Classification of Rauzy-Veech groups: proof of the Zorich conjecture
Rodolfo Guti\'errez-Romo
TL;DR
This paper classifies Rauzy-Veech groups across all strata of translation surfaces, proving they are related to arithmetic lattices and confirming Zorich's conjecture on their Zariski-density.
Contribution
It provides a complete classification of Rauzy-Veech groups and proves the Zorich conjecture on their Zariski-density in the context of translation surfaces.
Findings
Rauzy-Veech groups are commensurable to arithmetic lattices of symplectic groups
Confirmed Zorich's conjecture on Zariski-density of these groups
Classification applies to all connected components of all strata
Abstract
We classify the Rauzy-Veech groups of all connected components of all strata of the moduli space of translation surfaces in absolute homology, showing, in particular, that they are commensurable to arithmetic lattices of symplectic groups. As a corollary, we prove a conjecture of Zorich about the Zariski-density of such groups.
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Classification of Rauzy–Veech groups: proof of the Zorich conjecture
Rodolfo Gutiérrez-Romo
Institut de Mathématiques de Jussieu – Paris Rive Gauche, UMR 7586, Bâtiment Sophie Germain, 75205 PARIS Cedex 13, France.
Abstract.
We classify the Rauzy–Veech groups of all connected components of all strata of the moduli space of translation surfaces in absolute homology, showing, in particular, that they are commensurable to arithmetic lattices of symplectic groups. As a corollary, we prove a conjecture of Zorich about the Zariski-density of such groups.
1. Introduction
The Kontsevich–Zorich conjecture states the simplicity of the Lyapunov spectra of almost all interval exchange transformations or translation flows with respect to the Masur–Veech measures. After an important partial progress by Forni in \citeyearF:deviations [For02], it was established in full generality by Avila and Viana in \citeyearAV:KZ_conjecture [AV07]. Nevertheless, their methods leave open a stronger conjecture of Zorich from \citeyearZ:wind: the Zariski-density of the Rauzy–Veech groups [Zor99]. Indeed, the main ingredient of Avila and Viana’s proof is the fact that Rauzy–Veech groups are pinching and twisting, which is implied by Zariski-density by the work of Benoist [Ben97]. However, the converse is not true: there are known examples of pinching and twisting groups with small Zariski closure [AMY17, Appendix A].
The pioneering work of Avila, Matheus and Yoccoz [AMY17] shows that this conjecture holds for the particular case of hyperelliptic Rauzy–Veech groups. Their methods also laid the groundwork for further results in this direction.
In this article we answer this conjecture affirmatively by establishing a stronger statement. Indeed, our main result is the following theorem:
Theorem 1.1**.**
At the level of absolute homology, the Rauzy–Veech group of any connected component of any stratum of the moduli space of genus translation surfaces is an explicit finite-index subgroup of . In particular, if and the stratum is connected, it is equal to its entire ambient symplectic group.
This theorem its proved in various steps along the article. A summary of our results is presented in 6.8.
We obtain a similar result for the monodromy group, since it contains the Rauzy–Veech group. The Zariski-density was known in this case [Fil17, Corollary 1.7].
Corollary 1.2**.**
The monodromy group of any connected component of any stratum of the moduli space of genus translation surfaces is commensurable to . In particular, if and the stratum is connected, it is equal to its entire ambient symplectic group.
To prove 1.1, we reduce the general conjecture to the case of minimal strata. Indeed, we show in 6.5 that the Rauzy–Veech group of any connected component of any stratum contains the Rauzy–Veech group of specific connected components of minimal strata of surfaces of the same genus. For the case of minimal strata, we prove the following theorem:
Theorem 1.3**.**
The Rauzy–Veech group of a non-hyperelliptic connected component of a minimal stratum of the moduli space of genus translation surfaces is the preimage of the orthogonal group by the modulus-two reduction , where is the usual quadratic form on modulus-two homology.
An immediate corollary is that these groups have finite index in . The proof of 1.3 is elementary and is divided into two parts. First, we show that the level-two congruence subgroup of (that is, the kernel of the modulus-two reduction) is contained in such Rauzy–Veech groups in 3.6 and 3.14. Then, we show that the modulus-two reduction is surjective onto in 4.8. Both parts of the proof rely on constructing explicit sets of generators inductively, first for the level-two congruence subgroup and then for .
Our methods also allow us to describe the Rauzy–Veech groups of the non-hyperelliptic connected components of for at the level of relative homology explicitly. Indeed, we show in 5.1 that they can be written as suitable subgroups of semi-direct products.
Besides giving a new proof of Avila and Viana’s theorem establishing the Kontsevich–Zorich conjecture, our main result has other applications: for instance, Magee [Mag16] proved that the Zorich conjecture implies the uniform spectral gap property (and the uniform rate of mixing of the Teichmüller geodesic flow) of congruence covers of connected components of the strata of the moduli space of translation surfaces.
The article is organized as follows. In Section 2 we review the basic definitions and background. Sections 3 and 4 contain the two parts of the proof of 1.3: respectively, that the Rauzy–Veech group of a minimal stratum contains the level-two congruence subgroup and that its modulus-two reduction is surjective onto the corresponding orthogonal group. In Section 5 we give an explicit description of the Rauzy–Veech groups of the non-hyperelliptic components of for at the level of relative homology. In Section 6 we complete the proof of the main theorem by showing that the general conjecture can be reduced to the case of minimal strata. Finally, we make some computations explicit for the base cases of the induction used to prove 1.3 in the Appendix.
2. Rauzy–Veech groups
2.1. Rauzy induction
In this section we will briefly recall the Rauzy induction algorithm on permutations, the definition of Kontsevich–Zorich matrices and Rauzy–Veech groups. We refer the reader to the lecture notes by Yoccoz [Yoc10] and the survey by Viana [Via06] for more details.
Let be a finite set of cardinality . Let be a pair of bijections , which can be interpreted as a permutation. We write as
[TABLE]
where for and . We define an alternate form indexed by as:
[TABLE]
We say that is irreducible if for any . Moreover, we say that is degenerate if there exists such that one of the following three conditions holds:
[TABLE]
[TABLE]
or
[TABLE]
Otherwise, is said to be nondegenerate. From now on, we will always assume that a pair of bijections is both irreducible and nondegenerate.
The Rauzy induction algorithm consists of two operations on , which we call top and bottom. The top operation is defined as:
[TABLE]
where satisfies . In this case, we call the winner and the loser of the algorithm.
The bottom operation is defined as:
[TABLE]
where satisfies . In this case, we call the winner and the loser of the algorithm.
Observe that if is irreducible and nondegenerate, then and are irreducible and nondegenerate.
Consider now the directed graph whose vertices are the irreducible, nondegenerate pairs and such that is an edge if or . The connected (or, equivalently, strongly connected) components of this graph are called Rauzy classes. They are in a finite-to-one correspondence with the connected components of the strata of the moduli space of translation surfaces as was proven by Veech [Vee82].
2.2. Rauzy–Veech groups in homology
Let be a Rauzy class. We consider an undirected version of : for each arrow we add a reversed arrow . Now, let be an arrow in . Let and be, respectively, the winner and loser of the operation sending to . We define the Kontsevich–Zorich matrix indexed by as , where is the identity matrix and has only one non-zero coefficient, equal to , at position . Analogously, we define . Now consider a walk in starting at and ending at . We define , which satisfies \Omega_{\pi^{\prime}}=B_{\gamma}\Omega_{\pi}B_{\gamma}^{\mbox{\raisebox{0.99458pt}{\scalebox{0.6}{\intercal}}}}. In particular, if (that is, if is a cycle), one has that (acting on row vectors) belongs to . The Rauzy–Veech group of is the group generated by matrices of this form:
Definition 2.1**.**
Let be a Rauzy class and be a fixed vertex. We define the Rauzy–Veech group of as the set of matrices of the form where is a cycle on with endpoints at . We will always consider the action of on row vectors unless explicitly stated.
Observe if are vertices of the same Rauzy class , then and are isomorphic, so we can define the Rauzy–Veech group of a Rauzy class. Indeed, if is any walk joining and , then the conjugation by is an isomorphism between and and between and . This shows, in particular, that the Rauzy–Veech group of a Rauzy class has a well-defined index inside its ambient symplectic group.
Remark 2.2*.*
One could also define Rauzy–Veech monoids by considering loops on the directed graph instead of on the undirected graph . Nevertheless, the resulting object has the same Zariski closure as the Rauzy–Veech group, since the Zariski closure of a monoid is a group.
2.3. Rauzy–Veech groups in homotopy
In this section we follow the general discussion about Rauzy classes and Dehn twists by Avila, Matheus and Yoccoz [AMY17, Section 4].
Consider a Rauzy class . For every there is a canonical way to obtain a translation surface from an irreducible permutation. Namely, we consider the translation surface with length data and suspension data for each . We denote the resulting polygon by , the resulting surface by , its set of marked points by \mbox{\scalebox{1.05}{\Sigma}}_{\pi} and the pure mapping class group of relative to \mbox{\scalebox{1.05}{\Sigma}}_{\pi} by \mathrm{Mod}(M_{\pi},\mbox{\scalebox{1.05}{\Sigma}}_{\pi}). We equip with a basepoint and we set . We denote by \mbox{\scalebox{1.05}{\Sigma}}_{\pi}^{*} the set consisting of and the midpoints of the sides of .
For each arrow in there exists a homeomorphism respecting the naming of the sets of marked points \mbox{\scalebox{1.05}{\Sigma}}_{\pi}, \mbox{\scalebox{1.05}{\Sigma}}_{\pi^{\prime}} [AMY17, Section 4.1.2]. We denote by its isotopy class from (M_{\pi},\mbox{\scalebox{1.05}{\Sigma}}_{\pi}\cup\mbox{\scalebox{1.05}{\Sigma}}_{\pi}^{*}) to (M_{\pi},\mbox{\scalebox{1.05}{\Sigma}}_{\pi^{\prime}}\cup\mbox{\scalebox{1.05}{\Sigma}}_{\pi^{\prime}}^{*}) relative to \mbox{\scalebox{1.05}{\Sigma}}_{\pi}\cup\mbox{\scalebox{1.05}{\Sigma}}_{\pi}^{*}. We also define for the arrow . Now consider a walk in starting at and ending at . Similarly, we define . We can then define the modular Rauzy–Veech group in an analogous way to the homological case:
Definition 2.3**.**
Let be a Rauzy class and be a fixed vertex. We define the modular Rauzy–Veech group of as the set of mapping classes of the form [H_{\gamma}]\in\mathrm{Mod}(M_{\pi},\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) where is a cycle on with endpoints at .
The action of the map on the fundamental groups, where is an arrow in , can be made explicit. Indeed, let and be the winner and loser, respectively, of the Rauzy induction. Let be a simple closed curve starting from , then going to the midpoint of the top -side of and then coming back to , oriented upwards. It is easy to see that the homotopy classes of generate \pi_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi},\ast_{\pi}) and that their homology classes are a basis of H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}). We call the analogous curves for . Then, the induced homomorphism \pi_{1}(\gamma)\colon\pi_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi},\ast_{\pi})\to\pi_{1}(M_{\pi^{\prime}}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi^{\prime}},\ast_{\pi^{\prime}}) of on the fundamental groups is (up to homotopy): for every and
[TABLE]
The induced homomorphism of , written in terms of the homology classes of the curves and , is exactly . Therefore, the induced action of on homology is precisely . We will identify the matrix with a map H_{1}(M_{\pi^{\prime}}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi^{\prime}})\to H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) by using these bases.
2.3.1. Dehn twists
For each , we have that at least one of the Dehn twists along belongs to . These Dehn twists will be useful to generate .
Lemma 2.4**.**
Let be a vertex of a Rauzy class . Then, the left or right Dehn twist along belongs to for every .
Proof.
Fix and let be the isotopy class of . Let be a (possibly empty) walk on starting at ending on a vertex such that or , and such that is the first vertex of with this property. Such a walk exists because the Rauzy classes are strongly connected.
Let be the pure cycle at having as the winner. That is, all the arrows of are different and their winner is . Assume that it is a pure cycle of top type. We have that is equal to the left Dehn twist along , which we denote . Indeed, let be the isotopy class of . It easy to see that , since it is computed along a pure cycle [AMY17, Section 4.1.6]. We obtain that , where the last equality is straightforward by definition of Dehn twist. Our hypothesis on and the the previous discussion about the action of on the fundamental groups show that maps to , since is not the loser of any arrow of . We conclude that . If is a pure cycle of bottom type, similar computations show that is equal to the right Dehn twist along . ∎
Finally, the induced action T_{\alpha}\colon H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi})\to H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) of the (either left or right) Dehn twist has a simple expression in the basis consisting of the homology classes of . Indeed, the alternate form is defined as the intersection form of such curves, so we have that for any u\in H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}), where is the homology class of and is the bilinear form induced by . Since we will use the basis , we identify with the canonical vector whose only non-zero coordinate, equal to , is at position .
2.4. General properties of Rauzy–Veech groups
In this section we state some general properties of Rauzy–Veech groups. Recall that we consider the action on row vectors. First, we state the definition of a symplectic transvection:
Definition 2.5**.**
For a vector v\in H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}), we define the symplectic transvection along as:
[TABLE]
Observe that for any v\in H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}).
We have previously shown that for each . Conversely, for each primitive vector v\in H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) we can choose a simple closed curve on M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi} whose homology class is . If is the isotopy class of , then it is easy to see that the (either left or right) Dehn twist T_{\xi}\in\mathrm{Mod}(M_{\pi},\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) acts in homology as .
The following lemma allows us to construct more symplectic transvections from a set of generators.
Lemma 2.6**.**
Let v,w\in H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) such that . Then,
- •
;
- •
.
Proof.
It is a well-known fact that if f\in\mathrm{Mod}(M_{\pi},\mbox{\scalebox{1.05}{\Sigma}}_{\pi}), then [FM12, Fact 3.7]. The proof follows directly from this fact and the previous discussion, since the equality implies that and are primitive. ∎
We have that the group generated by symplectic transvections along canonical vectors is an invariant of the Rauzy class:
Lemma 2.7**.**
Let , be vertices of a Rauzy class . For , define and as the symplectic transvections along the homology classes of and , respectively. Let and be the subgroups generated by and , respectively. If is any walk in joining and , then the isomorphism defined by restricts to an isomorphism between and .
Proof.
By induction, we can assume that is the arrow . It is enough to show that, for each , there exists such that . Observe that for any v\in H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}). Therefore, if , we can choose and, if , we can choose , which is equal to either or depending on the type of . ∎
We will end up showing that, in absolute homology, the group in the previous lemma is the entire Rauzy–Veech group.
To formalize the idea of generating symplectic transvections from a set of generators using the previous lemma, we introduce the following definitions:
Definition 2.8**.**
For V\subseteq H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}), we consider the group generated by the symplectic transvections . We say that a set X\subseteq H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) is -closed if and implies for every . We denote by the smallest -closed set containing . By the previous lemma, .
We also need a way to generate squares of symplectic transvections along sum of vectors when their intersection form is zero. We have the following:
Lemma 2.9**.**
Let v_{1},v_{2},v_{3},v_{4}\in H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) such that
[TABLE]
then .
Proof.
We will first show that . For this, it is enough to observe that:
[TABLE]
We obtain that . Now, we have that:
[TABLE]
and, since , we also have that:
[TABLE]
Similarly, we have that:
[TABLE]
and that:
[TABLE]
We can prove that in a similar way by swapping the roles of and . ∎
The signs of the intersections are not important as shown by the following useful corollary:
Corollary 2.10**.**
Let v_{1}^{\prime},v_{2}^{\prime},v_{3}^{\prime},v_{4}^{\prime}\in H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) such that
[TABLE]
then .
Proof.
We apply the previous lemma with , and , where we first choose the signs of and so and then we choose the order of and so . ∎
Finally, we will prove that the Rauzy–Veech groups preserve the usual quadratic form on H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi};\mathbb{Z}/2\mathbb{Z}). We first recall the definition of this quadratic form \citesJ:spin[Appendix C]Z:representatives: for any u\in H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi};\mathbb{Z}/2\mathbb{Z}), we choose a smooth simple closed curve on M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi} whose modulus-two homology class is . We then define , where is the degree of the Gauss map (also known as index or turning number) of . For every we have that since the vector is the homology class of the curve , which has index [math]. For a vector we then have that:
[TABLE]
where the notation “” means “”.
Throughout the entire article, we denote the modulus-two reduction of an object by a bar above it: . We can now prove that the Kontsevich-Zorich matrices preserve these quadratic forms.
Lemma 2.11**.**
Let be an arrow of some Rauzy class. Then, we have that Q_{\pi^{\prime}}(u)=Q_{\pi}(u\hbox to0.0pt{\hskip 1.5434pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{B}}_{\gamma}) for every u\in H_{1}(M_{\pi^{\prime}}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi^{\prime}};\mathbb{Z}/2\mathbb{Z}).
Proof.
Let and be the winner and loser, respectively, for the Rauzy induction. Then, \hbox to0.0pt{\hskip 1.5434pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{B}}_{\gamma}=\hbox to0.0pt{\hskip 2.08333pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\mathrm{Id}}}+\hbox to0.0pt{\hskip 1.47914pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{E}}_{\alpha_{\mathrm{l}}\alpha_{\mathrm{w}}}. Thus,
[TABLE]
where denotes the bilinear form induced by .
On the other hand, since \Omega_{\pi^{\prime}}=B_{\gamma}\Omega_{\pi}B_{\gamma}^{\mbox{\raisebox{0.99458pt}{\scalebox{0.6}{\intercal}}}} an immediate computation yields:
[TABLE]
and, therefore,
[TABLE]
where we used that \hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}_{\alpha_{\mathrm{l}}\alpha_{\mathrm{w}}}=1. ∎
We obtain the following straightforward corollary:
Corollary 2.12**.**
The action on H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi};\mathbb{Z}/2\mathbb{Z}) of the Rauzy–Veech group preserves . In other words, , where O(Q_{\pi})\subseteq\mathrm{Sp}(\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}_{\pi},\mathbb{Z}/2\mathbb{Z}) is the orthogonal group induced by .
We will show that, for some connected components of strata, the previous corollary characterizes the Rauzy–Veech group. That is, the group is the preimage of for the modulus-two reduction. Nevertheless, this is not true for every connected component. In particular, for and every hyperelliptic connected component of or there exist orthogonal transvections (see Section 4) not belonging to the Rauzy–Veech group.
3. Generating the level-two congruence subgroup
The first part of our proof that the Rauzy–Veech group of minimal strata is the full preimage of consists of showing that \ker(\mathrm{Sp}(\Omega_{\pi},\mathbb{Z})\to\mathrm{Sp}(\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}_{\pi},\mathbb{Z}/2\mathbb{Z}))\subseteq\mathrm{RV}(\pi) where represents a non-hyperelliptic connected component of . That is, we have to prove that the level-two congruence subgroup is contained in .
For this section, we will use the following explicit permutation representatives of minimal strata computed by Zorich [Zor08, Proposition 3, Proposition 4]:
[TABLE]
for and
[TABLE]
for . One has that and that .
A finite set of generators of the level-two congruence subgroup is the following: given a symplectic basis , the squares of symplectic transvections and for every [Mum07, Appendix to Section 5]. Observe that this condition is redundant if , since . We will prove this fact for the family and separately.
3.1. Proof for the first family
Our objective is now proving that the level-two congruence subgroup can be generated for the family of irreducible permutations. We will denote for , \mbox{\scalebox{1.05}{\Sigma}}^{(g)} for \mbox{\scalebox{1.05}{\Sigma}}_{\tau^{(g)}} and for until explicitly stated.
We start by finding an appropriate symplectic basis:
Lemma 3.1**.**
If , a symplectic basis for (H_{1}(M^{(g)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(g)}),\langle\cdot,\cdot\rangle) is given by the following pairs of vectors:
[TABLE]
Proof.
The proof is a straightforward computation. ∎
We denote the last two vectors in the previous lemma by and , respectively. The next series of lemmas show that contains the desired squares of symplectic transvections for this basis. Recall that if v\in H_{1}(M^{(g)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(g)}) belongs to , then (and, therefore, ) by 2.4 and 2.6. For this reason, we will sometimes prove that . Moreover, if two vectors belong to and satisfy , then belongs to , so in these cases we get that “for free”. This is the case for the vectors and also for .
The following lemma is not a strict part of the proof, but will allow us to simplify many arguments.
Lemma 3.2**.**
If , then .
Proof.
Observe that . We continue inductively: for we have that:
[TABLE]
so . By continuing this way until we obtain the desired result. ∎
Lemma 3.3**.**
Let such that . Then, we have that .
Proof.
Let such that . We can use 2.10 with , , , . ∎
Lemma 3.4**.**
If , then and .
If , then .
Proof.
If , then by 3.2. We can use 2.10 with , , and to show that . Moreover, we have that since
[TABLE]
so .
If , we have that by 3.2. Observe that:
[TABLE]
so . Since and , we obtain that and belong to . We also obtain that since . ∎
The next lemma completes the proof:
Lemma 3.5**.**
Let . Then, .
Proof.
We consider two cases:
If , we have that by 3.2. Observe that since . We can prove that analogously.
If , let such that . We have that by the previous lemma. We conclude by 2.10 with , , and and with , , and . ∎
We obtain the following proposition:
Proposition 3.6**.**
For any , the symplectic transvections generate the level-two congruence subgroup \ker(\mathrm{Sp}(\Omega_{\tau^{(g)}},\mathbb{Z})\to\mathrm{Sp}(\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}_{\tau^{(g)}},\mathbb{Z}/2\mathbb{Z})). In particular, it is contained in .
3.2. Proof for the second family
We will now prove that the level-two congruence subgroup can be generated for the family of irreducible permutations. The proof is very similar to the one for the first family, although some computations are different so it is necessary to present it in full detail. We will denote for , \mbox{\scalebox{1.05}{\Sigma}}^{(g)} for \mbox{\scalebox{1.05}{\Sigma}}_{\sigma^{(g)}} and for until explicitly stated.
We start by finding an appropriate symplectic basis:
Lemma 3.7**.**
If , a symplectic basis for (H_{1}(M^{(g)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(g)}),\langle\cdot,\cdot\rangle) is given by the following pairs of vectors:
[TABLE]
Proof.
The proof is a straightforward computation. ∎
As before, we denote the last two vectors by and respectively. Observe that we have that and belong to “for free”.
We have a lemma analogous to 3.2:
Lemma 3.8**.**
If , then .
Proof.
We will prove that . Then, we can continue inductively as in the proof of 3.2. We have that:
[TABLE]
which shows what we wanted. ∎
The proofs of some lemmas are very similar to the ones for the first family (replacing 3.2 with 3.8) and in these cases we will just refer to such proofs. Indeed, one has that, for any , for any , so and can only differ for (and sometimes they are equal even in this case). We will implicitly exploit this fact to avoid having to repeat the proofs.
Lemma 3.9**.**
Let such that . Then, we have that .
Proof.
The proof is identical to that of 3.3. ∎
Lemma 3.10**.**
If , then and .
If , then .
Proof.
The proof is identical to that of 3.4. ∎
Lemma 3.11**.**
Let . Then, .
Proof.
The proof is identical to that of 3.5. ∎
Lemma 3.12**.**
We have that and for any .
Proof.
Observe that by 2.10 with , , and . Moreover, we have that:
[TABLE]
so . ∎
The following lemma completes the proof:
Lemma 3.13**.**
We have that for any and any . Moreover, .
Proof.
If , the result is obvious. If , it is enough to observe that
[TABLE]
If , let such that . If we can use that:
[TABLE]
to show that . The case can be proved similarly.
If , let by 3.8. We have that , so . We can use 2.10 with , , and to conclude that . The fact that can be proved similarly.
Finally, if we have that . Let . We will show that . If , we put , and . If , we put , and . Observe that , that and that:
[TABLE]
Since in any case, we obtain that . ∎
We obtain the following proposition:
Proposition 3.14**.**
For any , the symplectic transvections generate the level-two congruence subgroup \ker(\mathrm{Sp}(\Omega_{\sigma^{(g)}},\mathbb{Z})\to\mathrm{Sp}(\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}_{\sigma^{(g)}},\mathbb{Z}/2\mathbb{Z})). In particular, it is contained in .
4. Generating the orthogonal group
For this section and the next, we will use some other permutations representing the even and odd components of minimal strata. Indeed, we consider the following two families of irreducible permutations:
[TABLE]
for and
[TABLE]
for .
These two families of permutations are similar in many ways. For the sake of simplicity and brevity, for this section and the next we write to refer to either or and we set , \mbox{\scalebox{1.05}{\Sigma}}^{(d)}=\mbox{\scalebox{1.05}{\Sigma}}_{\pi^{(d)}}, and .
We have that the genus of is if is even and if is odd. In the latter case, the kernel of the symplectic form is generated by .
Moreover, using the notation from the classification of connected components [KZ03], we have that:
Lemma 4.1**.**
We have that for every :
[TABLE]
and for every :
[TABLE]
Proof.
Observe that if , then neither nor are hyperelliptic, so they belong to the only non-hyperelliptic connected component of the stratum.
For the rest of the cases, we can compute the Arf invariant of to determine the connected component in which lies. First, it is easy to see that if is odd, then the Arf invariant of is the same as that of , since any symplectic basis in dimensions is a maximal symplectic set in dimensions. Therefore, we only need to compute the Arf invariant for even .
We will use the definition of the Arf invariant as the value assumed most often by the quadratic form. We can count the number of non-singular vectors by establishing an appropriate recurrence. To this end, we define the following quantities:
- •
the number of non-singular vectors with an even number of s;
- •
the number of non-singular vectors with an odd number of s;
- •
the number of singular vectors with an even number of s;
- •
the number of singular vectors with an odd number of s.
Then, we have the following recurrence:
[TABLE]
Indeed, given a non-singular vector of with an even number of s we construct a non-singular vector of with an even number of s by setting the last coordinate to be zero. Moreover, for any non-singular vector of with an odd number of s we construct a non-singular vector of with an even number of s by setting the last coordinate to be . This proves that , and the other relations can be shown similarly.
We now assume . We have the following base cases for the recurrence (see the Appendix for computations): , , and . The linear recurrence can be then solved, yielding that:
[TABLE]
and therefore that the number of non-singular vectors is:
[TABLE]
Recall that we are assuming that is even, so is odd and . By analysing the sign of , we obtain that there are more non-singular than singular vectors if .
Now assume . We have the following base cases for the recurrence (see the Appendix for computations): , , and . We conclude that:
[TABLE]
so the number of non-singular vectors is:
[TABLE]
A similar analysis as in the previous case shows that there are more non-singular than singular vectors for if . ∎
Now, we need to prove that some specific elements of H_{1}(M^{(d)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(d)}) belong to .
Lemma 4.2**.**
If , then .
Proof.
First assume that . We have that:
[TABLE]
so . If , we conclude since and . We continue inductively: if with , we have that
[TABLE]
and we obtain that . Since and , we obtain the desired result.
Now assume that . Observe that:
[TABLE]
so . Now we will assume that .
We have that:
[TABLE]
so . If , we conclude since and . We continue inductively: if with , we have that
[TABLE]
and we obtain that . Since and , we obtain the desired result. ∎
For the rest of this section, we will fix until explicitly stated and write for ,
for \mbox{\scalebox{1.05}{\Sigma}}^{(d)}, etc. Recall that is the usual quadratic form on H_{1}(M\setminus\mbox{\scalebox{1.05}{\Sigma}};\mathbb{Z}/2\mathbb{Z}) and that:
[TABLE]
We also denote by the bilinear form on H_{1}(M\setminus\mbox{\scalebox{1.05}{\Sigma}};\mathbb{Z}/2\mathbb{Z}) induced by \overline{\hbox{}}$$\Omega.
Definition 4.3**.**
For v\in H_{1}(M\setminus\mbox{\scalebox{1.05}{\Sigma}};\mathbb{Z}/2\mathbb{Z}) such that , we define the orthogonal transvection along as:
[TABLE]
Observe that \hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{v}^{2}=\mathrm{Id} for each since \hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{v}^{2}(u)=\hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{v}(u+\langle u,v\rangle v)=u+\langle u,v\rangle v+\langle u,v\rangle v=u. Moreover, it is not hard to see that if v\in H_{1}(M\setminus\mbox{\scalebox{1.05}{\Sigma}}) then \hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{\text{\hskip 0.6779pt\hbox to0.0pt{\leavevmode\hbox{v}\hss}{\hskip-0.6779pt\leavevmode\hbox{\overline{\hbox{}}}}}} belongs to if and only if , so the set of orthogonal transvections coincides with the modulus-two reduction of the set of the symplectic transvections whose reductions preserve .
Lemma 4.4**.**
Let v,w\in H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi};\mathbb{Z}/2\mathbb{Z}) such that . Then, \hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{v}\hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{w}\hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{v}=\hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{w}\hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{v}\hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{w}=\hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{v+w}.
Proof.
Since , we obtain that . The proof then follows from 2.6, since we can find \tilde{v},\tilde{w}\in H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) such that and , . ∎
We denote the set of non-singular vectors by \mathrm{NS}(Q)\subseteq H_{1}(M\setminus\mbox{\scalebox{1.05}{\Sigma}};\mathbb{Z}/2\mathbb{Z}) and the set of singular vectors by \mathrm{S}(Q)\subseteq H_{1}(M\setminus\mbox{\scalebox{1.05}{\Sigma}};\mathbb{Z}/2\mathbb{Z}).
Definition 4.5**.**
For , let \hbox to0.0pt{\hskip 1.43123pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{G}}_{V}\subseteq O(Q) be the group spanned by the orthogonal transvections \{\hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{v}\}_{v\in V}. We say that a set is -closed if implies for every . We denote by the smallest -closed set containing . By the previous lemma, \{\hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{v}\}_{v\in V^{Q}}\subseteq\hbox to0.0pt{\hskip 1.43123pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{G}}_{V}.
Observe that this definition is analogous to 2.8. Indeed, the modulus-two reduction of a non-singular -closed set is a -closed set.
It is known from the theory of classical groups that orthogonal transvections generate if the dimension of the vector space is at least and the alternate form is nondegenerate [Gro02, Theorem 14.16]. This holds for every even . Since \hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{\alpha}=\hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{\text{\hskip 0.87032pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.87032pt\leavevmode\hbox{\overline{\hbox{}}}}}_{\alpha}} belongs to for every , it is enough to prove that \{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q}=\mathrm{NS}(Q) to obtain that . We will need one extra auxiliary lemma:
Lemma 4.6**.**
If , then v=\sum_{\alpha\in\mathcal{A}}\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha},v_{\beta}=\sum_{\alpha\neq\beta}\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\in\{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q} for each .
Proof.
We have that by 4.2. By taking the modulus-two reduction, this implies that v=\sum_{\alpha\in\mathcal{A}}\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\in\{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q}.
Now, for any we have that:
[TABLE]
so v_{\beta}\in\{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q}. ∎
Observe that \sum_{\alpha\in\mathcal{A}}\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\in\mathrm{NS}(Q^{(d)}) if and only if . We can now prove the final lemma of this section:
Lemma 4.7**.**
For each , we have \{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q}=\mathrm{NS}(Q)\setminus\ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}.
Proof.
The proof is by induction. The base cases, for and for , can be done computationally (see the Appendix for computations). The inductive step is exactly the same for both families of irreducible permutations.
We have that
[TABLE]
where \hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\sharp}=\sum_{\alpha<d}\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}. Observe that, for any u\in H_{1}(M^{(d)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(d)};\mathbb{Z}/2\mathbb{Z}),
[TABLE]
Let v\in H_{1}(M^{(d)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(d)};\mathbb{Z}/2\mathbb{Z}) be such that for and . We define p_{d-1}(u)\in H_{1}(M^{(d-1)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(d-1)};\mathbb{Z}/2\mathbb{Z}) to be the projection of by removing the -th coordinate. Then:
[TABLE]
Assume now that u\in\mathrm{NS}(Q^{(d)})\setminus\ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}. We consider several cases:
If , we have that . If is odd, then \hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d-1)} is invertible, so p_{d-1}(u)\notin\ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d-1)} and we obtain that u\in\{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q^{(d)}}. If, on the contrary, is even, then \ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d-1)}=\{0,\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\sharp}\}. If u=(\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\sharp}\ \ 0), then and can use the previous lemma to see that u\in\{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q^{(d)}}. If u\neq(\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\sharp}\ \ 0), then p_{d-1}(u)\notin\ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d-1)}, so we can use the induction hypothesis.
If , we have that u=v+\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{d}. If , or, equivalently, if , then . If is odd, then v\in\{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q^{(d)}} by induction hypothesis. If is even and p_{d-1}(u)=\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\sharp}, then and we can use the previous lemma to obtain that v\in\{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q^{(d)}}. Otherwise, we can use the induction hypothesis to obtain the same result. In any case, u=v+\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{d}\in\{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q^{(d)}} by -closedness. Finally, assume that , so . Thus,
[TABLE]
Since u\notin\ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}, there exists with \langle u,\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\beta}\rangle=1 so, Q^{(d)}(u+\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\beta})=\langle u,\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\beta}\rangle=1. We know that , since . Moreover, from \hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}_{\beta d}=1, we can see that:
[TABLE]
If is odd, then \hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d-1)} is invertible, so p_{d-1}(u+\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\beta}+\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{d})\notin\ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d-1)} and we conclude by induction hypothesis and -closedness since (u+\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\beta}+\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{d})+(e_{\beta}+\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{d})=u. If, on the contrary, is even, we conclude in the same way if u+\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\beta}+\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{d}\neq(\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\sharp}\ \ 0). Otherwise, and u=\sum_{\alpha\neq\beta}\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}, so we conclude by the previous lemma. ∎
We obtain the following proposition which completes the proof of Theorem 1.3:
Proposition 4.8**.**
For even , the orthogonal transvections \{\hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{\alpha}\}_{\alpha\in\mathcal{A}} generate the orthogonal group . In particular, it is contained in .
Proof.
By the previous lemma, we have that . Since the dimension of H_{1}(M^{(d)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(d)};\mathbb{Z}/2\mathbb{Z}) is at least , the orthogonal transvections generate . We conclude by 4.4. ∎
5. Non-hyperelliptic connected components of
As was done by Avila, Matheus and Yoccoz for the hyperelliptic connected components [AMY17], we can give an explicit description of the Rauzy–Veech groups for the remaining connected components of . We will continue using the representatives found in the previous section. We will prove the following theorem:
Theorem 5.1**.**
For any , the Rauzy–Veech group of a non-hyperelliptic connected component of is equal to the preimage of the orthogonal group by the modulus-two reduction \mathrm{Sp}(\Omega^{(d)},\mathbb{Z})\cap\mathrm{SL}(H_{1}(M^{(d)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(d)}))\to\mathrm{Sp}(\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)},\mathbb{Z}/2\mathbb{Z}), where . If is odd, it is isomorphic to . If is even, it is isomorphic to a finite-index subgroup of . Moreover, such groups are generated by .
Assume for the rest of this section that is odd. Let be the preimage in the statement of the theorem. We have that by 2.11. We will prove that every element of belongs to .
Recall that and that is generated by . Consider the -submodule of H_{1}(M^{(d)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(d)}) spanned by . Observe that is nondegenerate and that V\oplus\ker\Omega^{(d)}=H_{1}(M^{(d)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(d)}). We have the following decomposition:
Lemma 5.2**.**
For any , we have that acts as the identity on . Moreover, if and are the unique linear maps satisfying , then .
Proof.
The fact that follows from a straightforward computation:
[TABLE]
for any . Since is nondegenerate we obtain that . We have that preserves . In other words, is an eigenvector of . Therefore, we obtain that . Thus, , so . ∎
For any , we have that and that . In particular, is a subgroup of .
Lemma 5.3**.**
The group is isomorphic to \mathrm{Sp}(\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d-1)},\mathbb{Z}/2\mathbb{Z}) if and to if .
Proof.
If , then Q^{(d)}(\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\sharp})=1. Therefore, is regular, that is, the only element of is [math]. In this case, the map from to \mathrm{Sp}(\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}},\mathbb{Z}/2\mathbb{Z}) is an isomorphism [Gro02, Theorem 14.1, Theorem 14.2].
If , then Q^{(d)}(\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\sharp})=0, so is not regular. The restriction of to \overline{\hbox{}}$$V is a quadratic form for the nondegenerate symplectic form \hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}}. The natural projection p_{d-1}\colon\hbox to0.0pt{\hskip 1.52777pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}\to H_{1}(M^{(d-1)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(d-1)};\mathbb{Z}/2\mathbb{Z}) to the first coordinates is an isomorphism between Q^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}} and . In particular, the Arf invariants of Q^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}} and coincide. Observe that S^{1}\in O(Q^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}}) for any , since
[TABLE]
for any u\in\hbox to0.0pt{\hskip 1.52777pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}, so we obtain that \{S^{1}\ \mathbin{|}\ S\in O(Q^{(d)})\}\subseteq O(Q^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}}). By 4.7, we have that \{p_{d-1}(\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha})\}_{\alpha<d}^{Q^{(d-1)}}=\mathrm{NS}(Q^{(d-1)}) and therefore that \{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha<d}^{Q^{(d)}}=\hbox to0.0pt{\hskip 1.52777pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}\cap\mathrm{NS}(Q^{(d)}), showing that \{S^{1}\ \mathbin{|}\ S\in O(Q^{(d)})\}=O(Q^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}}), which is isomorphic to .
Furthermore, for any S^{1}\in O(Q|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}}) we can choose any linear map S^{0}\colon\hbox to0.0pt{\hskip 1.52777pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}\to\ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)} and define as the identity on \ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)} and as on \overline{\hbox{}}$$V. We have that since:
[TABLE]
for every .
Finally, for every S^{0},T^{0}\colon\hbox to0.0pt{\hskip 1.52777pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}\to\ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)} there exist unique elements of \overline{\hbox{}}$$V such that S^{0}(u)=\langle u,v\rangle\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\sharp} and T^{0}(u)=\langle u,v\rangle\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\sharp} for every u\in\hbox to0.0pt{\hskip 1.52777pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}. We identify with and with . From the equality , we obtain that
[TABLE]
where (S^{1})^{\mbox{\raisebox{0.99458pt}{\scalebox{0.6}{\intercal}}}} is the transpose of for the non-degenerate symplectic form \hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}}. This shows that is isomorphic to O(Q^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}})\ltimes\hbox to0.0pt{\hskip 1.52777pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}} and to . ∎
The following two propositions complete the proof of 5.1.
Proposition 5.4**.**
For any even , the Rauzy–Veech group of is equal to , where . It is isomorphic to a subgroup of of index and it is generated by .
Proof.
Let . We will prove that . First, we can find such that . This can be done since p_{d-1}\colon V\to H_{1}(M^{(d-1)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(d-1)}) is an isomorphism conjugating the action of and . Indeed, by 2.7 and the results of the previous two sections, the action of the subgroup of generated by the maps is conjugated by to . The group is maximal in in the sense that if is a group, then it is either or [BGP14, Theorem 3] 111This result uses the classification of finite simple groups.. Since \hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}_{d}^{1} does not preserve Q^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}}, we have that , showing that .
Now observe that, for any ,
[TABLE]
We set . Clearly , so and for any .
We will now show that for every . Indeed, we start by showing that for every . By 4.2, we have that belongs to . We will consider several cases.
We can assume that , since and represent the same connected component. If , then , so . We can use 2.10 with , , and , where if and if . We obtain that if . Now assume that . We have that
[TABLE]
so . We can use 2.10 by choosing , , and , so we obtain that . Finally, assume that . We can use 2.10 with , , and . We obtain that for every , so for every .
By writing , we get that for every .
Finally, let such that and . Since , we have that \hbox to0.0pt{\hskip 0.85414pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{S}}^{0}=\hbox to0.0pt{\hskip 1.11632pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{T}}^{0} since the map O(Q^{(d)})\to\mathrm{Sp}(\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)},\mathbb{Z}/2\mathbb{Z}) is injective. We obtain that , so there exists such that . Observe that and that , so .
The group is isomorphic to a subgroup of that we will soon describe. Let S^{1}\in\mathrm{Sp}(\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}},\mathbb{Z}/2\mathbb{Z}). There exists a unique linear map S^{0}\colon\hbox to0.0pt{\hskip 1.52777pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}\to\ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)} such that the map defined as the identity on \ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)} and as on \overline{\hbox{}}$$V belongs to . We can characterize as follows: is the unique element of satisfying [Gro02, Theorem 14.1]. We obtain that:
[TABLE]
If S^{1},T^{1}\in\mathrm{Sp}(\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}},\mathbb{Z}/2\mathbb{Z}), observe that for every u\in\hbox to0.0pt{\hskip 1.52777pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}} if and only if they belong to the same coset of \mathrm{Sp}(\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}},\mathbb{Z}/2\mathbb{Z})/O(Q^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}}). Let v_{S^{1}}\in\hbox to0.0pt{\hskip 1.52777pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}} be the unique element of \overline{\hbox{}}$$V such that S^{0}=\langle\cdot,v_{S^{1}}\rangle\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\sharp}. The image of the map consists of elements, depending on the Arf invariant of O(Q^{(d)}|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}}).
We can now describe up to isomorphism as follows: for each , we define V_{S^{1}}=\{w\in V\ \mid\ \hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{w}}=v_{\bar{S}^{1}}\}. Then, is isomorphic to , regarded as subgroup of . It has index in since a coset is determined by the unique vector v\in\hbox to0.0pt{\hskip 1.52777pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}} such that for every with \hbox to0.0pt{\hskip 1.21411pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{w}}=v. ∎
Proposition 5.5**.**
For any odd , the Rauzy–Veech group of is equal to , where . It is isomorphic to and it is generated by .
Proof.
The proof is very similar to that of 5.4. Let . We will prove that . First, we can find such that . This can be done since p_{d-1}\colon V\to H_{1}(M^{(d-1)}\setminus\mbox{\scalebox{1.05}{\Sigma}}^{(d-1)}) is an isomorphism conjugating the action of and by 2.7 and our previous results.
Now observe that, for any ,
[TABLE]
We set . Clearly , so and for any .
We will now show that for every . Indeed, we start by showing that for every . Indeed, by 4.2, belongs to . Since , we obtain that . We conclude that for every . By writing , we get that for every .
Finally, let such that and . Observe that and that , so . ∎
6. Rauzy–Veech groups of general strata
We start by stating in a more precise manner what the Zariski-density means in the case of strata with more than one marked point. Let be an irreducible permutation and let be any complement of inside H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}). We have that is a nondegenerate symplectic form on . For a map , we define to be the projection of on . Then, we define the Rauzy–Veech group on as the set , which is a subgroup of . If is another complement of and is any symplectic isomorphism, one has that conjugates the actions of and and the actions of and . Therefore, we say that a Rauzy–Veech group is Zariski-dense if is Zariski-dense inside .
Using the adjacency of strata, we will prove that contains the Rauzy–Veech group of a connected component of a minimal stratum. We will then analyse which strata are adjacent to each other, which will conclude the proof of 1.1 since the Rauzy–Veech group of any connected component of a minimal stratum is Zariski-dense by the classification of connected components [KZ03], the work on the hyperelliptic case by Avila, Matheus and Yoccoz [AMY17] and 1.3.
The following concepts were introduced by Avila and Viana [AV07, Section 5]:
Definition 6.1**.**
Let be an irreducible permutation on an alphabet . Let and let be the permutation on obtained by erasing the letter from the top and bottom rows of . If is irreducible, we say that it is a simple reduction of .
We also need a slightly stronger definition:
Definition 6.2**.**
Let be an irreducible permutation on an alphabet not containing . Let such that . We define the permutation on the alphabet by inserting just before in the top row and just before in the bottom row of . We say that is a simple extension of .
This definition can be extended, by the same rule, to the Rauzy class of . We denote by the extension map defined in this way.
If is a simple extension of , then is also irreducible [AV07, Lemma 5.4] and is a simple reduction of . Conversely, if is a simple reduction of whose omitted letter is not the last on the top or bottom row of , then is a simple extension of .
Simple extensions allow us to find copies of simpler Rauzy–Veech groups inside more complex ones. Indeed, we start by recalling the definition of the extension map. Let be an arrow in the Rauzy class of starting at . We define a path in the Rauzy class of starting at as follows: (1) if is a top arrow and the letter is added before the last letter on the bottom row of , then is constructed by applying two top operations to ; (2) if is a bottom arrow and the letter is added before the last letter on the top row of , then is constructed by applying two bottom operations to ; (3) otherwise, is constructed by applying one operation of the same type as to . In every case, starts at the image by of the start of and ends at the image by of the end of . This definition can be extended to any walk on the Rauzy class of by concatenation. We refer the reader to the work of Avila and Viana for more details [AV07, Section 5.2].
We restate a lemma [AV07, Lemma 5.6] from their work to match our context:
Lemma 6.3**.**
Let be a genus-preserving simple extension of and let be a complement of . Then, the injection \iota\colon H_{1}(M_{\pi^{\prime}}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi^{\prime}})\to H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) restricts to as a symplectic isomorphism between and such that is a complement of . Moreover, the map conjugates the actions of and , and also the actions of and a subgroup of .
Proof.
It is obvious from the definitions that is a symplectic isomorphism between and , so conjugates the actions of and . Since the genus is preserved, one has that is a complement of .
Now, let be an arrow in the Rauzy class of starting at and let . Assume that is constructed as case (1) in the definition. Let and be the last letters on the top and bottom rows of , respectively. Then, by definition, and . From these relations, it is easy to see that for every , so for every . Similar computations for cases (2) and (3) show that is a monomorphism mapping to a subgroup of . ∎
In particular, we obtain that .
We also have that genus-preserving simple extensions preserve the spin parity:
Lemma 6.4**.**
Let be a genus-preserving simple extension of . Then, the Arf invariants of and coincide.
Proof.
Let \iota\colon H_{1}(M_{\pi^{\prime}}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi^{\prime}})\to H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) be the natural injection. Then, one has that and for every v,w\in H_{1}(M_{\pi^{\prime}}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi^{\prime}}). Since the genus is preserved, we obtain that if is a maximal symplectic subset of H_{1}(M_{\pi^{\prime}}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi^{\prime}}), then is a maximal symplectic subset of H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) and we conclude that
[TABLE]
∎
We say that a permutation is standard if and . Observe that such a permutation is always irreducible. It was proven by Rauzy that standard permutations exist in every Rauzy class [Rau79]. The following lemma asserts the existence of some genus-preserving simple extensions for such permutations:
Lemma 6.5**.**
Let be a standard permutation from to . Assume that where and for every . Then, there exists an irreducible permutation such that and such that is a simple extension of , where are any integers satisfying .
Proof.
Let be the projection map obtained by identifying the sides of the polygon by translation.
Consider the bijection defined by:
- •
if ;
- •
;
- •
if ; and
- •
.
The bijection encodes the process of turning around a marked point in a clockwise manner. Indeed, the set corresponds to the top sides of , while the set corresponds to the bottom sides. For , let be its left endpoint. The orbit of by is equal to the set of top sides of whose left endpoints satisfy and the bottom sides of whose right endpoints satisfy .
We can use the orbit by to compute the conical angle of . Indeed, it is easy to see that such angle is equal to .
Now, let be a top vertex of such that is the conical singularity of order . By using that is standard, we can assume that . Indeed, if , then the top -side of joins [math] and a vertex such that . Let such that is the left endpoint of the top -side of , which exists since . We have that since .
Consider the set , ordered by the order its elements occur when applying to iteratively. Observe that an element is at an odd position if and only if it corresponds to a top side. Moreover, observe that its cardinality is equal to . Let be the -th element. We define the simple extension of by inserting a letter before in the top row and before in the bottom row. Let .
We will now prove that . Indeed, consider the bijection defined for in an analogous way as for . It is easy to see that:
- •
;
- •
;
- •
;
- •
if , then . Otherwise, let be the letter before in and then ;
and that coincides with elsewhere.
Let be the smallest natural number satisfying . If , then and, otherwise, . In both cases, . Moreover, since we have that we obtain that the orbit of by is:
[TABLE]
so, by the choice of , .
On the other hand, let be the smallest natural number satisfying . The orbit of by is:
[TABLE]
so .
These two orbits are disjoint, so the -side of joins two distinct conical singularities of orders and . Since coincides with outside of these orbits, the orders of the rest of the conical singularities are preserved. ∎
We obtain the following corollary:
Corollary 6.6**.**
Let be a standard permutation such that has genus . Let be a complement of . We have that:
- •
If and belongs to a connected stratum, then .
- •
If belongs to , where , and for every , then contains an isomorphic copy of the Rauzy–Veech group of and has index at most inside its ambient symplectic group.
Proof.
Assume first that belongs to a connected stratum. Let be a standard permutation representing some connected component of . By iterating the previous lemma and the fact that every Rauzy class contains standard permutations, there exists a sequence of simple extensions and Rauzy inductions taking to . Therefore, we obtain that contains an isomorphic copy of . Recall that the Rauzy–Veech groups of non-hyperelliptic components are maximal subgroups of [BGP14, Theorem 3]. If , the index of the Rauzy–Veech group of is 28, while the index of the Rauzy–Veech group of is 288, which is not divisible by 28. Therefore, in this case by maximality. If , contains isomorphic copies of the Rauzy–Veech groups of both non-hyperelliptic components of , so we also conclude by maximality.
Now, if belongs to let be a standard permutation representing . By iterating the previous lemma and using the fact that the Arf invariants of the quadratic forms are preserved by simple extensions, there exists a sequence of simple extensions and Rauzy inductions taking to , which completes the proof. ∎
The following lemma completes our classification of the Rauzy–Veech groups:
Lemma 6.7**.**
The Rauzy–Veech group restricted to a complement of the kernel of the symplectic form, where , and for every , is isomorphic to the Rauzy–Veech group of and has index inside its ambient symplectic group.
Proof.
We will use the explicit permutation representatives of such strata computed by Zorich [Zor08, Proposition 3, Proposition 3]. We start by proving the result for . Let and consider the representative:
[TABLE]
It is easy to see that is a basis of consisting of singular vectors for . Therefore, . Let V\subseteq H_{1}(M_{\tau}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\tau}) be the -module spanned by which is a complement of . For any we have that, for any u\in\hbox to0.0pt{\hskip 1.52777pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}},
[TABLE]
where is the projection of S|_{\text{\hbox to0.0pt{\hskip 0.31943pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{V}}}} on \ker\hbox to0.0pt{\hskip 1.1111pt\leavevmode\hbox{\overline{\hbox{}}}\hss}{\leavevmode\hbox{\Omega}}_{\tau}. Therefore, . By the previous corollary, we obtain that is isomorphic to and has the same index as . Now, a representative of any connected component of the form can be obtained by genus-preserving simple reductions of which consist of erasing some letters of the form for . These simple reductions are also simple extensions when reversed, so the index of the Rauzy–Veech group of cannot be larger than the index of the Rauzy–Veech group of . We conclude by the previous corollary.
The proof for the even components is completely analogous by using the representative:
[TABLE]
the basis of consisting of singular vectors for and the complement of defined as in the previous case. ∎
Finally, we will recall some concepts to state the classification theorem using a more common notation. We refer the reader to the survey by Viana [Via06] and the lecture notes by Yoccoz [Yoc10] for more details.
Let by an irreducible, nondegenerate permutation. There exist natural maps
[TABLE]
the former being surjective and the latter injective. The basis of H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) that we have thus far considered is dual to a basis of H_{1}(M_{\pi},\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) consisting on the sides of the polygon . The image of inside H_{1}(M_{\pi},\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) is the set spanned by (acting on column vectors). There is a natural nondegenerate symplectic form defined on by \langle\Omega_{\pi}v,\Omega_{\pi}w\rangle=v^{\mbox{\raisebox{0.99458pt}{\scalebox{0.6}{\intercal}}}}\Omega_{\pi}w.
The duality between the bases allows us to define an action of on H_{1}(M_{\pi},\mbox{\scalebox{1.05}{\Sigma}}_{\pi}), which can be understood as the action of on column vectors. This action preserves , so we can consider its restriction to , which is a subgroup of the symplectic group induced by the symplectic form on .
We have classified the Rauzy–Veech groups acting on H_{1}(M_{\pi}\setminus\mbox{\scalebox{1.05}{\Sigma}}_{\pi}) or, equivalently, its action on row vectors. Recalling 1.3, 5.1 and 2.7 and combining our results with those of Avila, Matheus and Yoccoz [AMY17] we can summarize the full classification of the group :
Theorem 6.8**.**
For a connected component of a stratum of genus translation surfaces, we denote by its associated Rauzy–Veech group and by the restriction of to the image of the symplectic form. We write for its ambient symplectic group. We have the following:
- •
If is hyperelliptic, then the group is characterized by preserving modulus two, as an action of column vectors, a specific finite set **[AMY17, Theorem 2.9]**.
- •
The group , where , and for every , is the subgroup of whose modulus-two reduction preserves, as an action on row vectors, a specific quadratic form whose Arf invariant is . It has index inside .
- •
For any other connected component , including every connected stratum for , one has that .
Moreover, is generated by symplectic transvections along canonical vectors.
Appendix
In this section we will explicitly state some computations for the base cases of the induction used to prove 1.3. Specifically, these facts were used to show that some permutations belong to the desired minimal strata and that the modulus-two reduction of the Rauzy–Veech groups are the entirety of the orthogonal groups.
It is possible to find all elements of for small values of by hand. This was done to find the base cases of the induction in 4.1. In particular:
Fact 1**.**
* consists of 36 elements, which, written on the basis (\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha})_{\alpha=1}^{6}, are:*
[TABLE]
Fact 2**.**
* consists of 120 elements, which, written on the basis (\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha})_{\alpha=1}^{8}, are:*
[TABLE]
Proving that \{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q^{(d)}}=\mathrm{NS}(\pi^{(d)}) for the base cases of the induction can be done as follows: we start with the set S_{1}=\{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}, which is contained in \{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q^{(d)}}. We define the set as the union of with the set of vectors of the form v+\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}, where and Q(v+\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha})=1. Clearly, S_{k}\subseteq\{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q^{(d)}} for each and, after a small number of iterations, it must be equal to \{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q^{(d)}}. We can then check that \{\hskip 0.17188pt\hbox to0.0pt{\leavevmode\hbox{e}\hss}{\hskip-0.17188pt\leavevmode\hbox{\overline{\hbox{}}}}_{\alpha}\}_{\alpha\in\mathcal{A}}^{Q^{(d)}}=\mathrm{NS}(\pi^{(d)}).
Acknowledgements: I am grateful to Giovanni Forni for his interesting questions and remarks which motivated the newer versions of the article. I’m also grateful to my advisors, Anton Zorich and Carlos Matheus.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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