sigma_{0,m^{\prime}})_{(\mathbf{i})})\subset
W((\Sigma_{0,n+1})_{(\mathbf{i})}).$

Moreover $\widetilde{\rho}_{p,(\mathbf{i})}$ preserves the two factors and
restrict to their $\widetilde{\rho}_{p,(\mathbf{i})}$ on their $T(P)$. It
follows that the separating order divides $e_{i}$ for $i\geq 2$. Hence the
relation reduces to $\rho_{p,(\mathbf{i})}(T_{c_{1}})^{m_{1}}=1.$ And the
separating order divides $m_{1}$ as desired.

The case $g=0$ being settled, the higher genus case follows by induction on
$g$. One uses the same argument as above cutting along non-separating edges.
One gets only a subspace

$W((\Sigma_{g,n})_{(\mathbf{i})})\subset W((\Sigma_{g+1,n})_{(\mathbf{i})}).$

the important point is the functoriality of the restriction of
$\widetilde{\rho}_{p,(\mathbf{i})}$ to this subspace.

The kernel of
$\rho_{p,(\mathbf{i})}:{\frac{{\rm{Mod}}(\Sigma_{g,n},P)}{T(P)[p]}}\longrightarrow
PGL(W((\Sigma_{g,n})_{(\mathbf{i})})$ is thus a subgroup $K$ which maps
isomorphically onto its image in $\bar{K}<{\rm{Aut}}(G)$. The action of
${\rm{Aut}}(G)$ on the admissible colorings of $G$ is then an invariant subset
of its action by $\rho_{p,(\mathbf{i})}$ on the projective space of conformal
blocks $\mathbb{P}(W((\Sigma_{g,n})_{(\mathbf{i})}))$. It follows that
$\bar{K}$ is contained in the subgroup of ${\rm{Aut}}(G)$ which fixes all
admissible colorings of $G$. Since one can always put the color 2 on all edges
except one, it follows that $\bar{K}$ fixes all edges of $G$.

Assume $k\in\bar{K}$ does not fix a generic vertex $v$. Then the three edges
emanating from $v$ arrive at $k.v$. It follows that $G$ is a theta graph.
Hence $(g,n)=(2,0)$. Assume $k$ fixes every generic vertex and does not fix a
tadpole vertex $v$. It follows that $k$ permutes 2 tadpole vertices joined by
an edge and $(g,n)=(2,0)$ too. Actually we reach a contradiction in this case
since it permutes the two tadpole loops.

In all other cases $k$ fixes all vertices and edges. Hence, $\bar{K}$ is a
subgroup of the $(\mathbb{Z}/2\mathbb{Z})^{t}$ corresponding to the tadpoles
hence to essential subsurfaces $\Sigma_{1,1}\subset\Sigma_{g,n}$. If we take
the $h\in Mod(\Sigma_{1,1})$ be the lift of the elliptic involution and push
it as a mapping class of $\Sigma_{g,n}$ we have $h^{2}=T_{c}$ where $c$ is the
boundary of $\Sigma_{1}^{1}$. Hence if this boundary is non peripheral we
deduce from $\rho_{p,1,(2)}(T_{c})\not=1$ that $\rho_{p,1,(2)}(h)\not=1$. We
conclude as above by induction on $g$. The excluded cases are exceptions, the
corresponding elements being central in the corresponding mapping class
groups. ∎

###  4.2. Isotropy groups of general stable curves

If $C$ is a stable curve of genus $g$ with $n$ marked punctures, its
automorphism group acts naturally on the dual graph $\Gamma_{C}$ of $C$, the
unoriented marked graph whose vertices are the irreducible components of $C$
marked by their genus and the marking of the punctures they contain and whose
edges are the double points connecting them, loops being allowed. A
topological model of $C$ is obtained from a closed oriented surface $S$ of
genus $g$ with $n$ marked points by pinching every loop of the multicurve
$\underline{\alpha}$ to a point. Then $\Gamma_{C}$ is isomorphic to the
weighted dual graph $\Gamma(\underline{\alpha})$ of the multicurve
$\underline{\alpha}$. Denote by $\widetilde{C}$ the normalization of $C$ along
with the node branches data.

######  Proposition 4.4.

Let $x:\\{pt\\}\to\overline{{\mathcal{M}}_{g}^{n\;an}}$ be a point
representing an $n$-pointed stable curve $C_{x}$ and
$B{\rm{Aut}}(C_{x})\to\overline{{\mathcal{M}}_{g}^{n\;an}}$ its residual
gerbe. Let $U$ be a neighborhood of $x$ such that $B{\rm{Aut}}(C_{x})\to U$ is
a deformation retract. Then there is a short exact sequence:

$1\to\mathbb{Z}^{\underline{\alpha}}\to\pi_{1}(U\cap{\mathcal{M}}_{g}^{n\;an},\ast)\to{\rm{Aut}}(C_{x})\to
1,$

where ${\rm{Aut}}(C_{x})$ acts through its natural action on the free abelian
group over the vertex set of the dual graph $\Gamma_{x}$ of $C_{x}$. Moreover,
if $C_{x}$ is maximally degenerate, then the exact sequence above is
isomorphic to the sequence in Lemma 4.1 for an appropriate pants
decomposition.

Let $\mathbf{p}=(p_{0},p_{1},\ldots,p_{r})$ be a ramification multi-index. If
$C$ is a stable curve whose topological type is obtained from the multicurve
$\underline{\alpha}$ let us define $\mu_{\mathbf{p}}(C)$ be the product
$\mu_{p_{i}}^{k_{i}}$, where $k_{i}$ is the number of loops in $\alpha$ of
type $i$.

######  Proposition 4.5.

Consider a point $x[\mathbf{p}]$ of
$\overline{{\mathcal{M}}_{g}^{n\;an}}[\mathbf{p}]$ mapping to a point $x$ of
$\overline{{\mathcal{M}}_{g}^{n\;an}}$ which corresponds to the stable curve
$C_{x}$. Then its isotropy group is given by the exact sequence:

$1\to\mu_{\mathbf{p}}(C_{x})\to\pi_{1}(\overline{{\mathcal{M}}_{g}^{n\;an}}[\mathbf{p}],x[\mathbf{p}])_{loc}\to{\rm{Aut}}(C_{x})\to
1.$

###  4.3. Faithfully representing isotropy groups

The purpose of this section is to find the image of the isotropy group under
suitable finite dimensional representations. Let $C$ be a stable curve of
genus $g$ with $n$ marked points with a topological model obtained from a
closed oriented surface $S$ with $n$ marked points by pinching every loop of a
multicurve $\underline{\alpha}$ to a point. Let
$\Gamma=\Gamma(\underline{\alpha})$ be the weighted dual graph of
$\underline{\alpha}$. Denote by $\widetilde{C}$ the normalization of $C$ along
with the node branches data. We then have an exact sequence:

$1\to\widetilde{{\rm{Aut}}}(\widetilde{C})\to{\rm{Aut}}(C)\stackrel{{\scriptstyle
B}}{{\to}}G{\rm{Aut}}(\Gamma(\underline{\alpha}))\to 1,$

where $\widetilde{{\rm{Aut}}}(\widetilde{C})$ is the group of automorphisms of
$\widetilde{C}$ preserving each node branch and
$G{\rm{Aut}}(\Gamma(\underline{\alpha}))$ is the subgroup of the group
${\rm{Aut}}(\Gamma(\underline{\alpha}))$ of automorphisms of the graph which
can be lifted to $C$.

Further let
$\widetilde{I}(C)\subseteq{\rm{PMod}}(S)={\rm{PMod}}(\Sigma_{g}^{n})$ denote
the lift of the automorphism group of the stable curve $C$ to the mapping
class group ${\rm{PMod}}(S)$, which fits into the exact sequence:

$1\to\mathbb{Z}^{\underline{\alpha}}\to\widetilde{I}(C)\stackrel{{\scriptstyle
Q}}{{\to}}{\rm{Aut}}(C)\to 1.$

Let $\theta_{p,S}:{\rm{Mod}}(S)\to{\rm{Aut}}(H_{1}(S;\mathbb{Z}/p\mathbb{Z}))$
be the homology representation for the surface $S$ on the homology with
$\mathbb{Z}/p\mathbb{Z}$ coefficients.

######  Lemma 4.6.

For every stable curve $C$ with associated multicurve $\underline{\alpha}$ and
$p\geq 3$ we have

$\ker(B\circ Q)\cap\ker(\theta_{p,S})\subset\mathbb{Z}^{\underline{\alpha}}.$

###### Proof.

Let $\widetilde{C}$ have the connected components $C_{i}$ which are of genus
$g_{i}$, $k_{i}$ marked points coming from the branched nodes and $r_{i}$
additional marked points inherited from the marked points of $C$. Note that
our initial surface $S$ is obtained from the union of surfaces
$\Sigma^{r_{i}}_{g_{i},k_{i}}$ by identifying boundary circles corresponding
to the pinched curves from $\underline{\alpha}$.

As the homological representation $\theta_{p,S}$ is functorial, the inclusion
induces a homomorphism
$H_{1}(\Sigma_{g_{i},k_{i}}^{r_{i}};\mathbb{Z}/p\mathbb{Z})\to
H_{1}(S;\mathbb{Z}/p\mathbb{Z})$ which is equivariant with respect to the
natural map:

$\Phi_{i}:{\rm{Mod}}(\Sigma_{g_{i},k_{i}}^{r_{i}})\longrightarrow{\rm{Mod}}(S,\underline{\alpha}).$

Set further:

$\Phi=\prod_{i}\Phi_{i}:\prod_{i}{\rm{Mod}}(\Sigma_{g_{i},k_{i}}^{r_{i}})\to{\rm{Mod}}(S,\underline{\alpha}).$

Let $\Lambda\subset H_{1}(S,\mathbb{Z}/p\mathbb{Z})$ and $\Lambda_{i}\subset
H_{1}(\Sigma_{g_{i},k_{i}}^{r_{i}};\mathbb{Z}/p\mathbb{Z})$ be the subgroups
generated by the homology classes of the components of the multicurves
$\underline{\alpha}$ and
$\underline{\alpha}_{i}=\underline{\alpha}\cap\Sigma_{g_{i},k_{i}}^{r_{i}}$,
respectively. These are isotropic subspaces for the intersection form which
are invariant by the image of $\Phi$ and
${\rm{Mod}}(\Sigma_{g_{i},k_{i}}^{r_{i}})$, respectively. Then, there is an
injective homomorphism

$I:\bigoplus_{i}H_{1}(\Sigma_{g_{i},k_{i}}^{r_{i}};\mathbb{Z}/p\mathbb{Z})/\Lambda_{i}\longrightarrow
H_{1}(S;\mathbb{Z}/p\mathbb{Z})/\Lambda,$

which is equivariant under the group homomorphism $\Phi$, where the groups act
via the homology representation.

Note that the action of the subgroup $\mathbb{Z}^{\underline{\alpha}_{i}}$ on
$H_{1}(\Sigma_{g_{i},k_{i}}^{r_{i}};\mathbb{Z}/p\mathbb{Z})/\Lambda_{i}$ is
trivial. Therefore the action of ${\rm{Mod}}(\Sigma_{g_{i},k_{i}}^{r_{i}})$ on
$H_{1}(\Sigma_{g_{i},k_{i}}^{r_{i}};\mathbb{Z}/p\mathbb{Z})$ induces an action
of ${\rm{Mod}}(\Sigma_{g_{i}}^{k_{i}+r_{i}})$ on the quotient
$H_{1}(\Sigma_{g_{i},k_{i}}^{r_{i}};\mathbb{Z}/p\mathbb{Z})/\Lambda_{i}$. The
latter action coincides, under the isomorphism induced by the inclusion
$H_{1}(\Sigma_{g_{i},k_{i}}^{r_{i}};\mathbb{Z}/p\mathbb{Z})\to
H_{1}(\Sigma_{g_{i}}^{k_{i}+r_{i}};\mathbb{Z}/p\mathbb{Z})$, with the reduced
homology representation of ${\rm{Mod}}(\Sigma_{g_{i}}^{k_{i}+r_{i}})$ on
$H_{1}(\Sigma_{g_{i}}^{k_{i}+r_{i}};\mathbb{Z}/p\mathbb{Z})/\Lambda_{i}$. Note
that the action of the pure mapping class group
${\rm{PMod}}(\Sigma_{g_{i}}^{k_{i}+r_{i}})$ on $\Lambda_{i}$ is trivial.

Although the elements of $\widetilde{I}(C)$ are not in the image of $\Phi$,
they do preserve $\Lambda$. Observe that the action of the subgroup
$\mathbb{Z}^{\underline{\alpha}}\subset\widetilde{I}(C)$ on
$H_{1}(S;\mathbb{Z}/p\mathbb{Z})/\Lambda$ is trivial, as well. By what
precedes, the group $\widetilde{I}(C)$ acts on
$\bigoplus_{i}H_{1}(\Sigma_{g_{i},k_{i}}^{r_{i}};\mathbb{Z}/p\mathbb{Z})/\Lambda_{i}$
via its quotient ${\rm{Aut}}(C)\subset{\rm{Aut}}(\widetilde{C})$. Observe that
the map $I$ is $\widetilde{I}(C)$-equivariant.

Furthermore there is natural map $\ker(B\circ
Q)\to\prod_{i}{\rm{Aut}}(C_{i})$. For $\phi\in\ker(B\circ Q)$ we denote by
$\phi_{i}$ the corresponding automorphism of $C_{i}$ viewed as a mapping class
in ${\rm{PMod}}(\Sigma_{g_{i}}^{k_{i}+r_{i}})$.

The well-known Serre lemma states that the action of a nontrivial finite order
element of ${\rm{Mod}}(\Sigma_{g}^{n})$ on $H_{1}(\Sigma_{g}^{n})$ is
nontrivial, provided $\Sigma_{g}^{n}$ is hyperbolic, i.e. $2g-2+n>0$. Since
the curve $C$ is stable, each component $C_{i}$ should be hyperbolic. If
$\theta_{p,S}(\phi)=1$, then $\phi_{i}$ acts trivially on
$H_{1}(\Sigma_{g_{i}}^{k_{i}+r_{i}};\mathbb{Z}/p\mathbb{Z})/\Lambda_{i}$. As
$\phi_{i}$ is a pure mapping class, its homological representation on
$H_{1}(\Sigma_{g_{i}}^{k_{i}+r_{i}};\mathbb{Z}/p\mathbb{Z})$ is trivial as
well. Therefore, Serre’s lemma enables to conclude that $\phi_{i}=1$.

∎

######  Remark 4.7.

The map $I$ extends naturally to an injective homomorphism:

$I:\bigoplus_{i}H_{1}(\Sigma_{g_{i},k_{i}}^{r_{i}},\partial\Sigma_{g_{i},k_{i}}^{r_{i}};\mathbb{Z}/p\mathbb{Z})\longrightarrow
H_{1}(S,\underline{\alpha};\mathbb{Z}/p\mathbb{Z})$

which is equivariant under the group homomorphism $\Phi$ and fits into the
exact sequence:

$1\to\bigoplus_{i}H_{1}(\Sigma_{g_{i},k_{i}}^{r_{i}},\partial\Sigma_{g_{i},k_{i}}^{r_{i}};\mathbb{Z}/p\mathbb{Z})\stackrel{{\scriptstyle
I}}{{\longrightarrow}}H_{1}(S,\underline{\alpha};\mathbb{Z}/p\mathbb{Z})\to
H_{1}(\Gamma(\underline{\alpha});\mathbb{Z}/p\mathbb{Z})\to 1.$

######  Lemma 4.8.

Let $C$ be a stable curve with $n$ marked points with associated multicurve
$\underline{\alpha}$, $x\in\widetilde{I}(C)$ and $p\geq 3$. If $B\circ
Q(x)\neq 1\in{\rm{Aut}}(\Gamma(\underline{\alpha}))$, then
$\rho_{p,{\mathbf{i}}}(x)\neq 1$, where $\mathbf{i}=(i_{1},\ldots,i_{n})$ is
any coloring of the marked points such that there exists a coloring
$(j_{1},\ldots,j_{q})$ of the nodes of $C$ so that for all $i$
$W((\Sigma_{g_{i},r_{i}+k_{i}})_{(\mathbf{i}_{i},\mathbf{j}_{i})})\not=0$
where $(\mathbf{i})_{i}$ is the restriction of $(\mathbf{i})$ to the $r_{i}$
marked points in $\Sigma_{g_{i},r_{i}+k_{i}}$ and $(\mathbf{j}_{i})$ is the
restriction of $(\mathbf{j})$ to the $k_{i}$ nodes in
$\Sigma_{g_{i},r_{i}+k_{i}}$.

###### Proof.

We can choose basis of the space of conformal blocks associated to $S$
corresponding to a pants decomposition containing the multicurve
$\mathbf{\alpha}$. Then the action of $\rho_{p}(x)$ on the conformal blocks
permutes the conformal blocks associated to the connected components of $C$
minus the nodes according to the nontrivial action of $B\circ Q(x)$. ∎

######  Proposition 4.9.

Let $\mathbf{i}=(2,2,\ldots,2)$, $g\geq 2,n\geq 0$, $(g,n)\not=(2,0)$
$2g-2+n>0$.

  1. (1)

If $p\geq 5$ is odd, then
$\rho_{p,(\mathbf{i})}\oplus\theta_{p,\Sigma_{g}^{n}}$ factors through
${\rm{Mod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[p]$ and sends
$\pi_{1}(\overline{{\mathcal{M}}_{g}^{n\;an}}[p],x[p])_{loc}$ isomorphically
onto its image.

  2. (2)

If $p\geq 10$ is even and $(g,n,p)\not=(2,0,12)$ then the representation
$\rho_{p,(\mathbf{i})}\oplus\theta_{2p,\Sigma_{g}^{n}}$ factors through
${\rm{Mod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[2p,p/{\rm
g.c.d.}(p,4)]$ and sends the isotropy group
$\pi_{1}(\overline{{\mathcal{M}}_{g}^{n\;an}}[2p,p/{\rm
g.c.d.}(p,4)],x[2p,p/{\rm g.c.d.}(p,4)])_{loc}$ isomorphically onto its image.

###### Proof.

It is well-known that $\ker\theta_{p}$ contains both $T_{\gamma}^{p}$, for
nonseparating $\gamma$ and $T_{\gamma}$, when $\gamma$ is separating. Then
Proposition 3.17 implies the claims concerning the factorization of the given
representations. From Lemmas 4.6 and 4.8 the kernel of these representations
is contained within $\mathbb{Z}^{\underline{\alpha}}$. Then the faithfulness
is a consequence of Proposition 4.3 which describes the kernel on
$\mathbb{Z}^{\underline{\alpha}}$ and Proposition 4.5 which describes the
isotropy group. ∎

###  4.4. $\overline{{\mathcal{M}}_{g}^{n\;an}}[p]$ is uniformizable

The last step in the proof of Theorem 1.1 is to show that the orbifolds we
consider are uniformizable.

######  Proposition 4.10.

Assume $g\geq 2,n\geq 0$ and $2g-2+n>0$.

If $p\geq 5$ is odd, then $\overline{{\mathcal{M}}_{g}^{n\;an}}[p]$ is
uniformizable. Further, for even $p\geq 10$, and $(g,n,p)\not=(2,0,12)$, the
orbifold $\overline{{\mathcal{M}}_{g}^{n\;an}}[2p,p/{\rm g.c.d.}(p,4)]$ is
uniformizable.

###### Proof.

Let $p$ be odd. Proposition 4.9 provided a projective representation of
$\pi_{1}(\overline{{\mathcal{M}}_{g}^{n\;an}}[p])$ which is faithful on the
isotropy subgroups. This can be easily converted into a linear representation
by first composing it with the adjoint representation of the projective
unitary group, as the later is centerfree. By Proposition 3.18 the image of
the linear representation is infinite and by the Selberg lemma it has a finite
index torsionfree subgroup $J$. By taking the quotient of the image by $J$ we
obtain a homomorphism from $\pi_{1}(\overline{{\mathcal{M}}_{g}^{n\;an}}[p])$
into a finite group $G$ in which all isotropy groups should inject. Therefore
$\overline{{\mathcal{M}}_{g}^{n\;an}}[p]$ is uniformizable, as claimed. When
$p$ is even, the proof is similar. ∎

######  Remark 4.11.

This gives an alternate and perhaps simpler version of the existence of smooth
finite Galois coverings of the moduli space of stable curves due to Looijenga,
Pikaart and Boggi. Letting $p\to\infty$, we also recover the universal
ramification of the Teichmüller tower ([9], corrected in [42]).

###  4.5. End of proof of Theorem 1.1

Proposition 4.10 and an argument from [17] implies that
$\pi_{1}(\overline{{\mathcal{M}}_{g}^{n\;an}}[p])$ is a Kähler group. In fact,
consider a Galois étale covering $M\to\overline{{\mathcal{M}}_{g}^{n\;an}}[p]$
with deck group $G$. Then $M$ can be chosen to be a smooth projective variety.
On the other hand there exists a projective surface $Y$ such that
$\pi_{1}(Y)=G$, as $G$ is finite. If $\widetilde{Y}$ denotes its universal
covering, then $\pi_{1}(\overline{{\mathcal{M}}_{g}^{n\;an}}[p])$ acts
diagonally on $\widetilde{M}\times Y$ freely and properly discontinuously.
Then the quotient is a smooth projective variety having fundamental group
$\pi_{1}(\overline{{\mathcal{M}}_{g}^{n\;an}}[p])$. The case of even $p$ is
similar.

This indeed shows that the groups
${\rm{Mod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[p]$, for odd $p\geq 5$,
and respectively
${\rm{Mod}}(\Sigma_{g}^{n})/{\rm{Mod}}(\Sigma_{g}^{n})[2p,p/{\rm
g.c.d.}(p,4)]$, for even $p\geq 10$ are Kähler and ends the proof of Theorem
1.1.

##  5\. Bogomolov-Katzarkov Surfaces

### 5.1.

Let $\pi:S\to C$ be a non isotrivial fibration of a smooth complex projective
algebraic surface $S$ onto a smooth curve $C$ such that:

  1. (1)

$\pi$ is a projective morphism with connected fibres,

  2. (2)

for generic $c\in C$, the fiber $\pi^{-1}(c)$ is a genus $g\geq 2$ curve

  3. (3)

the singular fibres of $\pi$ are reduced stable curves, in particular the
singular set consists of ordinary double points.

In a transcendental neighborhood of $\sigma\in S$ a node of a singular fiber
there are complex coordinates $(x,y)$ on $S$ and a complex coordinate $z$ on
$C$ near $\pi(\sigma)$ such that $z(\pi(x,y))=xy$.

By the definition of the stack $\overline{\mathcal{M}_{g}}$ of genus $g$
stable curves, we have a map $\psi:C\to\overline{\mathcal{M}_{g}}$, which
induces a holomorphic map $C^{an}\to\overline{\mathcal{M}^{an}_{g}}$. By
construction this map is covered by a map of stacks
$S\to\overline{\mathcal{M}^{1}_{g}}$ and we have a 2-commutative diagram:

$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathcal{M}^{1}_{g}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{1}}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\overline{\mathcal{M}_{g}}}$

The square is cartesian, i.e.: $S\simeq
C\times_{\psi,f_{1}}\overline{\mathcal{M}^{1}_{g}}$, $\pi$ being equivalent to
$\psi^{*}f_{1}$.

Fix an integer $N\geq 2$. If $N=2$ and $g(C)=0$ assume $\pi$ has at least 4
singular fibers, in general there at least 3 singular fibers. We will mostly
be interested in the case $N$ is large enough, meaning $N\geq 5$ if $N$ is
odd.

###  5.2.  The stack $\mathcal{S}_{\pi}[\mathbf{k}]$

For every multi-index $\mathbf{k}$ we define the following stack:

$\mathcal{S}_{\pi}[\mathbf{k}]:=S^{an}\times_{\overline{\mathcal{M}^{1\
an}_{g}}}\overline{\mathcal{M}^{1\ an}_{g}}[\mathbf{k}].$

According to the convention from the introduction we write
$\mathcal{S}_{\pi}[N]$ for $\mathcal{S}_{\pi}[N,N,N,\ldots,N]$ and
$\mathcal{S}_{\pi}[N,L]$ for $\mathcal{S}_{\pi}[N,L,L,\ldots,L]$.

######  Proposition 5.1.

The stack $\mathcal{S}_{\pi}[N]$ is a smooth compact Kähler orbifold with
moduli space $S$. There is a proper map
$\mathcal{S}_{\pi}[N]\to\mathcal{C}[N]$ where $\mathcal{C}[N]$ is a non-
elliptic orbicurve whose general fiber is a smooth curve of genus $g$ giving
rise to a fibration exact sequence :

$1\to
I_{N}(\mathbf{C}_{\pi})\to\pi_{1}(\mathcal{S}_{\pi}[N])\to\pi_{1}(\mathcal{C}[N])\to
1.$

where $I_{N}(\mathbf{C}_{\pi})$ is the image of the fundamental group of a
generic fiber into $\pi_{1}(\mathcal{S}_{\pi}[N]))$. Furthermore:

$I_{N}(\mathbf{C}_{\pi})\cong\frac{\pi_{1}(\Sigma_{g})}{\langle\langle c^{N},\
c\in\mathbf{C}_{\pi}\rangle\rangle}$

where $\mathbf{C}_{\pi}$ is a monodromy invariant family of conjugacy classes
in $\pi_{1}(\Sigma_{g})$ representing free homotopy classes of simple closed
oriented curves.

###### Proof.

This is in effect proved in [8]. For the convenience of the reader we will
rephrase their paper in our terminology.

The smoothness of $S$ translates into the statement that given an uniformizing
parameter $z$ for $C$ at $q\in C$ and $\eta$ a chart adapted to $\mathcal{D}$
centered at $\psi(q)$, $\eta^{-1}\psi$ assumes the form
$z\mapsto(z_{1},\dots,z_{3g-3})$ with $\frac{dz_{i}}{dz}(q)\not=0$ for $1\leq
i\leq k(\eta)=\\#Sing(\pi^{-1}(p))$. Furthermore, near the image $Q$ of a
singular point $q^{\prime}$ of $\pi^{-1}(q)$ we can find an étale chart of
$\overline{\mathcal{M}^{1\ an}_{g}}$ at $Q$ such that the map
$S\to\overline{\mathcal{M}^{1\ an}_{g}}$ takes the form
$(x,y)\mapsto(x,y,z_{2}(x,y),\ldots,z_{3g-3}(x,y))$ in local coordinates
$(x,y)$ of $S$ near $q^{\prime}$, whereas the map to $C$ is given by $t=xy$.
Using the étale charts in [15] in which $f_{1}$ takes the form
$(\xi_{1},\eta_{1},z_{2},\ldots,z_{3g-3})$ we also can arrange to have
$z_{k}(x,y)=z_{k}(xy)$ since $\pi\simeq\psi^{*}f_{1}$ .

So the map from $S$ to $\overline{\mathcal{M}^{1\ an}_{g}}$ is transverse to
$\mathcal{D}_{1}$ and the pull back of $\mathcal{D}_{1}$ is just the sum
$\Sigma:=\sum_{q\in B}S_{q}$ of the (reduced) singular fibres of $\pi$, $B$
being the set of critical values of $\pi$. Certainly the formation of the
smooth root stack of a normal crossing divisor commutes with transversal
mappings and we get:

$\mathcal{S}_{\pi}[N]\simeq S[N,\Sigma].$

There is a natural map $\mathcal{S}_{\pi}[N]\to\mathcal{C}[N]$ where
$\mathcal{C}[N]=C^{an}[\sqrt[N]{B}]$ according to ([10], Rem.2.2.2). Actually
$S^{an}\times_{C^{an}}\mathcal{C}[N]\simeq S[\sqrt[N]{\Sigma}]$ and
$\mathcal{S}_{\pi}[N]\to S[\sqrt[N]{\Sigma}]$ is the canonical stack. Indeed
the germ $S[\sqrt[N]{\Sigma}]_{q^{\prime}}$ is equivalent to the germ at the
origin of the stack with a cyclic quotient singularity of type
$\frac{1}{N}(1,-1)$:

$[\mu_{N}\backslash A_{N}]=[\mu_{N}\backslash\\{t^{N}=xy\\}].$

Now $\mathcal{C}[N]$ is an uniformizable orbicurve since otherwise $C$ were
rational and $\pi$ would have $\leq 2$ singular fibers, but this situation is
excluded by the big Picard theorem and the fact that the period domain for
ppav is a bounded domain. Let $C_{N}$ be a complex projective curve with an
action of a finite group $G$ such that $[G\backslash C_{N}]=\mathcal{C}[N]$,
namely the Galois cover $C_{N}\to C$ ramifies precisely over $B$ with index
$N$.

Consider now the following singular surface $S_{N}:=S\times_{C}C_{N}$. Then
$S_{N}$ is a normal surface with $A_{N-1}$ singularities and the canonical
stack $(S\times_{C}C_{N})^{can}$ is an étale covering of
$\mathcal{S}_{\pi}[N]$. Actually since $(S\times_{C}C_{N})^{can}$ is smooth
its fundamental group is isomorphic to the fundamental group of the open
substack obtained by deleting the finite substack of its orbifold points,
which is then precisely the open surface considered in [8]. There $C_{N}$ is
denoted by $R$, $S\times_{C}C_{N}$ by $S$ and the singular set of
$S\times_{C}C_{N}$ by $Q$ which is in a natural bijection withe the sets of
nodes.

Let $D$ be a small coordinate disk in $C_{N}$ centered at a singular fiber of
$S_{N}\to C_{N}$ which is a degree $N$ covering of a small coordinate disk
$D^{\prime}$ of $C$ centered at a critical point of $\pi$, such that $D$ only
ramifies over $D^{\prime}$ at that critical point. Let $T^{\prime}$ be a non
singular fiber of $\pi:S_{N}\times_{C_{N}}D\to D$ identified with a non
singular fiber of $\pi^{-1}(D^{\prime})$.

Then there is a collection $c^{\prime}_{1},\ldots,c^{\prime}_{q}$ of disjoint
simple curves in $T^{\prime}$ which appear as geometric vanishing cycles in a
Clemens contraction for the family of stable curves $S\times_{C}D^{\prime}\to
D^{\prime}$. Denote by $\bar{c}_{1},\ldots,\bar{c}_{q}$ their conjugacy
classes in $\pi_{1}(T^{\prime})$. Introduce base points denoted by
$\ast^{\prime}$ which are preserved by the maps under consideration.

######  Lemma 5.2.

The inclusion map
$(T^{\prime},\ast^{\prime})\hookrightarrow((S_{N}\times_{C_{N}}D)^{can},\ast^{\prime})$
induces at fundamental group level the quotient group homomorphism onto:

$\pi_{1}((S_{N}\times_{C_{N}}D)^{can},\ast^{\prime})=\pi_{1}(T^{\prime},\ast^{\prime})/\langle\langle\bar{c}_{1}^{N},\dots,\bar{c}_{q}^{N}\rangle\rangle,$

where $\langle\langle\\_\rangle\rangle$ means the normal subgroup generated by
the respective elements.

For the sake of simplicity we kept the notation
$\pi_{1}(T^{\prime},\ast^{\prime})$ instead of the more appropriate
$\pi_{1}(T^{\prime},\ast_{T^{\prime}})$.

###### Proof.

This is just a reformulation of ([8], Thm.2.1), using Van Kampen for stacks. ∎

One has $g(C_{N})\geq 1$ and we denote the universal covering
$\widetilde{C_{N}^{univ}}\cong\mathbb{C}\ \mathrm{or}\ \Delta$ by $\mathbb{E}$
to emphasize it is homeomorphic to an euclidian plane. Consider the canonical
stack

$\mathcal{T}:=(S_{N}\times_{C_{N}}\mathbb{E})^{can}\simeq
S_{N}^{can}\times_{C_{N}}\mathbb{E}$

and introduce a coherent choice of base points $\ast$. We can assume that
$\ast_{\mathcal{T}}$ does not lie on a singular fiber of the projection to
$\mathbb{E}$ and has consequently a trivial inertia group. Let $T$ be the
fiber of
$\pi_{\mathbb{E}}:=\pi\times_{C}1_{\mathbb{E}}:\mathcal{T}\to\mathbb{E}$ at
$\ast_{\mathbb{E}}$.

The set of branch points $B(\mathbb{E})$ of $\pi_{\mathbb{E}}$ is the preimage
of branch points of $\pi$ and thus it is discrete in $\mathbb{E}$. Hence there
is a countable discrete subset $J\subset]0,+\infty[$ such that
$\partial\mathbb{B}_{\mathbb{E}}(\ast,R)$ does not intersect $B(\mathbb{E})$
when $0<R\not\in J$. For each such $R$ we can construct a tree
$\mathcal{A}_{R}$ rooted at $\ast$ in $\mathbb{B}_{\mathbb{E}}(\ast,R)$ and
ending at the various points of
$B(\mathbb{E})\cap\mathbb{B}_{\mathbb{E}}(\ast,R)$. Obviously we can arrange
so that $R^{\prime}\geq
R\Rightarrow\mathcal{A}_{R}\subset\mathcal{A}_{R^{\prime}}$ with equality
between two consecutive values of $J$. We may also assume that the lifts of
the various $\ast^{\prime}$ lie on $\mathcal{A}_{R}$ . Using Ehresmann lemma
(and an horizontal normal bundle to $\pi_{\mathbb{E}}|_{\mathbb{E}\setminus
B(\mathbb{E})}$ along the geodesic rays of $\mathcal{A}_{R}$ emanating from
$\ast_{\mathbb{E}}$ we can transport the geometric vanishing cycles at each
$\ast^{\prime}$ to a system of disjoint simple curves in $T$. Doing this for
all $\ast^{\prime}$ yields a family $\\{{c}_{\alpha}\\}_{\alpha\in A(R)}$ of
closed simple curves in $T$. This yields a family
$\\{\bar{c}_{\alpha}\\}_{\alpha\in A(R)}$ of conjugacy classes in
$\pi_{1}(T,\ast)$ and $R\mapsto A(R)$ is an increasing function. Taking the
union for all $R$ we get a family
$\mathbf{C}_{\pi}=\\{\bar{c}_{\alpha}\\}_{\alpha\in A}$ of conjugacy classes
in $\pi_{1}(T,\ast)$.

######  Lemma 5.3.

We have

$\pi_{1}(\pi_{\mathbb{E}}^{-1}(\mathbb{B}_{\mathbb{E}}(\ast,R)),\ast)=\pi_{1}(T,\ast)/\langle\langle\bar{c}_{\alpha}^{N},\
\alpha\in A(R)\rangle\rangle.$

Moreover, the inclusion map
$(T,\ast)\hookrightarrow(\pi_{\mathbb{E}}^{-1}(\mathbb{B}_{\mathbb{E}}(\ast,R)),\ast)$
induces the quotient group homomorphism. In particular, when $R=\infty$, we
obtain

$\pi_{1}(\mathcal{T},\ast)=\pi_{1}(T,\ast)/\langle\langle\bar{c}_{\alpha}^{N},\
\alpha\in A\rangle\rangle.$

###### Proof.

Van Kampen for stacks. ∎

It follows from the covering theory for topological stacks ([46]) that there
is an exact sequence:

$1\to\pi_{1}(\mathcal{T},\ast)\to\pi_{1}(S_{N}^{can},\ast)\to\pi_{1}(C_{N},\ast)\to
1,$

the normal subgroup being the image of $\pi_{1}(T,\ast)$ and similarly

$1\to\pi_{1}(\mathcal{T},\ast)\to\pi_{1}(\mathcal{S}[N],\ast)\to\pi_{1}(\mathcal{C}[N],\ast)\to
1.$

We set therefore $I_{N}(\mathbf{C}_{\pi})=\pi_{1}(\mathcal{T},\ast)$. This
completes the proof of Proposition 5.1. ∎

###  5.3. The fiber group $I_{N}(\mathbf{C}_{\pi})$

As argued in [8], the fiber group is a very interesting object. Consider the