Dense real Rel flow orbits and absolute period leaves

Our two main results address the topological dynamics of real Rel flows and the foliations obtained by “complexifying” these flows. We state these results in Theorems 1.1 and 1.5 below, and we outline our methods of proof.

Since the area of a holomorphic $1$-form is constant along a real Rel flow orbit, let $\widetilde{\Omega}_{1}\mathcal{M}_{g}(\kappa)$ be the area-$1$ locus in $\widetilde{\Omega}\mathcal{M}_{g}(\kappa)$. A basic question in topological dynamics is the existence of a dense orbit. Our first main result completely answers this question for real Rel flows on strata.

When $n>1$, the stratum $\Omega\mathcal{M}_{g}(\kappa)$ also admits a holomorphic foliation $\mathcal{A}(\kappa)$ called the absolute period foliation. Two holomorphic $1$-forms lie on the same leaf of $\mathcal{A}(\kappa)$ if and only if there is a path between them along which the integrals along closed loops are constant. Leaves of $\mathcal{A}(\kappa)$ have a natural locally Euclidean metric, recorded by integrals along paths between distinct zeros. Orbits of real Rel flows project to geodesics on leaves of $\mathcal{A}(\kappa)$, so our density result has consequences for the absolute period foliation.

In [Win], a stronger density result is proven for leaves of $\mathcal{A}(\kappa)$ in the case where $n>1$ and $\Omega\mathcal{M}_{g}(\kappa)$ is connected. The above results are not explicit. However, our proofs will involve explicit constructions and will provide explicit examples of dense leaves of $\mathcal{A}(\kappa)$ and of many foliations whose leaves are contained in leaves of $\mathcal{A}(\kappa)$. Our second main result will provide a stronger and explicit version of Corollary 1.2.

Our study of real Rel flows is based on a study of their centralizers and normalizers. In our setting, we are most interested in the action of $\operatorname{SL}(2,\mathbb{R})$ and its subgroups on $\widetilde{\Omega}\mathcal{M}_{g}(\kappa)$. The horocycle flow and the geodesic flow are defined by the actions of

respectively. Real Rel flows commute with the horocycle flow, and both are normalized by the geodesic flow.

One of the novelties in our approach to Theorem 1.1 is an application of a recent result in [For] to the study of real Rel flows. Suppose that the closure of $\operatorname*{Rel}_{\mathbb{R}v}(X,\omega)$ contains the horocycle through $(X,\omega)$. Since the geodesic flow normalizes both real Rel flows and the horocycle flow, this property is invariant under the geodesic flow. By Corollary 1.3 in [For], pushforwards of horocycle arcs under the geodesic flow equidistribute in the associated $\operatorname{SL}(2,\mathbb{R})$-orbit closure, outside a sequence of times of zero upper density. By applying $g_{t_{n}}$ for an appropriate sequence of times $t_{n}$, we obtain real Rel flow orbits that become more and more dense in an $\operatorname{SL}(2,\mathbb{R})$-orbit closure. A short argument using the Baire category theorem then yields the existence of a real Rel flow orbit whose closure contains an $\operatorname{SL}(2,\mathbb{R})$-orbit closure. Thus, we obtain a simple and general criterion for the existence of a dense real Rel flow orbit in an $\operatorname{SL}(2,\mathbb{R})$-orbit closure. We present this argument in Section 3.

Our second observation is that holomorphic $1$-forms $(X,\omega)$ with a periodic horizontal foliation can provide explicit examples for which the closure of $\operatorname*{Rel}_{\mathbb{R}v}(X,\omega)$ contains the horocycle through $(X,\omega)$. In this case, $X$ is covered by finitely many cylinders of horizontal closed geodesics along with finitely many horizontal geodesic segments between zeros. The horocycle through $(X,\omega)$ is given by twisting each horizontal cylinder. The heights and circumferences of the horizontal cylinders, and the lengths of horizontal geodesic segments, are constant along the horocycle. The heights and circumferences of horizontal cylinders are also constant along a real Rel flow orbit, but the lengths of certain horizontal geodesic segments between zeros may change. However, if $v_{i}-v_{j}=0$ whenever $Z_{i}$ and $Z_{j}$ are the endpoints of a horizontal geodesic segment, then $\operatorname*{Rel}_{\mathbb{R}v}(X,\omega)$ can also be described by twisting the horizontal cylinders. In this setting, both the horocycle flow and the real Rel flow can essentially be thought of as linear flows on a compact torus parametrized by twist parameters for the horizontal cylinders. This was previously described in [HW] in the case of holomorphic $1$-forms with exactly two zeros. In Section 4, we show that if every horizontal cylinder is twisted along $\operatorname*{Rel}_{\mathbb{R}v}(X,\omega)$, and if distinct horizontal cylinders do not contain homologous closed geodesics, then it is possible to perturb the horizontal cylinder circumferences so that $\operatorname*{Rel}_{\mathbb{R}v}(X,\omega)$ is dense in this twist torus and so that the $\operatorname{SL}(2,\mathbb{R})$-orbit of $(X,\omega)$ is dense in the area-$1$ locus of its stratum component. For the latter density property, we rely on the explicit density criterion for $\operatorname{GL}^{+}(2,\mathbb{R})$-orbits in strata from [Wri1]. Theorem 1.1 is thus reduced to the following construction which is carried out in Section 5.

The constructions in the proof of Theorem 1.4 are rather involved and occupy the bulk of this paper. Briefly, we first use an intricate connected sum of tori to establish the case where every $k_{j}\in\kappa$ satisfies $k_{j}\leq g-1$ and the components of $v$ are distinct, and we then deal with the remaining cases using well-known surgeries.

Our second main result is an explicit density result for “complexified” real Rel flow orbits. A real vector subspace $V$ of $\mathbb{C}^{n}_{0}$ determines a foliation $\mathcal{A}_{V}(\kappa)$ of $\widetilde{\Omega}\mathcal{M}_{g}(\kappa)$ whose leaves project into leaves of $\mathcal{A}(\kappa)$. Our second main result provides explicit dense leaves of many subfoliations of $\mathcal{A}(\kappa)$ with complex $1$-dimensional leaves in all stratum components with multiple zeros.

A novelty in our approach to Theorem 1.5 is that it is based on analyzing periodic foliations in multiple directions simultaneously. Much of the proof of Theorem 1.5 is contained in the proof of Theorem 1.4. In most cases, the holomorphic $1$-forms we construct here have a periodic vertical foliation with exactly analogous properties. In particular, we end up constructing explicit holomorphic $1$-forms $(X,\omega)$ such that the closure of $\operatorname*{Rel}_{\mathbb{R}v}(X,\omega)$ contains the horocycle through $(X,\omega)$, and such that the closure of $\operatorname*{Rel}_{i\mathbb{R}v}(X,\omega)$ contains the opposite horocycle through $(X,\omega)$.

We expect that the hypothesis that $v$ has distinct components can be removed. However, with our approach, it seems that doing so would greatly increase the length of this paper, and we decided not to pursue this. Part of our motivation for proving Theorem 1.5 is as a complement to Theorem 1.2 in [Win], which produces explicit full measure sets of dense leaves of $\mathcal{A}(\kappa)$ in $\Omega_{1}\mathcal{M}_{g}(\kappa)$. The holomorphic $1$-forms in these dense leaves of $\mathcal{A}(\kappa)$ are in some sense as far as possible from having a periodic foliation in any direction. We hope that the methods in these two papers can be combined to make progress toward classifying the closures of leaves of $\mathcal{A}(\kappa)$ and $\mathcal{A}_{V}(\kappa)$. Lastly, we remark that the criterion in Theorem 1.3 can be readily verified for other $\operatorname{SL}(2,\mathbb{R})$-orbit closures, and we briefly discuss this in Section 5 as well.

In [BSW] and [MW2], it is shown that the only obstruction to a real Rel flow orbit being well-defined for all time comes from horizontal saddle connections. Related completeness results for leaves of $\mathcal{A}(\kappa)$ are proven in [McM2] and [McM3]. In the terminology of [McM2], Theorem 1.1 implies the existence of dense relative period geodesics in the area-$1$ locus of every connected component of $\Omega\mathcal{M}_{g}(k_{1},\dots,k_{n})$ when $n>1$.

The study of real Rel flows and the horocycle flow on strata are often intertwined. In [BSW], real Rel flows play an important role in the classification of orbit closures and invariant measures for the horocycle flow on the eigenform loci in $\Omega\mathcal{M}_{2}(1,1)$. It would be interesting to see if closures of real Rel flow orbits in these eigenform loci can also be classified. In [MW1], it is shown that horocycles in strata do not diverge, and that almost every holomorphic $1$-form in a horocycle has a uniquely ergodic vertical foliation. Similar results for families of interval exchange transformations arising from horocycles are proven in [MW2]. In contrast, [HW] gives an example of a real Rel flow orbit that is well-defined for all time but diverges in its stratum. Moreover, in [HW], it is shown that the Arnoux-Yoccoz surface in genus $3$ gives an example for which there is a unique holomorphic $1$-form in a real Rel flow orbit with a uniquely ergodic vertical foliation, while all others have a periodic vertical foliation. Other exotic behaviors of vertical foliations along relative period geodesics are exhibited in [McM3] and [McM4]. In [CSW], it is shown that closures of horocycles in $\Omega\mathcal{M}_{2}(1,1)$ can have non-integer Hausdorff dimension. In light of [CSW], it would be interesting to see if closures of real Rel flow orbits in $\Omega\mathcal{M}_{2}(1,1)$ can also have non-integer Hausdorff dimension.

In [Ygo], a related criterion to our Theorem 1.3 is given for the density of leaves of the absolute period foliation in the special case of affine invariant manifolds of rank $1$ (defined in [Wri2]). The criterion in [Ygo] is based on finding horizontally periodic surfaces where the horizontal cylinders can be twisted while staying in the absolute period leaf and accumulating on a horocycle. Horocycle invariance of the leaf closure is promoted to $\operatorname{SL}(2,\mathbb{R})$-invariance, and thus density due to the rank $1$ condition, using the existence of holomorphic $1$-forms with a hyperbolic Veech group element in every absolute period leaf in rank $1$. A key difference in our criterion is the use of Corollary 1.3 in [For], which is needed to obtain a density result for real Rel flow orbits (as opposed to the full absolute period foliation) as well as to obtain a general result for $\operatorname{SL}(2,\mathbb{R})$-orbit closures. Additionally, hyperbolic Veech group elements do not play a role in our arguments. Jon Chaika and Barak Weiss have informed us of work in progress in which they prove that real Rel flows are ergodic on connected components of $\widetilde{\Omega}_{1}\mathcal{M}_{g}(\kappa)$ when $n>1$, conditional on an extension of the results of [EM] to products of strata. We remark that their ergodicity result does not imply Theorem 1.5, since holomorphic $1$-forms in a stratum with a periodic foliation form a measure zero subset.

The dynamics of $\mathcal{A}(\kappa)$ on $\Omega_{1}\mathcal{M}_{g}(\kappa)$ have been more extensively studied, for instance, in [CDF], [Ham], [HW], [McM3], [Win], [Ygo]. Ergodicity of $\mathcal{A}(1,\dots,1)$ on $\Omega_{1}\mathcal{M}_{g}(1,\dots,1)$ is proven in [McM3] for $g=2,3$, and for all $g\geq 2$ in [CDF], [Ham]. Moreover, a classification of closures of leaves of $\mathcal{A}(1,\dots,1)$ is given in [CDF]. In [HW] and [Ygo], examples of dense leaves are given in a certain connected component of $\Omega_{1}\mathcal{M}_{g}(g-1,g-1)$ and in $\Omega_{1}\mathcal{M}_{3}(2,1,1)$. In [Win], ergodicity of $\mathcal{A}(\kappa)$ on the area-$1$ locus of connected strata with $n>1$ and the non-hyperelliptic connected component of $\Omega\mathcal{M}_{g}(g-1,g-1)$ for $g$ even is proven. Moreover, in [Win], explicit full measure sets of dense leaves are given in these loci.

The author thanks Curt McMullen for encouragement and comments on earlier versions of this work. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under grant DGE-1144152.

Let $X$ be a closed Riemann surface of genus $g\geq 2$, and let $\omega$ be a nonzero holomorphic $1$-form on $X$. The zero set $Z(\omega)$ is finite, and the orders of the zeros form a partition $\kappa=(k_{1},\dots,k_{n})$ of $2g-2$. For each $x\in X\setminus Z(\omega)$, there is a simply connected open neighborhood $U$ of $x$ and an injective holomorphic map $\phi:U\rightarrow\mathbb{C}$ given by $\phi(z)=\int_{x}^{z}\omega$ and satisfying $\omega=\phi^{\ast}(dz)$. These maps provide an atlas of charts on $X\setminus Z(\omega)$ whose transition maps are translations. Translation-invariant structures on $\mathbb{C}$ can be pulled back to structures on $X$ with singularities at the zeros of $\omega$. In particular, associated to the pair $(X,\omega)$ is a flat metric with a cone point of angle $2\pi(k+1)$ at a zero of order $k$. Additionally, the foliations of $\mathbb{C}$ by horizontal and vertical lines determine a horizontal foliation and a vertical foliation of $(X,\omega)$ each with $2k+2$ singular leaves meeting at a zero of order $k$.

A saddle connection on $(X,\omega)$ is an oriented geodesic segment with endpoints in $Z(\omega)$ and otherwise disjoint from $Z(\omega)$. Any closed geodesic in $X\setminus Z(\omega)$ is contained in a maximal open cylinder given by a union of freely homotopic parallel closed geodesics. The boundary of a cylinder is a finite union of parallel saddle connections. Any cylinder $C$ is isometric to a unique Euclidean cylinder of the form $\mathbb{R}/w\mathbb{Z}\times(0,h)$ with $h,w>0$, and $h$ and $w$ are the height and circumference of $C$, respectively. A horizontal cylinder is a cylinder containing a closed geodesic $\alpha$ such that $\int_{\alpha}\omega\in\mathbb{R}_{>0}$. The orientation of $\alpha$ determines a top boundary of $C$ and a bottom boundary of $C$, which are not necessarily disjoint. Given a saddle connection $\gamma$ which crosses a horizontal cylinder $C$ from bottom to top, the twist parameter of $C$ with respect to $\gamma$ is defined by ${\rm Re}\int_{\gamma}\omega\in\mathbb{R}/w\mathbb{Z}$. We similarly define a vertical cylinder with $i\mathbb{R}_{>0}$ in place of $\mathbb{R}_{>0}$, as well as the left boundary, the right boundary, and the twist parameter of a vertical cylinder with respect to a saddle connection crossing the cylinder from right to left. Two cylinders are homologous if they contain closed geodesics that represent the same element of $H_{1}(X;\mathbb{Z})$. The horizontal foliation of $(X,\omega)$ is periodic if every leaf is compact. In this case, $(X,\omega)$ is a union of finitely many disjoint horizontal cylinders and finitely many horizontal saddle connections. Similarly for the vertical foliation of $(X,\omega)$.

The moduli space of holomorphic $1$-forms $\Omega\mathcal{M}_{g}$ classifies pairs $(X,\omega)$ as above. The space $\Omega\mathcal{M}_{g}$ is a union of strata $\Omega\mathcal{M}_{g}(\kappa)$ indexed by partitions $\kappa=(k_{1},\dots,k_{n})$ of $2g-2$. A holomorphic $1$-form is in $\Omega\mathcal{M}_{g}(\kappa)$ if and only if it has exactly $n$ distinct zeros of orders $k_{1},\dots,k_{n}$. The bundle of relative homology groups $H_{1}(X,Z(\omega);\mathbb{Z})$ is locally trivial over $\Omega\mathcal{M}_{g}(\kappa)$. Given $(X_{0},\omega_{0})\in\Omega\mathcal{M}_{g}(\kappa)$, we can define period coordinates on a small neighborhood of $(X_{0},\omega_{0})$ by $(X,\omega)\mapsto[\omega]\in H^{1}(X_{0},Z(\omega_{0});\mathbb{C})$. By choosing a basis $\gamma_{1},\dots,\gamma_{2g+n-1}$ for $H_{1}(X_{0},Z(\omega_{0});\mathbb{Z})$, we obtain a map

and the components $\int_{\gamma_{j}}\omega$ are called the period coordinates of $(X,\omega)$. Period coordinates give $\Omega\mathcal{M}_{g}(\kappa)$ the structure of a complex orbifold of dimension $2g+n-1$. The area of $(X,\omega)$ is given by $\frac{i}{2}\int_{X}\omega\wedge\overline{\omega}$, and the area-$1$ locus of $\Omega\mathcal{M}_{g}(\kappa)$ is denoted by $\Omega_{1}\mathcal{M}_{g}(\kappa)$.

We recall the classification of connected components of strata from [KZ]. Suppose $k_{1},\dots,k_{n}$ are even. Let $\gamma\subset X\setminus Z(\omega)$ be a smooth oriented closed loop. Using the translation structure on $X\setminus Z(\omega)$, the index $\operatorname{ind}(\gamma)$ is defined as $1/2\pi$ times the total change in angle along the loop $\gamma$. In other words, $\operatorname{ind}(\gamma)$ is the degree of the Gauss map $\gamma\rightarrow S^{1}$. Let $\{\alpha_{j},\beta_{j}\}_{j=1}^{g}$ be a collection of smooth oriented closed loops in $X\setminus Z(\omega)$ that represents a symplectic basis for $H_{1}(X;\mathbb{Z})$ with respect to the algebraic intersection form. The parity of the spin structure $\phi(\omega)$ is defined by

and is independent of the choice of symplectic basis of $H_{1}(X;\mathbb{Z})$ and the choice of representative loops. Moreover, $\phi(\omega)$ is an invariant of the connected component of $(X,\omega)$ in $\Omega\mathcal{M}_{g}(\kappa)$. A connected component is even or odd if it consists of holomorphic $1$-forms $(X,\omega)$ with $\phi(\omega)=0$ or $\phi(\omega)=1$, respectively. A connected component is hyperelliptic if it consists of holomorphic $1$-forms on hyperelliptic Riemann surfaces with a unique zero, or if it consists of holomorphic $1$-forms on hyperelliptic Riemann surfaces with exactly two zeros that have equal order and are exchanged by the hyperelliptic involution. A connected component that is not hyperelliptic is nonhyperelliptic.

Let $\operatorname{GL}^{+}(2,\mathbb{R})$ be the group of automorphisms of the vector space $\mathbb{R}^{2}$ with positive determinant. The standard $\mathbb{R}$-linear action of $\operatorname{GL}^{+}(2,\mathbb{R})$ on $\mathbb{C}$ induces an action on $\Omega\mathcal{M}_{g}$, and this action preserves each stratum. The action of the subgroup $\operatorname{SL}(2,\mathbb{R})$ also preserves the area-$1$ locus. For $s,t\in\mathbb{R}$, define

The action of the diagonal subgroup $g_{t}$ is the geodesic flow, and the action of the unipotent subgroup $u_{s}$ is the horocycle flow. The subgroups $u_{s}$ and $v_{s}$ generate $\operatorname{SL}(2,\mathbb{R})$.

In [EMM], rigidity theorems are proven for $\operatorname{GL}^{+}(2,\mathbb{R})$-orbit closures in strata. In particular, orbit closures are properly immersed suborbifolds locally defined by homogeneous $\mathbb{R}$-linear equations in period coordinates. Building off of this work, explicit full measure subsets of dense $\operatorname{GL}^{+}(2,\mathbb{R})$-orbits in connected components of strata are given in [Wri1]. Let $\overline{\mathbb{Q}}\subset\mathbb{C}$ be the algebraic closure of $\mathbb{Q}$.

See [Wri1] for a more general result. We only stated the special case that we will use.

Given $(X,\omega)\in\Omega\mathcal{M}_{g}(\kappa)$, the projection from relative to absolute cohomology

defines a holomorphic submersion on a neighborhood of $(X,\omega)$ in $\Omega\mathcal{M}_{g}(\kappa)$. The fibers have complex dimension $n-1$ and form the leaves of a holomorphic foliation $\mathcal{A}(\kappa)$ called the absolute period foliation of $\Omega\mathcal{M}_{g}(\kappa)$. Holomorphic $1$-forms on the same leaf of $\mathcal{A}(\kappa)$ have the same area. The action of $\operatorname{GL}^{+}(2,\mathbb{R})$ sends leaves of $\mathcal{A}(\kappa)$ to leaves of $\mathcal{A}(\kappa)$.

Consider the finite cover

obtained by labelling the zeros $Z_{1},\dots,Z_{n}$ of holomorphic $1$-forms in $\Omega\mathcal{M}_{g}(\kappa)$. The foliation $\mathcal{A}(\kappa)$ lifts to a foliation of $\widetilde{\Omega}\mathcal{M}_{g}(\kappa)$, which we call the absolute period foliation of $\widetilde{\Omega}\mathcal{M}_{g}(\kappa)$. Consider the quotient vector spaces

where $(1,\dots,1)$ is the constant vector. The labelling of zeros provides a canonical identification of the kernel of $p$ with $\mathbb{C}^{n}_{0}$. Note that for $v\in\mathbb{C}_{0}^{n}$, the difference between two components $v_{i}-v_{j}$ is well-defined. The corresponding element of the kernel of $p$ evaluates to $v_{i}-v_{j}$ on any path from $Z_{j}$ to $Z_{i}$. Associated to a real vector subspace $V\subset\mathbb{C}^{n}_{0}$ is then a foliation $\mathcal{A}_{V}(\kappa)$ of $\widetilde{\Omega}\mathcal{M}_{g}(\kappa)$ whose leaves are contained in leaves of the absolute period foliation of $\widetilde{\Omega}\mathcal{M}_{g}(\kappa)$. In the case where $v\in\mathbb{R}^{n}_{0}$ and $V=\mathbb{C}v$, the action of $\operatorname{GL}^{+}(2,\mathbb{R})$ sends leaves of $\mathcal{A}_{V}(\kappa)$ to leaves of $\mathcal{A}_{V}(\kappa)$.

Let $L$ be the leaf of the absolute period foliation of $\widetilde{\Omega}\mathcal{M}_{g}(\kappa)$ through $(X_{0},\omega_{0})$. Given $q\in X$ and paths $\gamma_{j}$ from $q$ to $Z_{j}$, the relative period map

provides local coordinates on a neighborhood of $(X_{0},\omega_{0})$ in $L$. The map $\rho$ does not depend on the choice of $q$, but different choices of paths may translate the components of $\rho$ by absolute periods, which are constant on $L$. These local coordinates give $L$ a translation structure modelled on $\mathbb{C}^{n}_{0}$, and in particular a straight-line flow associated to each nonzero $v\in\mathbb{C}^{n}_{0}$. This straight-line flow gives rise to a partially defined flow on $\widetilde{\Omega}\mathcal{M}_{g}(\kappa)$ called a Rel flow. When $v\in\mathbb{R}^{n}_{0}$, this partially defined flow is called a real Rel flow. Orbits of Rel flows are not always well-defined for all time, since distinct zeros may collide in finite time. Rel flows will be denoted by $\operatorname*{Rel}_{tv}(X,\omega)$ with $t\in\mathbb{R}$ when well-defined. The map

is well-defined and continuous on an open subset of $\widetilde{\Omega}\mathcal{M}_{g}(\kappa)\times\mathbb{R}$. We denote by

the maximal domain of definition of the map $t\mapsto\operatorname*{Rel}_{tv}(X,\omega)$. For real Rel flows, the only obstructions to having $I(v,\omega)=\mathbb{R}$ come from horizontal saddle connections with distinct endpoints.

See also [MW2] and [McM2]. In particular, if $(X,\omega)$ does not have a horizontal saddle connection with distinct endpoints, then $\operatorname*{Rel}_{\mathbb{R}v}(X,\omega)$ is well-defined.

The horocycle flow commutes with real Rel flows, and the geodesic flow normalizes real Rel flows. For $v\in\mathbb{R}^{n}_{0}$ nonzero, $(X,\omega)\in\widetilde{\Omega}\mathcal{M}_{g}(\kappa)$, and $s,t\in\mathbb{R}$, we have

where in each case the left-hand side is well-defined if and only if the right-hand side is.

We describe a local surgery for changing the zero orders of a holomorphic $1$-form, or a collection of holomorphic $1$-forms on disjoint surfaces, without changing any absolute periods. See Sections 8-9 in [EMZ], Section 5 in [CDF], and Section 3 in [McM2] for related discussions.

Let $(X_{j},\omega_{j})$, $j=1,\dots,n$, be a collection of holomorphic $1$-forms. Choose a nonzero $z\in\mathbb{C}$, choose points $p_{j}\in X_{j}$, and choose $1\leq r_{j}\leq k_{j}+1$, where $k_{j}\geq 0$ is the order of $\omega_{j}$ at $p_{j}$. Suppose there are oriented geodesic segments $s_{j,1},\dots,s_{j,r_{j}}$ in $(X_{j},\omega_{j})$ with $\int_{s_{j,k}}\omega_{j}=z$, starting at $p_{j}$ and disjoint from $Z(\omega_{j})$ except possibly at $p_{j}$, and such that the counterclockwise angle around $p_{j}$ from $s_{j,1}$ to $s_{j,k}$ is $2\pi(k-1)$. We view the segments

as being cyclically ordered. For notational simplicity, we rename the segments

respectively. We refer to the following surgery as applying the slit construction to the segments $s_{1},\dots,s_{N}$. For $j=1,\dots,N$, slit $s_{j}$ to obtain a pair of oriented segments $s_{j}^{+},s_{j}^{-}$, corresponding to the left and right sides of $s_{j}$, respectively. Glue $s_{j}^{+}$ to $s_{j+1}^{-}$ isometrically and respecting the orientations, where indices are taken modulo $N$. The result is a connected topological surface, and the complex structure and the given holomorphic $1$-forms extend over the slits to give a holomorphic $1$-form $(X,\omega)$.

The starting points of the segments yield a zero of $\omega$ of order $-1+\sum_{j=1}^{n}(k_{j}-r_{j}+2)$, and the ending points of the segments yield a zero of order $-1+\sum_{j=1}^{n}r_{j}$. In the special case where $n=1$ and $1<r_{1}<k_{1}+1$, the zero $p_{1}$ of order $k_{1}$ is split into two zeros of orders $k_{1}-r_{1}+1$ and $r_{1}-1$. In the special case where $k_{j}=0$ for all $j$, the starting points of the segments yield a zero of $\omega$ of order $n-1$, and the ending points of the segments yield another zero of $\omega$ of order $n-1$.

Suppose $(X,\omega)\in\widetilde{\Omega}\mathcal{M}_{g}(\kappa)$ has a periodic horizontal foliation, and let $C_{1},\dots,C_{m}$ be the horizontal cylinders on $(X,\omega)$. Let $h_{j}$ and $w_{j}$ be the height and circumference of $C_{j}$, respectively. Choose a saddle connection $\gamma_{j}\subset C_{j}\cup Z(\omega)$ crossing $C_{j}$ from bottom to top, and let $t_{j}\in\mathbb{R}/w_{j}\mathbb{Z}$ be the twist parameter of $C_{j}$ with respect to $\gamma_{j}$. Let $\alpha_{j}\subset C_{j}$ be a closed geodesic with $\int_{\alpha_{j}}\omega\in\mathbb{R}_{>0}$. Cutting $C_{j}$ along $\alpha_{j}$, twisting to the left by $a_{j}\in\mathbb{R}$, and regluing, changes the twist parameter of $C_{j}$ from $t_{j}$ to $t_{j}+a_{j}$. The lengths of the horizontal saddle connections and the heights and circumferences of the horizontal cylinders on $(X,\omega)$ are unchanged in the process. In this way, we obtain a continuous map $\prod_{j=1}^{m}\mathbb{R}/w_{j}\mathbb{Z}\rightarrow\widetilde{\Omega}\mathcal{M}_{g}(\kappa)$. For convenience, we rescale the inputs to obtain a horizontal twist map