2.1 Volume

We now discuss the notion of volume which plays an important role in the theory. For our purposes, we shall mostly interested in the algebraic volume which is a topological invariant associated to a representation $\chi $ .

Let us recall this notion in the holomorphic case. For a symplectic basis $\{\alpha _1,\beta _1,\dots ,\alpha _g,\beta _g\}$ of $\mathrm {H}_1(S_{g},\mathbb {Z})$ we define the volume of a representation $\chi \colon \mathrm {H}_1(S_{g},\mathbb {Z})\longrightarrow {\mathbb C}$ as the quantity

(14) $$ \begin{align} \text{vol}(\,\chi)=\sum_{i=1}^g \Im\big(\,\overline{\chi(\alpha_i)}\,\chi(\beta_i)\big), \end{align} $$

where $\Im (\overline {z}w)$ is the usual symplectic form on ${\mathbb C}$ . As a consequence, this algebraic definition of volume of a character $\chi $ is invariant under precomposition with any automorphisms in $\mathrm {Aut}\,\big (\mathrm {H}_1(S_{g},\mathbb {Z})\big )\cong \mathrm {Sp}(2g,{\mathbb Z})$ . The image of $\chi $ , provided it has rank $2g$ , turns out to be a polarized module.

Haupt’s conditions. As already alluded in the introduction, there are some obstructions for realizing $\chi $ as the period character of some holomorphic differential $\omega $ on a compact Riemann surface X and both of these concern the volume. We shall recall them here for the reader’s convenience. The first requirement is that the volume of $\chi $ has to be positive with respect to some symplectic basis of $\mathrm {H}_1(S_{g},\mathbb {Z})$ . Indeed, one can show that this equals the area of the surface X endowed with the singular Euclidean metric induced by $\omega $ . There is a second obstruction that applies in the case $g\geq 2$ and only when the image of $\chi \colon \mathrm {H}_1(S_{g},\mathbb {Z})\to {\mathbb C}$ is a lattice, say $\Lambda $ in ${\mathbb C}$ . In fact, one can show that in this special case $(X,\omega )$ arises from a branched cover of the flat torus ${\mathbb C}/\Lambda $ . In particular, the following inequality must hold: $\mathrm {vol}(\chi )\ge 2\,\text {{Area}}(\mathbb C/\Lambda )$ . Haupt’s Theorem says that these are the only obstructions for realizing $\chi $ as the period of some holomorphic differential.

Remark 2.6. We provide here an alternative and more geometric definition of volume. Let X be a compact Riemann surface and let $\omega \in \Omega (X)$ be a (possibly meromorphic) differential with period character $\chi $ . Let E be the complex line bundle over X canonically associated to $\chi $ and let $f\colon X\longrightarrow E$ be a section. By lifting the map f to $\widetilde {f}\colon \widetilde {X}\longrightarrow \widetilde {X}\times \mathbb C$ and projecting onto the second coordinate, the natural volume form $\frac {i}{2}dz\wedge d\overline {z}$ on $\mathbb C$ pulls back to a volume form over X. The quantity given by the integration of this form over X, that is

(15) $$ \begin{align} \frac{i}{2}\int_X f^*\Big( \,dz\wedge d\overline{z}\,\Big)=\frac{i}{2}\int_X \omega\wedge\varpi, \,\,\, \text{ where } \omega=f^*dz \end{align} $$

is the area of the singular Euclidean metric determined by $\omega $ on X. In particular, by means of Riemann’s bilinear relations one can show that it agrees with the algebraic definition of volume as in Equation (14).

We now extend the notion of volume to meromorphic differentials. In this case, the notion of volume relies on a choice of a splitting of $S_{g,n}$ as the connected sum of the closed surface $S_{g}$ and the n-punctured sphere. Let $\gamma $ be a simple closed separating curve in $S_{g,n}$ that bounds a subsurface homeomorphic to $S_{g,1}$ . There is a natural embedding $S_{g,1}\hookrightarrow S_{g,n}$ and thence an injection $\imath _g\colon \mathrm {H}_1(S_{g},\mathbb {Z}) \longrightarrow \mathrm {H}_1(S_{g,n},\,\mathbb {Z})$ because the curve $\gamma $ is trivial in homology. In the same fashion, $\gamma $ bounds a subsurface homeomorphic to $S_{0,n+1}$ and the natural embedding $S_{0,n+1}\hookrightarrow S_{g,n}$ yields an injection $\imath _n\colon \mathrm {H}_1(S_{0,n},\,\mathbb Z)\to \mathrm {H}_1(S_{g,n},\,\mathbb {Z})$ . Therefore, given a splitting as above, any representation $\chi \colon \mathrm {H}_1(S_{g,n},\,\mathbb {Z})\longrightarrow {\mathbb C}$ gives rise to two representations $\chi _g=\chi \circ \imath _g$ and $\chi _n=\chi \circ \imath _n$ that determine $\chi $ completely.

We can now introduce the following:

2.2 Further invariants on translation surfaces

In Section §2.1 we have introduced the volume as an algebraic invariant naturally attached to a representation $\chi $ . In the present section we are going to introduce geometric invariants which we shall use to distinguish the connected components of strata whenever they fail to be connected.

2.2.1 Rotation number

In this subsection $\overline {X}$ is a compact Riemann surface of genus one, that is, an elliptic curve $\mathbb C/\Lambda $ with $\Lambda $ a lattice in $\mathbb C$ . We will introduce an invariant very specific to genus-one translation surfaces with poles.

Let X be a complex structure on $S_{1,n}$ , where $n\ge 1$ , and let $\omega \in \Omega (X)$ be an abelian differential on X with finite order poles at the punctures. Recall that away from the zeros and poles of $\omega $ there is a well-defined horizontal direction and hence a nonsingular horizontal foliation on $X\setminus \{\text { zeros of }\omega \,\}$ . Such a foliation extends to a singular horizontal foliation over the zeros of $\omega $ . In fact, for a zero, say P, of order k there are exactly $k+1$ horizontal directions leaving from P. Let $\gamma $ be a simple smooth curve parametrized by the arc length. Assume in addition that $\gamma $ avoids all the zeros and poles of $\omega $ . We shall define the index of $\gamma $ as a numerical invariant given by the comparison of the unit tangent field $\dot \gamma (t)$ and the unit vector field along $\gamma $ given by unit vectors tangent to the horizontal direction. More precisely, let us denote by $u(t)$ the unit vector at $\gamma (t)$ tangent to the leaf of the horizontal foliation through $\gamma (t)$ , then the assignment

(16) $$ \begin{align} t\longmapsto \theta(t)=\angle\big(\dot\gamma(t),\,u(t)\big) \end{align} $$

defines a mapping $f_\gamma \colon \mathbb S^1\longrightarrow \mathbb S^1$ .

Definition 2.12. The index of $\gamma $ is defined as $\deg (f_\gamma )$ and denoted by $\mathrm {Ind}(\gamma )$ .

Convention. We agree that $\mathbb S^1$ is counterclockwise oriented. As a consequence we agree that the index of $\mathrm {Ind}(\gamma )=\deg (f_\gamma )$ is positive is $\gamma $ is counterclockwise oriented.

The index $\mathrm {Ind}(\gamma )$ of a closed curve $\gamma $ measures the number of times the unit tangent vector field spins with respect to the direction given by the horizontal foliation. Since for translation surfaces the parallel transport yields a trivial homomorphism; see Remark 2.4, the total change of the angle is $2\pi \,\mathrm {Ind}(\gamma )$ .

We now allow perturbations of $\gamma $ in its homotopy class. The index of any curve remains unchanged if, while deforming it, we do not cross a zero or pole of $\omega $ . On the other hand, whenever we cross a zero or pole of order k the index of $\gamma $ changes by $\pm k$ . As a consequence, the index of a curve is well defined for homotopy classes in $\pi _1(\overline {X}\setminus \{\text {zeros and poles of }\omega \})$ . We can now define the rotation number.

Definition 2.13. Let $(X,\omega )\in \mathcal {H}_1(m_1,\dots ,m_k; -p_1,\dots ,-p_n)$ be a genus-one translation surface with poles. Let $\{\alpha ,\beta \}$ be a symplectic basis of $\mathrm {H}_1(\overline {X},\,\mathbb Z)$ , the first homology group of $\overline {X}$ . Let $\gamma _\alpha $ and $\gamma _\beta $ be the representatives of $\alpha $ and $\beta $ respectively. Assume they are both parametrized by the arc length and avoid the zeros and poles of $\omega $ . The rotation number of $(X,\omega )$ is defined as

(17) $$ \begin{align} \mathrm{rot}(X,\omega)=\gcd\big(\mathrm{Ind}(\gamma_\alpha),\,\mathrm{Ind}(\gamma_\beta),\,m_1,\dots,m_k,p_1,\dots,p_n\big). \end{align} $$

In the next subsection we briefly introduce the rotation number in a more algebraic way that we shall use only in the final Section §8. Notice that the well-known properties of $\gcd $ make the rotation number $\mathrm {rot}(X,\omega )$ well defined for a given element $(X,\omega )\in \mathcal {H}_1(m_1,\dots ,m_k; -p_1,\dots ,-p_n)$ . As we shall see in Section §2.3, the rotation number is used to distinguish the connected components of a generic stratum of genus-one differentials $\mathcal {H}_1(m_1,\dots ,m_k; -p_1,\dots ,-p_n)$ .

2.2.2 Hyperelliptic translation surface

For a complex structure X on $S_{g,n}$ we shall denote by $\overline {X}$ the compact Riemann surface obtained by filling the punctures with complex charts. Let us now introduce the following:

Definition 2.14. A translation surface $(X,\omega )$ (possibly with poles) is said to be hyperelliptic if the Riemann surface $\overline {X}$ is hyperelliptic and $\omega $ is anti-invariant under the hyperelliptic involution $\tau $ , that is, $\tau ^*\omega =-\omega $ ).

Remark 2.15. For a hyperelliptic translation surface $(X, \omega )$ , the hyperelliptic involution $\tau $ realizes an isometry between the singular Euclidean structures determined by the differentials $\omega $ and $-\omega $ on $\overline {X}$ . Therefore, not all strata can admit hyperelliptic translation surfaces. In fact, if $\omega $ has two zeros (or two poles) that are swapped by $\tau $ then they must have the same order, and if a zero (or a pole) is fixed by $\tau $ then it must have even order.

Note that there are no nonzero holomorphic one-forms on the sphere. As a result, every holomorphic differential $\omega $ on a hyperelliptic Riemann surface X is anti-invariant under the hyperelliptic involution, and hence, in this case $(X,\omega )$ is always a hyperelliptic translation surface. For meromorphic differentials such a scenario is no longer true as shown by the following example.

Example 2.16. Let us consider the translation structure with a single pole on the Riemann sphere $\mathbb {C}\mathrm {\mathbf {P}}^1$ determined by the meromorphic differential $dz$ . Let $l_1,\,l_2,\,l_3 \subset \mathbb {C}\mathrm {\mathbf {P}}^1$ be three geodesic segments for the standard Euclidean structure and slit $\mathbb {C}\mathrm {\mathbf {P}}^1$ along them. The resulting slit surface is homeomorphic to a sphere with three open disjoint disks removed. Geometrically, the resulting surface is a pair of pants equipped with a Euclidean structure and piece-wise geodesic boundary. In fact, each boundary component comprises two geodesic segments, say $l_{ij}$ with $i=1,2,3$ and $j=1,2$ , and two corner points each one of angle $2\pi $ . We consider two copies of this latter surface and we glue them by identifying the boundary segments having the same label. The final surface turns out to be a compact Riemann surface X of genus two equipped with a meromorphic differential $\omega $ with six zeros of order $1$ and two poles of order $2$ . The Riemann surface X, being compact of genus two, admits a hyperelliptic involution $\tau $ . We want to show with a direct computation that $(X,\omega )$ is not hyperelliptic.

Figure 1 A $2$ -fold branched covering $\pi : X\longrightarrow \mathbb {C}\mathrm {\mathbf {P}}^1$ .

Let $\pi \colon X\longrightarrow \mathbb {C}\mathrm {\mathbf {P}}^1$ be the $2$ -fold branched covering that naturally arises from our construction. The differential $\omega $ satisfies the equation $\omega =\pi ^*(dz)$ . According to Definition 2.14 above, $(X,\omega )$ is hyperelliptic if and only if $\tau ^*(\omega )=-\omega $ . On the other hand, being of genus two, the hyperelliptic involution $\tau $ is an automorphism of X and commutes with the projection $\pi \colon X\longrightarrow \mathbb {C}\mathrm {\mathbf {P}}^1$ in the sense that $\pi \circ \tau =\pi $ . Therefore, $\omega = \pi ^{*}(dz) = \tau ^{*}(\omega )$ and hence

(18) $$ \begin{align} \tau^*(\omega)=-\omega \iff \omega=0 \end{align} $$

which provides the desired contradiction.

2.2.3 Spin structure

The last geometric invariant we shall introduce is the spin structure. Let X be a compact Riemann surface and let P be a principal $\mathbb S^1$ -bundle over X. A spin structure on X is a choice of a linear functional $\xi \colon \mathrm {H}_1(P,\, \mathbb Z_2)\to \mathbb Z_2$ having nonzero value on the cycle representing the fiber $\mathbb S^1$ of P. This is equivalent to a choice of a double covering $Q\longrightarrow P$ whose restriction to each fiber of P is a $2$ -fold covering of $\mathbb S^1$ .

Recall that an element of $\mathrm {H}_1(P,\, \mathbb Z_2)$ can be regarded as a framed closed curve in X, by which we mean a closed curve in X and a smooth unit vector field along it. For a simple closed curve $\gamma $ in X there is a preferred frame given by its unit tangent vector field. We shall denote such a framed closed curve as $\vec {\gamma }$ . This is well defined in the sense that two homologous closed curves $\gamma ,\,\delta \in \mathrm {H}_1(X,\, \mathbb Z_2)$ yield homologous framed closed curves $\vec {\gamma },\,\vec {\delta }$ ; see [Reference JohnsonJoh80, Section 3]. The canonical lift of a simple closed curve $\gamma $ is defined as $\widetilde {\gamma }=\vec {\gamma }+\textbf {z}$ where $\textbf {z}\in \mathrm {H}_1(P,\, \mathbb Z_2)$ is the homology class represented by the fiber $\mathbb S^1$ . The assignment $\gamma \longmapsto \widetilde {\gamma }$ defines a mapping of sets $\mathrm {H}_1(X,\, \mathbb Z_2)\to \mathrm {H}_1(P,\, \mathbb Z_2)$ which fails to be a homomorphism; see [Reference JohnsonJoh80]. Nevertheless, post-composition of such a map with a spin structure $\xi $ yields a quadratic form $\Omega _\xi \colon \mathrm {H}_1(X,\, \mathbb Z_2)\to \mathbb Z_2$ defined as

(19) $$ \begin{align} \Omega_\xi(\gamma)=\xi\big(\,\widetilde{\gamma}\,\big)=\langle \,\xi,\,\widetilde{\gamma}\,\rangle. \end{align} $$

According to [Reference ArfArf41] and [Reference JohnsonJoh80, Section 5] we shall introduce the following:

Definition 2.17. For a symplectic basis $\{\alpha _1,\beta _1,\dots ,\alpha _g,\beta _g\}$ of $\mathrm {H}_1(X,\, \mathbb Z_2)$ the Arf-invariant of $\Omega _\xi $ is defined by the formula

(20) $$ \begin{align} \sum_{i=1}^g \Omega_\xi(\alpha_i)\Omega_\xi(\beta_i) \,\,(\text{mod}\,2). \end{align} $$

The parity of a spin structure $\xi $ is defined as the Arf-invariant of $\Omega _\xi $ . In particular, the parity of $\xi $ does not depend on the choice of the symplectic basis.

Let X be a complex structure on $S_{g,n}$ , with $g\ge 1$ , and $\omega \in \Omega (X)$ be an abelian differential on X with poles of finite order at the punctures. Recall that we can regard $\omega $ as a meromorphic differential on a compact Riemann surface $\overline {X}$ . Whenever $(X,\omega )\in \mathcal {H}_g(2m_1,\dots ,2m_k;-2p_1,\dots ,-2p_n)$ , then $\omega $ defines a natural spin structure on $\overline {X}$ as follows.

Let $\gamma $ be a simple closed curve parametrized by the arc length, and let $\vartheta $ be a unit vector field along $\gamma $ . The couple $(\gamma , \vartheta )$ defines a framed closed curve, that is, an element of $\mathrm {H}_1(P,\,\mathbb Z_2)$ . Recall that $\omega $ defines a nonsingular horizontal foliation away from zeros and poles, and hence, the unit vector field $\vartheta $ can be compared to the unit vector field along $\gamma $ tangent to the horizontal foliation. In fact, as in Section §2.2.1, this can be done by means of a mapping $f_{(\gamma ,\vartheta )}\colon \mathbb S^1\to \mathbb S^1$ defined as in Equation (16). The spin structure induced by the meromorphic differential $\omega $ on $\overline {X}$ is then defined as the $\mathbb Z_2$ -value linear functional

(21) $$ \begin{align} \xi\colon \mathrm{H}_1(P,\,\mathbb Z_2)\longrightarrow \mathbb Z_2,\,\, \text{ defined as }\,\, \xi(\gamma,\vartheta)=\deg(\,f_{(\gamma,\vartheta)}\,)\,\,(\text{mod}\,2). \end{align} $$

A direct application of Equation (21) shows that, for the canonical lift $\widetilde {\gamma }$ of $\gamma $ , the spin structure determined by $\omega $ satisfies the property

(22) $$ \begin{align} \xi(\,\widetilde{\gamma}\,)=\langle\,\xi,\,\widetilde{\gamma}\,\rangle=\langle\,\xi,\,\vec{\gamma}+\textbf{z}\,\rangle= \text{Ind}(\gamma)+1\,\,(\text{mod}\,2). \end{align} $$

As a consequence, we can apply formula (20) to compute the parity $\varphi (\omega )$ of the spin structure determined by a meromorphic differential $\omega $ . Given a symplectic basis $\{\alpha _1,\beta _1,\dots ,\alpha _g,\beta _g\}$ of $\mathrm {H}_1(X,\, \mathbb Z_2)$ , it follows that

(23) $$ \begin{align} \varphi(\omega)=\sum_{i=1}^g \big(\,\text{Ind}(\alpha_i)+1\big)\big(\,\text{Ind}(\beta_i)+1\big) \,\,(\text{mod}\,2). \end{align} $$

It is straightforward to check that $\varphi (\omega )$ does not depend on the choice of the symplectic basis. Finally, it follows from works of Atiyah in [Reference AtiyahAti71] and Mumford in [Reference MumfordMum71] that the spin parity is invariant under continuous deformations.

Remark 2.18. Let $(X,\omega )\in \mathcal {H}_g(2m_1,\dots ,2m_k,-1,-1)$ be a genus-g meromorphic differential with two simple poles. The meromorphic differential $\omega $ does not define any spin structure as in Equation (21) because, for any curve $\gamma $ , the index $\mathrm {Ind}(\gamma )$ is not longer well defined as an element of $\mathbb Z_2$ ; see formula (22). Nevertheless, we can still define an invariant that turns out to be the spin parity of some structure $(Y,\xi )$ in the stratum $\mathcal {H}_{g+1}(2m_1,\dots ,2m_k)$ . Recall that a neighborhood of a simple pole is an infinite cylinder. Around a puncture of $(X,\omega )$ , we can find a simple closed geodesic curve. In fact, there are infinitely many of such curves. Choose a waist geodesic curve on each cylinder and truncate $(X,\omega )$ along them. Since $(X,\omega )$ has only two simple poles and their residues are opposite (and nonzero), the curves above are isometric. We glue them and the resulting object is a translation structure on a closed surface of genus $g+1$ . We define this latter as $(Y, \xi )$ , and it lies in the stratum $\mathcal {H}_{g+1}(2m_1,\dots ,2m_k)$ by construction. Therefore, it makes sense to consider a parity of the spin structure for differentials in $\mathcal {H}_g(2m_1,\dots ,2m_k,-1,-1)$ . See [Reference BoissyBoi15, Section 5.3] for more details.

2.3 Moduli spaces and connected components

As alluded in the introduction, the moduli space $\Omega \mathcal {M}_{g,n}$ admits a natural stratification given by unordered partitions $(m_1,\dots ,m_k;-p_1,\dots ,-p_n)$ of $2g-2$ , where negative integers are allowed only in the case of meromorphic differentials. In the present subsection, we aim to recall for the reader’s convenience the classifications of connected components of the strata both in the holomorphic and meromorphic cases.

Let us discuss first the case of holomorphic differentials which has been studied by Kontsevich–Zorich in [Reference Kontsevich and ZorichKZ03, Theorems 1 and 2]. For low-genus surfaces, that is, $g=2,3$ , we have the following:

For surfaces of genus $g\ge 4$ , the following classification holds.

Let us discuss the case of meromorphic differentials and begin from the case of genus-one meromorphic differentials. Let $\mathcal {H}_{1}(m_1,\dots ,m_k;-p_1,\dots ,-p_n)$ be a stratum in $\Omega \mathcal {M}_{1,n}$ , and let d be the number of positive divisors of $\gcd (m_1,\dots ,m_k,p_1,\dots ,p_n)$ . Recall that a stratum is nonempty as soon as the Gauss–Bonnet condition holds for some nonnegative integer g; see Remark 2.3, and $\sum p_i>1$ . In [Reference BoissyBoi15, Theorem 1.1], Boissy showed the following.

Common divisors of the zero and pole orders in the above result are called rotation numbers. It is worth mentioning that connected components of the strata in genus one can also be classified by another invariant from the algebraic viewpoint; see [Reference Chen and CoskunCC14, Section §3.2]. For a stratum $\mathcal H_1(m_1,\dots ,m_k;\,-p_1,\dots ,-p_n)$ of genus-one differentials, let l be any positive divisor of $\gcd (m_1,\dots ,m_k,p_1,\dots ,p_n)$ – with the only exception being $l=n$ for signatures of the form $(n;-n)$ . We shall say that $(X,\omega )\in \mathcal H_1(m_1,\dots ,m_k;\,-p_1,\dots ,-p_n)$ has torsion number l if l is the largest positive integer such that

(24) $$ \begin{align} \sum_{j=1}^k \left(\frac{m_j}{l}\right)\,Z_j\,-\,\sum_{i=1}^n\left(\frac{p_i}{l}\right)\,P_i \,\sim \,0, \end{align} $$

where $\{Z_i,\dots ,Z_n\}$ and $\{P_1,\dots ,P_n\}$ are, respectively, the set of zeros and poles of $\omega $ . Here, $\sim $ stands for linear equivalence which means there exists a meromorphic function on the underlying Riemann surface with zeros at $\{Z_1,\dots , Z_k\}$ and poles at $\{P_1,\dots ,P_n\}$ with orders given by the coefficients in the relation. We shall use this characterisation in Section §8 to show a few lemmas concerning the exceptional cases listed in Theorem B.

For genus-g meromorphic differentials with $g\ge 2$ , the classifications of connected components is similar to that of holomorphic differentials. Let $\mathcal {H}_{g}(m_1,\dots ,m_k;-p_1,\dots ,-p_n)$ be a stratum in $\Omega \mathcal {M}_{g,n}$ . In [Reference BoissyBoi15], Boissy has showed that a stratum admits a hyperelliptic component if and only if it is one of the following:

(25) $$ \begin{align} \mathcal{H}_{g}(2m,-2p),\,\, \mathcal{H}_{g}(m, m,-2p),\,\, \mathcal{H}_{g}(2m,-p, -p),\,\, \mathcal{H}_{g}(m, m, -p, -p), \end{align} $$

for some $m\ge p\ge 1$ , otherwise there is no hyperelliptic component. According to Boissy, we introduce the following.

We now state the following result due to Boissy concerning connected components of the strata of genus-g meromorphic differentials with $g\ge 2$ .

This concludes the classification of the strata of meromorphic differentials.

3.1 Breaking a zero

The surgery we are going to describe has been introduced by Eskin–Masur–Zorich in [Reference Eskin, Masur and ZorichEMZ03, Section 8.1] and it literally ‘breaks up’ a zero in two, or possibly more, zeros of lower orders. Complex-analytically this can be thought as the analogue to the classical Schiffer variations for Riemann surfaces; see [Reference NagNag85]. This surgery only modifies a translation surface on a contractible neighborhood of the initial zero. In particular, after the surgery the resulting surface has the same genus as the former one but the type of zero orders is changed. Moreover, the new translation surface we obtain after the surgery has the same period character as the original one. As a consequence, this operation produces small deformation of the original translation structure in the same isoperiodic fiber.

Let us now explain this surgery in more detail. Let $(X,\omega )$ be a translation surface possibly with poles. Breaking a zero is a procedure that takes place at the $\varepsilon $ -neighbourhood of some zero of order m of the differential on which it looks like the pull-back of the form $dz$ via a branched covering $z\mapsto z^{m+1}$ . The differential is then modified by a surgery inside this $\varepsilon $ -neighbourhood. Once this surgery is performed, we obtain a new translation structure with two zeros of order $m_1$ and $m_2$ such that $m_1+m_2 = m$ . Furthermore, the translation structure remains unchanged outside the $\varepsilon $ -neighbourhood of such a zero of order m. The idea is to consider the $\varepsilon $ -neighbourhood of a zero of order m as $m+1$ copies of a disc D of radius $\varepsilon $ whose diameters are identified in a specified way. We can see this family of discs as a collection of $m+1$ upper half-discs and $m+1$ lower half-discs. See figure 2.

Figure 2 An $\varepsilon $ -neighbourhood of a zero of order $2$ .

We now break a zero of order m into two zeros of order $m_1$ and $m_2$ . To break a zero consists of identifying the diameters of the starting $m+1$ discs in a different way as follows. In order to do this, we modify the labelling on the upper half-disc indexed by $0$ , the lower half-disc indexed by $m_1$ , and all upper and lower half-discs with index more than $m_1$ accordingly. The modified labelling is shown below in Figure 3 for the case of splitting a zero of order $2$ into two zeros of order $1$ .

Figure 3 New labelling for breaking up a zero of order $2$ in two zeros of order $1$ .

We now identify $ul_i$ with $ll_i$ and $lr_i$ with $ur_{i+1}$ as before with the added identification of $um$ with $lm$ . This identification gives two zeros A and B, where A is a zero of the differential of order $k_1$ and B is a zero of order $k_2$ . We also get a geodesic line segment joining A and B. Given $c \in \mathbb {C}\setminus \{0\}$ with length less than $2\varepsilon $ , we can perform the surgery in such a way that the line segment joining A and B is c. It is also clear that such a deformation of the translation structure is only local. This procedure can be repeated multiple times to obtain zeros of orders $m_1, \dots , m_k$ from a single zero of order $m_1 + \cdots + m_k$ . We shall frequently rely on this procedure of breaking a zero in the remaining part of this paper. The following lemmas are easy to establish.

3.2 Bubbling a handle with positive volume

We now describe a second surgery introduced by Kontsevich and Zorich in [Reference Kontsevich and ZorichKZ03]. In their paper, this surgery is defined for holomorphic differentials, but, as already observed by Boissy in [Reference BoissyBoi15], the same surgery can be defined for meromorphic differentials because this is a local surgery. Topologically, bubbling a handle with positive volume consists of adding a handle, say $\Sigma $ to a given surface. Let us see how this can be done metrically. Let $(X,\omega )\in \mathcal {H}_g(m_1,\dots ,m_k;-p_1,\dots ,-p_n)$ be a translation surface, possibly with finite order poles. Let $l\subset (X,\omega )$ be a geodesic segment with distinct end points.

Figure 4 Bubbling a handle with positive volume.

We slit the translation surface $(X,\omega )$ along l, and we label the side that has the surface on its left as $l^+$ and the other side as $l^-$ . We then paste the extremal points of the geodesic segment, and the resulting surface has two geodesic boundary components having (by construction) the same length. Let $\mathcal {P}\subset \mathbb C$ be a parallelogram such that the two opposite sides are both parallel to l (via the developing map) – the handle $\Sigma $ is obtained by gluing the opposite sides of $\mathcal {P}$ . We can paste such a parallelogram to the slit $(X,\omega )$ , and the resulting topological surface is homeomorphic to $S_{g+1,n}$ . Metrically, we have a new translation surface $(Y,\xi )$ . We can distinguish three mutually disjoint possibilities:

1. 1. Both of the extremal points of $l\subset (X,\omega )$ are regular, then $(Y,\xi )\in \mathcal {H}_g(2,m_1,\dots ,m_k;-p_1,\dots ,-p_n)$ ,

2. 2. One of the extremal points of $l\subset (X,\omega )$ is a zero of $\omega $ of order $m_i$ , where $1\le i\le k$ , then $(Y,\xi )$ belongs to the stratum $\mathcal {H}_g(m_1,\dots ,m_i+2,\dots ,m_k;-p_1,\dots ,-p_n)$ ,

3. 3. Both of the extremal points are zeros of $\omega $ of orders $m_i,\,m_j$ , respectively, then $(Y,\xi )$ belongs to the stratum $\mathcal {H}_g(m_1,\dots ,\widehat {m}_i,\dots ,\widehat {m}_j,\dots ,m_i+m_j+2,\dots ,m_k;-p_1,\dots ,-p_n)$ .

The following holds.

Lemma 3.5. Let $(X,\omega )$ be a translation surface obtained from $(Y,\xi )\in \mathcal {H}_{g}\big (2m_1,\dots ,2m_k;-2p_1,\dots ,-2p_n\big )$ by bubbling a handle along a slit. Let $2\pi (l+1)$ be the angle around one of the extremal points of the slit, where $l=2m_i\ge 2$ for some $i=1,\dots ,k$ or $l=0$ . Then, the spin structures determined by $\omega $ and $\xi $ are related as follows

(26) $$ \begin{align} \varphi(\omega)-\varphi(\xi)=l \,\,\,(\mod 2). \end{align} $$

In particular, bubbling a handle does not alter the spin parity.

3.3 Bubbling a handle with nonpositive volume

We have seen above how to glue a handle with positive volume on a given translation surface $(X,\omega )$ . Here, we briefly describe a way to glue a handle with nonpositive volume and nontrivial periods; we shall describe a way to add handles with trivial periods in subsection §8.1. Topologically, this surgery deletes the interior of a parallelogram with distinct vertices on $(X,\omega )$ whose sides are given by the absolute periods of the handle we want to glue.

This includes the case in which the parallelogram is degenerate, that is, with empty interior, and the surgery reduces to slit along a segment. Since this construction is a local surgery, we explain how the gluing works in the complex plane because the same argument applies to any simply connected open neighborhood of $(X,\omega )$ . Let $a,b\in \mathbb C$ be real-collinear complex numbers with argument $\theta $ . Let l be a segment in $\mathbb C$ with slope $\theta $ and length equal to $|a|+|b|$ . Denote by $P_1$ and $P_3$ the end points of l. Slit $\mathbb C$ along l, and denote the resulting sides $l^{\pm }$ . More precisely, let $l^+$ denote the side of l leaving $\mathbb C$ on the right, and let $l^-$ denote the side of l leaving $\mathbb C$ on the left. On $l^+$ , let $P_2$ be the point at distance $|a|$ from $P_1$ . In the same fashion, on $l^-$ let $P_4$ be the point at distance $|a|$ from $P_3$ . We next identify the oriented segment $\overline {P_1\,P_2}$ with the oriented segment $\overline {P_4\,P_3}$ . Finally, identify the oriented segment $\overline {P_2\,P_3}$ with the oriented segment $\overline {P_1\,P_4}$ . The result is a handle with absolute periods $a,\,b\in \mathbb C$ and, by construction, it has volume zero; see volume formula (14) and Figure 5.

Figure 5 Bubbling a handle with zero volume.

We shall need the following lemmas.

Lemma 3.8. If $(X,\omega )$ is a translation surface of even type and $(Y,\xi )$ is a translation surface obtained by bubbling a handle with nonpositive volume, then

(27) $$ \begin{align} \varphi(\omega)-\varphi(\xi)=0\,\,\,(\mod 2). \end{align} $$

In particular, bubbling a handle with nonpositive volume does not alter the spin parity.

Proof. Let $(X,\omega )$ be a translation surface, and let $\mathcal {P}$ be an embedded parallelogram. Choose a collection of $2g$ oriented simple curves $\{\alpha _i,\,\beta _i\}_{1\le i\le g}$ on $(X,\omega )$ representing a symplectic basis for $\mathrm {H}_1(S_{g,n},\,\mathbb {Z})$ . Up to deforming the paths inside their isotopy classes, we can make them stay away from some neighborhood, say U, of $\mathcal {P}$ . In other words, we can assume that bubbling a handle with nonpositive volume can be performed inside a neighborhood U. In particular, the surgery does not affect the initial collection of paths. We now complete the former collection of curves by adding two simple curves, say $\alpha _{g+1},\,\beta _{g+1}$ obtained after bubbling. These curves can be taken so that they both lie inside the neighborhood U. Since $(X,\omega )$ is a structure of even type, the vertices of $\mathcal {P}$ all have even orders. As a consequence, $\mathrm {Ind}(\alpha _{g+1})$ and $\mathrm {Ind}(\beta _{g+1})$ are odd positive integers; see Figure 6.

Figure 6 An embedded parallelogram $\mathcal {P}$ on a translation surface of even type. The vertices of $\mathcal {P}$ are points of orders $2m_i$ , for $i=1,\dots ,4$ and $m_i=0$ for regular points. A unit vector field along $\alpha _{g+1}$ winds $2m_1+2m_2+1$ times and a unit vector field along $\beta _{g+1}$ winds $2m_2+2m_3+1$ times. According to Definition 2.12, the indices of $\alpha _{g+1},\,\beta _{g+1}$ are equal to their winding numbers – notice that both curves turn counterclockwise.

A direct computation shows that

$$ \begin{align*} \varphi(\xi)&=\sum_{i=1}^{g+1} \big(\text{Ind}(\alpha_i)+1\big)\big(\text{Ind}(\beta_i)+1\big) \,\,(\text{mod}\,2)\\ & = \sum_{i=1}^{g} \big(\text{Ind}(\alpha_i)+1\big) \big(\text{Ind}(\beta_i)+1\big)+ \big(\mathrm{Ind}(\alpha_{g+1})+1\big)\big(\mathrm{Ind}(\beta_{g+1}) +1\big)\,\,(\text{mod}\,2)\\ &= \varphi(\omega) + \big(\mathrm{Ind}(\alpha_{g+1})+1\big)\big( \mathrm{Ind}(\beta_{g+1}) +1\big)\,\,(\text{mod}\,2)\\ & =\varphi(\omega)\,\,(\text{mod}\,2) \end{align*} $$

as desired.

3.4 Gluing surfaces along rays

In what follows, we shall need to glue translation surfaces along infinite rays. Recall, in fact, that our strategy to prove Theorem A is based on constructing translation surfaces with prescribed data (periods and geometric invariants) by gluing surfaces of lower complexity. We begin by introducing the following.

As special cases, we have the following gluings.

We can note that in the constructions above at most one new branch point is introduced arising from the extremal points of the rays after identification. Furthermore, bubbling a plane does not change the topology of the underlying surface; in particular, the resulting structure after bubbling has the same periods as the former one. Another special case worth of interest for us is the following.

The following results hold.

The proofs of these lemmas are easy to establish and left to the reader.

4.1 Mapping class group action

In order to realize a representation $\chi $ as the period character of some translation surface with poles, in what follows it will be convenient to replace, if necessary, a given system of handle generators with a more suitable one. This replacement can be done by precomposing $\chi $ with an automorphism induced by a mapping class transformation, that is, an element of the mapping class group $\mathrm {Mod}(S_{g,n})$ , where

(28) $$ \begin{align} \mathrm{Mod}(S_{g,n})= \frac{\text{Aut}\left(\pi_1(S_{g,n})\right)}{\text{Inn}\left(\pi_1(S_{g,n})\right)}\cong\frac{\text{Homeo}^+(S_{g,n})}{\text{Homeo}_o(S_{g,n})}=\pi_o\left(\text{Homeo}(\,S_{g,n})\,\right). \end{align} $$

Thus, to prove our main Theorem A for a given $\chi $ , it is sufficient to construct a genus g meromorphic differential on $S_{g,n}$ for which the induced representation is $\chi \circ \phi $ for some $\phi \in \mathrm {Mod}(S_{g,n})$ . It is worth mentioning that this replacement is legitimized by the following:

(29) $$ \begin{align} \chi \in \mathrm{Per}\left( \mathcal{H}_g(m_1,\dots,m_k;-p_1,\dots,-p_n)\right) \,\,\Longleftrightarrow \,\, \chi\circ\phi \in \mathrm{Per}\left( \mathcal{H}_g(m_1,\dots,m_k;-p_1,\dots,-p_n)\right) \end{align} $$

for any $\phi \in \mathrm {Mod}(S_{g,n})$ , and $\mathrm {Per}$ is the period mapping defined in Equation (7). In fact, if $\phi $ is induced by a homeomorphism, say f, then the pull-back of the translation structure by f defines a new translation surface with poles in the same stratum and with period character $\chi \circ \phi $ ; see also [Reference Le FilsLF22, Lemma 3.1] for the holomorphic case.

4.2 Nontrivial systems of generators exist

We aim to list a few lemmas about the mapping class group action for finding systems of handle generators such that any element has a nontrivial period. Most of them have already been proved in [Reference Chenakkod, Faraco and GuptaCFG22, Section §11], and here, we report a sketch of the proof for the reader’s convenience. The following lemmas show that the image of every handle generator under $\chi $ can be assumed to be nonzero whenever $\chi $ is a nontrivial representation.

Proof. Let $\chi $ be a nontrivial representation, and let $\{\alpha _i,\,\beta _i\}_{1\le i\le g}$ be a system of handle generators. We can assume that $\chi (\alpha _1) \neq 0$ . If $\chi (\beta _1)=0$ , then replace $\beta _1$ with $\alpha _1\beta _1$ and observe that $\chi (\alpha _1\beta _1)\neq 0$ . Suppose there is an index i such that $\chi (\alpha _i) = \chi (\beta _i) = 0$ . We consider a mapping class $\phi $ such that

(30) $$ \begin{align} \phi(\alpha_1)=\alpha_1\alpha_i, \,\,\, \phi(\beta_1)=\beta_1, \,\,\, \phi(\alpha_i)=\alpha_i, \,\,\, \phi(\beta_i)=\beta_1\beta_i^{-1}, \end{align} $$

and $\phi $ is the identity on the other handle generators. Finally, we replace the generator $\alpha _i$ with $\alpha _i\beta _1\beta _i^{-1}$ . Observe that $\{\alpha _i\,\beta _1\beta _i^{-1},\, \beta _1\beta _i^{-1}\}$ is a pair of handle generators and $\chi (\alpha _i\,\beta _1\beta _i^{-1})=\chi (\beta _1\beta _i^{-1})\neq 0$ by construction. By iterating this process finitely many times, we get the desired result.

Furthermore, for a representation $\chi $ we can also assume that the $\chi _g$ -part induced by a splitting is nontrivial whenever the $\chi _n$ -part is nontrivial.

4.3 $\mathrm {GL}^+(2,\mathbb {R})$ -action

We now consider an action on the representation space given by postcomposition with elements of $\mathrm {GL}^+(2,\mathbb {R})$ . This action can be combined with the mapping class group action (see Section §4.1 above), and it is easy to check that these actions commute. In the sequel, it will be sometimes useful to consider this action in order to realize representations as the period of meromorphic differentials in a given connected component of a stratum. This is in fact legitimated by the fact

(31) $$ \begin{align} \chi \in \mathrm{Per}\left( \mathcal{H}_g(m_1,\dots,m_k;-p_1,\dots,-p_n)\right) \,\,\Longleftrightarrow \,\, A\,\chi\in \mathrm{Per}\left( \mathcal{H}_g(m_1,\dots,m_k;-p_1,\dots,-p_n)\right) \end{align} $$

for any $A\in \mathrm {GL}^+(2,\mathbb {R})$ , and $\mathrm {Per}$ is the period mapping defined in Equation (7).

4.4 System of handle generators with positive volume

Notice that a mapping class $\phi \in \mathrm {Mod}(S_{g,n})$ does not need to preserve any splitting in general. Let us now consider again the notion of volume for meromorphic differentials already introduced in Section §2.1. For a representation $\chi \colon \mathrm {H}_1(S_{g,n},\,\mathbb {Z})\longrightarrow {\mathbb C}$ , the volume of $\chi $ (see Definition 2.10) is well defined if and only if it is of trivial-ends type. In other words, for any representation $\chi $ of trivial-ends type and for any mapping class $\phi \in \mathrm {Mod}(S_{g,n})$ the equation

(32) $$ \begin{align} \mathrm{vol}(\chi_g)=\mathrm{vol}\left( (\chi\circ\phi)_g\right), \end{align} $$

where $\chi _g$ and $(\chi \circ \phi )_g$ are the representations induced by the systems of handle generators $\{\alpha _i,\,\beta _i\}_{1\le i\le g}$ and $\{\phi (\alpha _i),\,\phi (\beta _i)\}_{1\le i\le g}$ , respectively. For a representation $\chi $ of nontrivial-ends type, the volume is no longer well defined, and hence, it is no longer an invariant because the representation $\chi _g$ does depend on the splitting and therefore also $\text {vol}(\chi _g)$ depends on it. We shall take advantage of this caveat to prove the following result which will be exploited in the sequel. We begin with introducing the following.

Proof. Let $\chi $ be a representation of nontrivial-ends type, and let $\{\alpha _i,\,\beta _i\}_{1\le i\le g}$ be a system of handle generators. From Section §2.1, recall that the curve $\gamma =[\alpha _1,\beta _1]\cdots [\alpha _g,\beta _g]$ bounds a subsurface $\Sigma $ homeomorphic to $S_{g,1}$ , the embedding $i\colon \Sigma \hookrightarrow S_{g,n}$ yields an injection $\imath _g\colon \text {H}_1(\Sigma ,\,\mathbb {Z})\cong \mathrm {H}_1(S_{g},\mathbb {Z})\to \mathrm {H}_1(S_{g,n},\,\mathbb {Z})$ , and the representation $\chi _g$ is defined as $\chi \circ \imath _g$ . In the case $\mathrm {vol}(\chi _g)>0$ , there is nothing to prove and we are done, so let us suppose that $\mathrm {vol}(\chi _g)\le 0$ . Being $\chi $ of nontrivial-ends type, there is a puncture, say p, with nonzero residue and let $\gamma $ be a simple closed curve around p. Clearly, $\chi (\gamma )=\chi _n(\gamma )\neq 0$ . Since $\chi $ is not real-collinear, we choose a pair of handle generators $\{\alpha ,\beta \}$ in the given system such that $\chi (\beta )$ is not collinear with $\chi (\gamma )$ . Replace it with the pair of generators $\{\alpha ',\,\beta '\}$ , where $\beta '=\beta $ and $\alpha '=\alpha \,\beta ^{-n}\,\gamma \,(\beta \,\gamma )^n$ for $n\in \mathbb Z$ . This is a simple closed nonseparating, curve and we can easily compute that

(33) $$ \begin{align} \chi\left(\alpha\,\beta^{-n}\,\gamma\,(\beta\,\gamma)^n\,\right)=\chi(\alpha)+(n+1)\chi(\gamma). \end{align} $$

Up to relabelling all handles, we can assume $\{\alpha ,\,\beta \}=\{\alpha _1,\,\beta _1\}$ . The new system of handle generators will be $\{\alpha ',\,\beta ',\alpha _2,\beta _2,\dots ,\alpha _g,\,\beta _g\}$ . Such a system determines a different splitting, namely a different separating curve $\gamma '$ that bounds a subsurface $\Sigma '$ homeomorphic to $S_{g,1}$ . The embedding $i'\colon \Sigma '\hookrightarrow S_{g,n}$ yields a representation $\chi ^{\prime }_g=\chi \circ \imath ^{\prime }_g$ , where $\imath ^{\prime }_g$ is the injection in homology induced by $i'$ . The volume $\mathrm {vol}(\chi ^{\prime }_g)$ can be explicitly computed as follow:

(34) $$ \begin{align} \mathrm{vol}(\chi^{\prime}_g) & = \Im\left(\,\overline{\chi(\alpha')}\,\chi(\beta')\,\right)+\sum_{i=2}^g \Im \left(\,\overline{\chi(\alpha_i)}\,\chi(\beta_i)\,\right)\qquad\qquad\qquad\qquad\quad \ \end{align} $$

(35) $$ \begin{align} & = \Im\left(\,\overline{\chi(\alpha_1)+(n+1)\chi(\gamma)}\,\chi(\beta_1)\,\right)+\sum_{i=2}^g \Im \left(\,\overline{\chi(\alpha_i)}\,\chi(\beta_i)\,\right) \end{align} $$

(36) $$ \begin{align} &= (n+1)\Im\left(\,\overline{\chi(\gamma)}\,\chi(\beta_1)\,\right)+ \sum_{i=1}^g \Im \left(\,\overline{\chi(\alpha_i)}\,\chi(\beta_i)\,\right)\qquad\quad \ \end{align} $$

(37) $$ \begin{align} &= (n+1)\Im\left(\,\overline{\chi(\gamma)}\,\chi(\beta_1)\,\right)+\mathrm{vol}(\chi_g). \qquad\qquad\qquad\qquad \end{align} $$

Since $\mathrm {vol}(\chi _g)$ is constant and since $\chi (\gamma )$ and $\chi (\beta _1)$ are not collinear, by choosing $n\in \mathbb Z$ judiciously, we can make $\mathrm {vol}(\chi ^{\prime }_g)>0$ and hence obtain the desired result.

By combining Lemma 4.9 with Kapovich’s results in [Reference KapovichKap20], we can infer the following result on which we shall rely in the sequel.

4.5 Discrete and dense representations

In the sequel, we shall distinguish three kinds of representations according to the following.

We have the following lemmas.

Proof. Let $\chi $ be a representation of nontrivial-ends type such that $\mathrm {Im}(\chi )\subset \mathbb R$ . Recall that any system of handle generators $\{\alpha _i,\,\beta _i\}_{1\le i\le g}$ induces a splitting and hence a representation $\chi _n$ . Notice that, since $\chi $ is not discrete of rank one, at least one generator of $\mathrm {H}_1(S_{g,n},\,\mathbb {Z})$ has an irrational period. If $\mathrm {Im}(\chi _n)\subset \mathbb Q$ , then the result follows from [Reference Chenakkod, Faraco and GuptaCFG22, Lemma 11.5]; notice that this is always the case when $n=2$ (up to rescaling). Let us now assume $\mathrm {Im}(\chi _n)\not \subset \mathbb Q$ and $n\ge 3$ . Up to rescaling, we can assume one puncture, say $P_1$ , has a rational residue, say $r_1$ . Necessarily, there is a puncture $P_2$ with irrational residue $r_2$ . We can assume that at least one handle generator has an irrational period, otherwise we can change the system of handle generators above by replacing $\alpha _1$ with $\alpha _1\gamma _2$ , where $\gamma _2$ is a loop around the puncture $P_2$ . Notice that such a replacement can be performed with a mapping class transformation. Therefore, we can assume that $\alpha _1$ has an irrational period. There exists $\phi _i\in \mathrm {Mod}(S_{g,n})$ such that $\chi \circ \phi _i(\alpha _i)$ is also irrational. This mapping class can be explicitly written as

(38) $$ \begin{align} \phi_i(\alpha_i)=\alpha_1\,\alpha_i, \quad \phi_i(\beta_1)=\beta_1\,\beta_i^{-1}, \quad \phi_i(\delta)=\delta \quad \text{ for }\,\, \delta\notin\{\alpha_i,\beta_1\}. \end{align} $$

By composing all these $\phi _i$ ’s, the resulting mapping class $\phi $ provides a system of handle generators such that $\chi (\alpha _i)$ is irrational for any $i=1,\dots ,g$ . We now proceed as follows. If $\chi (\alpha _i)$ and $\chi (\beta _i)$ are linearly independent over ${\mathbb Q}$ , we can assume that both of them are positive after Dehn twists and we apply the Euclidean algorithm for making them arbitrarily small. In the case $\chi (\alpha _i)$ and $\chi (\beta _i)$ are linearly dependent over ${\mathbb Q}$ (hence $\chi (\beta _i)$ is also irrational), we replace $\beta _i$ with $\beta _i\gamma _1$ . Since $\chi (\gamma _1)\in \mathbb Q$ , then $\chi (\alpha _i)$ and $\chi (\beta _i\,\gamma _1)$ are linearly independent over ${\mathbb Q}$ . Again, we can assume that both of them are positive after Dehn twists and we apply the Euclidean algorithm for making them arbitrarily small.

5.1 Strata with poles of order two and zero residue

In this section, we shall prove the following lemma for the case where all poles have order exactly two and zero residue. We propose a proof in which the structures are explicitly constructed with prescribed periods and rotation number. As we shall see, all the other cases follow as simple variations of the constructions in this special case.

Let $\chi \colon \mathrm {H}_1(S_{1,\,n},\mathbb {Z})\longrightarrow {\mathbb C}$ be a nontrivial representation, and assume $\chi $ can be realized as the period character of a genus-one meromorphic differential with poles in the stratum $\mathcal {H}_1(2n;-2,\dots ,-2)$ . Recall that any such a stratum has two connected components according to the rotation number with the only exception being the stratum $\mathcal {H}_1(2,-2)$ which is connected and hence already handled in [Reference Chenakkod, Faraco and GuptaCFG22]. Therefore, we assume without loss of generality that $n\ge 2$ .

5.1.1 Realizing representations with rotation number one

We begin with realizing $\chi $ as the period character of some genus-one meromorphic differential with rotation number one. We shall distinguish two cases according to the volume of the representation $\chi $ .

5.1.1.1 Positive volume

Let $\alpha ,\beta $ be a pair of handle generators, and let $\mathcal {P}\subset \mathbb {C}$ be the parallelogram defined by the chain

(40) $$ \begin{align} P_0\mapsto P_0+\chi(\alpha)\mapsto P_0+\chi(\alpha)+\chi(\beta)=Q_0\mapsto P_0+\chi(\beta)\mapsto P_0, \end{align} $$

where $P_0\in {\mathbb C}$ is any point. According to our convention above, let us denote by $a_0^+$ (respectively $a_0^-$ ) the edge of $\mathcal {P}$ parallel to $\chi (\alpha )$ that bounds the parallelogram on its right (respectively left). In the same fashion, let us denote by $b_0^+$ (respectively $b_0^-$ ) the edge of $\mathcal {P}$ parallel to $\chi (\beta )$ that bounds the parallelogram on its right (resp. left). Recall that $(\mathbb {C},\, dz)$ is a genus-zero translation surface with trivial period character, no zeros and one pole of order $2$ at the infinity with zero residue. We slit $(\mathbb {C},\,dz)$ along a segment $a_n$ , and we denote the resulting segments as $a_n^+$ and $a_n^-$ . Let $P_n$ and $Q_n=P_n+\chi (\alpha )$ be the extremal points of $a_n$ . We next consider other $n-1$ copies of $(\mathbb {C},\, dz)$ , and we slit each of them along a segment b. For any $i=1,\dots ,n-1$ , let $P_i$ and $Q_i=P_i+\chi (\beta )$ be the extremal points of $b_i$ and denote as $b_i^+$ and $b_i^-$ the resulting segments after slitting. We are now ready to glue these n copies of $({\mathbb C},dz)$ and the parallelogram $\mathcal {P}$ all together. The desired structure is then obtained by identifying the segment $a_0^+$ with $a_n^-$ , the segment $a_0^-$ with $a_n^+$ , the segment $b_i^+$ with $b_{i+1}^-$ , where $i=1,\dots ,n-2$ and, finally, $b_0^-$ with $b_{n-1}^+$ and $b_0^+$ with $b_1^-$ ; see Figure 8.

Figure 8 Realizing a genus-one translation surface with poles of order $2$ , positive volume and rotation number equal to $1$ . In this case, all poles are assumed to have zero residue.

We will show that the resulting genus-one meromorphic differential, say $(X,\omega )$ , has rotation number one. We first find a closed loop representing the desired handle generator $\alpha $ in the following way. For any $i=1,\dots ,n-1$ , we consider a small metric circle of radius $\varepsilon $ centered at $P_i$ . After the cut and paste process just described, these $n-1$ circles define a smooth path with starting point on $b_0^-$ and ending point on $b_0^+$ . These two points on $\mathcal {P}$ differ by $\chi (\alpha )$ , so we can join them with a geodesic segment parallel to a. The resulting curve is a simple close curve on $(X,\omega )$ ; see Figure 8. We can see that $\text {Ind}(\alpha )= 1-n$ , with period equal to $\chi (\alpha )$ . In a similar way, we can find a closed loop representing the other handle generator $\beta $ . We consider a small metric circle of radius $\varepsilon $ centered at $P_n$ . After the cut and paste process, the extremal points of such an arc are on $a_0^\pm $ and differ by $\chi (\beta )$ on $\mathcal {P}$ . We join them with a geodesic segment and the resulting curve is simple and closed in $(X,\omega )$ . By construction, it has index one with period equal to $\chi (\beta )$ . By checking their (self) intersection numbers, the two loops form a pair of handle generators with desired periods. Therefore, according to the formula (2.13) we have that $\gcd (\, 1-n,\,1,\,2n,\,2\,)=1$ , and hence, the structure has rotation number one.

5.1.1.2 Nonpositive volume

We now assume that $\chi $ has nonpositive volume. This case is similar to the case of positive volume. As above, let $\alpha ,\beta $ be a set of handle generators, and let $\mathcal {P}\subset \mathbb {C}$ be the closure of the exterior of the parallelogram

(41) $$ \begin{align} P_0\mapsto P_0+\chi(\alpha)\mapsto P_0+\chi(\alpha)+\chi(\beta)=Q_0\mapsto P_0+\chi(\beta)\mapsto P_0, \end{align} $$

where $P_0\in {\mathbb C}$ is any point. Notice that $\mathcal {P}$ itself is a topological quadrilateral on $\mathbb {C}\mathrm {\mathbf {P}}^1$ . We denote the sides of $\mathcal {P}$ as $a^{\pm }$ and $b_0^{\pm }$ according to our convention. We next consider $n-1$ copies of $(\mathbb {C},\, dz)$ and we slit each of them along a segment b. For any $i=1,\dots ,n-1$ , we denote as $b_i^+$ and $b_i^-$ the resulting segments after slitting and let $P_i$ and $Q_i=P_i+\chi (\beta )$ be the extremal points of $b_i$ . The desired structure is then obtained by identifying the segment $a^+$ with $a^-$ , the segment $b_{i}^-$ with $b_{i+1}^+$ for $i=1,\dots ,n-2$ , the segment $b_0^-$ with $b_{1}^+$ and the segment $b_0^+$ with $b_{n-1}^-$ ; see Figure 9. In this case, it is still easy to observe that the final structure $(X,\omega )$ has rotation number one. We can find a closed loop representing $\alpha $ similarly as before. For $\beta $ , we can choose any segment in $\mathcal {P}$ parallel to $b_0^+$ with extremal points R and $R+\chi (\beta )$ . We join R (respectively $R+\chi (\beta )$ ) with any point in $a^-$ (respectively $a^+$ ) by means of an embedded arc. The resulting curve close up to a simple closed curve on $(X,\omega )$ and has index one. By checking intersection numbers the two loops form a pair of handle generators with desired periods. Therefore, the structure $(X,\omega )$ has rotation number one.

Figure 9 Realizing a genus-one translation surface with poles of order $2$ , nonpositive volume and rotation number equal to $1$ . In this case, all poles are assumed to have zero residue.