# The flow group of rooted abelian or quadratic differentials

Mark Bell  Independent
UK<EMAIL_ADDRESS>,  Vincent Delecroix  CNRS - Universitsé de
Bordeaux
351, cours de la Libération
33400 Talence<EMAIL_ADDRESS>,  Vaibhav Gadre  School of
Mathematics and Statistics
University of Glasgow
University Place, Glasgow, G128QQ UK<EMAIL_ADDRESS>,  Rodolfo
Gutiérrez-Romo  Centro de Modelamiento Matemático, CNRS-IRL 2807, Universidad
de Chile, Beauchef 851, Santiago, Chile<EMAIL_ADDRESS>and  Saul Schleimer
Department of Mathematics
University of Warwick
Coventry, CV47AL UK<EMAIL_ADDRESS>
###### Abstract.

We define the flow group of any component of any stratum of rooted abelian or
quadratic differentials (those marked with a horizontal separatrix) to be the
group generated by almost-flow loops. We prove that the flow group is equal to
the fundamental group of the component. As a corollary, we show that the plus
and minus modular Rauzy–Veech groups are finite-index subgroups of their
ambient modular monodromy groups. This partially answers a question of Yoccoz.

Using this, and recent advances on algebraic hulls and Zariski closures of
monodromy groups, we prove that the Rauzy–Veech groups are Zariski dense in
their ambient symplectic groups. Density, in turn, implies the simplicity of
the plus and minus Lyapunov spectra of any component of any stratum of
quadratic differentials. Thus, we establish the Kontsevich–Zorich conjecture.

###### Key words and phrases:

Moduli of Riemann surfaces, Quadratic differentials, Teichmüller dynamics,
Monodromy groups, Kontsevich–Zorich cocycle, Rauzy–Veech groups, Lyapunov
spectra

###### 2010 Mathematics Subject Classification:

Primary 30F60; Secondary 37D40, 37G15, 37A20

This work is in the public domain.

##  1\. Introduction

Moduli spaces of abelian or quadratic differentials consist of Riemann
surfaces endowed with abelian differentials or, respectively, meromorphic
quadratic differentials with at most simple poles. By integration along
appropriate relative cycles, these moduli spaces are endowed with local
complex (orbifold) charts known as _period coordinates_. The usual
identification of $\mathbb{C}$ with $\mathbb{R}^{2}$ gives rise to a natural
$\operatorname{SL}(2,\mathbb{R})$-action on period coordinates. The resulting
diagonal action is known as the _Teichmüller flow_.

Moduli spaces are a meeting ground of many mathematical disciplines. A very
deep example of this, which is moreover relevant for our work, is as follows.
Fix any differential $q$ and form its $\operatorname{SL}(2,\mathbb{R})$ orbit
closure. This is dynamically defined, yet is an algebraic variety [Fil16]. In
period coordinates the orbit closure is cut out by homogeneous linear
equations, with (real) algebraic coefficients [EMM15].

In general, one stratifies a space of differentials by fixing various
topological and combinatorial data such as the genus of the underlying surface
$S$, the number and character of the singularities, and so on. The resulting
strata are not necessarily connected; the classification of their components
is known [KZ03, Lan08, CM14].

Suppose now that $\mathcal{C}$ is such a _stratum component_ of abelian or
quadratic differentials. There is a forgetful map from $\mathcal{C}$ to
$\mathcal{M}(S)$: the moduli space of Riemann surface structures on $S$. Both
$\mathcal{C}$ and $\mathcal{M}(S)$ are orbifolds; both have manifold covers
which we will need.

For $\mathcal{M}(S)$ this story is classical. Briefly, points in
$\mathcal{M}(S)$ are in fact equivalence classes; taking the universal cover
breaks these classes apart. This gives the Teichmüller space $\mathcal{T}(S)$
which is homeomorphic to an open ball in $\mathbb{R}^{6g-6}$; here
$g=\operatorname{genus}(S)$. The deck group of this covering is the mapping
class group $\mathrm{Mod}(S)$.

To obtain a manifold cover of $\mathcal{C}$ we consider _rooted_
differentials. A choice of root is simply a horizontal unit tangent vector at
a singularity. The choice of root removes any symmetry of the differential and
so unwraps the orbifold locus. The resulting finite cover is a manifold which
may not be connected. For instance, differentials with roots at zeroes of
different orders lie in different components. We fix a connected component of
this cover and denote it by $\mathcal{C}^{\mathrm{root}}$.

The maps from the previous paragraphs give us the following sequence of
homomorphisms:

$\pi_{1}(\mathcal{C}^{\mathrm{root}})\to\pi_{1}^{\mathrm{orb}}(\mathcal{C})\to\pi_{1}^{\mathrm{orb}}(\mathcal{M}(S))=\mathrm{Mod}(S)\xrightarrow{\rho}\operatorname{Aut}(H_{1}(S;\mathbb{Z}))\cong\operatorname{Sp}(2g,\mathbb{Z})$

Here the third map, $\rho$, is the symplectic representation of the mapping
class group: the action of $\mathrm{Mod}(S)$ on the homology of $S$. We call
the image of $\pi_{1}^{\mathrm{orb}}(\mathcal{C})$, inside the mapping class
group, the _modular monodromy group_. The image of the modular monodromy group
under $\rho$ is known as the _monodromy group_.

The (modular) monodromy groups are “topological offspring” of the stratum
component $\mathcal{C}$.

In an attempt to relate the topology and dynamics of $\mathcal{C}$ we ask the
following: to what extent can $\pi_{1}^{\mathrm{orb}}(\mathcal{C})$ be
“detected” by the Teichmüller flow? More precisely, let
$U\subseteq\mathcal{C}$ be a contractible open set missing the orbifold locus.
Fix a base-point $q_{0}\in U$. Consider the Teichmüller trajectories that
start and end in $U$. For each, we connect its endpoints to $q_{0}$ inside of
$U$ to get a loop based at $q_{0}$. As $U$ is contractible, the resulting
based homotopy class is independent of the choices we made inside of $U$. We
call these based homotopy classes _almost-flow loops_. The _flow group_ of
$\mathcal{C}$ associated with the pair $(U,q_{0})$ is the subgroup of
$\pi_{1}^{\mathrm{orb}}(\mathcal{C})$ generated by all such loops. The
question can then be stated as:

######  Question 1.1.

For any stratum component $\mathcal{C}$ of the moduli space of abelian or
quadratic differentials, is the flow group of $\mathcal{C}$ equal to
$\pi_{1}^{\mathrm{orb}}(\mathcal{C})$?

This is a version of a question of Yoccoz [Yoc10, Section 9.3].111Yoccoz asks
if the image of the flow group in $\mathrm{Mod}(S)$ is all of
$\mathrm{Mod}(S)$. However, what is meant here is the modular monodromy group
[Mat21].

This question can also be stated for rooted differentials by defining the flow
group analogously. As $\mathcal{C}^{\mathrm{root}}$ is a manifold, the open
set $U$ can be any contractible open set in $\mathcal{C}^{\mathrm{root}}$. A
positive answer to this question is our main result: Let
$\mathcal{C}^{\mathrm{root}}$ be any component of a stratum of the moduli
space of rooted abelian or quadratic differentials. Let
$q_{0}\in\mathcal{C}^{\mathrm{root}}$ be any base-point and $U$ any
contractible open set containing $q_{0}$. Then the flow group of
$\mathcal{C}^{\mathrm{root}}$ associated with the pair $(U,q_{0})$ is equal to
$\pi_{1}(\mathcal{C}^{\mathrm{root}},q_{0})$. Since
$\mathcal{C}^{\mathrm{root}}$ is a finite cover of $\mathcal{C}$, this theorem
shows that the answer to 1.1 is yes, at least up to finite index.

Through the zippered rectangles construction, the Teichmüller flow on
$\mathcal{C}^{\mathrm{root}}$ can be coded combinatorially by the reduced
Rauzy diagram. In the course of the proof of Theorem 5.6, we prove Let
$\mathcal{D}^{\mathrm{red}}$ be the reduced Rauzy diagram for
$\mathcal{C}^{\mathrm{root}}$. Then the natural homomorphism
$\pi_{1}(\mathcal{D}^{\mathrm{red}})\to\pi_{1}(\mathcal{C}^{\mathrm{root}})$
is surjective. Taking a further image to $\mathrm{Mod}(S)$, this answers up to
finite index, a weaker version of Yoccoz’s question. The flow group and some
of its applications to Teichmüller dynamics are also discussed by Hamenstädt
[Ham18, Section 4.2].

For strata of abelian differentials, previous work by Calderon and
Calderon–Salter also allows us to explicitly compute the image of the flow
group inside of $\mathrm{Mod}(S)$ and of
$\operatorname{Aut}(H_{1}(S;\mathbb{Z}))$ (or some larger group, such as
$\operatorname{Aut}(H_{1}(S,Z;\mathbb{Z}))$), up to finite index [Cal20, CS19,
CS19a, CS20].

### Cocycles

Fix $\mathcal{C}$, a stratum component. Given a bundle over $\mathcal{C}$, the
Teichmüller flow gives us a natural cocycle. The most studied of these is the
_Kontsevich–Zorich cocycle_. This can be lifted to a connected component
$\mathcal{T}\mathcal{C}$ of the Teichmüller space of abelian or quadratic
differentials (the choice of $\mathcal{T}\mathcal{C}$ is, in general, not
unique [Cal20, CS20]).

In more detail, we define a vector bundle over $\mathcal{T}\mathcal{C}$ with a
suitable fibre. In the abelian case, this fibre is the first cohomology of the
underlying topological surface; in the quadratic case, it is the first
cohomology of the orientation double cover. By Poincaré-duality, it is also
possible to use the corresponding homology groups as the fibre.

The $\operatorname{SL}(2,\mathbb{R})$-action induces a trivial dynamical
cocycle on this vector bundle. By modding out by the mapping class group, the
vector bundle descends to a bundle over $\mathcal{C}$ known as the _Hodge
bundle_ ; similarly the cocycle descends to the _Kontsevich–Zorich cocycle_
[KZ97, Kon97]. In the quadratic case, the cocycle then naturally splits into
two distinct symplectically orthogonal blocks, usually referred to as the
_plus_ (or _invariant_) and _minus_ (or _anti-invariant_) pieces.

Many interesting dynamical properties of abelian or quadratic differentials
can be written in terms of the Lyapunov exponents of the Kontsevich–Zorich
cocycle. An important example are the deviations of ergodic averages of the
linear flow on almost every abelian or quadratic differential [Zor97, EKZ14].
In fact, when the Lyapunov spectrum of the Kontsevich–Zorich cocycle is
simple, these deviations can be precisely described.

Kontsevich–Zorich conjectured that the Lyapunov spectrum is simple for all
abelian stratum components [Conjecture 2]Zor97[page 1499]Zor99. Their
conjecture extends naturally to the quadratic case as follows. We form the
branched orientation double cover. The homology of the cover splits into the
plus and minus eigenspace for the involution; the
$\operatorname{SL}(2,\mathbb{R})$ action preserves this splitting. Simplicity
is conjectured in both pieces [Zor18].

For the abelian case, this conjecture was established in the famous work by
Avila–Viana [AV07a]. The quadratic case is known for many stratum components
but not in full generality [Tre13, Gut17].

Our paper establishes simplicity in all cases; as discussed below, our proof
relies on certain machinery of these previous authors, but is independent of
their theorems.

### Rauzy–Veech groups

The Rauzy–Veech groups are subgroups of the symplectic group generated by the
matrices (in a preferred basis) induced by evaluating these cocycles over
based loops in the Rauzy diagram. It follows from Theorem 5.2 that Rauzy–Veech
groups have finite index in the corresponding monodromy groups. We leverage
this finiteness with standard techniques of splitting zeroes [AV07a, Gut19] to
extend from simpler strata to more complicated ones in order to prove the
following.

The Rauzy–Veech groups for all components of all abelian strata are Zariski
dense in their ambient symplectic groups. The same holds for the plus and
minus Rauzy–Veech groups for all components of all quadratic strata.

The groups of Theorem 9.1 that arise, by splitting singularities, from abelian
strata are known to be finite index inside the ambient symplectic groups (over
$\mathbb{Z}$) and hence Zariski dense. This was done by Avila–Matheus–Yoccoz
[AMY18] for abelian hyperelliptic components and by the fourth author [Gut19,
Gut17] for all other components mentioned above. Using our techniques, and
again replying on certain machinery from previous work, our Theorem 9.1 gives
Zariski density in all cases.

By the work of Benoist [Ben97], Zariski density of an appropriate Rauzy–Veech
group implies that the monoids associated with the Kontsevich–Zorich cocycles
are “rich” in the sense of the simplicity criterion of Avila–Viana [AV07,
AV07a]. As a consequence of Theorem 9.1, we can apply the Avila–Viana
criterion to prove the Kontsevich–Zorich simplicity conjecture.

The Kontsevich–Zorich cocycle has a simple spectrum for all components of all
strata of abelian differentials. The plus and minus Kontsevich–Zorich cocycles
also have a simple spectrum for all components of all strata of quadratic
differentials.

As mentioned before, simplicity was known for all abelian [AV07a] and some
quadratic stratum components [Gut17]. It is also known for the principal
stratum of quadratic differentials by different methods through the recently
announced solution by Eskin–Mirzakhani–Rafi of the Furstenberg problem for
random walks on the mapping class group. However, we have claimed the known
results as our proof is self-contained and is uniform across all stratum
components.

With Theorem 5.6 in hand, we can compute the Kontsevich–Zorich cocycle over
any loop in $\mathcal{C}^{\mathrm{root}}$ and not just along the Teichmüller
flow. This additional flexibility implies that, for Zariski density, we can
always consider a monodromy group instead of a Rauzy–Veech group.

For the monodromy groups of abelian differentials, and also for the monodromy
groups induced by the minus piece of the cocycle for quadratic differentials,
we directly apply some of Filip’s results to obtain Zariski density [Fil17,
Corollary 1.7]. For the monodromy groups induced by the plus piece of the
cocycle, we need to discuss _algebraic hulls_.

The algebraic hull of the Kontsevich–Zorich cocycle restricted to a linear
invariant suborbifold can be thought of as the smallest algebraic group into
which the cocycle can be measurably conjugated. As such, the hull is both an
algebro-geometric and an ergodic-theoretic object. Eskin–Filip–Wright showed
that the algebraic hull is as large as it can be, namely it equals the
stabiliser of the tautological plane (that is, the cohomology classes spanned
by the real and imaginary parts of the differential) in the Zariski closure of
the monodromy group [EFW18, Theorem 1.1].

The plus piece of the Kontsevich–Zorich cocycle does not meet the tautological
plane. The stabiliser then equals the Zariski closure of the monodromy, and
hence so does the algebraic hull. This result, together with Filip’s
classification of the possible Lie algebra representations of algebraic hulls
[Fil17, Theorem 1.2], allows us to show that the Zariski closure of the
monodromy group corresponding to the plus piece is
$\operatorname{Sp}(2g,\mathbb{R})$ by a simple dimension count.

Finally, we remark that, just as Eskin–Filip–Wright’s theorem shows that the
algebraic hull is as large as it can be, Theorem 5.6 shows that the flow group
is also as large as it can be. Thus, for stratum components, Theorem 5.6 can
be considered as a dynamical analogue of Eskin–Filip–Wright’s result.

### Acknowledgements

The authors are immensely grateful to Carlos Matheus for countless
illuminating conversations. We also thank Giovanni Forni, Maxime Fortier
Bourque, Erwan Lanneau and Alex Wright for their helpful comments on an
earlier version of this article.

The fourth author is grateful to the ANID AFB-170001, the FONDECYT Iniciación
11190034, and the MATHAMSUD 21-MATH-07 grants.

##  2\. Strategies

We outline the steps and the key ideas in our proofs.

#### Passing to rooted differentials

The dynamical issues considered here are stable under passing to a finite
cover of the given stratum component $\mathcal{C}$. We pass to the space
$\mathcal{C}^{\mathrm{root}}$ of _rooted differentials_ : differentials marked
with a horizontal separatrix (or, equivalently, a horizontal unit tangent
vector at a marked point). The reasons are two-fold. Unlike $\mathcal{C}$, the
cover $\mathcal{C}^{\mathrm{root}}$ is a manifold, which simplifies various
transversality (and fundamental group) arguments. Also, a generic rooted
differential admits a description via a zippered rectangles construction.

#### Zippered rectangles and the based loop theorem

The _zippered rectangles construction_ is originally due to Veech [Vee82] for
abelian stratum components and due to Boissy–Lanneau [BL09] for quadratic
stratum components. Parameter spaces of zippered rectangles, where the length
of the base-arc is normalised, define contractible open sets in
$\mathcal{C}^{\mathrm{root}}$ which we call _polytopes_. The union of the
polytopes is dense in $\mathcal{C}^{\mathrm{root}}$. However, the complement
of their union is complicated; the polytopes do not give a cell structure on
$\mathcal{C}^{\mathrm{root}}$. For instance, there are compact arcs in
$\mathcal{C}^{\mathrm{root}}$ that intersect polytope faces infinitely many
times. See Section A.1 for an explicit example and relevant discussions. As a
result, the _based loop theorem_ , which we explain below, cannot be deduced
from naïve transversality arguments.

Fortunately, as discussed by Yoccoz [Yoc10, Proposition in Section 9.3], the
subset of rooted differentials that do _not_ admit any zippered rectangle
construction is contained in the (codimension two) set of differentials that
have both a vertical and a horizontal saddle connection. Thus, any based loop
$\gamma\colon[0,1]\to\mathcal{C}^{\mathrm{root}}$ can be homotoped to be
disjoint from such differentials.

After this homotopy, we can cover the image of $\gamma$ by finitely many
reasonably nice charts. Unfortunately, these may not be contained in the
interior of any of the polytopes defined above. We arrange matters so that the
boundaries of these charts are codimension-one embedded submanifolds. A
further homotopy makes $\gamma$ transverse to these boundaries remaining
covered by the charts. Since a chart may not be contained in the interior of a
polytope, the lengths of the base-arcs in these charts need not be normalised.
We fix the required normalisation as follows.

Given a sufficiently small subsegment of $\gamma$, where the base-arcs are not
normalised, we apply the (forward or backward, as needed) Teichmüller flow.
This replaces (via homotopy) the subsegment of $\gamma$ by two segments
contained in the flow and one segment contained in the interior of a polytope.
Doing this finitely many times, we homotope $\gamma$ to be a concatenation of
segments which are forward or backward Teichmüller segments or completely
contained inside a polytope. This is our _based loop theorem_ , namely Theorem
4.23.

#### Rauzy induction and the Teichmüller flow

The combinatorial information of a zippered rectangles construction is an
irreducible generalised permutation [BL09]. Also, there are various associated
parameters such as the dimensions of the rectangles and the heights of the
various zippers. The combinatorics together with the parameters uniquely
specify the differential.

If we apply the forward Teichmüller flow, the base-arc grows until it violates
the normalisation. At this point we pass to the largest base-arc strictly
contained in the original base-arc. In this way we obtain a new irreducible
generalised permutation as well as new parameters. We call a single such
operation a _Rauzy–Veech move_.

The collection of all these moves gives a renormalisation procedure known as
the _Rauzy–Veech renormalisation_ or the _Rauzy–Veech induction_. It was
originally defined by Rauzy and Veech for abelian differentials [Rau79, Vee82]
and by Boissy–Lanneau [BL09] for quadratic differentials. Applying the
Teichmüller flow, we obtain a sequence of pairs of combinatorics and
parameters. Thus, the Rauzy–Veech renormalisation gives a coding for the
Teichmüller flow.

We encode this as an “automaton” (a directed graph) as follows. The vertices
are equivalence classes of irreducible generalised permutations suited to
$\mathcal{C}^{\mathrm{root}}$. Two permutations $\pi$ and $\pi^{\prime}$ are
equivalent if we can precompose with a permutation $\sigma$ to obtain
$\pi\circ\sigma=\pi^{\prime}$. There is a directed edge from $[\pi]$ to
$[\rho]$ if some representative of the latter arises from a single Rauzy–Veech
move. This automaton is called the _reduced Rauzy diagram_. Since the
Teichmüller flow is ergodic, as shown by Masur [Mas82] and Veech [Vee82,
Vee86], it follows that the reduced Rauzy diagram is _strongly connected_ :
there is a directed path from any vertex to any other vertex. By accelerating
the renormalisation, we can derive a coding that has the properties that the
Avila–Viana criterion stated below requires.

#### Flow groups and the fundamental group

There is a natural homomorphism from the fundamental group of the reduced
Rauzy diagram (as an undirected graph) to the fundamental group of
$\mathcal{C}^{\mathrm{root}}$. By leveraging the based loop theorem and
Rauzy–Veech sequences for Teichmüller segments, we show that the homomorphism
is surjective. This partially answers a question of Yoccoz [Yoc10, Remark in
Section 9.3].

We use the based loop theorem, and the above surjectivity, to show that the
flow group is equal to the fundamental group of $\mathcal{C}^{\mathrm{root}}$.
See Theorem 5.5 and Theorem 5.6. In other words, at the level of the
fundamental group, the Teichmüller flow captures the topology of
$\mathcal{C}^{\mathrm{root}}$, and hence the topology of $\mathcal{C}$, up to
finite index.

#### Cocycle simplicity

By a criterion of Avila–Viana [AV07, AV07a], simplicity of natural integrable
cocycles, such as the Kontsevich–Zorich cocycle, boils down to the existence
of a coding with an almost product structure and a notion of “richness” of the
cocycle. As we indicated earlier, a coding with the required integrability and
distortion properties can be achieved by accelerating the Rauzy–Veech
renormalisation. This was done by Avila–Gouëzel–Yoccoz [AGY06] for abelian
differentials and by Avila–Resende [AR12] for quadratic differentials. See
Section 6 for more details. The remaining task, and the crux of the problem,
is to show the richness of the cocycle. The required richness was established
by Avila–Viana [AV07a] for abelian stratum components by a direct computation.

In general, to obtain the richness condition for a symplectic cocycle it is
enough to establish the Zariski density of an appropriate group inside the
symplectic group; using work of Benoist [Ben97] it implies the above notion of
richness (Zariski density is, in fact, strictly stronger [AMY18, Appendix A]).

For the Kontsevich–Zorich cocycle, the relevant group is the Rauzy–Veech
group. Its Zariski density was proved for hyperelliptic components by
Avila–Matheus–Yoccoz. In fact, their result is stronger as they show it to be
an explicit finite index subgroup of its ambient symplectic group [AMY18,
Theorem 1.1]. This finite index result was extended by the fourth author to
all abelian stratum components and to quadratic stratum components that have
abelian components on their boundary [Theorem 1.1]Gut19[Theorem 1.1]Gut17.

Our main result, namely Theorem 5.6 stating that the flow group equals the
fundamental group of $\mathcal{C}^{\mathrm{root}}$, is crucial to achieve the
Zariski density of the Rauzy–Veech group of any stratum component. Indeed, it
allows us to compute the cocycle along any loop in
$\mathcal{C}^{\mathrm{root}}$ instead of only along almost T. This extra
flexibility is significant since we do not have to restrict to directed loops
in the reduced Rauzy diagram. In recent work [Fil17], Filip gives a finite
list of possible Zariski closures of the monodromy of a linear invariant
suborbifold. From this description, he also derives the fact that the Zariski
closure of the monodromy restricted to the symplectic block that contains the
tautological plane is the full symplectic group for this block. Combined with
this fact, our Theorem 5.6 directly yields simplicity for abelian components.

A quadratic stratum component lifts to a linear invariant suborbifold of its
orientation double-cover and hence Filip’s result applies to this situation.
The involution on the orientation double-cover splits the Kontsevich–Zorich
cocycle into two symplectically orthogonal blocks, usually referred to as the
_plus_ (or _invariant_) piece and the _minus_ (or _anti-invariant_) pieces.
The minus piece contains the tautological plane. Again by Filip’s corollary,
the Zariski closure for the minus cocycle is the full symplectic group.
Simplicity of the minus cocycle follows directly from combining this with
Theorem 5.6.

It remains to tackle the plus cocyle. Here, we exploit our extra flexibility
to build a dimension argument that eliminates all but the full symplectic
group as the Zariski closure. We carry out the dimension argument first for
components of minimal strata and hyperelliptic components with two zeros to
conclude Zariski density for the monodromy groups of these components. This
implies the Zariski density of their Rauzy–Veech groups as they are finite
index in the monodromy groups (a consequence of Theorem 5.2). We then deal
with a few remaining low genera components by using a well-known criterion for
Zariski density [PR14]. Finally, we extend the density to Rauzy–Veech groups
of all quadratic components by standard techniques of surgery/splitting
zeroes. The density allows us to apply the Avila–Viana criterion to conclude
the proof of the Kontsevich–Zorich conjecture in full generality.

##  3\. Preliminaries

###  3.1. Moduli spaces of abelian and quadratic differentials

A connected, oriented surface $S$ of finite type, that is, with finite genus
and finitely many marked points, can be equipped with a conformal/complex
structure by charts to the complex plane and holomorphic transition functions.
The Teichmüller space of $S$ is the space of marked conformal structures on
$S$. The mapping class group $\mathrm{Mod}(S)$ is the group of orientation
preserving diffeomorphisms of $S$ modulo isotopy. The mapping class group acts
on the Teichmüller space by changing the marking. The quotient $\mathcal{M}$
is the moduli space of Riemann surfaces homeomorphic to $S$.

The cotangent bundle to Teichmüller space is the space of (marked) meromorphic
quadratic differentials on $S$ with at most simple poles. The zeroes or poles
of the differential must lie at the marked points. The quotient by the mapping
class group is the moduli space $\mathcal{Q}$ of quadratic differentials. The
space $\mathcal{Q}$ is stratified by the orders at the marked points and
components of the strata are classified by the following combinatorial and
algebraic invariants:

  1. (1)

The singularity data which can be encapsulated as follows. Let $Z\subseteq S$
be a non-empty and finite set of points; we set $n=|Z|$. Let $\kappa\colon
Z\to\\{-1,0\\}\cup\mathbb{N}$ be any function so that $\sum\kappa(z)=4g-4$.
The points $z\in Z$ with $\kappa(z)=-1$ are called _simple poles_ and these
have to be the marked points of $S$. The points with $\kappa(z)=0$ are called
_regular points_. To ensure generality, we allow finitely many additional
points in $S$ to be marked as regular points.

  2. (2)

Abelian or quadratic, that is, whether the vertical foliations of
differentials in the component are orientable or not. In the abelian case, the
function $\kappa$ is even at every $z\in Z$, so it is common to consider
$\kappa/2$ instead of $\kappa$ as the function giving the singularity data. We
will follow this convention.

  3. (3)

Hyperelliptic or non-hyperelliptic (when possible), that is, whether
differentials in the component have some rotational symmetry of order two with
$2g+2$ fixed points [Lan04].

  4. (4)

Odd or even spin (only for abelian components for which $\kappa(z)$ is even
for each $z\in Z$), which is defined as the Arf invariant of a specific
quadratic form Joh[Appendix C]Zor08.

  5. (5)

Regular or irregular (when possible), which can be distinguished by the
dimension of a cohomology group corresponding to a specific divisor [CM14].

For the reader’s convenience, we state the complete classification of abelian
and quadratic stratum components in Section 8.1.

We note that, in general, a stratum component is an _orbifold_. We refer to
the book by Boileau–Maillot–Porti [BMP03] for background on orbifolds and
their fundamental groups, although in most of our exposition we will only
consider the fundamental groups of actual manifolds.

######  Remark 3.2.

The extent to which $q\in\mathcal{C}$ is _marked_ varies in different
expositions. For us, we are assuming that $\mathcal{C}$ is as small as
possible; so we have forgotten the marking by $S$ and forgotten the marking by
$Z$. However, this does not mean that we erase marked regular points—we only
erase their names, as well as the names of all poles and zeros. Thus,
travelling around a loop in $\mathcal{C}$, a pair of points $z,z^{\prime}\in
Z$ may be permuted. We deduce that, while there is no map from
$\pi_{1}^{\mathrm{orb}}(\mathcal{C})\to\operatorname{Sym}(Z)$, for a loop in
$\mathcal{C}$ there is a well-defined conjugacy class in
$\operatorname{Sym}(Z)$.

###  3.3. $\operatorname{SL}(2,\mathbb{R})$-action

We fix a stratum component $\mathcal{C}$ and let $q$ be a differential in
$\mathcal{C}$. By integrating a square-root of $q$ we get charts from $S$ to
$\mathbb{C}$ with transition functions that are translations (or half-
translations), that is, transition functions that are of the form $z\to z+c$
(or $z\to\pm z+c)$.

The action of the group $\operatorname{SL}(2,\mathbb{R})$ on
$\mathbb{R}^{2}=\mathbb{C}$ can be restricted to the charts. As the transition
functions are translations (or half-translations), the
$\operatorname{SL}(2,\mathbb{R})$-action preserves the form of the transition
functions. As a result, it descends to an action on the differentials. As the
classifying invariants are also preserved, the
$\operatorname{SL}(2,\mathbb{R})$ orbit of any differential in $\mathcal{C}$
is contained in $\mathcal{C}$.

Fixing a basis for the relative homology of $(S,Z)$, we can compute periods.
The period of a basis element is the integral of a square root of $q$ over an
arc in the homology class of the element. The periods define local charts on
$\mathcal{C}$. By the famous work of Eskin–Mirzakhani–Mohammadi [EMM15],
closures of $\operatorname{SL}(2,\mathbb{R})$-orbits inside $\mathcal{C}$ are
suborbifolds cut out by linear equations (with real coefficients and no
constant terms) in the period coordinates. Such an orbit closure is called a
_linear invariant suborbifold_.

The diagonal part of the $\operatorname{SL}(2,\mathbb{R})$-action defines _the
Teichmüller flow_.

###  3.4. Monodromy groups

The canonical projection $\mathcal{C}\to\mathcal{M}$ associates to a
differential the underlying conformal structure. So we may consider in
$\pi_{1}^{\mathrm{orb}}(\mathcal{M})=\mathrm{Mod}(S)$ the image of
$\pi_{1}^{\mathrm{orb}}(\mathcal{C})$ under the induced map on the orbifold
fundamental groups. The image group $\operatorname{MMon}(\mathcal{C})$ is
called the _modular monodromy group_ of $\mathcal{C}$.

The mapping class group $\mathrm{Mod}(S)$ has a natural action on the
(absolute) integral homology $H_{1}(S;\mathbb{Z})$. The action preserves the
symplectic form on $H_{1}(S;\mathbb{Z})$ given by the algebraic intersection.
As a result, $\mathrm{Mod}(S)$ admits a representation to the automorphism
group of $H_{1}(S;\mathbb{Z})$ that preserves the symplectic form. The
restriction of this symplectic representation to
$\operatorname{MMon}(\mathcal{C})$ gives us a subgroup of the symplectic group
which we call the _monodromy group_ $\operatorname{Mon}(\mathcal{C})$ of
$\mathcal{C}$.

###  3.5. Rooted differentials

For this article, we need to pass to a finite manifold cover of $\mathcal{C}$.
We begin as follows.

######  Definition 3.6.

Suppose that $q\in\mathcal{C}$ is a differential. Let $z$ be a zero, regular
point, or pole of $q$. Let $v$ be a unit tangent vector at $z$ pointing along
the horizontal foliation. We call the pair $(q,v)$ a _rooted_ differential.

The usual difference between the order of a point and the total angle at a
point allows us to show that the number of rootings of $q$ is $4g-4+2|Z|$.
However, some rootings of $q$ may be equivalent to others when $q$ has a
symmetry.

Rooted differentials are intended to reproduce the notion of a _marked
translation surface_ that is widely used in the literature [Yoc10, Boi20]. We
use $\mathcal{C}^{\mathrm{root}}$ to denote the space of rooted differentials.

##  4\. Zippered rectangles

We now outline a procedure to pass from flat geometry to combinatorics with
parameters. To do so, we exhibit a generic rooted quadratic differential as a
collection of rectangles with gluings. This is a well-known construction,
originally due to Veech [Vee82], called _zippered rectangles_. As we are
interested in the topology (fundamental group) of
$\mathcal{C}^{\mathrm{root}}$ and not just in the dynamics of the Teichmüller
flow on $\mathcal{C}^{\mathrm{root}}$, we will present the full details of the
construction for greater clarity.

There are several useful systems of parameters. We make use of the
_singularity_ parameters but there are other commonly used parameters such as
_zipper_ parameters introduced by Veech. We define the parameters and discuss
how to move between them.

###  4.1. The combinatorics

A _saddle connection_ for a quadratic differential is a flat geodesic that
connects a pair of possibly distinct points in $Z$ and is otherwise disjoint
from $Z$. We say that a quadratic differential $q$ has a _vertical vanishing
coordinate_ (respectively, _horizontal vanishing coordinate_) if $q$ has a
horizontal (respectively, vertical) saddle connection. Let
$\mathcal{V}\subseteq\mathcal{C}^{\mathrm{root}}$ be the set of such. So
$\mathcal{V}$ is a countable union of codimension-one loci.

Continuing in this way, we say that a quadratic differential $q$ is _doubly
vanishing_ if $q$ has both a horizontal and a vertical saddle connection. Let
$\mathcal{W}\subseteq\mathcal{C}^{\mathrm{root}}$ be the set of such
differentials. So $\mathcal{W}$ is a countable union of codimension-two loci.

A quadratic differential is said to be _vertically non-vanishing_
(respectively, _horizontally non-vanishing_) if it has no horizontal
(respectively, vertical) saddle connections. We will call a quadratic
differential _doubly non-vanishing_ if it has neither horizontal nor vertical
saddle connections.

Given a rooted quadratic differential $(q,v)$, let $I_{v}$ be the horizontal
separatrix defined by $v$, which may be finite if $q$ has a vertical vanishing
coordinate. Given a point $x\in I_{v}$, let $I(x)$ be the subarc of $I_{v}$
from the base of the root to $x$.

######  Definition 4.2.

We say that $I(x)$ is a _base-arc_ if

  1. (1)

the interior of $I(x)$ meets every leaf of the vertical foliation, and

  2. (2)

at least one of the two rays (going “up” or “down”) perpendicular to $I_{v}$
and starting at $x$ hit a singularity in $Z$ before hitting $I(x)$ a second
time.

The proof of the following lemma is analogous to the one given by Yoccoz in
the abelian case [Yoc10, Proposition 5.6].

######  Lemma 4.3.

Let $(q,v)$ be a rooted quadratic differential. If $q$ is not doubly
vanishing, then it admits a base-arc.

###### Proof.

Assume first that $q$ has no vertical saddle connection. Thus the vertical
foliation for $q$ is minimal (as otherwise the closure of a vertical leaf
would be a subsurface with boundary, containing saddle connections). Thus, the
interior of any horizontal subarc of $I_{v}$ will meet every leaf of the
vertical foliation, so condition (1) in Definition 4.2 is met. Shortening the
arc as needed, we can arrange condition (2) in Definition 4.2.

Assume instead that $q$ has no horizontal saddle connection. Now the
horizontal foliation for $q$ is minimal. Thus, we can and do take a
sufficiently long subarc of $I_{v}$ so that its interior meets every vertical
leaf. Then, condition (1) in Definition 4.2 is met. Making the arc longer as
needed, we can arrange condition (2) in Definition 4.2. ∎

Since $I$ is simply connected, we can orient the vertical foliation in a small
neighbourhood of $I$. We so orient the vertical foliation (locally) so that
the “upwards direction” crosses $I$ from right to left.

We consider the first return map to $I$, in $q$, defined by travelling along
the vertical foliation. We travel in both directions (up and down) to find
both the first return map and its “inverse”. Let $S_{\mathrm{t}}$ be the
(finite) set of points $x\in I$ where the upward leaf from $x$ runs in to a
singularity in $Z$ before returning to $I$. We define $S_{\mathrm{b}}$
similarly. If $q$ is horizontally non-vanishing, the sets $S_{\mathrm{t}}$ and
$S_{\mathrm{b}}$ are disjoint, but this is not true in general. To distinguish
between the points in $S_{\mathrm{t}}$ and $S_{\mathrm{b}}$, we will add the
labels $\mathrm{t}$ and $\mathrm{b}$.

Let $I_{\text{t}}$ be the components of $I-S_{\mathrm{t}}$. We call these the
_top intervals_. Similarly, we define the _bottom intervals_ $I_{\text{b}}$ to
be the components of $I-S_{\mathrm{b}}$. Again, since if $q$ is horizontally
non-vanishing, we have $I_{\text{t}}\cap I_{\text{b}}=\varnothing$, but this
is not true in general. Thus, we will distinguish them by the labels
$\mathrm{t}$ and $\mathrm{b}$.

There is a fixed-point free involution $\tau$ on
$I_{\text{t}}\times\\{\mathrm{t}\\}\sqcup I_{\text{b}}\times\\{\mathrm{b}\\}$
as follows. Any interval $(J,\ast)$, where
$\ast\in\\{\mathrm{t},\mathrm{b}\\}$, pairs with the interval
$(J^{\prime},\ast^{\prime})=\tau(J,\ast)$ so that the first return map takes
$(J,\ast)$ to $(J^{\prime},\ast^{\prime})$. Thus
$|I_{\text{t}}|+|I_{\text{b}}|$ is even. We write
$2d=|I_{\text{t}}|+|I_{\text{b}}|$.

We capture the above information, combinatorially, as follows. Let
$\mathcal{A}$ be a set of $d$ _letters_. Let $\ell=|I_{\text{t}}|$ and
$m=|I_{\text{b}}|$. Note that the sets $I_{\text{t}}$ and $I_{\text{b}}$ are
ordered by how the intervals appear along $I$. We use this to index the
intervals: $I_{\text{t}}\times\\{\mathrm{t}\\}=\\{J_{i}\\}_{i=1}^{\ell}$ and
$I_{\text{b}}\times\\{\mathrm{b}\\}=\\{J_{i}\\}_{i=\ell+1}^{\ell+m}$. We
define $\pi\colon\\{1,2,\ldots,2d\\}\to\mathcal{A}$ to be any two-to-one map
with the following property: for all $a\in\mathcal{A}$, if
$\\{i,j\\}=\pi^{-1}(a)$ then $\tau(J_{i})=J_{j}$. Associated with $\pi$ is a
fixed-point free involution $\sigma$ of $\\{1,2,\ldots,2d\\}$ where
$\sigma(i)=j$ implies $\pi(i)=\pi(j)$. Maps $\pi$ of the above type were first
considered by Danthony–Nogueira [DN88, DN90], and by Boissy–Lannaeu [BL09] and
are known as _generalised permutations_. As shown by Boissy–Lanneau [BL09,
Theorems A–D], the generalised permutations $\pi$ that arise in the above
construction are _irreducible_. Moreover, the set of generalised permutations
that arise from $\mathcal{C}^{\mathrm{root}}$ is known as the _Rauzy class_ of
$\mathcal{C}^{\mathrm{root}}$. We denote the Rauzy class by
$\mathcal{R}(\mathcal{C}^{\mathrm{root}})$. Moreover, as we vary over all
stratum components and choices of rootings, all irreducible generalised
permutations arise from this construction. See the article by Boissy–Lanneau
[BL09] for a combinatorial definition of irreducibility and a proof of these
facts.

We refer to the letters $\pi(1),\ldots,\pi(\ell)$ as the _top letters_ for
$\pi$. Similarly, we refer to the letters $\pi(\ell+1),\ldots,\pi(\ell+m)$ as
the _bottom letters_. Any letter that is both a top letter and a bottom letter
is called a _translation letter_. Any letter that is only a top letter (or
only a bottom letter) is called a _flip letter_. We explain the terminology
below.

We say that $\pi$ is a _abelian permutation_ if it has no flip letters. We say
that $\pi$ is a _quadratic permutation_ it has (at least one) top flip letter
and (at least one) bottom flip letter. All generalised permutations that arise
in the construction above are of one of this two types.

From now on, will eschew the terminology “generalised permutation” and
collectively refer to abelian and quadratic permutations simply as
_permutations_.

###  4.4. The rectangles

Let $\alpha\in\mathcal{A}$ and let $\pi^{-1}(\alpha)=\\{i,\sigma(i)\\}$. There
is an associated rectangle $R=R_{\alpha}$ with the following properties.

  1. (1)

The horizontal sides of $R$ are exactly $R\cap I=J_{i}\cup J_{\sigma(i)}$.

  2. (2)

With the exception of one rectangle, each vertical side of $R$ contains
exactly one singularity. The exceptional rectangle is either $R_{\pi(\ell+m)}$
or $R_{\pi(\ell)}$ depending on whether the right endpoint of $I$ is in
$S_{\mathrm{t}}$ or $S_{\mathrm{b}}$, respectively.

###  4.5. The zippers

We now define the zippers. Let $p\in S_{\mathrm{t}}$. By definition, the
perpendicular ray that goes up from $p$ hits a singularity before it can
return to $I$. We call the resulting vertical segment $Z(p)$ a _top zipper_.
Similarly, we define _bottom zippers_ to be the segments of the perpendicular
rays that go down from points in $S_{\mathrm{b}}$ and hit a singularity before
they return to $I$.

###  4.6. Singularity parameters

We have fixed an orientation on the surface. Let $R=R_{\alpha}$ be the
rectangle with letter $\alpha\in\mathcal{A}$. Recall that every rectangle $R$
has two vertical sides and two horizontal sides. Laying $R$ out in the plane
we call its sides the _east_ , _north_ , _west_ , and _south_ sides. By
construction, the east and west sides of $R$ lie in vertical leaves
$\ell_{\text{E}}$ and $\ell_{\text{W}}$ that meet $Z$, the set of
singularities, in exactly one point before returning to $I$. We also lay out
$\ell_{\text{E}}$ and $\ell_{\text{W}}$ in the plane. Let $z_{\text{E}}$ and
$z_{\text{W}}$ be the images of the singularities in $\ell_{\text{E}}$ and
$\ell_{\text{W}}$, as they lie in the plane. Also by construction, at least
one of $z_{\text{E}}$ or $z_{\text{W}}$ lies in (the closure of) a vertical
side of $R$.

Breaking symmetry, suppose that $z_{\text{W}}$ lies in the west side of $R$,
not just in $\ell_{\text{W}}$. Let $m$ be the horizontal spanning arc of $R$,
which has one endpoint at $z_{\text{W}}$. Let $p$ be the endpoint of $m$ lying
in $\ell_{\text{E}}$. We call $p$ the _projection_ of $z_{\text{W}}$ to
$\ell_{\text{E}}$. We define the _singularity width_ of the letter $\alpha$ to
be

$x_{\alpha}=|m|$

that is, the unsigned length of $m$. Note that $x_{\alpha}$ is exactly the
width of $R=R_{\alpha}$.

If $p=z_{\text{E}}$, we define the _singularity height_ of the letter $\alpha$
to be $y_{\alpha}=0$. Otherwise, let $\ell$ be the bounded segment of
$\ell_{\text{E}}-\\{p,z_{\text{E}}\\}$. We orient the path $\gamma=m\cup\ell$
away from $z_{\text{W}}$. Note that $\gamma$ turns right or left at $p$
depending on the position of $z_{\text{E}}$ in $\ell_{\text{E}}$. This turning
is defined due to the orientation on $q$; also it is independent of the
choices made. We now define $y_{\alpha}$ the _singularity height_ of the
letter $\alpha$. We take the magnitude of $y_{\alpha}$ to be

$|y_{\alpha}|=|\ell|$

We take the sign of $y_{\alpha}$ to be positive if and only if $\gamma$ turns
left at $p$

(a) The curve $\gamma$ turns left at $p$ and can be tightened to a saddle
connection.

(b) The curve $\gamma$ turns right at $p$ and can be tightened to a saddle
connection.

(c) The curve $\gamma$ may not be able to be tightened to a saddle connection.

Figure 4.7. Three cases of singularity coordinates.

######  Remark 4.8.

The singularity height $y_{\alpha}$ is _not_ , in general, the height of
$R=R_{\alpha}$. We discuss this point further below.

######  Remark 4.9.

For all but one rectangles $R=R_{\alpha}$, the points $z_{\text{E}}$ and
$z_{\text{W}}$ lie in its east and west sides, respectively. When this
happens, the path $\gamma$ defined above can be tightened to give a saddle
connection in $q$. The parameter $x_{\alpha}+iy_{\alpha}$ is then (up to
global change of sign) the period of $\gamma$. However, there are (abelian and
quadratic) differentials where $\gamma$ is not homotopic (relative to its
endpoints) to a saddle connection Figure 4.7. This accounts for the complexity
of the definition of the singularity height $y_{\alpha}$.

Note that the horizontal edges of rectangles representing the top letters
(correctly repeating the flip letters) are arcs whose union is exactly the
base-arc $I$. The same holds for the bottom letters. We deduce the _width
equality_

(4.10) $\sum_{k=1}^{\ell}x_{\pi(k)}=\sum_{k=\ell+1}^{\ell+m}x_{\pi(k)}.$

Again, the left sum is over the top while the right is over the bottom. Since
every translation letter appears exactly once on each side, we deduce from the
width equality that $\sum x_{\alpha}=\sum x_{\beta}$; here $\alpha$ ranges
over the top flip letters and $\beta$ ranges over the bottom flip letters.

###  4.11. Zipper parameters

Breaking symmetry, let $p\in S_{\mathrm{t}}\times\\{\mathrm{t}\\}$ where we
assume $p\neq r\times\\{\mathrm{t}\\}$ if $r\in S_{\mathrm{t}}$. Let $Z(p)$ be
a top zipper based at $p$. By a slight abuse of notation, think of $p$ as a
point in $I$. Let $R_{\pi(i)}$ for $i\leqslant\ell$ be the rectangle to the
left of $Z(p)$. Then the horizontal coordinate of $p$, that is the distance of
$p$ from the left-endpoint of $I$, is given by

$x(p)=\sum_{j=1}^{i}x_{\pi(j)}.$

The height of $Z(p)$ is given by

$h(Z(p))=\sum_{j=1}^{i}y_{\pi(j)}.$

and we require this to be positive. This gives us the _top zipper
inequalities_

(4.12) $\sum_{j=1}^{i}y_{\pi(j)}>0$

for all $i<\ell$.

Similarly, if $Z(p)$ for $p\in S_{\mathrm{b}}\times\\{\mathrm{b}\\}$ and
$p\neq r\times\\{\mathrm{b}\\}$ if $r\in S_{\mathrm{b}}$ is a bottom zipper
and $R_{\pi(i)}$ for $i\geqslant\ell+1$ then the horizontal coordinate is

$x(p)=\sum_{j=\ell+1}^{i}x_{\pi(j)}$

and the height is

$h(Z(p))=\sum_{j=\ell+1}^{i}y_{\pi(j)}.$

Here we require the height $h(Z(p))$ to be negative. This gives us the _bottom
zipper inequalities_

(4.13) $\sum_{j=\ell+1}^{i}y_{\pi(j)}<0$

for all $\ell+1\leqslant i<\ell+m$.

It remains to consider the right endpoint $r$. The zipper height of $Z(r)$
gives us a linear relation in the $y$ parameters. Note that the above
equalities express the height $h(Z(r))$ in two ways; namely

$h(Z(r))=\sum_{j=1}^{\ell}y_{\pi(j)}$

and

$h(Z(r))=\sum_{j=\ell+1}^{\ell+m}y_{\pi(j)}.$

We deduce the _height_ equality

(4.14) $\sum_{k=1}^{\ell}y_{\pi(k)}=\sum_{k=\ell+1}^{\ell+m}y_{\pi(k)}.$

This is equivalent to $\sum y_{\alpha}=\sum y_{\beta}$, where $\alpha$ ranges
over the top flip letters and $\beta$ ranges over the bottom flip letters.

The height and width equalities are essentially identical. Thus, the
dimensions of the space of $x$ and $y$ parameters are equal; they are
$|\mathcal{A}|$ in the abelian case and $|\mathcal{A}|-1$ in the quadratic
case.

###  4.15. Rectangle parameters

For all rectangles $R=R_{\alpha}$, at least one of the points $z_{\text{E}}$
and $z_{\text{W}}$ lie in its east and west sides, respectively. Breaking
symmetry, suppose that $\alpha$ is a top letter and $z_{\text{E}}$ lies in its
east side. Let $Z(p)$ for $p\in S_{\mathrm{t}}\times\\{\mathrm{t}\\}$ be the
zipper with end point $z_{\text{E}}$. If $\alpha$ is a translation letter then
there is a zipper $Z(p^{\prime})$ for $p^{\prime}\in
S_{\mathrm{b}}\times\\{\mathrm{b}\\}$ with endpoint $z_{E}$ such that the
union $Z(p)\cup Z(p^{\prime})$ is the east side of $R_{\alpha}$. Recall that
the heights of bottom zippers are negative. Hence, the height $h(R_{\alpha})$
satisfies

(4.16) $h(R_{\alpha})=h(Z(p))-h(Z(p^{\prime})).$

If $\alpha$ is a flip letter instead then there is a zipper $Z(p^{\prime})$
for $p^{\prime}\in S_{\mathrm{t}}\times\\{\mathrm{t}\\}$ with end point
$z_{E}$ such that the union $Z(p)\cup Z(p^{\prime})$ is the east side of
$R_{\alpha}$. The height $h(R_{\alpha})$ is then

(4.17) $h(R_{\alpha})=h(Z(p))+h(Z(p^{\prime})).$

A similar discussion follows if $\alpha$ is a bottom letter.

###  4.18. Polytopes in $\mathcal{C}^{\mathrm{root}}$

Because of the flexibility in choosing base-arcs, a rooted differential can
have (infinitely) many different zippered rectangles constructions. For
example, suppose that $q$ is a doubly non-vanishing rooted differential. Then
the vertical foliation for $q$ is minimal; to see this, note that otherwise
the closure of a vertical leaf would be a subsurface with boundary, containing
saddle connections. Thus, for this $q$ any subarc $I\subseteq I_{v}$
satisfying condition (2) in Definition 4.2 can serve as a base-arc.

To remove this ambiguity from the combinatorics, an additional base-arc
normalisation must be imposed, as follows. Let $\mathcal{R}$ be the Rauzy
class of $\mathcal{C}^{\mathrm{root}}$. Fix an irreducible permutation
$\pi\in\mathcal{R}$.

######  Definition 4.19.

We define the set $P_{\pi}$ of _parameters_ for $\pi$ to be the pairs
$(x,y)\in\mathbb{R}^{\mathcal{A}}\times\mathbb{R}^{\mathcal{A}}$ satisfying

  1. (1)

the width and height equalities (4.10) and (4.14),

  2. (2)

the positivity condition $x_{\alpha}>0$ for all $\alpha$,

  3. (3)

the zipper inequalities (4.12) and (4.13), and

  4. (4)

the _base-arc normalisation_

$1<\sum_{i=1}^{\ell}x_{\pi(i)}<1+\min\\{x_{\pi(\ell)},x_{\pi(\ell+m)}\\}.$

The pair of inequalities in (4) are the promised restrictions on the length of
the base-arc. Any zippered rectangles construction arising from parameters in
this way is called _(base-arc) normalised_.

Let $q$ be a doubly non-vanishing rooted differential. Let $Z(q,v)$ be the
subset of $x$ along the separatrix $I_{v}$ such that $I(x)$ is a base-arc.

######  Lemma 4.20.

Let $q$ be a doubly non-vanishing rooted differential. Then the base of the
root is the only accumulation point of $Z(q,v)$ in $I_{v}$.

###### Proof.

Suppose that a point $x\in Z(q,v)$ is an accumulation point of $Z(q,v)$ and
that $x$ is not the root. As $q$ is doubly non-vanishing, there is no upper
bound on the lengths of possible base-arcs. Hence, $Z(q,v)$ contains a point
$x^{\prime}$ such that the base-arc $I(x^{\prime})$ is longer than $I(x)$. But
then the first return map to $I(x^{\prime})$ along the vertical has infinitely
many intervals, which is a contradiction. See Yoccoz’s lectures notes for more
details [Yoc10, Section 3.1]. ∎

It follows that $Z(q,v)$ contains a point $r$ whose base-arc is the shortest
among those whose length is at least one. The base-arc $I(r)$ is then the
unique subarc of $I_{v}$ whose zippered rectangle construction yields an
irreducible permutation with parameters that satisfy condition (4) of
Definition 4.19.

The above procedure applies to rooted quadratic differentials off of a
(somewhat complicated) measure zero set. For each such, it gives a pair
(combinatorics, parameter).

The opposite direction is provided by Boissy and Lanneau [BL09, Lemma 2.12].
Suppose that $\pi$ in $\mathcal{R}(\mathcal{C}^{\mathrm{root}})$ is an
irreducible permutation. Suppose that $(x,y)$ is any parameter in $P_{\pi}$.
Then, by placing a marked point at the origin of $\mathbb{C}$, by laying out
an arc on the positive real axis, by laying down rectangles, and gluing
according to the associated zipper lengths, Boissy and Lanneau build a
quadratic differential; the details are somewhat subtle.

We call this differential $q_{\pi}(x,y)$; we use $q_{\pi}\colon
P_{\pi}\to\mathcal{C}^{\mathrm{root}}$ to denote the resulting map. We call
the image
$\mathcal{C}_{\pi}=q_{\pi}(P_{\pi})\subseteq\mathcal{C}^{\mathrm{root}}$ a
_polytope_. For any doubly non-vanishing rooted differential $q$ in
$\mathcal{C}_{\pi}$, if $(\pi,(x,y))$ are its normalised combinatorics and
parameters, then $q_{\pi}(x,y)=q$.

On the other hand, it may happen that the polytopes arising from distinct
permutations coincide as sets. More precisely, consider the following
equivalence relation on permutations. Two permutations $\pi$ and
$\pi^{\prime}$ are _equivalent through re-indexing_ if there is a permutation
$p\in\operatorname{Sym}(\mathcal{A})$ such that $\pi^{\prime}=p\circ\pi$. As
letter re-indexing by $p$ does not affect the geometric construction, it
follows that if $\pi$ and $\pi^{\prime}$ are equivalent then
$\mathcal{C}_{\pi}=\mathcal{C}_{\pi^{\prime}}$.

######  Lemma 4.21.

For any irreducible permutation $\pi\in\mathcal{R}$, the map $q_{\pi}$ is a
homeomorphism from $P_{\pi}$ onto $\mathcal{C}_{\pi}$. Moreover, if $\pi$ and
$\pi^{\prime}$ are not equivalent then
$\mathcal{C}_{\pi}\cap\mathcal{C}_{\pi^{\prime}}=\varnothing$. The union of
the sets $\mathcal{C}_{\pi}$ is dense in $\mathcal{C}^{\mathrm{root}}$.

###### Proof.

Fix $\pi$ and a parameter $(x_{\alpha},y_{\alpha})_{\alpha\in\mathcal{A}}$.
For every letter $\alpha\in\mathcal{A}$ we are given an arc $\gamma_{\alpha}$
connecting a pair of singularities whose period is exactly
$x_{\alpha}+iy_{\alpha}$. Boissy–Lanneau show that these periods give
coordinates in $\mathcal{C}^{\mathrm{root}}$ [BL09, Lemma 2.12]. Since
$q_{\pi}$ is linear in these periods, it is continuous and injective. Since
$P_{\pi}$ and $\mathcal{C}_{\pi}$ have the same dimension, the map $q_{\pi}$
is a homeomorphism onto its image.

Thus, for any non-equivalent $\pi$ and $\pi^{\prime}$ the intersection
$\mathcal{C}_{\pi}\cap\mathcal{C}_{\pi^{\prime}}$ is open. Hence, if it is
non-empty, it must contain a doubly non-vanishing differential. However, this
contradicts that uniqueness of the permutation given by the zippered
rectangles construction.

Furthermore, every doubly non-vanishing differential $q$ in
$\mathcal{C}^{\mathrm{root}}$ lies in some $\mathcal{C}_{\pi}$, so we are
done.

∎

The above lemma shows that we can pass unambiguously from a typical
differential (such as a doubly non-vanishing differential) to combinatorics
and parameters.

###  4.22. Based loops in $\mathcal{C}^{\mathrm{root}}$

The main result of this article is the following theorem stating that every
based loop in $\mathcal{C}^{\mathrm{root}}$ can almost be straightened out
into a concatenation of Teichmüller geodesic segments.

######  Theorem 4.23.

Let $\mathcal{C}^{\mathrm{root}}$ be a stratum component of the moduli space
of rooted quadratic differentials. Let $q_{0}$ be a base-point in
$\mathcal{C}^{\mathrm{root}}$. Let
$\gamma\colon[0,1]\to\mathcal{C}^{\mathrm{root}}$ be a loop based at $q_{0}$.
Then, up to a homotopy relative to the base-point, the loop $\gamma$ can be
written as a finite concatenation of paths that are either

  * •

(forward or backward) Teichmüller geodesic segments; or

  * •

contained inside some polytope.

###### Proof.

We fix a base-point $q_{0}$ in $\mathcal{C}^{\mathrm{root}}$. For convenience,
we assume that $q_{0}$ is doubly non-vanishing and hence contained in some
polytope.

Recall that the set $\mathcal{W}\subseteq\mathcal{C}^{\mathrm{root}}$ of
doubly vanishing rooted differentials is a countable union of codimension-two
loci. Moreover, if $q\in\mathcal{C}^{\mathrm{root}}-\mathcal{W}$, then it
admits a base-arc by Lemma 4.3.

Let $q$ be a differential in $\mathcal{C}^{\mathrm{root}}-\mathcal{W}$ and $I$
be a base-arc. This gives a permutation $\pi$ and singularity parameters
$(x,y)$.

Since the singularity parameters are coordinates for
$\mathcal{C}^{\mathrm{root}}-\mathcal{W}$, there exists an open set in
$\mathcal{C}^{\mathrm{root}}$ containing $q$ given by the zippered rectangles
construction with underlying permutation $\pi$. Note that the base-arc $I$ may
not be normalised, so $q$ may not belong to $\mathcal{C}_{\pi}$. However, the
only condition that the parameters $(x,y)$ may not satisfy to belong to
$\mathcal{C}_{\pi}$ is

$1<\sum_{k=1}^{\ell}x_{\pi(k)}<1+\min\\{x_{\pi(\ell)},x_{\pi(\ell+m)}\\},$

which can be forced to hold by applying the Teichmüller flow. Thus, there
exists $t(q)\in\mathbb{R}$ such that $g_{t(q)}q\in\mathcal{C}_{\pi}$.

We now define $U(q)\subseteq\mathcal{C}^{\mathrm{root}}$ to be a contractible
open set around $q$ obtained by varying the parameters $(x,y)$ by a very small
amount so that $g_{t(q)}U(q)\subseteq\mathcal{C}_{\pi}$. That is, with respect
to the parameters $(x,y)$, the set $U(q)$ is a _box_ with sides parallel to
the coordinate planes. Therefore, $\partial U(q)$ is a union of finitely many
codimension-one embedded submanifolds (with boundary) in
$\mathcal{C}^{\mathrm{root}}$.

The locus $\mathcal{V}$ of vanishing rooted differentials, that is, rooted
differentials with a horizontal or vertical saddle connection, can be covered
by countably many relatively open codimension-one charts. Hence, we may apply
a homotopy (relative to $q_{0}$) to arrange that $\gamma$ is transverse to
$\mathcal{V}$. This can be done by using standard techniques in differential
topology [Hir94, Theorem 2.5, page 78]. After this, the loop $\gamma$ is
disjoint from $\mathcal{W}$.

Now, the boxes $(U({\gamma(s)}))_{s}$ cover $\gamma$, so, by compactness,
there exists a finite collection $s_{0},\dotsc,s_{n}\in[0,1]$ such that
$(U({\gamma(s_{j})}))_{j}$ covers $\gamma$. Let $U_{j}=U(\gamma(s_{j}))$.

We now perform a further homotopy (relative to $q_{0}$) supported in the union
of the boxes. Again appealing to standard techniques [Hir94, Theorem 2.5, pp.
78], we now have that $\gamma$ is transverse to the sides of the boxes $U_{j}$
and is again transverse to $\mathcal{V}$.

We obtain that $\gamma$ intersects $\partial U_{j}$ only finitely many times
and, therefore, that $\gamma^{-1}(U_{j})$ is a finite union of intervals in
$[0,1]$ for each $k$. All such intervals are open, except for possibly two
intervals $[0,s)$ and $(s^{\prime},1]$. If these two intervals exist in
$\gamma^{-1}(U_{j})$, we replace them by their union
$[0,s)\cup(s^{\prime},1]$.

Now, we select a minimal subcollection $J_{0},\dotsc,J_{m}$ of these sets that
covers $[0,1]$. Thus, $\gamma(J_{k})$ is contained inside some $U_{j}$, which
we denote by $V_{k}$. Observe that the list $V_{0},\dotsc,V_{m}$ may contain
repetitions. By setting $J_{m+1}=J_{0}$, we assume that the indices are chosen
so that $J_{k}\cap J_{k+1}$ is a non-empty open interval or a set of the form
$[0,s)\cup(s^{\prime},1]$. Without loss of generality, we can assume that
$0\in J_{m}\cap J_{0}$, since this can be arranged by covering $J_{0}$ and
$J_{m}$ with smaller intervals and rearranging the indices.

Since $\gamma$ is transverse to $\mathcal{V}$, we have that $\gamma(s)$ lies
in $\mathcal{V}$ for at most countably many $s\in[0,1]$. Thus, there exists a
doubly non-vanishing quadratic differential $q_{k+1}$ in the image of each set
$\gamma(J_{k}\cap J_{k+1})$ (with $q_{m+1}=q_{0}$). Hence, we obtain a
sequence of times $0=s_{0}\leqslant s_{1}\leqslant\dotsb\leqslant s_{m}=1$
such that the closed intervals $[s_{k},s_{k+1}]\subseteq[0,1]$ cover $[0,1]$
and $\gamma(s_{k})=q_{k}$.

Figure 4.24. Illustration of the proof of Theorem 4.23. Part of the loop
$\gamma$ is depicted as a solid curve. The dotted lines represent the
boundaries of the polytopes. Unlike the boxes $V_{k-1}$ and $V_{k+1}$, the box
$V_{k}$ is not contained inside a polytope, so the Teichmüller flow must be
applied to it. The resulting segment $\delta_{k}$ is shown as dashed curve.

Since $V_{k}$ is equal to one of the $U_{j}$ by construction, there exists a
real number $t_{k}$ such that $g_{t_{k}}V_{k}$ is completely contained inside
some polytope $\mathcal{C}_{\pi_{k}}$. Let $\delta_{k}$ be the path starting
at $q_{k}$ and ending at $q_{k+1}$ given by the concatenation of the paths

  * •

$g_{t}q_{k}$ for $t\in[0,t_{k}]$;

  * •

$g_{t_{k}}\gamma(s)$ for $s\in[s_{k},s_{k+1}]$; and

  * •

$g_{-t}q_{k+1}$ for $t\in[-t_{k},0]$.

if $t_{k}\geqslant 0$ or

  * •

$g_{-t}q_{k}$ for $t\in[0,-t_{k}]$;

  * •

$g_{t_{k}}\gamma(s)$ for $s\in[s_{k},s_{k+1}]$; and

  * •

$g_{t}q_{k+1}$ for $t\in[t_{k},0]$.

if $t_{k}\leqslant 0$. See Figure 4.24 for an illustration of this proof.

The union of the arcs $\delta_{k}$ and $\gamma_{k}=\gamma|[s_{k},s_{k+1}]$
bounds a disc in $\mathcal{C}^{\mathrm{root}}$ foliated by the arcs
$g_{t}\gamma_{k}$ where $t\in[0,t_{k}]$. In particular, $\delta_{k}$ is
homotopic to $\gamma_{k}$, relative to the endpoints.

Let $\delta$ be the concatenation of the paths $(\delta_{k})_{k}$. By
construction, $\delta$ is a closed curve, homotopic to $\gamma$. Moreover, the
pieces $g_{t_{k}}\gamma(s)$ for $s\in[s_{k},s_{k+1}]$ in the concatenation are
the only paths that are not (forward or backward) Teichmüller geodesic
segments. This concludes the proof of the theorem. ∎

######  Remark 4.25.

We do not attempt to make optimal choices to reduce the length of geodesic
pieces in the concatenation for $\delta$. A simple way to reduce these lengths
is to choose the normalised zippered rectangle construction whenever $q$
admits one. Thus, if $q\in\mathcal{C}^{\mathrm{root}}-\mathcal{W}$ is
contained in some polytope, then we can choose the box $U(q)$ to be contained
inside the same polytope and set $t(q)=0$ for such boxes. On the other hand,
when $q$ does not lie in any polytope we can choose the length of the base-arc
to be as close to $1$ as possible, so $t(q)$ is as small as possible. See
Section A.4 for a concrete example of the construction.

###  4.26. Rauzy–Veech induction and the Teichmüller flow

We will now define Rauzy–Veech induction on zippered rectangles. The induction
is defined by passing to the smaller base-arc with length
$|I|-\min\\{x_{\pi(\ell)},x_{\pi(\ell+m)}\\}$.

Let $\pi$ be an irreducible permutation in $\mathcal{R}$. Let $(x,y)$ be
singularity parameters for a zippered rectangle construction with underlying
permutation $\pi$. Let $\alpha=\pi(\ell)$ and $\beta=\pi(\ell+m)$. Since $\pi$
is irreducible, $\alpha\neq\beta$ and we will assume that $x_{\alpha}\neq
x_{\beta}$. Breaking symmetry, suppose that $x_{\alpha}>x_{\beta}$. In this
case, we say that the _top letter wins_. We set the new width parameters as

$x^{\prime}_{\alpha}=x_{\alpha}-x_{\beta},$

and $x^{\prime}_{\rho}=x_{\rho}$ for all $\rho\neq\alpha$. Similarly we set
the new height parameters as

$y^{\prime}_{\beta}=y_{\beta}+y_{\alpha},$

and $y^{\prime}_{\rho}=y_{\rho}$ for all $\rho\neq\beta$. The parameter
transformations can be encoded in terms of a matrix. Let
$E=(e_{rs})_{r,s\in\mathcal{A}}$ be the $\mathcal{A}\times\mathcal{A}$
elementary matrix with ones along the diagonal, $e_{\alpha\beta}=1$ and all
other entries zero. Then, $Ex^{\prime}=x$ and $E^{T}y=y^{\prime}$.

To define the new permutation we consider the two cases:

  1. (1)

$\alpha$ is a translation letter; or

  2. (2)

$\alpha$ is a flip letter.

Suppose $\alpha$ is a translation letter and let $\pi(j)=\alpha$ for some
$\ell+1\leqslant j<\ell+m$. We then set

  * •

$\pi^{\prime}(i)=\pi(i)$ for all $i\leqslant j$,

  * •

$\pi^{\prime}(j+1)=\beta$, and

  * •

$\pi^{\prime}(i)=\pi(i-1)$ for all $i>j+1$.

Suppose now that $\alpha$ is a flip letter and let $\pi(j)=\alpha$ for some
$1\leqslant j<\ell$. We set the top indices to range from $1$ to $\ell+1$ and
the bottom indices to range from $\ell+2$ to $\ell+m$ and then set

  * •

$\pi^{\prime}(i)=\pi(i)$ for all $i<j$,

  * •

$\pi^{\prime}(j)=\beta$,

  * •

$\pi^{\prime}(i)=\pi(i-1)$ for all $i>j$.

With the above definitions, We set
$R_{\mathrm{t}}(\pi,x,y)=(\pi^{\prime},x^{\prime},y^{\prime})$. If
$x_{\alpha}<x_{\beta}$ instead, then we say that the _bottom letter wins_. The
definition of $R_{\mathrm{b}}$ is analogous.

The codimension-one locus $x_{\alpha}=x_{\beta}$ is contained in
$\mathcal{V}$. The induction is undefined for it.

Rauzy–Veech induction makes the Rauzy class
$\mathcal{R}=\mathcal{R}(\mathcal{C}^{\mathrm{root}})$ into a directed graph.
The vertices of $\mathcal{D}$ are irreducible permutations in $\mathcal{R}$
with an arrow from permutation $\pi$ to $\pi^{\prime}$ if
$\pi^{\prime}=R_{\mathrm{t}}(\pi)$ or $R_{\mathrm{b}}(\pi)$. A component
$\mathcal{D}$ of this graph is called a _Rauzy diagram_.

We now explain the coding of the Teichmüller flow using normalised parameters
and Rauzy–Veech induction.

Let $\mathcal{C}_{\pi}=q_{\pi}(P_{\pi})$ be the polytope given by normalised
singularity parameters for an irreducible permutation $\pi$. Let
$\alpha=\pi(\ell)$ and $\beta=\pi(\ell+m)$.

We say that $q$ is a _forward-tied_ differential for $\pi$ if there exists a
sequence $q_{n}=q_{\pi}(\pi,x_{n},y_{n})$ converging to $q$ for which the
length of the base-arc $I$ tends to
$1+\min\\{x_{\pi(\ell)},x_{\pi(\ell+m)}\\}$ from below. Similarly, we say $q$
is _backward-tied_ for $\pi$ if it is the limit of a sequence where the length
of the base-arc tends to $1$ from above.

The period of the base-arc is linear in period coordinates. Therefore, the
backward-tied differentials are contained in a codimension-one locus in
$\mathcal{C}^{\mathrm{root}}$. It will become clear using Rauzy–Veech
induction that the forward-tied differentials are also contained in a
codimension-one locus in $\mathcal{C}^{\mathrm{root}}$.

Let $\mathcal{F}(\pi)$ and $\mathcal{B}(\pi)$ be the set of forward-tied and
backward-tied differentials for $\pi$. We call these the _flow faces_. Let $q$
be a forward-tied differential. Suppose that there is a sequence
$q_{n}=q_{\pi}(\pi,x_{n},y_{n})$ converging to $q$ for which the parameters
$x_{n}$ and $y_{n}$ are bounded away from zero and infinity respectively. Then
the sequence $(x_{n},y_{n})$ converges to some $(x,y)$ in the closure of
$P_{\pi}$ such that all its widths $x_{\alpha}>0$ and heights $y_{\alpha}$ are
bounded. It follows that $q$ is contained in the interior of
$\mathcal{F}(\pi)$ and the map $q_{\pi}$ extends from $P_{\pi}$ to such
parameters. We can similarly characterise differentials in the interior of
$\mathcal{B}(\pi)$ and extend $q_{\pi}$ to such parameters.

Let $q$ be a forward-tied differential in the interior of $\mathcal{F}(\pi)$
and further suppose that $x_{\alpha}>x_{\beta}$. The Rauzy–Veech induction on
$(\pi,x,y)$ is then defined and let
$(\pi^{\prime},x^{\prime},y^{\prime})=R_{\mathrm{t}}(\pi,x,y)$. Let $I$ be the
base-arc for $q=q_{\pi}(\pi,x,y)$ and $I^{\prime}$ the base-arc for
$q=q_{\pi^{\prime}}(\pi^{\prime},x^{\prime},y^{\prime})$. Note that

$|I^{\prime}|=|I|-x_{\beta}=1+\min\\{x_{\alpha},x_{\beta}\\}-x_{\beta}=1.$

This means that $q^{\prime}$ is a backward-tied differential for
$\pi^{\prime}$.

We conclude that the interiors of
$\mathcal{F}(\pi)\cap\\{x_{\alpha}>x_{\beta}\\}$ and
$\mathcal{F}(\pi)\cap\\{x_{\alpha}<x_{\beta}\\}$ are identified with
corresponding subsets of $\mathcal{B}(R_{\mathrm{t}}(\pi))$ and
$\mathcal{B}(R_{\mathrm{b}}(\pi))$, respectively.

Let $q\in\mathcal{C}^{\mathrm{root}}-\mathcal{V}$ and let $g_{t}q$ for
$t\geqslant 0$ be the Teichmüller geodesic ray through $q$. Since $q$ is
doubly non-vanishing, it is contained in some polytope $\mathcal{C}_{\pi}$ and
its parameters satisfy $x_{\pi(\ell)}\neq x_{\pi(\ell+m)}$. Let $I$ be the
base-arc in $q$. Then the length of the base-arc in $g_{t}q$ is $e^{t}|I|$.
Therefore, there is some time $t_{1}>0$ such that
$g_{t_{1}}q\in\mathcal{F}(\pi)$. By Rauzy induction, $g_{t_{1}}q$ is also
contained in $\mathcal{B}(\pi^{\prime})$ where $\pi=R_{\ast}(\pi)$ where
$\ast=\mathrm{t}$ or $\ast=\mathrm{b}$, depending on which of $x_{\pi(\ell)}$
and $x_{\pi(\ell+m)}$ is larger. Let
$\alpha^{\prime},\beta^{\prime}\in\mathcal{A}$ be the last top and bottom
letters of $\pi^{\prime}$, respectively. As $g_{t_{1}}q$ is also doubly non-
vanishing, the widths corresponding to $\alpha^{\prime}$ and $\beta^{\prime}$
do not coincide. So there is time $t_{2}>t_{1}$ such that $g_{t_{2}}q$ is
contained in $\mathcal{F}(\pi^{\prime})$. This description continues
iteratively.

It is possible that the Rauzy diagram $\mathcal{D}$ contains permutations
$\pi$ and $\pi^{\prime}$ that are equivalent under some re-indexing
$p\in\operatorname{Sym}(\mathcal{A})$. For such permutations,
$\mathcal{C}_{\pi}=\mathcal{C}_{\pi^{\prime}}$. Note then that the
permutations arising from $\pi$ and $\pi^{\prime}$ by Rauzy–Veech induction
are also pairwise equivalent under the same re-indexing $p$. It follows that
$p$ induces a symmetry of $\mathcal{D}$ as a directed graph. We call the
quotient of $\mathcal{D}$ by all such symmetries the _reduced Rauzy diagram_
and denote it by $\mathcal{D}^{\mathrm{red}}$. It is then clear that we should
use the quotient graph $\mathcal{D}^{\mathrm{red}}$ to code Teichmüller flow
on $\mathcal{C}^{\mathrm{root}}$.

The above coding of Teichmüller flow has an immediate combinatorial
consequence. By the work of Masur and Veech [Mas82, Vee82], the Teichmüller
flow on $\mathcal{C}^{\mathrm{root}}$ is ergodic for the Masur–Veech measure.
By ergodicity, a positive measure set in $\mathcal{C}_{\pi}-\mathcal{V}$,
visits every $\mathcal{C}_{\sigma}$ under Teichmüller flow. This implies that
the reduced Rauzy diagram $\mathcal{D}^{\mathrm{red}}$ is a strongly connected
graph, that is, there is a directed path between any pair of vertices. It then
follows that each component of $\mathcal{D}$ is also strongly connected. See
the article by Boissy–Lanneau for more details [BL09].

The following three lemma are standard facts in the theory of Rauzy–Veech
sequences and are often implicitly used. We include proof sketches for
completeness.

######  Lemma 4.27.

For any doubly non-vanishing rooted differential $(q,v)$ and any $T>0$ the
geodesic segment $[q,g_{T}q]$ crosses finitely many flow faces.

###### Proof.

By applying Teichmüller flow, we may assume that $(q,v)$ is contained in some
$\mathcal{B}_{0}=\mathcal{B}(\pi)$. Let
$\mathcal{B}_{1},\mathcal{B}_{2},\dots$ be the sequence of backward faces the
geodesic segment $[q,g_{T}q]$ crosses. Let $0<t_{k}\leqslant T$ be the
monotonically increasing sequence of times such that $g_{t_{k}}q$ is contained
in $\mathcal{B}_{k}$.

Recall that $Z(q,v)$ is the set of points $x$ in $I_{v}$ such that $I(x)$ is a
base-arc. Identifying $I_{v}$ with the positive real axis, it follows from the
definitions that each point $e^{-t_{k}}$ is contained in $Z(q,v)$.

By Lemma 4.20, the intersection $Z(q,v)\cap[e^{-T},1)$ is finite. Hence, the
sequence $t_{k}$ is finite and we are done. ∎

######  Remark 4.28.

It follows from the previous lemma that the curve $\delta$ we construct in the
proof of Theorem 4.23 crosses finitely many flow faces.

Let $\zeta$ be a finite Rauzy–Veech sequence that starts at $\pi$ and ends at
$\pi^{\prime}$. Let $P_{\zeta}\subseteq P_{\pi}$ be the parameters of
differentials in $\mathcal{C}_{\pi}$ whose Rauzy–Veech sequence begins with
$\zeta$. By inductively using the definition of Rauzy–Veech induction, it
follows that the set $P_{\zeta}$ is a convex open subset of $P_{\pi}$. In
particular, $\mathcal{C}_{\zeta}=q_{\pi}(P_{\pi})$ is path connected.

We say that a Teichmüller segment $[q,g_{t}q]$ is a $\zeta$-segment if
$q\in\mathcal{C}_{\zeta},g_{t}q\in\mathcal{C}_{\pi^{\prime}}$ and the
Rauzy–Veech sequence of $[q,g_{t}q]$ is $\zeta$.

######  Lemma 4.29.

Let $\zeta$ be a finite Rauzy–Veech sequence that starts at $\pi$ and ends at
$\pi^{\prime}$. Then any pair of $\zeta$-segments are isotopic in
$\mathcal{C}^{\mathrm{root}}$ through $\zeta$-segments.

###### Proof.

As $[q,g_{t}q]$ is a $\zeta$-segment, it follows that there exists an open set
$U$ in $q_{\pi}(P_{\zeta})$ centred at $q$ such that $g_{t}U$ is contained in
$\mathcal{C}_{\pi^{\prime}}$. Any $\zeta$-segment with an endpoint in $U$ is
thus homotopic to $[q,g_{t}q]$. Indeed, for any $q^{\prime}$ in $U$, we can
connect $q$ to $q^{\prime}$ by an arc contained in $U$. We flow the arc for
time $t$ to get an arc in $\mathcal{C}_{\pi^{\prime}}$. We thus have a
homotopy between $[q,g_{t}q]$ and $[q^{\prime},g_{t}q^{\prime}]$. We then do a
further homotopy from $[q^{\prime},g_{t}q^{\prime}]$ and
$[q^{\prime},g_{t^{\prime}}q^{\prime}]$. The lemma then follows from the path
connectedness of $q_{\pi}(P_{\zeta})$. ∎

######  Lemma 4.30.

Let $U$ be an open set contained in some polytope $\mathcal{C}_{\pi}$. Then
there exists a Rauzy–Veech sequence $\theta$ (that depends on $U$) starting
from $\pi$ such that, for every $q\in\mathcal{C}_{\theta}$, the Teichmüller
segment in $\mathcal{C}_{\pi}$ containing $q$ intersects $U$.

###### Proof.

Let $\Delta$ be the standard simplex in $\mathbb{R}^{\mathcal{A}}$. Let
$p:P_{\pi}\to\Delta$ be the projection $(x,y)\to x/\|x\|_{1}$.

By iteration, the transformation on parameters induced by a Rauzy–Veech
sequence $\zeta$ is encoded by a non-negative matrix $B_{\zeta}$, that is, if
$(x,y)$ is in $P_{\zeta}$ then the new parameters $x^{(\zeta)}$ are related to
$x$ by $B_{\zeta}x^{(\zeta)}=x$. Suppose $\zeta$ ends at $\pi^{\prime}$. It
also follows that $p(B_{\zeta}P_{\pi^{\prime}})=p(P_{\zeta})$.

It then suffices to show that there is a Rauzy–Veech sequence $\theta$ such
that $p(B_{\theta}P_{\pi^{\prime}})$ is contained in $p(q_{\pi}^{-1}U)$, where
$\pi^{\prime}$ is the permutation that $\theta$ ends at. This is a standard
fact but we will include a brief justification for completeness.

By Masur’s [Mas82] and Veech’s [Vee82] solution of the Keane conjecture or
even more strongly by Kerckhoff–Masur–Smillie [KMS86], we can find $q\in U$
with a uniquely ergodic vertical foliation or more strongly giving a recurrent
Teichmüller ray. Let $\zeta_{n}$ be the Rauzy–Veech sequence of length $n$ for
$q$ and $B_{n}$ the corresponding matrix. Also, let $\pi_{n}$ denote the
permutation at its end. Since the vertical foliation is uniquely ergodic, the
nested sequence $p(B_{n}P(\pi_{n}))$ converges to $p(q_{\pi}^{-1}(q))$ as
$n\to\infty$. Since all such sets are polytopes inside the standard simplex
$\Delta$, there is some $n$ large enough such that $p(B_{n}P_{\pi_{n}})$ is
contained inside $p(q_{\pi}^{-1}U)$. We make take $\theta$ to be $\zeta_{n}$
to conclude the proof.

∎

##  5\. The flow group is the fundamental group

Let $\pi_{1}(\mathcal{D}^{\mathrm{red}},\pi)$ be the fundamental group based
at $\pi$ of $\mathcal{D}^{\mathrm{red}}$ as an undirected graph. Let $q_{0}$
be a point in $\mathcal{C}_{\pi}$.

To simplify notation, we will fix a component of $\mathcal{D}$ evenly covering
$\mathcal{D}^{\mathrm{red}}$. We will continue to refer to a vertex in
$\mathcal{D}^{\mathrm{red}}$ as an irreducible permutation when in fact it is
an equivalence class of vertices related by letter re-indexing. Note that
every directed path in $\mathcal{D}^{\mathrm{red}}$ is realised by an actual
Rauzy–Veech sequence in $\mathcal{D}$. Thus, we will use the actual
Rauzy–Veech sequences in $\mathcal{D}$ to concatenate sensibly.

######  Proposition 5.1.

There is a natural homomorphism

$\pi_{1}(\mathcal{D}^{\mathrm{red}},\pi)\to\pi_{1}(\mathcal{C}^{\mathrm{root}},q_{0}).$

###### Proof.

Let $\sigma$ be any irreducible permutation in $\mathcal{D}^{\mathrm{red}}$.
As $\mathcal{D}^{\mathrm{red}}$ is strongly connected, we can choose a
directed path $\xi(\sigma)$ from $\pi$ to $\sigma$ and a directed path
$\xi^{\prime}(\sigma)$ from $\sigma$ to $\pi$. We choose empty paths for
$\xi(\pi)$ and $\xi^{\prime}(\pi)$.

Every loop $\kappa$ in $\mathcal{D}^{\mathrm{red}}$ based at $\pi$ can be
written as a concatenation of alternating forward and backward paths. Breaking
symmetry, suppose the odd indexed paths are forward paths and the even indexed
paths are backward paths. We may then write the concatenation as
$\kappa_{1}\kappa_{2}^{-1}\kappa_{3}\kappa_{4}^{-1}\dotsb{}$.

Let $\sigma_{i}$ and $\tau_{i}$ be the beginning and ending permutations for
$\kappa_{i}$. Then we have the string of relations
$\pi=\sigma_{1},\tau_{1}=\sigma_{2},\tau_{2}=\sigma_{3}$, etc. The
concatenation for $\kappa$ is then equal to the concatenation
$\lambda_{1}\lambda_{2}^{-1}\lambda_{3}\lambda_{4}^{-1}\dotsb{}$, where each
$\lambda_{i}$ is a loop based at $\pi$ given by
$\lambda_{i}=\xi(\sigma_{i})\,\kappa_{i}\,\xi^{\prime}(\tau_{i})$.

It thus suffices to associate a loop in $\mathcal{C}^{\mathrm{root}}$ based at
$q_{0}$ with a directed loop $\kappa$ in $\mathcal{D}^{\mathrm{red}}$ based at
$\pi$. Observe that an actual Rauzy–Veech sequence in $\mathcal{D}$
representing $\kappa$ might end at a permutation $\pi^{\prime}$ equivalent to
$\pi$. However, this will not matter as
$\mathcal{C}_{\pi^{\prime}}=\mathcal{C}_{\pi}$ in that case. Let $q$ be a
point in $\mathcal{C}_{\pi}$. As $\mathcal{C}_{\pi}$ is a polytope and hence
contractible, any pair of paths in $\mathcal{C}_{\pi}$ from $q_{0}$ to $q$ are
homotopic relative to the endpoints. We fix one such path and call it
$\eta_{q}$. By $\eta_{q}^{-1}$, we mean the reverse path from $q$ to $q_{0}$.
We now choose any $\kappa$-segment $\gamma_{\kappa}$. By Lemma 4.29, any two
choices of $\gamma_{\kappa}$ are homotopic. Let $q$ and $q^{\prime}$ in
$\mathcal{C}_{\pi}$ be the beginning and end points of $\gamma_{\kappa}$. We
then map $\kappa$ to the based loop
$\eta_{q}\,\gamma_{\kappa}\,\eta_{q^{\prime}}^{-1}$ in
$\mathcal{C}^{\mathrm{root}}$.

It remains to show that this map is a homomorphism. Let $\kappa^{\prime}$ and
$\kappa^{\prime\prime}$ be two directed loops based at $\pi$. Let
$\kappa=\kappa^{\prime}\kappa^{\prime\prime}$ and let $\gamma_{\kappa}$ be a
$\kappa$-segment. Let $q$ and $q^{\prime\prime}$ in $\mathcal{C}_{\pi}$ be the
beginning and the end points of $\gamma_{\kappa}$. We deduce that there exists
a point $q^{\prime}$ in $\mathcal{C}_{\pi}$ on $\gamma_{\kappa}$ such that if
we write $\gamma_{\kappa}=[q,q^{\prime}]\cup[q^{\prime},q^{\prime\prime}]$
then $\gamma_{\kappa^{\prime}}=[q,q^{\prime}]$ is a $\kappa^{\prime}$-segment
and $\gamma_{\kappa^{\prime\prime}}=[q^{\prime},q^{\prime\prime}]$ is a
$\kappa^{\prime\prime}$-segment. Then
$\eta_{q}\,\gamma_{\kappa}\,\eta_{q^{\prime\prime}}^{-1}$ is homotopic to
$(\eta_{q}\,\gamma_{\kappa^{\prime}}\,\eta_{q^{\prime}}^{-1})(\eta_{q^{\prime}}\gamma_{\kappa^{\prime\prime}}\eta_{q^{\prime\prime}}^{-1})$
and so we conclude that the map is a homomorphism.

∎

As a consequence of Theorem 4.23, we prove

######  Theorem 5.2.

Let $\mathcal{C}^{\mathrm{root}}$ be a component of a stratum of rooted
abelian or quadratic differentials. Let $\pi$ be a permutation in
$\mathcal{D}^{\mathrm{red}}$. Let $q_{0}$ be a base-point in
$\mathcal{C}^{\mathrm{root}}$ contained in the polytope $\mathcal{C}_{\pi}$.
Then the natural homomorphism

$\pi_{1}(\mathcal{D}^{\mathrm{red}},\pi)\to\pi_{1}(\mathcal{C}^{\mathrm{root}},q_{0})$

is surjective.

###### Proof.

By Theorem 4.23, a loop $\gamma$ based at $q_{0}$ is homotopic to a finite
concatenation of paths $\gamma_{i}$ where each $\gamma_{i}$ is either a
(forward or backward) Teichmüller geodesic segment or is contained inside a
polytope. Breaking symmetry, we may assume $\gamma$ is the concatenation
$\gamma_{1}\gamma_{2}\dotsb\gamma_{k}$ where the odd indexed $\gamma_{i}$ are
contained in a polytope and the even indexed $\gamma_{i}$ are (forward or
backward) Teichmüller segments. By Lemma 4.27, the Teichmüller segments
$\gamma_{2i}$ give us finite Rauzy–Veech sequences $\zeta_{2i}$ such that

  * •

$\zeta_{2}$ starts at $\pi$; and

  * •

successive $\zeta_{2i}$ can be concatenated as undirected paths in
$\mathcal{D}$.

The concatenation $\zeta_{2}\zeta_{4}\dotsb{}$ descends to a loop $\kappa$ in
$\mathcal{D}^{\mathrm{red}}$ based at $\pi$. As before, the actual sequence in
$\mathcal{D}$ might end at a permutation $\pi^{\prime}$ equivalent to $\pi$
but that will not matter as $\mathcal{C}_{\pi^{\prime}}=\mathcal{C}_{\pi}$. As
in the proof of Proposition 5.1, the loop $\kappa$ can be written as a
concatenation of (forward and backward) loops based at $\pi$. The surjectivity
follows. ∎

######  Remark 5.3.

As the components of $\mathcal{D}$ evenly cover $\mathcal{D}^{\mathrm{red}}$,
it follows from Theorem 5.2 that there is a finite cover of
$\mathcal{C}^{\mathrm{root}}$ that corresponds to components of $\mathcal{D}$.
We denote this cover by $\mathcal{C}^{\mathrm{lab}}$. This cover is easy to
describe intrinsically in the case of an abelian stratum component but its
description for a quadratic stratum component is an interesting question.

Let $q_{0}$ be a base-point in $\mathcal{C}^{\mathrm{root}}$ and $U$ be a
contractible open set around $q_{0}$. For every $q\in U$, we choose a path
$\eta_{q}$ from $q_{0}$ to $q$ inside $U$. As $U$ is contractible, any choice
of $\eta_{q}$ is homotopic to any other choice relative to their end-points.
We take the convention that $\eta_{q}^{-1}$ is $\eta_{q}$ in reverse
connecting $q$ to $q_{0}$.

Let $\gamma$ be a Teichmüller segment that begins at some $q\in U$ and ends at
some $q^{\prime}\in U$. The concatenation
$\eta_{q}\gamma\eta_{q^{\prime}}^{-1}$ is a loop in
$\mathcal{C}^{\mathrm{root}}$ based at $q_{0}$. We call such loops _almost-
flow loops_ (based at $q_{0})$.

######  Definition 5.4.

The flow group $G(U,q_{0})$ is the subgroup of
$\pi_{1}(\mathcal{C}^{\mathrm{root}},q_{0})$ generated by the almost-flow
loops based at $q_{0}$.

We first prove the following theorem.

######  Theorem 5.5.

Let $\mathcal{C}^{\mathrm{root}}$ be a component of a stratum of rooted
abelian or quadratic differentials. Let $q_{0}$ be a base-point contained in
some polytope $\mathcal{C}_{\pi}$. Then

$G(\mathcal{C}_{\pi},q_{0})=\pi_{1}(\mathcal{C}^{\mathrm{root}},q_{0}).$

###### Proof.

In the proof of Theorem 5.2, we showed that images of directed loops in
$\pi_{1}(\mathcal{D}^{\mathrm{red}},\pi)$ generate
$\pi_{1}(\mathcal{C}^{\mathrm{root}},q_{0})$. So it suffices to show that
every directed loop $\zeta$ in $\mathcal{D}^{\mathrm{red}}$ based at $\pi$ is
realised by an almost-flow loop in $G(\mathcal{C}_{\pi},q_{0})$. We choose any
$q$ in $\mathcal{C}_{\zeta}$ and let $[q,g_{t}q]$ be a $\zeta$-segment that it
gives. By definition, $q^{\prime}=g_{t}q$ is also contained in
$\mathcal{C}_{\pi}$. Then the based loop
$\eta_{q}\gamma\eta_{q^{\prime}}^{-1}$ realises the loop $\zeta$. This
concludes the proof of the theorem. ∎

As a corollary, we obtain one of our main results.

######  Theorem 5.6.

For any base-point $q_{0}$ in $\mathcal{C}^{\mathrm{root}}$ and any
contractible open set $U$ containing $q_{0}$

$G(U,q_{0})=\pi_{1}(\mathcal{C}^{\mathrm{root}},q_{0}).$

###### Proof.

Let $q$ be a point in $U$. As $U$ is contractible, it follows that
$G(U,q)\cong G(U,q_{0})$. Since the union of polytopes is dense in
$\mathcal{C}^{\mathrm{root}}$, we may then assume that $q_{0}$ is contained in
some polytope $\mathcal{C}_{\pi}$. Suppose $V\subseteq U$ is a smaller
contractible open set that contains $q_{0}$. By definition of the flow groups,
$G(V,q_{0})$ is a subgroup of $G(U,q_{0})$. So we may assume that $U$ is also
contained in $\mathcal{C}_{\pi}$. It now suffices to show that any directed
loop $\zeta$ in $\mathcal{D}^{\mathrm{red}}$ based at $\pi$ can be written as
a word in almost-flow loops in $G(U,q_{0})$.

By Lemma 4.30, there is a Rauzy–Veech sequence $\theta$ starting from $\pi$
such that for any $q\in\mathcal{C}_{\theta}$ the Teichmüller segment in
$\mathcal{C}_{\pi}$ containing $q$ intersects $U$. Note then that the same is
true for any finite extension $\theta\zeta$.

As $\mathcal{D}$ is strongly connected, we may extend $\theta$ to assume that
it also ends at $\pi$. If $\pi^{\prime}$ is equivalent to $\pi$, we also get a
loop $\theta^{\prime}$ based at $\pi^{\prime}$ so that $\theta$ and
$\theta^{\prime}$ descend to the same loop in $\mathcal{D}^{\mathrm{red}}$.
Let $q$ be a differential in $\mathcal{C}_{\theta\theta}$. As the Teichmüller
segment in $\mathcal{C}_{\pi}$ containing $q$ intersects $U$, we may assume
$q$ is in $U$.

By definition, there exists a time $t>0$ such that the Teichmüller segment
$[q,g_{t}q]$ is a $\theta\theta$-segment. We may decompose $[q,g_{t}q]$ as
$[q,g_{s}q]\cup[g_{s}q,g_{t}q]$ such that both segments $[q,g_{s}q]$ and
$[g_{s}q,g_{t}q]$ are $\theta$-segments. In particular, $q^{\prime}=g_{s}q$ is
also contained in $\mathcal{C}_{\theta}$. As the Teichmüller segment in
$\mathcal{C}_{\pi}$ containing $q^{\prime}$ intersects $U$, by tweaking $s$ we
may assume that $q^{\prime}$ is also contained in $U$. Let
$\gamma_{u}=[q,q^{\prime}]$. Then $\eta_{q}\gamma_{u}\eta_{q^{\prime}}^{-1}$
is an almost-flow loop in $G(U,q_{0})$ that realises $\theta$.

Now, let $\zeta$ be an oriented loop in $\mathcal{D}^{\mathrm{red}}$ based at
$\pi$. We consider the oriented loop $\xi=\theta\zeta\theta$. As a sequence in
$\mathcal{D}$, the loop $\zeta$ could terminate at some $\pi^{\prime}$
equivalent to $\pi$. In this case we mean the concatenation
$\theta\zeta\theta^{\prime}$, where $\theta^{\prime}$ is a loop in
$\mathcal{D}$ based at $\pi^{\prime}$ that descends to the same loop in
$\mathcal{D}^{\mathrm{red}}$ as $\theta$. We will assume first that $\zeta$ is
a loop based at $\pi$ in $\mathcal{D}$. The argument below extends to the case
where $\zeta$ ends instead at $\pi^{\prime}$ by concatenating the appropriate
arcs.

Let $\gamma_{\xi}=[q,g_{t}q]$ be a $\xi$-segment. As the Teichmüller segment
in $\mathcal{C}_{\xi}$ containing $q$ intersects $U$, we may choose $q$ to be
in $U$. We then write $\gamma_{\xi}$ as a concatenation
$[q,g_{s}q]\cup[g_{s}q,g_{t}q]$ where $[q,g_{s}q]$ is a $\theta\zeta$-segment
and $[g_{s}q,g_{t}q]$ is a $\theta$-segment. Let $q^{\prime}=g_{s}q$ and
$q^{\prime\prime}=g_{t}q$. As $[q^{\prime},q^{\prime\prime}]$ is a
$\theta$-segment, $q^{\prime}$ is contained in $\mathcal{C}_{\theta}$. The
Teichmüller segment inside $\mathcal{C}_{\pi}$ containing $q^{\prime}$
intersects $U$. So by changing $s$, we may assume that $q^{\prime}$ is also
contained in $U$.

The Teichmüller segment $\gamma^{\prime}=[q,q^{\prime}]$ then begins and ends
in $U$. So the directed loop $\theta\zeta$ is realised by the almost-flow loop
$\eta_{q}\gamma^{\prime}\eta_{q^{\prime}}^{-1}$ in $G(U,q_{0})$.

As we already established that $\theta$ is realised by an element in
$G(U,q_{0})$, we deduce that $\zeta$ must also be realised by an element in
$G(U,q_{0})$. This concludes the proof of the corollary.

∎

##  6\. Dynamics of the Teichmüller flow

###  6.1. Coding formalism

Let $\Pi$ be a finite or countable set. We consider the symbolic space
$\Sigma=\Pi^{\mathbb{Z}}$ endowed with the left shift map S. Suppose
$w\in\Pi^{m}$ is a finite word. The (forward) cylinder $\Sigma(u)$ induced by
$u$ is defined as $\Sigma(u)=\\{a\in\Sigma\text{ such that }a_{k}=u_{k}\text{
for }k=0,\dotsc,m-1\\}$. Given another word $v\in\Pi^{n}$, we write
$uv\in\Pi^{m+n}$ for the concatenation of $u$ and $v$.

######  Definition 6.2.

We say that an S-invariant probability measure $\mu$ has _bounded distortion_
if there exists a constant $K>0$ such that, for any finite words $u\in\Pi^{m}$
and $v\in\Pi^{n}$,

$\frac{1}{K}\mu(\Sigma(u))\mu(\Sigma(v))\leqslant\mu(\Sigma(uv))\leqslant
K\mu(\Sigma(u))\mu(\Sigma(v)).$

The bounded distortion property allows us to treat the symbolic space “almost”
as a Bernoulli shift. For this reason, it is also called an _approximate
product structure_. Since $\mu$ is shift-invariant, the previous definition
would not change if we used backward or centred cylinders instead of forward
cylinders.

We will now describe how the Teichmüller flow can be coded by such a symbolic
setup. Let $\pi$ be an irreducible generalised permutation in
$\mathcal{D}^{\mathrm{red}}$. For a backward-tied differential $q$ in
$\mathcal{B}(\pi)$, let $(q,g_{t}q)$, for some $t>0$, be the longest
Teichmüller segment entirely contained in $\mathcal{C}_{\pi}$. It follows that
$g_{t}q$ is contained in $\mathcal{F}(\pi)$. Let $\zeta$ be a finite
Rauzy–Veech sequence starting at $\pi$. Let $\mathcal{S}_{\zeta}$ be the
differentials $q$ in $\mathcal{B}(\pi)$ for which the Teichmüller segment
above is contained in $\mathcal{C}_{\zeta}$.

Recall that $\Delta$ is the standard simplex in $\mathbb{R}^{\mathcal{A}}$ and
$p:P_{\pi}\mapsto\Delta$ is the projection $(x,y)\to x/\|x\|_{1}$. Let
$B_{\theta}$ denote the matrix of a Rauzy–Veech sequence $\theta$. As
indicated in the proof of Lemma 4.30, it is possible to find a sequence
$\theta$ from $\pi$ to $\pi$ such that $p(B_{\theta}P_{\pi})$ is compactly
contained in $p(P_{\pi})$. Let $\overline{p}:(x,y)\mapsto y$ be the map that
records the height parameters. By extending $\theta$ to a longer loop based at
$\pi$, we may also assume that $\overline{p}(P_{\pi})$ is compactly contained
in $\overline{p}(B_{\theta}P_{\pi})$. We fix this $\theta$ once and for all
and consider $\mathcal{S}_{\theta}$. By finessing $\theta$ further, we may
assume that it is _neat_ , that is, if $\theta=\zeta\eta$ and
$\theta=\eta^{\prime}\zeta$ then $\zeta=\theta$. In the coding that we
consider, the set $\mathcal{S}_{\theta}$ will serve as a transverse section to
the Teichmüller flow.

We recall the bounded distortion theorem for Rauzy–Veech induction. This
theorem states that that there exists a constant $K\geqslant 1$, that depends
only on the topology of the surface, and a countable collection of finite
Rauzy–Veech sequences $\zeta$ from $\pi$ to $\pi$ such that

  * •

for the map $p(q_{\pi}^{-1}(\mathcal{B}(\pi)))\to
p(q_{\pi}^{-1}(\mathcal{B}(\pi)))$ given by $x\mapsto
B_{\zeta}\,x/\|B_{\zeta}\,x\|_{1}$, its Jacobian $\mathcal{J}$ satisfies

$\frac{1}{K}\mathcal{J}(x_{1})<\mathcal{J}(x_{2})<K\mathcal{J}(x_{1})$

for any pair of points $(x_{1},y_{1}),(x_{2},y_{2})\in
q_{\pi}^{-1}(\mathcal{B}(\pi))$;

  * •

no $\zeta$ contains $\theta$, that is, $\zeta$ cannot be written as a
concatenation $\eta\theta\eta^{\prime}$; and

  * •

up to excising a set of differentials of measure zero,

$\bigcup_{\zeta}\mathcal{B}(\zeta)=\mathcal{B}(\pi).$

For more details, we refer the reader to the article by Avila–Gouëzel–Yoccoz
[AGY06, Section 4] for abelian differentials, and by Avila–Resende [AR12,
Sections 4 and 5] for quadratic differentials.

We stress that the Rauzy–Veech sequences considered here are sequences in
$\mathcal{D}^{\text{red}}$. To be precise with constraints that involve
concatenations, they should be imposed over all lifts in $\mathcal{D}$. For
instance, fixing a lift of $\pi$ in $\mathcal{D}$, the second constraint
should be read as finding a loop $\zeta$ in $\mathcal{D}$ that returns to
$\pi$ and does not contain a lift of $\theta$. Alternatively, the coding we
describe is a coding of the flow lifted to $\mathcal{C}^{\mathrm{lab}}$ that
is “equivariant” with respect to the Deck group for the covering
$\mathcal{C}^{\mathrm{lab}}\to\mathcal{C}^{\mathrm{root}}$.

With this setup, the first return maps under Teichmüller flow to
$\mathcal{S}_{\theta}$ are given by Rauzy–Veech sequences of the form
$\theta\zeta\theta$. Note then that, by tweaking the constant $K$, the
Jacobian of the map $p(q_{\pi}^{-1}(\mathcal{S}_{\theta}))\mapsto
p(q_{\pi}^{-1}(\mathcal{S}_{\theta}))$ given by $x\mapsto
B_{\theta\zeta\theta}\,x/\|B_{\theta\zeta\theta}\,x\|_{1}$ satisfies

$\frac{1}{K}\mathcal{J}(x_{1})<\mathcal{J}(x_{2})<K\mathcal{J}(x_{1})$

for each $\zeta$ and for any pair of points $(x_{1},y_{1}),(x_{2},y_{2})\in
q_{\pi}^{-1}(\mathcal{S}_{\theta})$. In fact, the Jacobian property is
enforced by the stronger property of the matrix that, up to a constant, that
depends only on the topology of the surface, all columns have the same norm.
More precisely,
$\|B_{\theta\gamma\theta}\,x\|_{1}\asymp\|B_{\theta\gamma\theta}\,x^{\prime}\|_{1}$
for any $(x,y),(x^{\prime},y^{\prime})\in q_{\pi}^{-1}(\mathcal{S}_{\theta})$.

Moreover, we have that, up to excising a set of zero measure,

(6.3) $\mathcal{S}_{\theta}=\bigcup_{\zeta}\mathcal{S}_{\theta\zeta\theta}.$

We thus have a countable full measure partition of $\mathcal{S}_{\theta}$ into
sets $\mathcal{S}_{\theta\zeta\theta}$ such that each
$\mathcal{S}_{\theta\zeta\theta}$ is the image of smooth map
$\phi_{\zeta}\colon\mathcal{S}_{\theta}\mapsto\mathcal{S}_{\theta}$. Each map
$\phi_{\zeta}$ is diffeomorphic onto its image and its inverse is a _uniformly
expanding Markov map_ in the sense of Avila–Gouëzel–Yoccoz [AGY06, Definition
2.2]. We assemble the inverses of $\phi_{\zeta}$ into a map $\Phi$ on
$\mathcal{S}_{\theta}$.

For such an expanding map $\Phi$, there exists a unique $\Phi$-invariant
absolutely continuous probability measure $\nu$, which is automatically
ergodic and even mixing Aar[Section 2]Avi-Gou-Yoc. In the case of the
Teichmüller flow, the measure $\nu$ is the restriction of the Masur–Veech
measure on $\mathcal{C}^{\mathrm{root}}$.

We now set $\Pi$ as the countable set of sequences $\zeta$ above. The
partition given by (6.3) induces an equivariant bijection between
$(\Sigma,\text{{S}})$ and
$(\bigcup_{\zeta\in\Pi}\mathcal{S}_{\theta\zeta\theta},\Phi)$. The measure
$\mu$ that we will consider on $\Sigma$ is the unique S-invariant probability
measure rendering this bijection a measure-theoretic conjugation. The bounded
distortion inherited by $\nu$ from the Jacobians becomes equivalent to the
bounded distortion of $\mu$ given by Definition 6.2.

###  6.4. Return times

A function $\xi:\Sigma\mapsto\mathbb{R}$ is _Hölder_ if there exists a non-
negative exponent $\alpha<1$ and a constant $C>0$ such that for any finite
sequence $u$ and any $a,b\in\Sigma(u)$

$|\xi(a)-\xi(b)|\leqslant C\alpha^{m},$

where $m$ is the length of $u$.

A _roof function_ is a Hölder function
$\xi\colon\Sigma\mapsto\mathbb{R}_{>0}$. A suspension of the symbolic space is
a space that is homeomorphic to $\Sigma\times[0,1]$ with the identification
$(a,0)\sim(\text{{S}}(a),1)$. The roof function equips the suspension with a
flow $\psi$ which flows in the interval direction and satisfies
$\psi_{\xi(a)}(a,0)=(\text{{S}}(a),1)$.

######  Definition 6.5.

A roof function is said to have _exponential tails_ if there exists $h>0$ such
that

$\int_{\Sigma}e^{h\xi}d\mu<\infty.$

The property of having exponential tails implies, in particular, that the
volume of the suspension with respect to the local product measure $d\mu\,dt$
is finite. We will discuss this integrability in more detail when we talk of
cocycles.

Note that under the measure-theoretic conjugacy, the section
$\mathcal{S}_{\theta}$ gets identified with $\Sigma\times\\{0\\}$. The
function $\xi$ we are interested in is the return time function to
$\mathcal{S}_{\theta}$ under Teichmüller flow. It is easy to check that the
return time on $\mathcal{S}_{\theta\zeta\theta}$ is given by
$\xi(x)=\log\|B_{\theta\zeta\theta}\,x\|_{1}$. The measure $d\nu\,dt$ is the
Masur–Veech measure on $\mathcal{C}^{\mathrm{root}}$. The coding of the
Teichmüller flow can be summarised as follows.

######  Theorem 6.6.

There exists a countable set $\Pi$ whose full shift $(\Sigma,\text{{S}})$,
$\Sigma=\Pi^{\mathbb{Z}}$, carries an S-invariant probability measure with
bounded distortion and a roof function $\xi$ with exponential tails such that
there exists a measure-theoretic conjugacy
$f\colon\Sigma\times[0,1]\to\mathcal{C}^{\mathrm{root}}$ (where
$\mathcal{C}^{\mathrm{root}}$ is equipped with the Masur–Veech measure) that
satisfies $f\circ\psi_{t}=g_{t}\circ f$ for all $t\in\mathbb{R}$.

###  6.7. Cocycles

A linear cocycle with values in $\operatorname{SL}(m,\mathbb{R})$ for the
Teichmüller flow is a map
$C\colon\mathcal{C}\times\mathbb{R}\mapsto\operatorname{SL}(m,\mathbb{R})$
satisfying

  * •

$C(q,0)=I$ where $I$ is the identity matrix, and

  * •

$C(q,s+t)=C(g_{s}q,t)\,C(q,s)$ for all $q\in\mathcal{C}$ and
$s,t\in\mathbb{R}$.

The most well known example is the Kontsevich–Zorich cocycle which records the
dynamics of the flow on the integral first homology group of the surface.
Choosing charts on $\mathcal{C}$, one can choose a basis for a trivialisation
of the surface homology in these charts. As the Teichmüller flow has Poincaré
recurrence, one can consider the change of basis matrices as a flow trajectory
returns to a chart. This is the Kontsevich–Zorich cocycle. As the flow
preserves the intersection form on the homology, the cocycle takes values in
the symplectic group over $\mathbb{Z}$.

Another example is the Rauzy–Veech cocycle defined on
$\mathcal{C}^{\mathrm{lab}}$, which is the finite cover of
$\mathcal{C}^{\mathrm{root}}$ corresponding to the cover
$\mathcal{D}\to\mathcal{D}^{\text{red}}$ defined in Remark 5.3. Here, the
polytopes $\mathcal{C}_{\pi}$ carry preferred coordinates through the
normalised width and height parameters. The itinerary through polytopes of a
typical flow trajectory is recorded by the Rauzy–Veech sequence. The
coordinate transformation of a Rauzy–Veech sequence is linear. This defines
the Rauzy–Veech cocycle. For an abelian stratum component, one can associate
to an irreducible permutation a natural spanning set for the absolute homology
of the surface. With respect to these spanning sets the Kontsevich–Zorich
cocycle lifted to $\mathcal{C}^{\mathrm{lab}}$ is the same as the Rauzy–Veech
cocycle. Thus, the two can be studied simultaneously. This structure is not
available for stratum components of quadratic differentials.

A linear cocycle for the Teichmüller flow is said to be _integrable_ with
respect to a finite flow invariant measure if for all $t$ the functions
$q\mapsto\log\|C(q,t)\|$ and $q\mapsto\log\|C(q,t)^{-1}\|$ are $L^{1}$ with
respect to the measure. As our cocycles are integer valued, the condition
$q\mapsto\log\|C(q,t)\|$ being $L^{1}$ suffices. The flow invariant measure we
are interested in is the Masur–Veech measure on $\mathcal{C}$ and its lifts to
$\mathcal{C}^{\mathrm{root}}$ and $\mathcal{C}^{\mathrm{lab}}$. The lift to
$\mathcal{C}^{\mathrm{root}}$ is exactly the measure $d\nu\,dt$.

The Teichmüller flow is ergodic with respect to the Masur–Veech measure. Thus,
if a cocycle is integrable, then Oseledets theorem applies: for almost every
$q$ and every non-zero vector $v\in\mathbb{R}^{m}$ the limit

$\lim_{t\to\infty}\frac{1}{t}\log\frac{\|C(q,t)v\|_{1}}{\|v\|_{1}}$

exists and depends only on $v$ and not on $q$. Moreover, the limit can achieve
up to $m$ values:
$\lambda_{1}\geqslant\lambda_{2}\geqslant\dots\geqslant\lambda_{m}$. This set
of numbers is known as the _Lyapunov spectrum_.

We say a cocycle on $\mathcal{C}^{\mathrm{root}}$ is _locally constant_ if the
cocycle is constant over the cylinder sets in our coding. By choosing a
trivialisation of the homology over each polytope $\mathcal{C}_{\pi}$, the
Kontsevich–Zorich cocycle lifted to $\mathcal{C}^{\mathrm{root}}$ is locally
constant. The Rauzy–Veech cocycle is locally constant by construction.

We normalise the invariant measure $\mu$ on the section $\mathcal{S}_{\theta}$
to be a probability measure. Let $\zeta$ be a symbol in $\Pi$ and let
$B_{\theta\zeta\theta}$ be the associated Rauzy matrix. By standard methods of
computing volumes of images of projective linear maps, there exists a constant
$C>1$ that depends only on the topology of the surface such that

$\frac{1}{C\|B_{\theta\zeta\theta}\|_{1}^{d-1}}<\mu(\mathcal{S}_{\theta\zeta\theta})<\frac{C}{\|B_{\theta\zeta\theta}\|_{1}^{d-1}}.$

See our previous article for more details [Bel+19, Lemma 5.11]. Thus, for the
discrete Rauzy–Veech cocycle $q\mapsto B(q)$ on $\mathcal{S}_{\theta}$, we get
that, up to a similar uniform multiplicative constant,

$\int_{\mathcal{S}_{\theta}}\log\|B(q)\|_{1}\,d\mu(q)\asymp\sum_{\zeta\in\Pi}\frac{\log\|B_{\theta\zeta\theta}\|_{1}}{\|B_{\theta\zeta\theta}\|_{1}^{d-1}}.$

To estimate the integral above, we organise the sequences $\zeta\in\Pi$ by the
$L^{1}$ norms of the matrices $B_{\theta\zeta\theta}$ considered on a
multiplicative scale on $\mathbb{R}_{+}$. Recurrence estimates in the bounded
distortion theorem for Rauzy–Veech sequences show that there exist constants
$M>1$ and $0<c<1$ that depend only on the topology of the surface such that,
for the set
$\Pi_{n}=\\{\zeta\in\Pi\,:\,\|B_{\theta\zeta\theta}\|_{1}\in[1,M^{n})\\}$,

$\sum_{\zeta\in\Pi_{n}}\mu(\mathcal{S}_{\theta\zeta\theta})>1-c^{n}.$

It follows that
$\sum_{\zeta}\log\|B_{\theta\zeta\theta}\|_{1}/\|B_{\theta\zeta\theta}\|_{1}^{d-1}$
is dominated by $\sum nc^{n}$ and, hence, that the discrete Rauzy–Veech
cocycle is integrable. Note that up to an additive constant that depends only
on the surface, the first return time $\xi(q)$ at any $q$ in
$\mathcal{S}_{\theta\zeta\theta}$ is $\log\|B_{\theta\zeta\theta}\|_{1}$. So
the integrability of the Rauzy–Veech cocycle is equivalent to the finiteness
of the Masur–Veech volume of $\mathcal{C}^{\mathrm{root}}$.

Using the above estimates for $\gamma\in\Pi_{n}$, it is straightforward to
derive that, if a locally constant cocycle with values in
$\operatorname{SL}(m,\mathbb{Z})$ is integrable with respect to the
Masur–Veech measure, then it is integrable with respect to
$(\mathcal{S}_{\theta},\mu)$. As a result, the integrability over
$(\mathcal{S}_{\theta},\mu)$ of the plus and minus cocycles (that we define in
Section 9) can be deduced from their integrability with respect to the
Masur–Veech measure. In any case, we will give a direct verification of the
integrability over $(\mathcal{S}_{\theta},\mu)$ of the plus and minus cocycles
in Section 10. To do that, we will need the following lemma.

We say that a cocycle $C$ is _dominated by_ a cocycle $C^{\prime}$ if there is
a constant $K>0$ that depends only on the surface such that the $L^{1}$-norms
satisfy $\|C\|_{1}\leqslant K\|C^{\prime}\|_{1}$.

######  Lemma 6.8.

Suppose that $C$ is a locally constant cocycle dominated by the Rauzy–Veech
cocycle. Then $C$ is integrable in either sense.

###### Proof.

The lemma follows directly from integrability of the Rauzy–Veech cocycle. ∎

Recall that a cocycle into $\operatorname{SL}(m,\mathbb{R})$ has a _simple_
Lyapunov spectrum if its Lyapunov spectrum consists on $m$ distinct numbers.
We will now state a weaker version of the Avila–Viana criterion for simplicity
of the Lyapunov spectrum. The actual criterion is more general Avi-
Via07a[Theorem 7.1]Avi-Via07b, but we state it specifically for our context.

Let $\gamma_{1}$ and $\gamma_{2}$ be almost-flow loops given by directed
Rauzy–Veech sequences $\eta_{1}$ and $\eta_{2}$. Then the concatenation
$\gamma_{1}\gamma_{2}$ is also realised by an almost-flow loop given by the
Rauzy–Veech sequence $\eta_{1}\eta_{2}$. Thus, the almost-flow loops give us a
monoid. By evaluating a locally constant cocycle for each almost-flow loop, we
get a representation of the monoid into $\operatorname{SL}(m,\mathbb{R})$. As
the cocyles we consider here are defined over $\mathbb{Z}$ and preserve a
symplectic structure, this representation has an image in the symplectic
group.

######  Criterion 6.9.

Let $C$ be a locally constant integrable cocycle for the Teichmüller flow. If
the associated monoid is Zariski dense in the symplectic group, then the
Lyapunov spectrum is simple.

As was previously mentioned, the criterion stated by Avila–Viana [AV07a,
Theorem 7.1] does not require Zariski density. Instead, it has the weaker
hypothesis of requiring the presence of _pinching_ and _twisting_ elements in
the group generated by the monoid. It is a classical fact that a Zariski dense
monoid gives a group that contains a pinching element. It follows from the
work of Benoist [Ben97] that this group also contains elements that are
twisting relative to the pinching element. As we directly establish Zariski
density, we will omit the precise definitions of pinching and twisting, which
are technical to state.

Furthermore, the criterion stated by Avila–Viana does not, strictly speaking,
require the cocycle to be symplectic; it requires the cocycle to take values
in the special linear group and satisfy pinching and twisting. Nevertheless,
we state the criterion for symplectic cocycles as all of the cocycles that we
consider are symplectic.

##  7\. Rauzy–Veech groups

Recall that $\mathcal{C}^{\mathrm{lab}}$ is the cover of
$\mathcal{C}^{\mathrm{root}}$ corresponding to the covering
$\mathcal{D}^{\mathrm{red}}\to\mathcal{D}$, as defined in Remark 5.3. In the
definitions that follow, we operate in $\mathcal{C}^{\mathrm{lab}}$.

###  7.1. Rauzy–Veech groups of abelian components

Let $\mathcal{C}$ be an abelian stratum component. There is a natural spanning
set for the absolute homology of the surface that one can associate to any
permutation $\pi$ in its Rauzy diagram $\mathcal{D}$ [AV07a, AMY18, Gut19].
For any loop $\delta$ in $\mathcal{D}$ based at $\pi$, one can define a matrix
in $\operatorname{Sp}(2g,\mathbb{Z})$ by computing the linear action on
absolute homology induced by $\delta$, in terms of the preferred spanning set.
The _Rauzy–Veech group_ of $\pi$, denoted $\operatorname{RV}(\pi)$, is the
subgroup of $\operatorname{Sp}(2g,\mathbb{Z})$ generated by such matrices. The
matrix associated with any loop $\delta$ coincides with the Rauzy–Veech matrix
$B_{\delta}$ but this is special for abelian stratum components.

###  7.2. Minus and plus pieces for quadratic stratum components

Let $\mathcal{C}$ be a component of a stratum of quadratic differentials.
There is a branched double cover $\widetilde{S}$ of $S$ such that the lifts
$\widetilde{q}$ for $q\in\mathcal{C}$ are abelian differentials on
$\widetilde{S}$. This is often called _the orientation double cover of the
quadratic differential_. The cover is branched over every zero of $q$ with odd
order and every pole of $q$. We will give the construction of the orientation
double cover of a typical rooted quadratic differential shortly.

The differential $\widetilde{q}$ is symmetric with respect to an involution
and the quotient is $q$. Viewed as an involution of $\widetilde{S}$, the
induced linear action on $H^{1}(\widetilde{S},\widetilde{Z};\mathbb{Z})$ has
eigenvalues $\\{1,-1\\}$. The $(+1)$-eigenspace is usually referred to as the
plus (or invariant) piece and the $(-1)$-eigenspace is usually referred to as
the minus (or anti-invariant) piece. By Poincaré-duality, we can also consider
it as a splitting of the homology
$H_{1}(\widetilde{S},\widetilde{Z};\mathbb{Z})$.

The Teichmüller flow on $\mathcal{C}$ defines a cocycle by its action on
$H^{1}(\widetilde{S},\widetilde{Z};\mathbb{Z})$. The cocyle preserves the plus
and minus eigenspaces. The plus Kontsevich–Zorich cocycle is its restriction
to the plus piece. Similarly, we also get the minus Kontsevich–Zorich cocycle.

###  7.3. Plus Rauzy–Veech groups and integrability of the plus cocycle

As the absolute part of the plus piece is invariant under the involution, it
is isomorphic to the absolute homology of $S$. Hence, the _plus Rauzy–Veech
group_ $\operatorname{RV}^{+}(\pi)$ can be defined in a similar way to the
abelian case by associating to each irreducible quadratic permutation a
preferred spanning set for the absolute homology of $S$. Then, for any loop
$\delta$ in the Rauzy diagram $\mathcal{D}$ based at $\pi$, we may associate a
matrix for the homology action induced by $\delta$ using the preferred
spanning set. This matrix does not coincide in general with the Rauzy–Veech
matrix $B_{\delta}$ and so we prefer to give a direct proof of the
integrability of the plus cocycle.

More precisely, for each quadratic permutation in $\mathcal{D}$ there is a
choice of a spanning set $\\{c_{\alpha}\\}_{\alpha\in\mathcal{A}}$ for the
plus piece such that the matrix for the plus cocycle has a simple form in each
Rauzy–Veech move. See the work by the fourth author [Gut17, Section 4.1] for
the description of the spanning set.

The simple form of the matrix has the following description. Suppose $\alpha$
and $\beta$ are top and bottom letters in $\pi$. Breaking symmetry, suppose
that $x_{\alpha}>x_{\beta}$. Let $\delta=\sigma\to\tau$ be the Rauzy–Veech
move dictated by the width constraint. If $c_{\alpha}$ and $c_{\beta}$ have
non-zero algebraic intersection then the matrix satisfies
$C_{\delta}=I+M_{\alpha,\beta}$, where $M_{\alpha,\beta}$ is the matrix whose
$(\alpha,\beta)$-entry is one and zero otherwise. On the other hand, if
$c_{\alpha}$ and $c_{\beta}$ have zero algebraic intersection, then
$C_{\delta}=I-M_{\alpha,\beta}-2M_{\alpha,\alpha}$.

The explicit matrices give us a direct proof of the integrability of the plus
cocycle. Inductively, the matrix for the cocycle can be defined for any finite
Rauzy–Veech sequence $\delta$ as a product of matrices for individual
Rauzy–Veech moves.

######  Lemma 7.4.

For any finite Rauzy–Veech sequence $\delta$

$\|C_{\delta}\|_{1}\leqslant\|B_{\delta}\|_{1},$

where $B_{\delta}$ is the Rauzy–Veech matrix for $\delta$.

###### Proof.

For any matrix $C$ with coefficients $c_{rs}$, let $|C|$ be the non-negative
matrix with coefficients $|c_{rs}|$.

Note that for an individual Rauzy move $\delta$ the plus cocycle satisfies
$|C_{\delta}|=B_{\delta}$. Now let
$\delta=\delta_{1}\delta_{2}\dotsc\delta_{k}$ be a finite Rauzy–Veech
sequence. We observe that

$\|C_{\delta}\|_{1}=\|C_{\delta_{1}}\dotsc
C_{\delta_{k}}\|_{1}\leqslant\||C_{\delta_{1}}|\dotsc|C_{\delta_{k}}|\|_{1}=\|B_{\delta_{1}}\dotsc
B_{\delta_{k}}\|_{1}=\|B_{\delta}\|_{1}$

∎

As the above lemma shows, the plus cocycle is dominated by the Rauzy–Veech
cocycle. The integrability of the plus cocycle now follows from Lemma 6.8.

(a) Original double cover.

(b) Double cover after a Rauzy move.

Figure 7.5. Example of the spanning set for the minus piece rendering the
linear transformations coming from Rauzy moves equal to the Rauzy–Veech
matrices. The original permutation is $\big{(}\begin{smallmatrix}1&2&1&2&3\\\
&3&4&4\end{smallmatrix}\big{)}$ representing the stratum
$\mathcal{Q}(2,-1,-1)$, which becomes $\big{(}\begin{smallmatrix}1&2&1&2\\\
3&3&4&4\end{smallmatrix}\big{)}$ after one bottom Rauzy move. In this case,
the cycles in the spanning set can be tightened to saddle connections, so they
are drawn in this manner. The general case is similar.

###  7.6. Minus Rauzy–Veech groups and integrability of the minus cocycle

The minus piece is in the kernel of the map induced on homology by the
branched covering $\widetilde{S}\to S$. As a result, the minus cocycle has to
be analysed directly in the orientation double cover of a quadratic
differential. Here again, for each irreducible quadratic permutation there is
a natural choice for a spanning set for the minus piece. Using this preferred
set, the _minus Rauzy–Veech group_ $\operatorname{RV}^{-}(\pi)$ can now be
defined in a similar way to the other types of Rauzy–Veech groups. For rooted
quadratic differentials that admit a zippered rectangles construction, we will
explicitly construct their orientation double cover and then precisely
describe the resulting matrices. These matrices preserve a specific
alternating form defined by Avila–Resende [AR12, Equation 9].

Consider the arcs between singularities that we used to define singularity
parameters. These arcs are a spanning set in the relative homology. Using
these arcs, we can build a spanning set for the minus piece as follows.

To construct the orientation double cover of the rooted differential, we take
two copies of the zippered rectangles. Let $1\leqslant i\leqslant\ell+m$. As
notation, a rectangle $R_{i}$ will be denoted as $R_{i}^{(1)}$ in the first
copy and $R_{i}^{(2)}$ in the second copy. The gluings are now constructed as
follows.

  * •

If $\pi(i)$ is a translation letter, then $R_{i}^{(1)}$ is glued to
$R_{\sigma(i)}^{(1)}$ and $R_{i}^{(2)}$ is glued to $R_{\sigma(i)}^{(2)}$ as
before.

  * •

If $\pi(i)$ is a flip letter then $R_{i}^{(1)}$ is glued to
$R_{\sigma(i)}^{(2)}$ by a translation.

The resulting abelian differential is the orientation double cover of the
original quadratic differential. The involution rotates each rectangle by 180
degrees and maps it to the corresponding rectangle in the other copy.

Let $\alpha\in\mathcal{A}$ be a letter. Let $a_{\alpha}$ be the arc in the
original quadratic differential oriented so that its period is
$x_{\alpha}+y_{\alpha}$ where $x_{\alpha},y_{\alpha}$ are the singularity
parameters. Let $a_{\alpha}^{(1)}$ and $a_{\alpha}^{(2)}$ be the lifts of
$a_{\alpha}$ to the double cover. The spanning set for the minus piece in the
relative homology of the orientation double cover is now defined as follows:

  * •

Suppose $\alpha$ is a translation letter. Then let
$A_{\alpha}=a_{\alpha}^{(1)}+a_{\alpha}^{(2)}$.

  * •

Suppose $\alpha$ is a flip letter. Then let
$A_{\alpha}=a_{\alpha}^{(1)}-a_{\alpha}^{(2)}$.

See Figure 7.5 for an illustration of these cycles.

It is straightforward to check that in a Rauzy–Veech move the linear change of
these spanning sets is exactly encoded by the Rauzy–Veech matrix. Thus the
Kontsevich–Zorich cocycle on the minus piece of the relative cohomology (by
duality) coincides with the Rauzy–Veech cocycle. In particular, the
$L^{1}$-norm of the restriction to the minus piece in absolute cohomology is
dominated by the $L^{1}$-norm of the Rauzy–Veech matrix. Thus, the minus
cocycle is integrable.

###  7.7. Modular Rauzy–Veech groups

By considering mapping classes instead of the homological actions induced by
loops in the Rauzy diagram, we can define the _modular Rauzy–Veech groups_ ,
the _plus modular Rauzy–Veech groups_ and the _minus modular Rauzy–Veech
groups_. These groups are then subgroups of mapping class groups and their
images by the symplectic representations coincide with the corresponding
Rauzy–Veech groups. They are denoted $\operatorname{MRV}(\pi)$,
$\operatorname{MRV}^{+}(\pi)$ and $\operatorname{MRV}^{-}(\pi)$, respectively.

###  7.8. Rauzy–Veech groups and monodromy groups

Elementary theory of covering spaces together with our main theorems Theorem
4.23 and Theorem 5.5 implies that Rauzy–Veech groups are finite-index
subgroups of appropriate monodromy groups. More precisely:

######  Corollary 7.9.

Let $\mathcal{C}$ be a component of a stratum of abelian or quadratic
differentials and let $\pi$ be an irreducible permutation that represents
$\mathcal{C}$. Let $\widetilde{\mathcal{C}}$ be any finite manifold cover of
$\mathcal{C}$ (which, in particular, can be taken to be either
$\mathcal{C}^{\mathrm{root}}$ or $\mathcal{C}^{\mathrm{lab}}$). Then, the
following groups are finite-index subgroups inside the following larger
groups:

  1. (1)

$\pi_{1}(\widetilde{\mathcal{C}})$ inside
$\pi_{1}^{\mathrm{orb}}(\mathcal{C})$;

  2. (2)

the (modular) monodromy group of $\widetilde{\mathcal{C}}$ inside the
(modular) monodromy group of $\mathcal{C}$;

  3. (3)

if $\pi$ is abelian, the (modular) Rauzy–Veech group of $\pi$ inside the
(modular) monodromy group of $\mathcal{C}$ of $\pi$;

  4. (4)

if $\pi$ is quadratic, the (modular) plus (respectively, minus) Rauzy–Veech
group of $\pi$ inside the (modular) monodromy group of $\mathcal{C}$
corresponding to the plus (respectively, minus) piece of the homology.

###### Proof.

Since $\widetilde{\mathcal{C}}$ is a finite cover of $\mathcal{C}$, parts (1)
and (2) follow from elementary theory of covering spaces.

Suppose that $\pi$ is abelian. Then the modular Rauzy–Veech group of $\pi$ is
a subgroup of the modular monodromy group of $\mathcal{C}^{\mathrm{lab}}$. The
push-forward of the modular Rauzy–Veech group to $\mathcal{C}^{\mathrm{root}}$
is exactly the image of the flow group $G(\mathcal{C}_{\pi},q_{0})$ inside the
mapping class group. By Theorem 5.5, this image equals the modular monodromy
group of $\mathcal{C}^{\mathrm{root}}$. By part (2), it is a finite-index
subgroup of the modular monodromy group of $\mathcal{C}$. The rest of part (3)
is obtained by applying the symplectic representation. Part (4) is obtained
analogously. ∎

Part (3) of the previous corollary provides a partial answer (that is, up to
finite index) to a question of Yoccoz [Yoc10, Remark in Section 9.3]. Part (4)
extends these partial answers to analogous questions for quadratic stratum
components.

######  Remark 7.10.

For abelian components, the overall structure of how various groups that we
have considered fit together can be organised in the following commutative
diagram:

${\pi_{1}(\mathcal{D})}$${\pi_{1}(\mathcal{C}^{\mathrm{lab}})}$${\operatorname{MRV}(\pi)}$${\operatorname{RV}(\pi)}$${\pi_{1}(\mathcal{D}^{\mathrm{red}})}$${G(\mathcal{C}_{\pi})=\pi_{1}(\mathcal{C}^{\mathrm{root}})}$${\operatorname{MMon}(\mathcal{C}^{\mathrm{root}})}$${\operatorname{Mon}(\mathcal{C}^{\mathrm{root}})}$${\pi_{1}^{\mathrm{orb}}(\mathcal{C})}$${\operatorname{MMon}(\mathcal{C})}$${\operatorname{Mon}(\mathcal{C})}$${\mathrm{Mod}(S)}$${\operatorname{Sp}(2g,\mathbb{Z})}$f.i.f.i.f.i.f.i.f.i.f.i.f.i.

where “f.i.” stands for “finite index”, and recall that $\operatorname{MMon}$
is the modular monodromy group and that $\operatorname{Mon}$ is the monodromy
group. This diagram actually allows us to define the “most general” version of
a Rauzy–Veech group, which is the image of the group homomorphism
$\pi_{1}(\mathcal{D})\to\pi_{1}(\mathcal{C}^{\mathrm{lab}})$. Theorem 5.2
shows that this group is actually equal to
$\pi_{1}(\mathcal{C}^{\mathrm{lab}})$. Thus, any other version of Rauzy–Veech
group can be obtained as the image of $\pi_{1}(\mathcal{C}^{\mathrm{lab}})$ by
an appropriate group homomorphism.

Similar commutative diagrams can also be stated for quadratic components by
considering the images into the plus and minus pieces separately.

Combining Corollary 7.9 with the work of Calderon and Calderon–Salter [Cal20,
CS19, CS19a, CS20], we obtain a classification of the modular Rauzy–Veech
groups and the Rauzy–Veech groups in relative homology, up to finite index,
for genus at least five for non-hyperelliptic components. More precisely:

######  Corollary 7.11.

Let $S$ be a topological surface of genus at least five. Let $\mathcal{C}$ be
a non-hyperelliptic component of a stratum of abelian differentials on $S$
whose set of marked points is $Z$. Let $\phi$ is the absolute framing induced
by the horizontal vector field of a surface in $\mathcal{C}$. We have that:

  1. (1)

The modular Rauzy–Veech group in $\mathrm{Mod}(S,Z)$ is a finite-index
subgroup of $\mathrm{Mod}(S,Z)[\phi]$, that is, of the stabiliser of $\phi$
inside $\mathrm{Mod}(S,Z)$.

  2. (2)

The Rauzy–Veech group in $\operatorname{PAut}(H_{1}(S,Z;\mathbb{Z}))$ is a
finite-index subgroup of the kernel of the crossed homomorphism
$\Theta_{\phi}\colon\operatorname{PAut}(H_{1}(S,Z;\mathbb{Z}))\to
H^{1}(S,\mathbb{Z}/2\mathbb{Z})$ defined by Calderon–Salter [CS19a, Section
4].

This classification was already known for hyperelliptic components, and in
this case the index is known to be one [AMY18].

##  8\. Classification of components and a reduction strategy

###  8.1. Classification of the components of strata of abelian and quadratic
differentials

For the reader’s convenience, we restate the complete classification of the
components of abelian and quadratic strata.

######  Theorem 8.2 ([KZ03]).

The following is the classification of the components of the strata of abelian
differentials (up to regular marked points).

  * •

In genus one, the only stratum is $\mathcal{H}(0)$. It is non-empty, connected
and hyperelliptic.

  * •

In genus two, the only strata are $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$.
They are non-empty, connected and hyperelliptic.

  * •

In genus three, the strata $\mathcal{H}(4)$ and $\mathcal{H}(2,2)$ have two
components. One of them is hyperelliptic and the other one corresponds to odd
spin structures. Every other stratum is non-empty and connected.

  * •

Finally, for genus $g$ at least four:

    * –

The stratum $\mathcal{H}(2g-2)$ has three components. One of them is
hyperelliptic, and the other two are distinguished by even and odd spin
structures.

    * –

The stratum $\mathcal{H}(g-1,g-1)$ can have two or three components depending
on the parity of $g$. If $g$ is even, it has two components. One of them is
hyperelliptic, and the other one is not. If $g$ is odd, it has three
components. One of them is hyperelliptic, and the other two are distinguished
by even and odd spin structures.

    * –

All other strata of the form $\mathcal{H}(2\kappa_{1},\dotsc,2\kappa_{n})$
have two components, distinguished by even and odd spin structures.

    * –

The remaining strata are non-empty and connected.

######  Theorem 8.3 ([Lan08, CM14]).

The following is the classification of the components of the strata of
quadratic differentials (up to regular marked points).

  * •

In genus zero, every stratum is non-empty and connected.

  * •

In genus one, the strata $\mathcal{Q}(0)$ and $\mathcal{Q}(1,-1)$ are empty.
All other strata are nonempty and connected.

  * •

In genus two, the strata $\mathcal{Q}(4)$ and $\mathcal{Q}(3,1)$ are empty.
Moreover, the stratum $\mathcal{Q}(2,2)$ is non-empty, connected and
hyperelliptic.

  * •

In genus three, the strata $\mathcal{Q}(9,-1)$, $\mathcal{Q}(6,3,-1)$ and
$\mathcal{Q}(3,3,3,-1)$ have two components, known as _regular_ and
_irregular_ components.

  * •

In genus four, the strata $\mathcal{Q}(6,6)$, $\mathcal{Q}(6,3,3)$ and
$\mathcal{Q}(3,3,3,3)$ have three components. One of them is hyperelliptic,
and the other two are known as _regular_ and _irregular_ components. Moreover,
the strata $\mathcal{Q}(12)$ and $\mathcal{Q}(9,3)$ have two components, known
as _regular_ and _irregular_ components.

  * •

Finally, for genus at least two:

    * –

The strata of the form $\mathcal{Q}(4j+2,4k+2)$, $\mathcal{Q}(4j+2,2k-1,2k-1)$
and $\mathcal{Q}(2j-1,2j-1,2k-1,2k-1)$ for $j,k\geqslant 0$ not contained in
the previous list have two components. One of them is hyperelliptic and the
other one is not.

    * –

The remaining strata are non-empty and connected.

###  8.4. Adjacency of strata and a reduction strategy

For abelian differentials [AV07a], the adjacency between components of
different strata was exploited to show that Rauzy–Veech groups of simpler
components are contained inside the Rauzy–Veech groups of more complex ones.
We will exploit the same strategy to obtain the containment of Rauzy–Veech
groups of quadratic stratum components.

We start with the notions of _simple extensions_.

######  Definition 8.5.

Let $\sigma$ be an irreducible permutation. We say that a permutation $\tau$
is a _type preserving simple extension_ of $\sigma$ if $\tau$ is quadratic
(respectively, abelian) when $\sigma$ is quadratic (respectively, abelian) and
$\tau$ can be obtained from $\sigma$ by inserting a single letter $\alpha$ in
such a way that:

  * •

at most one occurrence of $\alpha$ is at the beginning of a row in $\tau$; and

  * •

no occurrence of $\alpha$ is at the end of a row in $\tau$.

Similarly, we have

######  Definition 8.6.

Let $\sigma$ be an irreducible abelian permutation. We say that $\tau$ is a
_type changing simple extension_ of $\sigma$ is $\tau$ is quadratic and $\tau$
can be obtained from $\sigma$ by inserting a single top flip letter $\alpha$
and a single bottom flip letter $\beta$ such that

  * •

at most one occurrence of $\alpha$ (respectively, $\beta$) is at the beginning
of a row in $\tau$; and

  * •

no occurrence of $\alpha$ (respectively, $\beta$) is at the end of a row in
$\tau$.

As irreducible quadratic permutations already possess flip letters, there are
no type changing extensions from a quadratic permutation to an abelian one.

The notion of simple extensions was original introduced by Avila–Viana [AV07a]
in their proof of the Kontsevich–Zorich conjecture for abelian differentials.
By expanding it to type changing extensions, the notion was extended to
quadratic differentials by the fourth author [Gut17].

Note that if $\sigma$ is an irreducible permutation and $\tau$ is a (type
preserving or changing) simple extension of $\sigma$, then $\tau$ is also
irreducible [Gut17, Lemma 3.2]. Also, note that the genera of the embodying
strata of $\sigma$ and $\tau$ are the same.

Let $\tau$ be a (type preserving or changing) simple extension of $\sigma$ and
suppose that $\zeta$ is a directed loop based at $\sigma$ in the Rauzy diagram
$\mathcal{D}$ that contains $\sigma$. It is then possible to shadow $\zeta$ by
a Rauzy–Veech sequence starting from $\tau$ by requiring that the added letter
(or letters) always lose if they participate in a Rauzy move [Gut17, Section
3]. It also follows from this description that the shadowing Rauzy–Veech
sequence also returns to $\tau$. This implies that $\operatorname{RV}(\sigma)$
is a subgroup of $\operatorname{RV}(\tau)$.

We now explain the geometric content underlying simple extensions. A type
preserving simple extension allows us to split a singularity into a pair of
singularities [Gut17, Lemma 5.1]. A type changing simple extension allows us
to split a singularity in to three singularities at least one of which has odd
order [Gut17, Corollary 5.2].

For quadratic stratum components which are our focus, the numerical invariant
$\kappa$ gets re-organised as follows;

  * •

If $\kappa_{1}\geqslant 1$, $\kappa_{2},\dotsc,\kappa_{n}\geqslant-1$ are
integers and $\kappa_{1,1},\kappa_{1,2}\geqslant-1$ are integers satisfying
$\kappa_{1,1}+\kappa_{1,2}=\kappa_{1}$, then there exists a permutation
$\sigma$ whose embodying stratum is
$\mathcal{Q}(\kappa_{1},\dotsc,\kappa_{n})$ and a permutation $\tau$ whose
embodying stratum is
$\mathcal{Q}(\kappa_{1,1},\kappa_{1,2},\kappa_{2},\dotsc,\kappa_{n})$ such
that $\tau$ is a simple extension of $\sigma$.

  * •

If $\kappa_{1}\geqslant 0$, $\kappa_{2}\dotsc,\kappa_{n}\geqslant 1$ are
integers and $\kappa_{1,1},\kappa_{1,2},\kappa_{1,3}\geqslant-1$ are integers
which are not all even and satisfy
$\kappa_{1,1}+\kappa_{1,2}+\kappa_{1,3}=2\kappa_{1}$, then there exists a
permutation $\sigma$ whose embodying stratum is
$\mathcal{H}(\kappa_{1},\dotsc,\kappa_{n})$ and a permutation $\tau$ whose
embodying stratum is
$\mathcal{Q}(\kappa_{1,1},\kappa_{1,2},\kappa_{1,3},2\kappa_{2},\dotsc,2\kappa_{n})$
such that $\tau$ is a simple extension of $\sigma$.

These facts suggest the following strategy to show the Zariski density of
every Rauzy–Veech group corresponding to the plus piece of quadratic
differentials:

  1. (1)

Prove Zariski density for minimal quadratic strata, that is, for strata of the
form $\mathcal{Q}(4g-4)$ for $g\geqslant 3$ with the exception of
$\mathcal{Q}(12)^{\mathrm{reg}}$ and $\mathcal{Q}(12)^{\mathrm{irr}}$ (with
$g=4$) whose density has to be proved separately for technical reasons. Then
use simple extensions to extend the density to every connected stratum with
$g\geqslant 3$. For strata in $g\geqslant 3$ that have two components, one
non-hyperelliptic and the other hyperelliptic, the non-hyperelliptic component
can also be reached by such simple extensions from minimal strata. Note that a
hyperelliptic component cannot arise by a simple extension of a non-
hyperelliptic one. So we pass to the next case in our strategy.

  2. (2)

Prove Zariski density for every hyperelliptic component that has the form
$\mathcal{Q}(4j+2,4k+2)$ for $j,k\geqslant 1$. The rest of the hyperelliptic
components arise by a string of simple extensions of these and hence we can
extend density, except for those containing poles. Nevertheless, these last
components arise as a string of simple extensions from the hyperelliptic
component of minimal abelian strata.

The remaining cases are all in genus four and lower and we will outline the
strategy for those.

  3. (3)

The regular and irregular components in genus four arise by a simple extension
from $\mathcal{Q}(12)$ or from $\mathcal{H}(6)$; the extensions from
$\mathcal{H}(6)$ are already treated in previous work by the fourth author
[Gut17, Table 1]. The density then extends to these.

  4. (4)

Prove Zariski density explicitly for $\mathcal{Q}(9,-1)^{\mathrm{irr}}$ in
genus three. The remaining regular and irregular components in genus three can
be handled by exhibiting a simple extension from $\mathcal{Q}(8)$ or from
$\mathcal{H}(4)$; the extensions from $\mathcal{H}(4)$ are again already
treated in previous work by the fourth author [Gut17, Table 1].

  5. (5)