Prove Zariski density explicitly for $\mathcal{Q}(5,-1)$ in genus two. The
remaining non-hyperelliptic components all have at least three singularities
and not all of these have even orders. So we may extend density from
$\mathcal{H}(2)$ by using simple extensions.

  6. (6)

All quadratic strata in genus one have at least three singularities and not
all of the singularities can have even orders. So we may extend density from
$\mathcal{H}(0)$ by using simple extensions.

For abelian differentials, a similar strategy gives containment of Rauzy–Veech
groups. As the Rauzy–Veech groups can be explicitly figured out in the base
cases, they can then also be classified in all abelian cases using the
containment. They turn out to be either the full symplectic group, or certain
special finite index subgroups of the symplectic group. See the work by
Avila–Matheus–Yoccoz [AMY18] for hyperelliptic abelian components and by the
fourth author [Gut19] for general abelian components. Plus and minus
Rauzy–Veech groups for quadratic stratum components that can arise by a string
of simple extensions from abelian base cases are also more tractable even
though here a complete classification is not yet achieved [Gut17].

##  9\. Zariski density

In this section we prove one of our main results.

######  Theorem 9.1.

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.

Being the full symplectic group or a finite index subgroup, Rauzy–Veech groups
for abelian strata and quadratic components that arise by simple extensions
from abelian strata are Zariski dense.

The Rauzy–Veech groups of the quadratic base cases are harder to track
directly and here we bypass them. Instead, we leverage Filip’s results [Fil17]
to prove Zariski density of their monodromy groups. By Corollary 7.9,
Rauzy–Veech groups are finite index in the monodromy groups. We deduce that
they are Zariski dense. For clarity, we again refer the reader to the
commutative diagram in Section 7. We then extend the density to all quadratic
stratum components by simple extensions.

The proof presented here is self-contained and works for any (abelian or
quadratic) connected component. A key ingredient is the work [Fil17] of Filip
that gives a list of possible Zariski closures for algebraic hulls of linear
invariant suborbifolds.

###  9.2. Filip’s results

We briefly describe Filip’s results [Fil17, Theorem 1.2 and Corollary 1.7] for
the possible Zariski closures of the monodromy and algebraic hulls of a linear
invariant suborbifold. Let $\mathcal{N}$ be a linear invariant suborbifold.

  * •

If $p(T\mathcal{N})$ is the subbundle of the Kontsevich–Zorich cocycle that
contains the tangent space to $\mathcal{N}$, then the monodromy has no zero
exponents on $p(T\mathcal{N})$ and the closure of the monodromy for the action
on $p(T\mathcal{N})$ is the full symplectic group [Fil17, Corollary 1.7]. This
readily implies the Zariski density of the monodromy group of any abelian
component, and, as detailed in Section 9.4, also implies the Zariski density
of the monodromy group of the minus piece of any quadratic component.

  * •

For strongly irreducible subbundles that do not contain the tautological
plane, Filip shows that the Lie algebra representation of the corresponding
piece of the algebraic hull must be, up to compact factors, one from the
following list [Fil17, Theorem 1.2]:

    1. (1)

$\mathfrak{sp}(2g,\mathbb{R})$ in the standard representation,

    2. (2)

$\mathfrak{su}(p,q)$ in the standard representation or $\mathfrak{su}(p,1)$ in
any exterior power representation,

    3. (3)

$\mathfrak{so}(2n-1,2)$ in the spin representation,

    4. (4)

$\mathfrak{so}^{\ast}(2n)$ in the standard representation, or
$\mathfrak{so}(2n-2,2)$ in either of the spin representations.

On the other hand, Eskin–Filip–Wright show that the algebraic hull coincides
with the Zariski closure of the monodromy group for subbundles that do not
contain the tautological plane [EFW18, Theorem 1.1]. Therefore, the previous
theorem also classifies the Lie algebra representations of the Zariski closure
of such monodromy groups.

###  9.3. Zariski density for abelian components

The Zariski density of the monodromy group for all abelian components follows
directly from Filip’s results, as the subbundle $p(T\mathcal{N})$ is the
entire Hodge bundle.

###  9.4. Zariski density for the minus piece

Assume that $\mathcal{C}$ is a component of a stratum of the moduli space of
quadratic differentials. Let $\widetilde{\mathcal{C}}$ be the linear invariant
suborbifold consisting on the abelian differentials obtained by the
orientation double cover of the quadratic differentials in $\mathcal{C}$. Let
$\widetilde{S}$ be the underlying topological surface of the elements of
$\widetilde{\mathcal{C}}$.

Let $p\colon H^{1}(\widetilde{S},\widetilde{Z};\mathbb{Z})\to
H^{1}(\widetilde{S};\mathbb{Z})$ be the restriction map to the absolute
homology. By Filip’s classification, the monodromy group acting on
$p(T\widetilde{\mathcal{C}})$ is Zariski dense [Fil17, Corollary 1.7] in the
full symplectic group. On the other hand, we have that
$p(T\widetilde{\mathcal{C}})$ is exactly
$H_{-}^{1}(\widetilde{S};\mathbb{Z})$, that is, the cohomology classes which
are anti-invariant with respect to the action of $\iota$. This can be seen by
observing that if $(X,\omega)$ is an element of $\widetilde{\mathcal{C}}$,
then $\iota^{*}\omega=-\omega$.

By Lefschetz-duality, this is equivalent to saying that the monodromy group is
Zariski dense when acting on
$H_{1}^{-}(\widetilde{S},\widetilde{Z};\mathbb{Z})$, so we obtain Zariski
density for the minus piece of the homology.

###  9.5. Zariski density for the plus piece

It remains to show the Zariski density for the plus piece. We will first prove
the Zariski density of the monodromy groups for minimal strata, hyperelliptic
components with two singularities and some sporadic components. As Theorem 5.2
implies that the Rauzy–Veech groups are finite index in monodromy groups, it
will follow that the Rauzy–Veech groups for these base components are also
Zariski dense. We will then extend the density to the Rauzy–Veech groups of
all components by simple extensions. In turn, this will imply that the
monodromy groups of all components are Zariski dense.

For the base components, we will work directly on $H_{1}(S;\mathbb{Z})$ as it
is isomorphic to $H_{1}^{+}(\widetilde{S};\mathbb{Z})$ in such a way that the
corresponding monodromy groups are conjugate.

First observe that the monodromies in $H_{1}^{+}(S;\mathbb{Z})$ and
$H_{+}^{1}(S;\mathbb{Z})$ are isomorphic by Poincaré duality. Let $M$ denote
this monodromy group. Note that $H_{+}^{1}(S;\mathbb{Z})$ does not contain the
tautological plane. By Eskin–Filip–Wright [EFW18, Theorem 1.1], the algebraic
hull coincides with the Zariski closure of the monodromy group $M$. This
ensures that we can directly apply Filip’s classification. Moreover, Treviño
[Tre13, Theorem 1] proved that the plus Lyapunov spectrum contains no zero
exponents.

Recall that the action of $M$ on $H_{1}(S;\mathbb{R})$ is said to be _strongly
irreducible_ if no finite-index subgroup of $M$ preserves a non-trivial vector
subspace of $H_{1}(S;\mathbb{R})$. We will later show that this action is
indeed strongly irreducible.

Assume for now that the action of $M$ is strongly irreducible and let
$\mathfrak{m}$ be the Lie algebra of the Zariski closure of $M$. Applying
Filip’s classification to $\mathfrak{m}$ and using the absence of zero
exponents, we deduce that the possibilities for $\mathfrak{m}$ are, up to
compact factors:

  1. (1)

$\mathfrak{sp}(2g,\mathbb{R})$ in the standard representation (degree $2g$,
dimension $g(2g+1)$);

  2. (2)

$\mathfrak{su}(p,p)$ in the standard representation (degree $4p$, dimension
$4p^{2}-1$);

  3. (3)

$\mathfrak{so}(2n-1,2)$ in the spin representation (degree $2^{n}$, dimension
$n(2n+1)$);

  4. (4)

$\mathfrak{so}(2n-2,2)$ in one of the spin representations (degree $2^{n-1}$,
dimension $n(2n-1)$); or

  5. (5)

$\mathfrak{so}^{*}(2n)$ in the standard representation for even $n$ (degree
$4n$, dimension $n(2n-1)$).

This list can be further refined by observing that the degree and dimension of
the representation must match because of the strong irreducibility. In each of
the above cases, we derive:

  1. (1)

$\dim_{\mathbb{R}}\mathfrak{sp}(2g,\mathbb{R})=g(2g+1)$;

  2. (2)

$p=g/2$, so $\dim_{\mathbb{R}}\mathfrak{su}(g/2,g/2)=g^{2}-1$;

  3. (3)

$n=\log_{2}(2g)$, so
$\dim_{\mathbb{R}}\mathfrak{so}(2n-1,2)=\log_{2}(2g)\log_{2}(8g^{2})$;

  4. (4)

$n=\log_{2}(4g)$, so
$\dim_{\mathbb{R}}\mathfrak{so}(2n-2,2)=\log_{2}(4g)\log_{2}(8g^{2})$; and

  5. (5)

$n=g/2$, so $\dim_{\mathbb{R}}\mathfrak{so}^{*}(g/2,g/2)=g(g-1)/2$.

Note that possibilities (2)–(4) require the genus to be even. Thus, if $g$ is
odd and if the action of $M$ is strongly irreducible then $\mathfrak{m}$ has
to be $\mathfrak{sp}(2g,\mathbb{R})$. Hence, for odd genus it suffices to
establish the strong irreducibility of the $M$-action. For even genus, along
with showing the strong irreducibility, we will eliminate all but the
symplectic representation by setting up a dimension count. Here, we exploit
the additional flexibility that Theorem 5.6 provides.

As we use Dehn twists to build a dimension count, we record the following
formula: suppose $c,c^{\prime}$ are oriented multi-curves. As an element of
the homology

(9.6) $T(c)(c^{\prime})=c^{\prime}+\omega(c^{\prime},c)c,$

where $T$ is the left Dehn twist and $\omega(\ast,\ast)$ is the algebraic
intersection number.

The next lemma shows how Dehn twists in $M$ can be used to prove the largeness
of a subspace of $H_{1}(S;\mathbb{R})$ invariant under some finite index
subgroup of $M$.

######  Lemma 9.7.

Let $V\neq\\{0\\}$ a subspace of $H_{1}(S;\mathbb{R})$ on which a finite-index
subgroup $N$ of $M$ acts irreducibly. Suppose that $T(v)\in M$ for some $v\in
H_{1}(S;\mathbb{R})$. If there exists $v^{\prime}\in V$ such that
$\omega(v,v^{\prime})\neq 0$, then $v\in V$.

###### Proof.

By Equation 9.6 and the hypothesis, the linear combination
$T(v)^{k}(v^{\prime})-v^{\prime}$ is then a non-zero multiple of $v$ for every
$k\geqslant 1$. Since $|M:N|<\infty$, $T(v)^{k}\in N$ for some $k\geqslant 1$.
The lemma follows.

∎

We now state and prove a strong irreducibility criterion that we will use
throughout the discussion that follows.

######  Lemma 9.8.

Let $B$ be a finite set of cycles in $H_{1}(S;\mathbb{R})$ such that

  * •

the span of $B$ is $H_{1}(S;\mathbb{R})$;

  * •

for any pair $u\neq u^{\prime}\in B$, there exists a _chain_
$u=u_{0},\dotsc,u_{k}=u^{\prime}$ such that $\omega(u_{j},u_{j+1})\neq 0$ for
all $0\leqslant j\leqslant k-1$; and

  * •

$T(u)$ is in $M$ for all $u\in B$.

Then, $M$ acts strongly irreducibly on $H_{1}(S;\mathbb{R})$.

###### Proof.

Let $V\neq\\{0\\}$ be a subspace on which a finite-index subgroup $N$ of $M$
acts irreducibly. By hypothesis, it suffices to show that $B$ is contained in
$V$.

Since $B$ spans $H_{1}(S;\mathbb{R})$, there exists $v\in V$, $u\in B$ such
that $\omega(v,u)\neq 0$. Then, by Lemma 9.7, $u\in V$. Let $u^{\prime}\neq u$
be an element of $B$. By hypothesis, there is a chain
$u=u_{0},\dotsc,u_{k}=u^{\prime}$ such that $\omega(u_{j},u_{j+1})\neq 0$ for
all $0\leqslant j\leqslant k-1$. By applying Lemma 9.7 inductively, it follows
that for all $j$ the cycles $u_{j}$ are contained in $V$. We conclude that $B$
is contained in $V$, so the lemma follows. ∎

######  Remark 9.9.

Explicit sets $B$ to which we will apply the previous lemma, will not always
be a basis of $H_{1}(S;\mathbb{R})$; extra cycles may be needed to satisfy the
chain hypothesis.

###  9.10. Minimal strata

For $g\geqslant 3$, let $d=2g$ and

$\pi_{d}=\setcounter{MaxMatrixCols}{11}\begin{pmatrix}1&2&1&3&4&5&6&7&8&\cdots&d-1\\\
2&4&3&5&4&8&7&\cdots&d&d-1&d\end{pmatrix}.$

The rooted differentials in $\mathcal{C}_{\pi_{d}}$ belong to
$\mathcal{Q}(4g-4)$.

As the first step, we will prove that the action of the monodromy group on
$H_{1}(S;\mathbb{R})$ is strongly irreducible. To do so, we will choose an
explicit set of cycles spanning the homology. The parity of $g$ will dictate a
choice of slightly different sets of cycles. See Figure 11(a) for a flat
surface in $\mathcal{Q}(16)$ for $g=5$, and Figure 11(b) and Figure 11(c) for
a flat surface in $\mathcal{Q}(12)$ for $g=4$. Note, however, that the stratum
$\mathcal{Q}(12)$ has two components, and that the figure depicts a quadratic
differential in $\mathcal{Q}(12)^{\mathrm{reg}}$. We will complete the proof
for $g=4$ in Appendix B, but for now we will only show the strong
irreducibility for $\mathcal{Q}(12)^{\mathrm{reg}}$. The pattern for the
cycles in even genus greater than four remains the same as in Figure 11(b) and
Figure 11(c), so we don’t need a separate figure.

In the second step, we will prove Zariski density of the monodromy group for
even genus (odd genera being directly covered by Filip’s result).

In our proofs, we will use a subset of the cycles (and their combinations)
that are core curves of cylinders on the flat surface. Dehn twists in such
cycles are in the monodromy group, so we may use them in our computations.

Let $M$ be the monodromy group and let $\mathfrak{m}$ be the Lie algebra of
its Zariski closure. We set $\varepsilon=(-1)^{g}$.

(a) Odd $g$.

(b) Even $g$ with the slope-$1$ curve $b$.

(c) Even $g$ with the modified slope-$1$ curve $b^{\prime}$.

Figure 9.11. Basis and useful curves.

We consider the collection of the following homology classes:

  * •

$c_{1}$ and $c_{d}$ are the homology classes of the dashed curves;

  * •

$c_{2}$ and $c_{d-1}$ are the homology classes of the dash-dotted curves; and

  * •

$c_{3},\dotsc,c_{d-2}$ are the homology classes of the solid curves.

These curves form a basis for $H_{1}(S;\mathbb{R})$ (which is symplectic if
ordered appropriately). The densely dotted slope-$1$ curve is
$b=\sum_{i=1}^{d}c_{i}$ and the loosely dotted horizontal curve is
$p=c_{1}+\varepsilon c_{d}$. In even genus, we also need the curve
$b^{\prime}=\sum_{i=1}^{d-4}c_{i}-c_{d-1}-c_{d}$ obtained by modifying $b$.
Note that the set $\\{c_{2},\dotsc,c_{d-1},b,p\\}$ is a basis for
$H_{1}(S;\mathbb{R})$ when $g$ is odd and the set
$\\{c_{2},\dotsc,c_{d-1},b,b^{\prime},p\\}$ is a spanning set for
$H_{1}(S;\mathbb{R})$ when $g$ is even.

######  Lemma 9.12.

The action of $M$ on $H_{1}(S;\mathbb{R})$ is strongly irreducible.

###### Proof.

We set

$B=\begin{cases}\\{c_{2},\dotsc,c_{d-1},b,p\\}&\text{if $g$ is odd}\\\
\\{c_{2},\dotsc,c_{d-1},b,b^{\prime},p\\}&\text{if $g$ is even}\\\
\end{cases}$

As directly seen from Figure 11(a) and Figure 11(b), the cycles in the set
$\\{c_{2},\dotsc,c_{d-1},b\\}$ are given by core curves of cylinders on the
corresponding flat surfaces. Now consider the cylinder with core curve
$c_{d-3}-c_{d-2}$. A left-handed shear applied inside of this cylinder makes
the four horizontal saddle connections have slope one. Equivalently, it
performs a one-quarter Dehn twist. This straightens the modified slope-one
curve $b^{\prime}$; see Figure 11(c). Thus $b^{\prime}$ is the core curve of a
cylinder on the deformed surface. We deduce that for any cycle $u$ in $B$ the
Dehn twist $T(u)$ is in $M$.

We also note that for any pair $u\neq u^{\prime}$ in $B$ there exists a chain
$u=u_{0},\dotsc,u_{k}=u^{\prime}$ such that $\omega(u_{j},u_{j+1})\neq 0$ for
all $0\leqslant j\leqslant k-1$.

It follows that the set $B$ satisfies the hypothesis of Lemma 9.8 and thus the
action of $M$ on $H_{1}(S;\mathbb{R})$ is strongly irreducible. ∎

The Dehn twist $T(c)$ along a homology cycle $c$ is also a symplectic
transvection. For the following calculation we will think of it as such. If
$T(c)^{k}\in M$, then $T(c)^{k}-\mathrm{Id}\in\mathfrak{m}$. As notation, let
$D(c)=T(c)-\mathrm{Id}$, $E(c)=T(c)^{2}-\mathrm{Id}$.

The cycles $c_{2},\dotsc,c_{d-1}$ and $p$ are realised as core curves of
cylinders on the surface. Hence, the Dehn twists $T(c_{i})$ for
$i=2,\dotsc,d-1$ and $T(p)$ are in $M$. Observe that:

$\begin{aligned} D(c_{2})(c_{1})&=-c_{2}\\\ D(c_{2i+1})(c_{2i+2})&=c_{2i+1}\\\
D(c_{2i+2})(c_{2i+1})&=-c_{2i+2}\\\ D(c_{d-1})(c_{d})&=-c_{d-1}\\\
D(p)(c_{2})=p,\,D(p)(c_{d-1})&=\varepsilon p\end{aligned}\qquad\begin{aligned}
D(c_{2})(c_{i})&=0\text{ for }i\neq 1\\\ D(c_{2i+1})(c_{j})&=0\text{ for
}j\neq 2i+2\\\ D(c_{2i+2})(c_{j})&=0\text{ for }j\neq 2i+1\\\
D(c_{d-1})(c_{j})&=0\text{ for }j\neq d\\\ D(p)(c_{j})&=0\text{ for
}j\notin\\{2,d-1\\}\end{aligned}$

All elements above belong to $\mathfrak{m}$.

Moreover, for each $1\leqslant i,j\leqslant g-2$ we have that the elements
$T(c_{2i+1}+c_{2j+1})^{2}$, $T(c_{2i+1}+c_{2j+2})^{2}$ and
$T(c_{2i+2}+c_{2j+2})^{2}$ all belong to $M$. Indeed, if $i=j$ then
$|\omega(c_{2i+1},c_{2j+2})|=1$, so $T(c_{2i+1}+c_{2j+2})\in M$. Otherwise,
this follows from a result by the fourth author using $b$ as the auxiliary
vector [Gut19, Corollary 2.8]. Observe that

$\displaystyle E(c_{2i+1}+c_{2j+1})(c_{2i+2})$
$\displaystyle=2(c_{2i+1}+c_{2j+1})$ $\displaystyle
E(c_{2i+1}+c_{2j+1})(c_{2j+2})$ $\displaystyle=2(c_{2i+1}+c_{2j+1})$
$\displaystyle E(c_{2i+1}+c_{2j+2})(c_{2i+2})$
$\displaystyle=2(c_{2i+1}+c_{2j+2})$ $\displaystyle
E(c_{2i+1}+c_{2j+2})(c_{2j+1})$ $\displaystyle=-2(c_{2i+1}+c_{2j+2})$
$\displaystyle E(c_{2i+2}+c_{2j+2})(c_{2i+1})$
$\displaystyle=-2(c_{2i+2}+c_{2j+2})$ $\displaystyle
E(c_{2i+2}+c_{2j+2})(c_{2j+1})$ $\displaystyle=-2(c_{2i+2}+c_{2j+2}).$

The number of elements of the form $D(\ast)$ is $d-1=2g-1$. The number of
elements of the form $E(\ast)$ are $\binom{2g-4}{2}$.

With basis $\\{c_{1},\dotsc,c_{d}\\}$, we identify the vector space of linear
transformations of $H_{1}(S;\mathbb{R})$ with $\text{Mat}_{d\times
d}(\mathbb{R})$. We may then associate a matrix to the elements $D(\ast)$ and
$E(\ast)$ considered above. Let $M_{i,j}$ be the matrix with the $(i,j)$-entry
one and all other entries zero. We may then write the matrices for $D(\ast)$
and $E(\ast)$ as linear combinations of $M_{i,j}$.

From the calculations above, we note that

  * •

The matrix for $D(c_{2i+1})$ is $M_{(2i+1),(2i+2)}$.

  * •

The matrix for $D(c_{2i+2})$ is $-M_{(2i+2),(2i+1)}$.

  * •

The matrix $M_{(2i+2),(2j+1)}$ features only in the linear combination of the
matrix for $E(c_{2i+1}+c_{2j+1})$.

  * •

The matrix $M_{(2i+2),(2j+2)}$ features only in the linear combination of the
matrix for $E(c_{2i+1}+c_{2j+2})$.

  * •

The matrix $M_{(2i+1),(2j+2)}$ features only in the linear combination of the
matrix for $E(c_{2i+2}+c_{2j+2})$.

It follows that all the elements considered above are linearly independent in
$\text{Mat}_{d\times d}(\mathbb{R})$. Thus,

$\dim_{\mathbb{R}}\mathfrak{m}\geqslant\binom{2g-4}{2}+2g-1=2g^{2}-7g+9.$

We conclude that if $g\geqslant 6$, we have that

$\displaystyle\dim_{\mathbb{R}}\mathfrak{m}$
$\displaystyle>\dim_{\mathbb{R}}\mathfrak{su}(g/2,g/2)=g^{2}-1$
$\displaystyle\dim_{\mathbb{R}}\mathfrak{m}$
$\displaystyle>\dim_{\mathbb{R}}\mathfrak{so}(2n-1,2)=\log_{2}(2g)\log_{2}(8g^{2})\text{
for }n=\log_{2}(2g)$ $\displaystyle\dim_{\mathbb{R}}\mathfrak{m}$
$\displaystyle>\dim_{\mathbb{R}}\mathfrak{so}(2n-2,2)=\log_{2}(4g)\log_{2}(8g^{2})\text{
for }n=\log_{2}(4g)$ $\displaystyle\dim_{\mathbb{R}}\mathfrak{m}$
$\displaystyle>\dim_{\mathbb{R}}\mathfrak{so}^{*}(g/2)=g(g-1)/2.$

Hence, in these cases, $\mathfrak{m}=\mathfrak{sp}(2g,\mathbb{R})$ and that
$M=\operatorname{Sp}(2g,\mathbb{R})$. Recall directly from the list that
$\mathfrak{m}$ is $\mathfrak{sp}(2g,\mathbb{R})$ when $g$ is odd, as the
action of $M$ on $H_{1}(S;\mathbb{R})$ is strongly irreducible.

Since minimal strata only occur for $g\geqslant 3$, the only remaining case is
$g=4$. We treat this case in Appendix B.

###  9.13. Hyperelliptic components with two singularities

Let $r,s\geqslant 1$ be odd integers. Consider the permutation $\pi_{r,s}$
defined by

$\setcounter{MaxMatrixCols}{11}\begin{pmatrix}A&0&1&2&\cdots&r-1&A&r&r+1&\cdots&r+s\\\
r+s&r+s-1&\cdots&r&B&r-1&r-2&r-3&\cdots&0&B\\\ \end{pmatrix}.$

The embodying component for this permutation is
$\mathcal{Q}(2r,2s)^{\mathrm{hyp}}$ and we can assume that $r\geqslant s$. See
Figure 9.14 for a flat surface in $\mathcal{Q}(6,2)^{\mathrm{hyp}}$. The genus
of the underlying surface is $g=(r+s+2)/2$. Let:

  * •

$c_{0},\dotsc,c_{r+s}$ be the homology classes of the solid curves;

  * •

$c_{A}$ and $c_{B}$ be the homology classes of the dashed curves; and

  * •

$c_{AB}$ be the homology class of the dotted curve.

These cycles form a spanning set of the relative homology
$H_{1}(S,Z;\mathbb{Z})$. As a relative cycle,
$c_{AB}=-c_{s}+c_{r+s}-c_{A}+c_{B}$.

In absolute homology, $c_{A}+c_{B}=0$. Excising $c_{B}$ (or $c_{A}$) we obtain
exactly $2g=r+s+2$ curves. So $\\{c_{0},\dotsc,c_{r+s}\\}\cup\\{c_{A}\\}$ is a
basis of $H_{1}(S;\mathbb{Z})$. Note that

  * •

all cycles in $\\{c_{0},\dotsc,c_{r+s}\\}\cup\\{c_{AB}\\}$ are represented by
core curves of cylinders and hence all $T(c_{j})$, for $0\leqslant j\leqslant
r+s$, and $T(c_{A})$ are in $M$; and

  * •

any pair of cycles in $\\{c_{0},\dotsc,c_{r+s}\\}$ intersect. Moreover,
$c_{AB}$ intersects $c_{0}$.

Thus, the basis satisfies the hypothesis of Lemma 9.8 which proves that the
action of $M$ on $H_{1}(S;\mathbb{R})$ is strongly irreducible.

We again consider the Dehn twist $T(c)$ in a cycle $c$ as a symplectic
transvection. If $T(c)^{k}\in M$, then $T(c)^{k}-\mathrm{Id}\in\mathfrak{m}$.
As notation, let $D(c)=T(c)-\mathrm{Id}$.

The cycles $c_{i}$, for $0\leqslant i\leqslant r+s$, and $c_{i}+c_{j}$, for
$0\leqslant i<j\leqslant r+s$, are core curves of cylinders on the flat
surface. Hence, $T(c_{i}),T(c_{i}+c_{j})\in M$ for each $0\leqslant
i<j\leqslant r+s$ and, thus, $D(c_{i}),D(c_{i}+c_{j})\in\mathfrak{m}$. As
before, we use the basis $\\{c_{0},c_{1},\dotsc,c_{r+s},c_{A}\\}$ to identify
linear transformations of $H_{1}(S;\mathbb{R})$ with $\text{Mat}_{2g\times
2g}(\mathbb{R})$. Again as before, for $0\leqslant i,j\leqslant r+s$ let
$M_{i,j}$ be the matrix with $(i,j)$-entry one and all remaining entries zero.
Similarly, we have the definitions for $M_{i,A}$ and $M_{A,A}$.

Figure 9.14. Curves for the hyperelliptic case.

We use the following notation. If $P(i,j)$ is a logical proposition on $i$ and
$j$, we define

$\llbracket P(i,j)\rrbracket=\begin{cases}1&\text{if }P(i,j)\text{ is true}\\\
0&\text{if }P(i,j)\text{ is false}.\end{cases}$

We then note that

$D(c_{i})=-\sum_{k<i}M_{i,k}+\sum_{k>i}M_{i,k}+\llbracket i<r\rrbracket
M_{i,A}.$

Similarly, with $i<j$ note that

$\displaystyle D(c_{i}+c_{j})=$
$\displaystyle{}-(M_{i,i}+M_{j,i})+(M_{i,j}+M_{j,j})$
$\displaystyle{}-2\sum_{k<i}(M_{i,k}+M_{j,k})+2\sum_{j<k}(M_{i,k}+M_{j,k})$
$\displaystyle{}+2\llbracket i<j<r\rrbracket(M_{i,A}+M_{j,A})$
$\displaystyle{}+\llbracket i<r\leqslant j\rrbracket(M_{i,A}+M_{j,A}).$

######  Lemma 9.15.

The collection of matrices $\\{D(c_{i})\\}\cup\\{D(c_{i}+c_{j})\\}_{i<j}$ are
linearly independent.

###### Proof.

Any linear combination can be regrouped as $\sum_{i=0}^{r+s}S_{i}$ where

$S_{i}=a_{i}D(c_{i})+\sum_{j=i+1}^{r+s}a_{i,j}D(c_{i}+c_{j}).$

In particular, when $i=r+s$ the summation is empty and
$S_{r+s}=a_{r+s}D(c_{r+s})$.

Note that the matrices in $S_{0}$ are the only ones in the whole linear
combination whose first row does not vanish. Inductively, the matrices in
$S_{i+1}$ are the only ones outside $S_{0},\dotsc,S_{i}$ whose $(i+1)$-th row
is empty. Hence, it suffices to argue that each such collection
$\\{D(c_{i}),D(c_{i}+c_{i+1}),\dotsc,D(c_{i}+c_{r+s})\\}$ is linearly
independent.

Let $0\leqslant i\leqslant r+s$ and let $W$ be the $i$-th row of $D(c_{i})$.
Moreover, for $i+1\leqslant j\leqslant r+s$, let $V_{j}$ be the $i$-th row of
$D(c_{i}+c_{j})$.

######  Claim 9.16.

The row vectors $W,V_{i+1},\dotsc,V_{r+s}$ are linearly independent.

###### Proof (Claim).

We index the standard basis in $\mathbb{R}^{2g}$ as
$\\{e_{0},\dotsc,e_{r+s},e_{A}\\}$. From the formulae above, we deduce

$W=-\sum_{k<i}e_{k}+\sum_{k>i}e_{k}+\llbracket i<r\rrbracket e_{A}.$

and for $i+1\leqslant j\leqslant r+s$

$\displaystyle V_{j}=$
$\displaystyle{}-2\sum_{k<i}e_{k}-e_{i}+e_{j}+2\sum_{k>j}e_{k}$
$\displaystyle{}+(2\llbracket i<j<r\rrbracket+\llbracket i<r\leqslant
j\rrbracket)e_{A}$

Consider $V_{r+s}=-2\sum_{k<i}e_{k}-e_{i}+e_{r+s}+\llbracket i<r\rrbracket
e_{A}$. Observe that the linear combination of canonical vectors realising
$V_{j}-2V_{r+s}$ does not feature $e_{r+s}$ for $i+1\leqslant j<r+s$. Thus, we
redefine $V_{j}$ to be $V_{j}-2V_{r+s}$ for such $j$ and continue inductively.
That is, the next step redefine $V_{j}$ to be $V_{j}-2V_{r+s-1}$ for every
$i+1\leqslant j<r+s-1$.

We obtain that

$\displaystyle V_{j}={}$ $\displaystyle
2(-1)^{j+1}\sum_{k<i}e_{k}+(-1)^{j+1}e_{i}+e_{j}$
$\displaystyle{}+2(-1)^{j}\llbracket i<j<r\rrbracket e_{A}$
$\displaystyle{}+(-1)^{j}\llbracket i<r\leqslant j\rrbracket e_{A}.$

We also apply a similar process to $W$, redefining it to be
$W-V_{r+s}-V_{r+s-1}-\dotsb-V_{i+1}$. Then, we obtain that

$W=\begin{cases}-\sum_{k<i}e_{k}+\llbracket i<r\rrbracket e_{A}&\text{ if $i$
is even}\\\ +\sum_{k\leqslant i}e_{k}-\llbracket i<r\rrbracket e_{A}&\text{ if
$i$ is odd}.\end{cases}$

We observe

  * •

for $i+1\leqslant j\leqslant r+s$, the vector $e_{j}$ is featured only in the
linear combination for $V_{j}$, and

  * •

the vector $W$ is in the kernel of the projection
$\mathbb{R}^{2g}\to\mathbb{R}^{r+s-i}$ given by
$(u_{0},\dotsc,u_{r+s})\mapsto(u_{i+1},\dotsc,u_{r+s})$.

We thus conclude the claim. ∎

By the claim, the collection $\\{D(c_{i})\\}\cup\\{D(c_{i}+c_{j})\\}_{i<j}$
for each fixed $i$ is linearly independent. The lemma then follows from our
grouping. ∎

By Lemma 9.15,

$\dim_{\mathbb{R}}\mathfrak{m}\geqslant\binom{2g-1}{2}+2g-1=g(2g-1).$

We conclude that if $g\geqslant 5$, we have that

$\displaystyle\dim_{\mathbb{R}}\mathfrak{m}$
$\displaystyle>\dim_{\mathbb{R}}\mathfrak{su}(g/2,g/2)=g^{2}-1$
$\displaystyle\dim_{\mathbb{R}}\mathfrak{m}$
$\displaystyle>\dim_{\mathbb{R}}\mathfrak{so}(2n-1,2)=\log_{2}(2g)\log_{2}(8g^{2})\text{
for }n=\log_{2}(2g)$ $\displaystyle\dim_{\mathbb{R}}\mathfrak{m}$
$\displaystyle>\dim_{\mathbb{R}}\mathfrak{so}(2n-2,2)=\log_{2}(4g)\log_{2}(8g^{2})\text{
for }n=\log_{2}(4g)$ $\displaystyle\dim_{\mathbb{R}}\mathfrak{m}$
$\displaystyle>\dim_{\mathbb{R}}\mathfrak{so}^{*}(g/2,g/2)=g(g-1)/2.$

Thus, we obtain that $\mathfrak{m}=\mathfrak{sp}(2g,\mathbb{R})$ and that
$M=\operatorname{Sp}(2g,\mathbb{R})$ for $g\geqslant 5$.

To extend to $g=4$, we include $D(c_{AB})$ in the collection of matrices and
prove linear independence. Note that

$\displaystyle D(C_{AB})={}$ $\displaystyle 3(M_{0,0}-M_{s,0}-M_{A,0})$
$\displaystyle{}+4\sum_{k<s}(M_{0,k}-M_{s,k}-M_{A,k})$
$\displaystyle{}+2\sum_{k=s+1}^{r-1}(M_{0,k}-M_{s,k}-M_{A,k}).$

We expand the collection $S_{0}$ in the proof of Lemma 9.15 to include
$D(C_{AB})$. Let $\\{W,V_{1},\dotsc,V_{r+s}$ be the vectors that arise from
$S_{0}$ in 9.16. It suffices to show that the first row of $D(C_{AB})$ cannot
be written as a linear combination of $\\{W,V_{1},\dots,V_{r-1}\\}$. This is
readily seen after the vectors $W,V_{1},\dotsc,V_{r-1}$ are redefined as per
the Gaussian elimination process described in the proof of 9.16.

For $g=3$, we have directly from the list that
$\mathfrak{m}=\mathfrak{sp}(2g,\mathbb{R})$ since the action of $M$ on
$H_{1}(S;\mathbb{R})$ is strongly irreducible.

The only remaining case is $g=2$. Here, we have that

$\displaystyle\dim_{\mathbb{R}}\mathfrak{m}\geqslant 6$
$\displaystyle>3=\dim_{\mathbb{R}}\mathfrak{su}(1,1)$
$\displaystyle\mathfrak{sp}(4,\mathbb{R})$
$\displaystyle\cong\mathfrak{so}(3,2)$
$\displaystyle\dim_{\mathbb{R}}\mathfrak{sp}(4,\mathbb{R})=10$
$\displaystyle<15=\dim_{\mathbb{R}}\mathfrak{so}(4,2),$

so the only possibility for $\mathfrak{m}$ is
$\mathfrak{sp}(4,\mathbb{R})\cong\mathfrak{so}(3,2)$.

###  9.17. Exceptional non-minimal strata

In Table 1, we exhibit an explicit simple extension from a component of a
minimal stratum to each non-hyperelliptic component of exceptional non-minimal
strata that was not already treated by the fourth author [Gut17, Table 1],
except for $\mathcal{Q}(9,-1)^{\mathrm{irr}}$. Indeed, there does not exist a
simple extension from $\mathcal{Q}(8)$ to this last component as noted by
Lanneau [Lan08], so we treat it separately in Appendix B. These computations
were performed by using the surface_dynamics package for SageMath [Ste+20].

Source | Target | Permutation
---|---|---
$\mathcal{Q}(8)$ | $\mathcal{Q}(9,-1)^{\mathrm{reg}}$ | $\begin{pmatrix}1&2&1&3&4&3&5&2\\\ &6&5&4&A&A&6\end{pmatrix}$
$\mathcal{Q}(12)^{\mathrm{reg}}$ | $\mathcal{Q}(6,6)^{\mathrm{reg}}$ | $\begin{pmatrix}1&2&A&3&4&5&A&6&7&5\\\ &2&4&6&8&7&8&3&1\end{pmatrix}$
$\mathcal{Q}(12)^{\mathrm{irr}}$ | $\mathcal{Q}(6,6)^{\mathrm{irr}}$ | $\begin{pmatrix}&1&2&3&4&3&5&6&7\\\ 8&1&6&8&A&4&2&7&A&5\end{pmatrix}$
$\mathcal{Q}(12)^{\mathrm{reg}}$ | $\mathcal{Q}(9,3)^{\mathrm{reg}}$ | $\begin{pmatrix}1&2&3&A&4&3&5&6&7\\\ 2&4&6&8&7&8&5&A&1\end{pmatrix}$
$\mathcal{Q}(12)^{\mathrm{irr}}$ | $\mathcal{Q}(9,3)^{\mathrm{irr}}$ | $\begin{pmatrix}1&2&3&4&5&A&4&A&6&7\\\ &2&6&8&5&3&7&8&1\end{pmatrix}$
Table 1. Explicit extensions into non-hyperelliptic components of exceptional
strata with less than three singularities. The permutation in the third column
represents the component in the second column. Erasing letters $A$ and $B$
produces a permutation representing the component of a minimal stratum in the
first column.

##  10\. Simplicity

We now prove the Kontsevich–Zorich conjecture.

######  Theorem 10.1.

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.

###### Proof.

Let $\mathcal{C}$ be any component of a stratum of abelian or quadratic
differentials. We have the following facts.

  * •

The Teichmüller flow on a finite cover of $\mathcal{C}$ admits a coding as a
countable shift with approximate product structure.

  * •

We have established that the plus and minus cocycles lifted to this cover are
locally constant and integrable.

  * •

We have also established that associated monoids for the plus and minus
cocycles (lifted to this cover) are Zariski dense in the corresponding
symplectic groups.

By 6.9, it follows that the Lyapunov spectra for the plus and minus cocycles
are simple. ∎

##  Appendix A Examples

###  A.1. The decomposition of $\mathcal{C}^{\mathrm{root}}$ is not polytopal

(a) $\gamma(0)$.

(b) $\gamma(1)$.

Figure A.2. A curve $\gamma\colon[0,1]\to\mathcal{H}(0,0)^{\mathrm{root}}$
that ends at $\mathcal{V}$. The real periods are deformed following the
arrows, while the imaginary periods remain constant.

In this section, we present an explicit example showing that the decomposition
of $\mathcal{C}^{\mathrm{root}}$ into the union of the
$\overline{\mathcal{C}_{\pi}}$, where $\pi\in\mathcal{R}$, is “not polytopal”.
Indeed, a compact arc in $\mathcal{C}^{\mathrm{root}}$ may intersect
$\mathcal{S}=\mathcal{C}^{\mathrm{root}}-\bigcup_{\pi\in\mathcal{R}}\mathcal{C}_{\pi}$
infinitely many times even if it is transverse to $\mathcal{V}$.

Let $\mathcal{C}=\mathcal{H}(0,0)$ and consider the curve shown in Figure A.2.
The rooted differential $\gamma(1)$ contains a “wide” vertical cylinder, that
is, a vertical cylinder such that an arc of length one emanating from the root
ends before it crosses the cylinder entirely. On the other hand, we assume
that that the rooted differential $\gamma(0)$ is doubly non-vanishing, and
that it admits a normalised zippered rectangles construction for the
underlying permutation $\pi=\big{(}\begin{smallmatrix}1&2&3\\\
3&2&1\end{smallmatrix}\big{)}$.

Starting from $s=0$ and as $s$ increases, the normalised base-arc shrinks
until its length is exactly equal to $1+\min\\{x_{1},x_{3}\\}$ at $s=s_{1}>0$.
Thus, $\gamma(s)$ admits a zippered rectangles construction for every
$0\leqslant s<s_{1}$, but $\gamma(s_{1})$ does not. Indeed, $\gamma(s_{1})$
belongs to a flow face. As $\gamma$ passes through this flow face, a forward
Rauzy move must be performed to again obtain a normalised zippered rectangles
construction. The winning letter is $3$, so the resulting permutation after
the Rauzy move is again $\pi$.

This process continues inductively. Indeed, starting from $s=s_{k}$, for any
integer $k\geqslant 1$, and as $s$ increases, the normalised base-arc
continues to shrink until the curve hits the flow face again at
$s=s_{k+1}>s_{k}$. Thus, $\gamma(s)$ admits a normalised zippered rectangles
construction for every $s_{k}<s<s_{k+1}$. A Rauzy move must be performed when
the curve crosses the flow face; the winning letter continues to be $3$.
Hence, the resulting permutation is again $\pi$.

In summary, there exists a countable collection
$0<s_{1}<\dotsb<s_{k}<s_{k+1}<\dotsb{}$ such that:

  1. (1)

$\gamma(s)$ admits a normalised zippered rectangles construction for
$0\leqslant s<s_{1}$ and every $s_{k}<s<s_{k+1}$ for $k\geqslant 1$;

  2. (2)

$\gamma(s_{k})$ belongs to a flow face for every $k\geqslant 1$.

Thus, $\gamma$ intersects $\mathcal{S}$ infinitely many times and there is no
finite Rauzy–Veech sequence shadowing $\gamma$. Moreover, as this process
unfolds, the width of the rectangle $R_{1}$ goes to zero, while its height
diverges grows indefinitely. Hence, the normalised zippered rectangles
constructions along $\gamma$ become more and more degenerate and do not
converge to a well-defined element of $P_{\pi}$. See Figure A.3 for an
illustration of this phenomenon.

(a) $\gamma(0)$.

(b) $\gamma(s)$ for $s_{1}<s<s_{2}$.

(c) $\gamma(s)$ for $s_{2}<s<s_{3}$.

Figure A.3. Normalised zippered rectangles construction of $\gamma(s)$.

Similar examples exist also for _toppling faces_ , that is, when either some
width $x_{\alpha}$ or some zipper height goes to zero. It is possible to find
a compact arc in $\mathcal{C}^{\mathrm{root}}$ transverse to $\mathcal{V}$
that intersects infinitely many toppling faces. Indeed, a simple way to obtain
such an example is to consider a horizontal slit $J$ that does not meet the
normalised base-arc $I$, and such that some vertical segments emanating from
$I$ meet $J$ before their first return. Then, the slit can be rotated until it
becomes vertical (in a way that it still does not meet the base-arc). This
forces the widths of some rectangles to hit zero infinitely many times before
the slit becomes vertical.

###  A.4. Crossing a toppling face

In this section, we present concretely the construction used in the based loop
theorem, namely Theorem 4.23, in which any loop in
$\pi_{1}(\mathcal{C}^{\mathrm{root}},q_{0})$ is written as a finite
concatenation of paths that are forward (or backward) Teichmüller segments or
are contained inside a polytope. The path in $\mathcal{C}^{\mathrm{root}}$
that we present is not closed, but it still illustrates the key point. For
more complicate paths or loops, this procedure has to be done several times.

(a) $\gamma(0)$.

(b) $\gamma(1/2)$.

(c) $\gamma(1)$.

Figure A.5. A curve $\gamma\colon[0,1]\to\mathcal{H}(0,0)^{\mathrm{root}}$
that passes through $\mathcal{V}$. The real periods are deformed following the
arrows, while the imaginary periods remain constant.

Let $\mathcal{C}=\mathcal{H}(0,0)$. Consider the path
$\gamma\colon[0,1]\to\mathcal{C}^{\mathrm{root}}$ illustrated in Figure A.5.
Assume that $\gamma(0)$ and $\gamma(1)$ are doubly non-vanishing, while
$\gamma(1/2)$ has a vertical saddle connection and, thus, belongs to
$\mathcal{V}$.

Figure A.6. Normalised zippered rectangles construction of $\gamma(0)$.

Since $\gamma(0)$ is doubly non-vanishing, it admits a normalised base-arc.
The resulting zippered rectangles construction, with underlying permutation
$\pi=\big{(}\begin{smallmatrix}1&2&3\\\ 3&2&1\end{smallmatrix}\big{)}$, is
shown in Figure A.6.

If $0\leqslant s<1/2$, a parameter $(x_{s},y_{s})\in P_{\pi}$ of this zippered
rectangles construction satisfies $q_{\pi}(x_{s},y_{s})=\gamma(s)$. As $s$
increases towards $1/2$, these parameters approach the boundary of $P_{\pi}$
and $\gamma(1/2)\notin\mathcal{C}_{\pi}$. In particular, as $s$ increases to
$1/2$, the width $x_{2}$ tends to zero while all other parameters stay bounded
away from zero and infinity, and thus $\gamma(1/2)$ can be said to lie on a
toppling face.

(a) Before cutting and pasting.

(b) After cutting and pasting by two backward Rauzy moves.

Figure A.7. Zippered rectangles construction of $\gamma(1/2)$ with a base-arc
of length at least $5/3$.

On the other hand, $\gamma(1/2)$ is not doubly vanishing. It thus admits a
base-arc. We take a base-arc of length at least $5/3$, since the interior of
any horizontal segment with length at least $5/3$ meets every leaf of the
vertical foliation. The resulting (unnormalised) zippered rectangles
construction, with underlying permutation
$\sigma=\big{(}\begin{smallmatrix}1&3&2\\\ 3&2&1\end{smallmatrix}\big{)}$, is
shown in Figure A.7. As Teichmüller flow by $T=-\log(5/3)$ normalises the
base-arc, we have that $g_{T}(\gamma(1/2))\in\mathcal{C}_{\sigma}$. Observe
that $\sigma$ is obtained from $\pi$ by two backward Rauzy moves.

(a) $\gamma(0)$.

(b) $\gamma(1)$.

Figure A.8. Zippered rectangles constructions with a base-arc of length at
least $5/3$.

For “large” deformations of parameters, the zippered rectangles construction
with a base-arc of length at least $5/3$ is contained in
$\mathcal{C}_{\sigma}$ after Teichmüller flow by $T=-\log(5/3)$. In
particular, we have that $g_{T}(\gamma(s))\in\mathcal{C}_{\sigma}$ for every
$s\in[0,1]$. Let $(x_{s}^{\prime},y_{s}^{\prime})\in P_{\sigma}$ such that
$q_{\sigma}(x_{s}^{\prime},y_{s}^{\prime})=g_{T}(\gamma(s))$. Figure A.8 shows
these zippered rectangles constructions for $\gamma(0)$ and $\gamma(1)$.

(a) Before cutting and pasting.

(b) After cutting and pasting by two backward Rauzy moves followed by two
forward Rauzy moves.

Figure A.9. Normalised zippered rectangles construction of $\gamma(1)$.

Finally, $\gamma(1)$ is also doubly non-vanishing. Thus it admits a normalised
base-arc. The resulting zippered rectangles construction, with underlying
permutation $\tau=\big{(}\begin{smallmatrix}1&2&3\\\
3&1&2\end{smallmatrix}\big{)}$, is shown in Figure A.9. Observe that $\tau$ is
obtained from $\pi$ by two backward Rauzy moves followed by two forward Rauzy
moves.

If $1/2<s\leqslant 1$, a parameter $(x_{s},y_{s})$ of this zippered rectangles
construction satisfies $q_{\tau}(x_{s},y_{s})=\gamma(s)$. As $s$ decreases
towards $1/2$, these parameters approach the boundary of $P_{\tau}$ and
$\gamma(1/2)\notin\mathcal{C}_{\tau}$.

Putting everything together, we obtain three open sets
$U_{0},U_{1/2},U_{1}\subseteq\mathcal{C}^{\mathrm{root}}$ satisfying:

  * •

$U_{0}=q_{\pi}(W_{0})$, where $W_{0}\subseteq P_{\pi}$ is an open set
containing $(x_{0},y_{0})$ whose closure is contained in $P_{\pi}$;

  * •

$U_{1/2}=g_{-T}(q_{\sigma}(W_{1/2}))$, where $W_{1/2}\subseteq P_{\sigma}$ is
an open set containing $(x_{s}^{\prime},y_{s}^{\prime})$ for every $s\in[0,1]$
whose closure is contained in $P_{\sigma}$; and

  * •

$U_{1}=q_{\tau}(W_{1})$, where $W_{1}\subseteq P_{\tau}$ is an open set
containing $(x_{1},y_{1})$ whose closure is contained in $P_{\tau}$.

Then, $\gamma$ is homotopic, relative to its endpoints, to the concatenation
of the paths:

  * •

$g_{t}\gamma(0)$ for $t\in[0,T]$;

  * •

$g_{T}\gamma(s)$ for $s\in[0,1]$; and

  * •

$g_{-t}\gamma(1)$ for $t\in[0,T]$.

Therefore, the combinatorial description of this concatenation is

$\begin{pmatrix}1&2&3\\\
3&2&1\end{pmatrix}\xrightarrow{\mathrm{b}^{-1}}\begin{pmatrix}1&3&2\\\
3&2&1\end{pmatrix}\xrightarrow{\mathrm{t}^{-1}}\begin{pmatrix}1&3&2\\\
3&2&1\end{pmatrix}\xrightarrow{\mathrm{b}}\begin{pmatrix}1&2&3\\\
3&2&1\end{pmatrix}\xrightarrow{\mathrm{t}}\begin{pmatrix}1&2&3\\\
3&1&2\end{pmatrix}$

which is the (undirected) Rauzy–Veech sequence shadowing $\gamma$.

##  Appendix B Zariski density of the remaining cases

In this section, we explicitly check the Zariski density for the plus piece of
the four remaining components, namely $\mathcal{Q}(5,-1)$,
$\mathcal{Q}(9,-1)^{\mathrm{irr}}$, $\mathcal{Q}(12)^{\mathrm{reg}}$ and
$\mathcal{Q}(12)^{\mathrm{irr}}$. We do this by using the following sufficient
criterion.

######  Criterion B.1 ([PR14, Theorem 9.10]).

Let $G$ be a subgroup of $\operatorname{Sp}(2g,\mathbb{Z})$. We have that $G$
is Zariski dense in $\operatorname{Sp}(2g,\mathbb{R})$ provided the Zariski
closure of $G$ is not a power of $\operatorname{SL}(2,\mathbb{R})$, and there
exist elements $A,B\in G$ satisfying:

  1. (1)

$A$ is Galois-pinching in the sense of Matheus–Möller–Yoccoz [MMY15]. That is,
all of its eigenvalues are real and have distinct moduli, and the Galois group
of its characteristic polynomial is maximal; and

  2. (2)

$B$ has infinite order and does not commute with $A$.

Since $A$ is symplectic, its characteristic polynomial $P$ is reciprocal.
Thus, the Galois group of $P$ is contained inside an appropriate
hyperoctahedral group. Hence, this group is maximal if and only if it has
order $2^{g}g!$. Moreover, if a monodromy group is a power of
$\operatorname{SL}(2,\mathbb{R})$, then it has more than one compact factor,
which is forbidden for strongly irreducible pieces [Theorem 1.2]Fil17[Theorem
1.1]Esk-Fil-Wri. Thus, if we can establish B.1 together with Lemma 9.8, we
obtain the Zariski density of $G$ inside $\operatorname{Sp}(2,\mathbb{R})$.

For all of the remaining components, we will follow the same strategy to show
that the hypotheses of B.1 hold. We will start with a specific permutation
$\pi$. We will then exhibit two cycles $\delta_{1}$ and $\delta_{2}$ based at
$\pi$ in the reduced Rauzy diagram such that their induced matrices $A_{1}$
and $A_{2}$ in a preferred basis (for the action on absolute homology) can be
combined to produce the matrices $A$ and $B$ that the criterion requires.
Specifically, in all cases, $B$ can be taken to be $A_{1}$ and $A$ can be
taken to be $A_{1}A_{2}$.

These cycles $\delta_{1}$ and $\delta_{2}$ were found by a randomised computer
search on the reduced Rauzy diagrams. We tried to choose cycles that are
relatively short to avoid the resulting matrices having entries that are
greater than $100$.

###  B.2. Zariski density of $\mathcal{Q}(5,-1)$

Let

$\pi=\begin{pmatrix}1&2&3&2&4\\\ 4&5&5&3&1\end{pmatrix}$

and

$\displaystyle\delta_{1}={}$
$\displaystyle\mathrm{b}^{3}\mathrm{t}^{2}\mathrm{b}^{3}\mathrm{t}\mathrm{b}^{2}\mathrm{t}\mathrm{b}^{3}\mathrm{t}^{2}\mathrm{b}^{3}$
$\displaystyle\delta_{2}={}$
$\displaystyle\mathrm{t}^{2}\mathrm{b}\mathrm{t}\mathrm{b}\mathrm{t}\mathrm{b}\mathrm{t}^{3}\mathrm{b}\mathrm{t}\mathrm{b}\mathrm{t}\mathrm{b}^{2}\mathrm{t}^{2}$

Figure B.3. Representative of $\mathcal{Q}(5,-1)$.

Consider the four curves $c_{1},\dotsc,c_{4}$ depicted in Figure B.3 as solid
or dashed lines. These cycles form a basis for the absolute homology as their
intersection matrix is

$\Omega=\begin{pmatrix}0&0&-1&-1\\\ 0&0&1&0\\\ 1&-1&0&-1\\\
1&0&1&0\end{pmatrix}$

that has determinant $1$. On the other hand, the cycle $v\in
H_{1}(S;\mathbb{R})$ depicted in Figure B.3 as dash-dotted vertical lines can
be written as $v=-c_{2}+c_{3}+c_{4}$. Thus, the set
$B=\\{c_{1},c_{3},c_{4},v\\}$ readily satisfies the hypotheses of Lemma 9.8,
so the $M$-action is strongly irreducible.

In the chosen basis, the matrices induced by $\delta_{1}^{2}$ and
$\delta_{2}^{2}$ are

$A_{1}=\begin{pmatrix}1&-2&-2&0\\\ 0&-1&-2&0\\\ 0&0&1&2\\\
0&0&0&-1\end{pmatrix},\quad A_{2}=\begin{pmatrix}-1&0&0&0\\\ 0&-2&2&-1\\\
1&2&-1&2\\\ -2&1&-2&0\end{pmatrix}$

Then, $A=A_{1}A_{2}$ has the form

$A=\begin{pmatrix}-1&0&1&-2\\\ 2&2&-4&3\\\ 2&6&-7&0\\\ 0&5&-4&-4\\\
\end{pmatrix}$

The characteristic polynomial $P$ of $A$ is
$P(t)=t^{4}+10t^{3}+22t^{2}+10t+1$. We verified in Magma [BCP97] that $A$ is
Galois pinching, that is, it satisfies condition (1) of B.1. Setting
$B=A_{1}$, we similarly check that $B$ satisfies condition (2) of the
criterion. Thus, the plus piece of $\mathcal{Q}(5,-1)$ is Zariski dense inside
$\operatorname{Sp}(4,\mathbb{R})$.

###  B.4. Zariski density of $\mathcal{Q}(9,-1)^{\mathrm{irr}}$

Let

$\pi=\begin{pmatrix}1&2&3&4&5&6&3\\\ 7&7&6&5&4&2&1\end{pmatrix}$

and

$\displaystyle\delta_{1}={}$
$\displaystyle\mathrm{b}^{4}\mathrm{t}^{5}\mathrm{b}^{3}\mathrm{t}\mathrm{b}^{5}\mathrm{t}\mathrm{b}^{6}\mathrm{t}^{2}$
$\displaystyle\delta_{2}={}$
$\displaystyle\mathrm{b}^{4}\mathrm{t}^{2}\mathrm{b}^{3}\mathrm{t}^{3}\mathrm{b}^{7}\mathrm{t}^{3}\mathrm{b}\mathrm{t}^{3}\mathrm{b}\mathrm{t}^{2}\mathrm{b}^{2}\mathrm{t}^{3}\mathrm{b}^{2}\mathrm{t}^{2}\mathrm{b}^{2}\mathrm{t}^{2}\mathrm{b}^{2}$

Consider the six curves $c_{1},\dotsc,c_{6}$ depicted in Figure B.5 as solid
or dashed lines. These cycles form a basis for the absolute homology, as their
intersection matrix is

$\Omega=\begin{pmatrix}0&-1&0&-1&-1&-1\\\ 1&0&0&-1&-1&-1\\\ 0&0&0&1&1&1\\\
1&1&-1&0&-1&-1\\\ 1&1&-1&1&0&-1\\\ 1&1&-1&1&1&0\end{pmatrix}$

that has determinant $1$. On the other hand, the cycle $v\in
H_{1}(S;\mathbb{R})$ depicted in Figure B.5 as dash-dotted vertical lines can
be written as $v=c_{2}+c_{3}$. Thus, the set
$B=\\{c_{1},c_{2},c_{4},c_{5},c_{6},v\\}$ readily satisfies the hypotheses of
Lemma 9.8, so the $M$-action is strongly irreducible.

Figure B.5. Representative of $\mathcal{Q}(9,-1)^{\mathrm{irr}}$.

In the chosen basis, the matrices induced by $\delta_{1}^{2}$ and
$\delta_{2}^{2}$ are

$A_{1}=\begin{pmatrix}0&2&-1&-1&-1&-2\\\ 0&-1&0&0&0&0\\\ -1&-1&0&-2&-2&-1\\\
0&0&0&-1&0&0\\\ 0&0&0&0&-1&0\\\ 1&3&-1&2&2&0\end{pmatrix},\quad
A_{2}=\begin{pmatrix}-3&-10&-2&-4&-6&-4\\\ 1&3&0&2&2&0\\\ 2&3&-2&0&-1&0\\\
0&-2&-2&-1&-2&-2\\\ -3&-7&1&-2&-2&0\\\ 0&0&0&0&0&-1\end{pmatrix}$

Then, $A=A_{1}A_{2}$ has the form

$A=\begin{pmatrix}-2&0&2&0&-1&-1\\\ -6&-1&3&-2&0&-3\\\ 7&-1&-2&2&3&1\\\
3&-3&2&1&3&-2\\\ 5&-3&3&2&3&-2\\\ 8&-2&-2&2&5&0\end{pmatrix}$

The characteristic polynomial $P$ of $A$ is
$P(t)=t^{6}+t^{5}-22t^{4}-52t^{3}-22t^{2}+t+1$. Again, we use Magma to check
that $A$ is Galois pinching. Setting $B=A_{1}$, we can readily check that $B$
satisfies condition (2) of the criterion. Thus, the plus piece of
$\mathcal{Q}(9,-1)^{\mathrm{irr}}$ is Zariski dense inside
$\operatorname{Sp}(6,\mathbb{R})$.

###  B.6. Zariski density of $\mathcal{Q}(12)^{\mathrm{reg}}$

Let

$\pi=\begin{pmatrix}1&2&1&3&4&5&6&7\\\ 2&4&3&6&5&8&7&8\end{pmatrix}$

and

$\displaystyle\delta_{1}={}$
$\displaystyle\mathrm{b}^{4}\mathrm{t}^{2}\mathrm{b}\mathrm{t}^{6}\mathrm{b}\mathrm{t}^{4}\mathrm{b}^{2}\mathrm{t}\mathrm{b}^{4}\mathrm{t}^{2}\mathrm{b}^{7}\mathrm{t}^{2}\mathrm{b}^{2}\mathrm{t}\mathrm{b}^{5}\mathrm{t}^{4}\mathrm{b}\mathrm{t}\mathrm{b}\mathrm{t}\mathrm{b}$
$\displaystyle\delta_{2}={}$
$\displaystyle\mathrm{b}\mathrm{t}\mathrm{b}\mathrm{t}^{3}\mathrm{b}\mathrm{t}^{6}\mathrm{b}^{2}\mathrm{t}^{4}\mathrm{b}\mathrm{t}^{3}\mathrm{b}^{2}\mathrm{t}^{3}\mathrm{b}^{2}\mathrm{t}\mathrm{b}^{4}\mathrm{t}^{2}\mathrm{b}\mathrm{t}^{3}\mathrm{b}^{2}\mathrm{t}^{2}\mathrm{b}^{3}\mathrm{t}\mathrm{b}^{5}$

Let $M$ be the monodromy group of $\mathcal{Q}(12)^{\mathrm{reg}}$. We have
that the $M$-action is strongly irreducible by Lemma 9.12.

Consider the six curves $c_{1},\dotsc,c_{6}$ shown in Figure B.7. Ordered
appropriately, these curves form a symplectic basis. In this basis, the
matrices induced by $\delta_{1}^{2}$ and $\delta_{2}^{2}$ are

$\displaystyle A_{1}$ $\displaystyle=\begin{pmatrix}2&5&-1&7&6&2&1&0\\\
-1&0&1&-1&-2&1&3&0\\\ -1&2&-1&2&1&1&-2&0\\\ -1&-4&1&-6&-5&-2&0&0\\\
1&1&2&1&0&2&5&0\\\ -1&-3&1&-4&-4&-1&3&0\\\ 0&0&0&0&0&0&-1&0\\\
-1&-5&7&-11&-10&2&11&-1\end{pmatrix}$ $\displaystyle A_{2}$
$\displaystyle=\begin{pmatrix}1&3&-5&1&6&-7&-3&-1\\\ 0&-2&2&0&-3&3&1&1\\\
4&-1&-1&2&3&-4&1&0\\\ -7&5&-3&-2&-1&3&-5&-1\\\ 5&-5&3&2&-2&1&5&2\\\
2&0&-1&0&2&-3&0&-1\\\ 8&-1&-4&3&7&-10&0&-1\\\
2&3&-5&1&6&-7&-3&-2\end{pmatrix}$

Then, $A=A_{1}A_{2}$ has the form

$A=\begin{pmatrix}17&-7&15&-17&5&11&36&20\\\ 23&-13&25&-38&8&24&62&33\\\
6&0&0&7&0&-7&-5&-2\\\ 33&-20&34&-50&6&37&89&51\\\ 28&-16&31&-47&9&33&81&44\\\
15&-7&12&-15&3&8&27&15\\\ 21&-6&5&12&-6&-6&7&11\\\
1&-1&0&1&-2&1&1&2\end{pmatrix}$

Figure B.7. Representative of $\mathcal{Q}(12)^{\mathrm{reg}}$.

The characteristic polynomial $P$ of $A$ is
$P(t)=t^{8}+20t^{7}-1686t^{6}-24t^{5}+36258t^{4}-24t^{3}-1686t^{2}+20t+1$.
Using Magma, we can check that $A$ is Galois pinching. Setting $B=A_{1}$, we
can also similarly check that $B$ satisfies condition (2) of the criterion.
Thus, the plus piece of $\mathcal{Q}(12)^{\mathrm{reg}}$ is Zariski dense
inside $\operatorname{Sp}(8,\mathbb{R})$.

###  B.8. Zariski density of $\mathcal{Q}(12)^{\mathrm{irr}}$

Let

$\pi=\begin{pmatrix}1&2&1&3&4&5&6&7\\\ 2&6&5&4&3&8&7&8\end{pmatrix}$

and

$\displaystyle\delta_{1}={}$
$\displaystyle\mathrm{b}^{3}\mathrm{t}\mathrm{b}^{2}\mathrm{t}^{5}\mathrm{b}\mathrm{t}^{5}\mathrm{b}^{2}\mathrm{t}^{3}\mathrm{b}\mathrm{t}^{2}\mathrm{b}^{2}\mathrm{t}^{6}\mathrm{b}\mathrm{t}^{2}\mathrm{b}^{2}\mathrm{t}\mathrm{b}\mathrm{t}^{3}\mathrm{b}^{2}\mathrm{t}^{6}\mathrm{b}^{4}$
$\displaystyle\delta_{2}={}$
$\displaystyle\mathrm{b}^{5}\mathrm{t}^{5}\mathrm{b}\mathrm{t}\mathrm{b}\mathrm{t}^{4}\mathrm{b}^{3}\mathrm{t}^{7}\mathrm{b}^{2}\mathrm{t}\mathrm{b}\mathrm{t}^{4}\mathrm{b}^{4}$

Consider the six curves $c_{1},\dotsc,c_{6}$ depicted in Figure B.9 as solid,
dashed or dash-dotted lines. These cycles form a basis for the absolute
homology as their intersection matrix is

$\begin{pmatrix}0&1&0&0&0&0&0&0\\\ -1&0&0&0&0&0&0&0\\\ 0&0&0&-1&-1&-1&0&0\\\
0&0&1&0&-1&-1&0&0\\\ 0&0&1&1&0&-1&0&0\\\ 0&0&1&1&1&0&0&0\\\ 0&0&0&0&0&0&0&1\\\
0&0&0&0&0&0&-1&0\end{pmatrix}$

and has determinant $1$. On the other hand, the cycle $b$, depicted as the
slope-$1$ densely dotted lines, can be written as
$b=c_{1}+c_{2}-c_{3}+c_{4}+c_{7}-c_{8}$, and the cycle $p$ depicted as loosely
dotted horizontal lines can be written as $p=c_{1}+c_{8}$. Thus, the set
$B=\\{c_{2},c_{3},c_{4},c_{5},c_{6},c_{7},b,p\\}$ readily satisfies the
hypotheses of Lemma 9.8, so the $M$-action is strongly irreducible.

Figure B.9. Representative of $\mathcal{Q}(12)^{\mathrm{irr}}$.

In the chosen basis, the matrices induced by $\delta_{1}^{2}$ and
$\delta_{2}^{2}$ are

$\displaystyle A_{1}$ $\displaystyle=\begin{pmatrix}-1&6&-2&6&11&8&6&-4\\\
2&-5&0&-6&-9&-6&-4&6\\\ -1&-3&-2&-6&-8&-5&-3&4\\\ -1&-5&-1&-9&-11&-7&-5&6\\\
0&0&0&0&-1&0&0&0\\\ 1&3&1&6&8&4&3&-4\\\ -2&-4&2&-4&-8&-6&-5&0\\\
0&-6&2&-6&-11&-8&-6&3\\\ \end{pmatrix}$ $\displaystyle A_{2}$
$\displaystyle=\begin{pmatrix}-2&1&1&1&-1&-2&-3&0\\\ 3&-5&3&3&3&0&-2&0\\\
0&0&-1&0&0&0&0&0\\\ 0&0&0&-1&0&0&0&0\\\ 0&2&-2&-2&-1&2&2&0\\\
-3&4&-3&-3&-3&-1&2&0\\\ 0&0&0&0&0&0&-1&0\\\
-7&13&-9&-9&-7&2&5&-1\end{pmatrix}$

Then, $A=A_{1}A_{2}$ has the form

$A=\begin{pmatrix}6&-15&1&1&6&12&2&43\\\ -19&27&3&5&4&-25&4&-43\\\
-7&-19&2&1&12&18&-2&51\\\ -33&11&6&9&22&-11&4&13\\\
-41&34&8&11&21&-33&8&-29\\\ -24&30&5&7&8&-28&6&-40\\\
-15&24&3&5&4&-23&5&-35\\\ 32&-12&-4&-6&-16&10&0&5\end{pmatrix}$

The characteristic polynomial $P$ of $A$ is
$P(t)=t^{8}-47t^{7}-794t^{6}+11691t^{5}-22022t^{4}+11691t^{3}-794t^{2}-47t+1$.
By using Magma again, we can explicitly check that $A$ is Galois pinching.
Setting $B=A_{1}$, we can similarly check that $B$ satisfies condition (2) of
the criterion. Thus, the plus piece of $\mathcal{Q}(12)^{\mathrm{irr}}$ is
Zariski dense inside $\operatorname{Sp}(8,\mathbb{R})$.

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