2.1 Ergodic theory of stretchquake disks

Perhaps the marquee results in Teichmüller dynamics are the Ratner-type theorems of Eskin–Mirzakhani [Reference Epstein and MardenEM06] and Eskin–Mirzakhani–Mohammadi [Reference Eskin, Mirzakhani and MohammadiEMM15], building off work of McMullen [Reference McMullenMcM07] in genus 2. These results allow for a complete description of the invariant measures, orbit closures, and the distribution of orbits of the P and $\mathsf {SL}_2 \mathbb {R}$ actions on strata of quadratic differentials, up to understanding the structure and classification of certain special subvarieties of $\mathcal {Q}^1\mathcal {M}_g$ .

An invariant subvariety $\mathcal L$ of a stratum of $\mathcal {QM}_g$ is an immersed suborbifold that is locally cut out by $\mathbb {R}$ -linear equations in period coordinates; in [Reference FilipFil16], Filip proved that invariant subvarieties are algebraic varieties defined over $\overline {\mathbb Q}$ . Let $\mathcal L^1$ denote the intersection of $\mathcal L$ with the unit-area locus $\mathcal {Q}^1\mathcal {M}_g$ ; this is also referred to as an invariant subvariety. The affine measure on period coordinates induces an (infinite) measure (up to scale) $\nu _{\mathcal L}$ on $\mathcal L$ which in turn gives rise to a probability measure $\nu _{\mathcal L}^1$ called the affine measure on $\mathcal L^1$ .

The following is an amalgam of [Reference Epstein and MardenEM06, Theorem 1.4], [Reference Eskin, Mirzakhani and MohammadiEMM15, Theorems 2.1 and 2.10], and [Reference FilipFil16, Theorem 1.1]; we direct the reader to the original papers for a number of further related results.

The conjugacy between $\mathcal {P}^1\mathcal {M}_g$ and $\mathcal {Q}^1\mathcal {M}_g$ allows one to extend the earthquake flow to a Borel measurable (and everywhere-defined) P action whose orbits are locally smooth submanifolds. The A orbits of this action are called dilation rays $\operatorname {\mathrm {Dil}}_t(X)$ , and are defined in terms of the shear-shape coordinates introduced in [Reference Calderon and FarreCF24]. The generic dilation ray for any P-invariant measure on $\mathcal {P}^1\mathcal {M}_g$ is a geodesic for the Lipschitz metric on Teichmüller space (i.e., a generalized stretch line) [Reference Calderon and FarreCF24, §§1.2 and 15]. To delineate the P action on $\mathcal {P}^1\mathcal {M}_g$ from the P action on $\mathcal {Q}^1\mathcal {M}_g$ , we call this the stretchquake action on $\mathcal {P}^1\mathcal {M}_g$ .

Our construction of the Borel isomorphism $\mathcal {O}$ in [Reference Calderon and FarreCF24] allowed us to pull back the measure classification part of Theorem 2.1, deducing that every stretchquake-invariant ergodic measure on $\mathcal {P}^1\mathcal {M}_g$ is the pullback of the affine measure on some invariant subvariety of $\mathcal {Q}^1\mathcal {M}_g$ (see Theorem C therein). Our results on the continuity of $\mathcal {O}$ now allow us to pull back the genericity part of Theorem 2.1.

Theorem 2.2 (Stretchquakes equidistribute)

For any $(X, \lambda ) \in \mathcal {P}^1\mathcal {M}_g$ , any $\phi \in C_c(\mathcal {P}^1\mathcal {M}_g)$ , and any $r \in \mathbb {R} \setminus \{0\}$ , we have that

$$\begin{align*}\lim_{T \to \infty} \frac{1}{T} \int_0^T \int_0^r \phi \left(\operatorname{\mathrm{Dil}}_t \operatorname{\mathrm{Eq}}_s (X,\lambda) \right) \, ds \, dt = \int_{\mathcal L} \phi \, d\mathcal{O}^*\nu_{\mathcal L}^1\end{align*}$$

where $\nu ^1_{\mathcal L}$ is the affine measure on $\mathcal L^1 = \overline {P \cdot \mathcal {O}(X, \lambda )}$ .

Theorem A does not provide enough continuity to pull back the orbit closure part of Theorem 2.1.

We can also pull back Theorem 2.3 of [Reference Eskin, Mirzakhani and MohammadiEMM15], which states that the condition of being P-ergodic defines a closed condition on the space of probability measures on $\mathcal {P}^1\mathcal {M}_g$ . This is analogous to a result of Mozes and Shah for unipotent flows acting on homogeneous spaces [Reference Mozes and ShahMS95].

2.2 Ergodic theory of the earthquake flow

In the homogeneous setting, Ratner’s theorems hold not only for actions by $\mathsf {SL}_2\mathbb {R}$ but for any subgroup generated by unipotents; in particular, they hold for any unipotent flow. There have been major recent efforts to investigate analogues of Theorem 2.1 for the Teichmüller horocycle flow.

In [Reference Chaika, Smillie and WeissCSW20], it was shown that there exist fractal orbit closures for the horocycle flow. The classification of ergodic invariant measures is a major open problem, with complete (Ratner-like) answers in only a few special cases [Reference Bainbridge, Smillie and WeissBSW22, Reference Calta and WortmanCW10, Reference Eskin, Marklof and Witte MorrisEMWM06]; see also [Reference Chaika, Weiss and YgoufCWY23]. In these cases, one also usually gets corresponding genericity results for the horocycle flow. Pulling these back (using a more general version of Theorem B), we also get corresponding statements for the earthquake flow. For example, we have the following:

However, it was recently shown in both [Reference Chaika, Smillie and WeissCSW20] and [Reference Chaika, Khalil and SmillieCKS25] that not every point is generic for the horocycle flow. Pulling back (a slightly modified version of) the result from [Reference Chaika, Khalil and SmillieCKS25] yields:

A proof follows at the end of this subsection. The nongenericity result of [Reference Chaika, Khalil and SmillieCKS25] is in fact a corollary of their result that the Mozes–Shah phenomenon (Theorem 2.3) does not hold for the horocycle flow alone. We conclude a similar statement for earthquakes:

Proof of Theorem 2.6

Let all notation be as in the proof of Theorem 2.7. Corollary 1.2 of [Reference Chaika, Khalil and SmillieCKS25] constructs a dense $G_\delta $ in $\mathcal {Q}^1$ such that for any q in this set, there are sequences $r_n, s_n \to \infty $ such that in the weak- $*$ topology

(2) $$ \begin{align} u_{[0, r_n]} q := \frac{1}{r_n}\int_0^{r_n} \delta_{u_s q} \, ds \to \nu \hspace{3ex} \text{ and } \hspace{3ex} u_{[0, s_n]} q := \frac{1}{s_n}\int_0^{s_n} \delta_{u_s q} \, ds \to \nu_{\mathcal{Q}}^1 \end{align} $$

where $\nu _{\mathcal {Q}}^1$ is the Masur–Smillie–Veech measure on $\mathcal {Q}^1$ and $\nu $ is some other measure. With a very slight modification to their proof, one can also ensure that $\nu $ is the convex combination of ergodic measures from the proof of Theorem 2.7.Footnote ² We outline the argument below.

Let $\{r_n\}$ denote the periods of the closed horocycles limiting to $\nu $ and consider the sets

$$\begin{align*}V_k:= \{ q \, |\, d(u_{[0, r_n]}q, \nu) < 1/k \text{ for some } r_n> k\} \end{align*}$$

where d is any metric on the space of probability measures on $\mathcal {Q}^1$ inducing the weak- $*$ topology. These sets are defined by inequalities, hence are open, while they also contain arbitrary long horocycles that limit to $\nu $ , hence each $V_k$ is dense since $\nu $ has full support. Taking the intersection of the $V_{k}$ ’s yields a dense $G_\delta $ where any point has some subsequence of times $r_{n_k} \to \infty $ along which $u_{[0, r_{n_k}]} q \to \nu $ . On the other hand, by the ergodicity of the horocycle flow with respect to Masur–Smillie–Veech measure $\nu _{\mathcal {Q}}^1$ there is another dense $G_\delta $ whose ergodic averages along a different sequence of times $\{s_n\}$ converges to $\nu _{\mathcal {Q}}^1$ ; taking any point in the intersection of these dense $G_\delta $ ’s gives a q satisfying (2) for $\nu $ as desired.

We may now pull back (2) to obtain points in $\mathcal {P}^1\mathcal {M}_2$ that are not generic for the earthquake flow. To get arbitrary genus, one needs only take covers of the appropriate degree.

2.3 Expanding horospheres

Theorems B and C also provide a new perspective on the equidistribution of certain interesting sets inside of $\mathcal {P}^1\mathcal {M}_g$ and $\mathcal {Q}^1\mathcal {M}_g$ .

In [Reference MirzakhaniMir07], Mirzakhani established the equidistribution of level sets for the hyperbolic length of a simple multicurve in $\mathcal {P}^1\mathcal {M}_g$ ; these are the analogues of expanding horospheres based at a cusp in a cusped hyperbolic manifold. In [Reference Arana-HerreraAH21] and [Reference LiuLiu22], Arana–Herrera and Liu independently proved similar results for the level sets of a broader class of functions of the lengths of individual components of a multicurve, and in [Reference Arana-Herrera and CalderonAHC22, §5], Arana–Herrera and the first author proved a similar result given constraints on the geometry of the complementary subsurface to the multicurve. Via averaging and unfolding arguments in the style of Margulis [Reference MargulisMar70], these equidistribution results can be turned into counting results for multicurves [Reference Arana-HerreraAH22, Reference LiuLiu22, Reference Arana-Herrera and CalderonAHC22]. The proofs of all three of the above-mentioned “expanding horosphere” results rely on the ergodicity of the earthquake flow and delicate absolute continuity arguments.

On the other hand, there are also equidistribution results for the level sets of extremal length (and more generally, pushes of pieces of the unstable foliation of $\mathcal {Q}^1\mathcal {M}_g$ under the Teichmüller geodesic flow), which play the role of expanding horospheres in $\mathcal {Q}^1\mathcal {M}_g$ . See [Reference ForniFor21, Theorem 1.6] and [Reference Eskin, Mirzakhani and MohammadiEMM22, Proposition 3.2], as well as [Reference Arana-Herrera and CalderonAHC23]. The map $\mathcal {O}$ is defined leafwise on the unstable leaves to take hyperbolic length to extremal length, and so in particular we see that it takes horospheres to horospheres. Theorems B and C now show that the equidistribution results of [Reference MirzakhaniMir07, Reference Arana-HerreraAH21, Reference LiuLiu22, Reference Arana-Herrera and CalderonAHC22] on $\mathcal {P}^1\mathcal {M}_g$ are equivalent to those of [Reference ForniFor21, Reference Eskin, Mirzakhani and MohammadiEMM22, Reference Arana-Herrera and CalderonAHC23] on the principal stratum of $\mathcal {Q}^1\mathcal {M}_g$ .

While expanding horospheres in $\mathcal {P}^1\mathcal {M}_g$ have only been studied for multicurves, the equidistribution results in $\mathcal {Q}^1\mathcal {M}_g$ hold for any leaf of the unstable foliation. Using Theorem B, we can deduce similar results in $\mathcal {P}^1\mathcal {M}_g$ . To state this result, let us first fix some notation.

For any $\lambda \in \mathcal {ML}$ , let $H_\lambda $ denote the 1–level set of its length function. Following [Reference ForniFor21], say that a measure $\mu $ on $H_\lambda $ is horospherical if it is in the Lebesgue class with continuous density and such that almost all of the conditional measures on earthquake flow lines are just restrictions of Lebesgue. Such a measure can be obtained, for example, by picking some nice set in $H_\lambda $ using shear coordinates for a completion of $\lambda $ , then restricting the natural affine measure coming from shear coordinates to that set.

Given a horospherical measure $\mu $ on $H_\lambda $ , one can lift it to live on the section $H_\lambda \times \{\lambda \} \subset \mathcal {P}^1\mathcal {T}_g$ and then take the pushforward of $\mu $ under the covering map to get a measure $\widehat {\mu }$ on $\mathcal {P}^1\mathcal {M}_g$ .

Work of Lindenstrauss and Mirzakhani [Reference Lindenstrauss and MirzakhaniLM08] implies that whenever $\lambda $ is arational (i.e., contains no simple closed curves), the projection of the level set $H_\lambda $ is dense in $\mathcal {M}_g$ . This is an analogue of a result of Dani that says that any horospere in a cusped hyperbolic manifold is either closed or dense [Reference DaniDan78]. In the homogeneous setting, horospheres arise as the orbits of a horospherical subgroup, and in fact every nonclosed orbit equidistributes. In $\mathcal {P}^1\mathcal {M}_g$ or $\mathcal {Q}^1\mathcal {M}_g$ there is no notion of a horospherical subgroup (and these spaces are completely inhomogeneous), but we expect that one should be able to use the techniques of this paper to prove an analogous result.

The results of [Reference ForniFor21] and [Reference Eskin, Mirzakhani and MohammadiEMM22] hold for nonprincipal strata as well; pulling these back, we can deduce equidistribution results for specific slices of expanding horospheres; the limiting measures will now be stretchquake-flow invariant ergodic measures which are singular to Mirzakhani measure. We remark more on this in a sequel paper, in which we also apply recent deep results in Teichmüller dynamics to address Mirzakhani’s twist torus conjecture [Reference Calderon and FarreCF25].

4.1 Geodesic laminations

Recall that a geodesic lamination $\lambda $ on a closed hyperbolic surface $X\in \mathcal {T}_g$ is a closed set foliated by simple complete geodesics called its leaves. We denote by $\mathcal {GL}(X)$ the space of all geodesic laminations contained in X equipped with the Hausdorff metric $d_X^H$ on closed subsets of X. It is well known that $\mathcal {GL}(X)$ is closed in the Hausdorff topology, hence is itself a compact metric space.

We are mostly interested in chain recurrent geodesic laminations, which are those that can be approximated by geodesic multi-curves; let $\mathcal {GL}^{cr}(X)$ denote the (closed) subspace of chain recurrent geodesic laminations (see [Reference ThurstonThu98, §§6 and 8]). We collect here a list of useful, well-known facts about geodesic laminations and their geometry.

A transverse measure $\mu $ on a geodesic lamination $\lambda $ assigns a finite Borel measure $\mu _k$ to every arc k transverse to $\lambda $ satisfying that the support of $\mu _k$ is equal to $k\cap \lambda $ . In addition, this assignment is required to be invariant under holonomy. Often, we abuse notation and use the symbol $\lambda $ for a geodesic lamination equipped with a transverse measure. Likewise, for a transverse arc k, we let $\lambda (k)\ge 0$ denote the total mass of the measure deposited on k from $\lambda $ .

We denote by $\mathcal {ML}_g$ the space of measured geodesic laminations (with the topology of measure convergence). There is a continuous, locally bi-linear geometric intersection form

$$\begin{align*}i: \mathcal{ML}_g \times \mathcal{ML}_g \to \mathbb{R}_{\ge 0}\end{align*}$$

extending the minimal geometric intersection number between simple multi-curves.

Since any measurable lamination is chain recurrent, there is a set-theoretic map $\mathcal {ML}_g \to \mathcal {GL}^{cr}(X)$ remembering only the support of a measured lamination $\lambda $ . While this map is not continuous, almost every point with respect to the Lebesgue measure on $\mathcal {ML}_g$ is a point of continuity; see Section 15.1 for details.

We denote by $\mathcal {PT}_g$ the space $\mathcal {T}_g\times \mathcal {ML}_g$ , which can (and will) be thought of either as a bundle over $\mathcal {T}_g$ or over $\mathcal {ML}_g$ . The mapping class group $\operatorname {\mathrm {Mod}}_g$ acts nicely with quotient $\mathcal {PM}_g := \mathcal {PT}_g/\operatorname {\mathrm {Mod}}_g$ . Thurston introduced a continuous, homogeneous length function $\mathcal {PT}_g \to \mathbb {R}_{>0}$ that extends the total hyperbolic length of simple multi-curves [Reference ThurstonThu88]. Let $\mathcal {P}^1\mathcal {T}_g$ and $\mathcal {P}^1\mathcal {M}_g$ denote the unit length loci.

4.2 Quadratic differentials

A quadratic differential q on a Riemann surface X is a section of the symmetric square of its cotangent bundle. Away from its zeros, a holomorphic quadratic differential q locally has the form $dz^2$ , and the expression $|dz|^2$ defines a flat cone metric on X whose cone points correspond to the zeros of q. We denote by $\mathcal {QT}_g$ the bundle of holomorphic quadratic differentials over $\mathcal {T}_g$ minus the zero section and $\mathcal {QM}_g = \mathcal {QT}_g/\operatorname {\mathrm {Mod}}_g$ the moduli space of such. We let $\mathcal {Q}^1\mathcal {T}_g$ and $\mathcal {Q}^1\mathcal {M}_g$ be the unit area loci of holomorphic quadratic differentials, that is, the area of the singular flat metric on S induced by such a differential is $1$ .

All of these spaces are naturally broken up into strata, sub-orbifolds that parametrize those q with fixed number and order of zeros. Strata are not necessarily connected, but their components have been classified [Reference Kontsevich and ZorichKZ03, Reference LanneauLan08, Reference Chen and MöllerCM14]. Apart from a few sporadic examples in genus four, the components of a stratum are always classified by whether or not every surface in them is hyperelliptic, whether or not they parametrize squares of abelian differentials, and by the Arf invariant of an associated spin structure (when one exists). Throughout the paper, we will use $\mathcal {Q}$ to refer to a component of a stratum and $\mathcal {Q}^1$ for its unit-area locus.

Strata can be given local period coordinates as follows. Every quadratic differential comes with an orientation cover $\widehat q \to q$ that is a double cover of q, branched over the zeros of odd order. This object is naturally an abelian differential corresponding to the square root of q. One can measure the periods of all relative cycles in order to associate to $\widehat {q}$ a map

$$\begin{align*}H_1(\widehat q, Z(\widehat q); \mathbb{Z}) \to \mathbb{C}\end{align*}$$

that is naturally anti-invariant under the covering involution of $\widehat {q}$ . Performing this construction in families therefore locally models a (marked) stratum component $\mathcal {Q}$ on $H^1(\widehat q; Z(\widehat q), \mathbb {C})^-$ , the $-1$ -eigenspace for the covering involution. The natural affine measure on this subspace pulls back to a measure called the Masur–Smillie–Veech measure $\nu _{\mathcal {Q}}$ on $\mathcal {Q}$ which can be used to induce a probability measure $\nu _{\mathcal {Q}}^1$ on $\mathcal {Q}^1$ that is ergodic for the $\mathsf {SL}_2(\mathbb {R})$ action [Reference MasurMas82, Reference VeechVee82].

4.3 Measured foliations

A (singular) measured foliation on S is a $C^1$ foliation $\mathcal F$ of $S\setminus Z$ , where Z is a finite set (the singular points), equipped with a transverse measure $\nu $ on arcs transverse to $\mathcal F$ . The transverse measure is required to be invariant under holonomy and every singularity is modeled on a standard k-pronged singularity. Isotopic measured foliations are viewed as identical. The space $\mathcal {MF}_g$ of Whitehead equivalence classes of singular measured foliations is equipped with its measure topology making the geometric intersection pairing $i:\mathcal {MF}_g \times \mathcal {MF}_g \to \mathbb {R}_{\ge 0}$ given by integrating the product of transverse measures continuous; see [Reference ThurstonThu88, Reference Fathi, Laudenbach and PoénaruFLP12] for details.

By straightening the nonsingular leaves of a measured foliation with respect to a choice of hyperbolic metric and taking the closure, one can obtain a geodesic lamination together with a (pushforward) transverse measure. This describes a natural map $\mathcal {MF}_g \to \mathcal {ML}_g$ which is in fact a homeomorphism commuting with geometric intersection [Reference LevittLev83]. As such, we often conflate measured foliations and measured laminations.

Important examples of singular measured foliations are the directional foliations coming from quadratic differentials. Building on work of Hubbard and Masur, Gardiner and Masur proved that a holomorphic quadratic differential q is determined uniquely by the Whitehead equivalence classes of its imaginary and real foliations, whose leaves are horizontal and vertical, respectively. The imaginary and real foliations of q bind the surface, a condition that is equivalent to the fact that the corresponding measured geodesic laminations cut S into compact disks. Moreover, every pair of measured foliations which bind S are realized as the vertical and horizontal foliations of some $q\in \mathcal {QT}_g$ , and this assignment yields a homeomorphism

(3) $$ \begin{align} \mathcal{QT}_g \cong \mathcal{MF}_g \times \mathcal{MF}_g \setminus \Delta \end{align} $$

where $\Delta $ denotes the set of pairs of measured foliations that do not together bind the surface [Reference Gardiner and MasurGM91, Reference Hubbard and MasurHM79].

This product structure gives rise to two natural foliations $\mathcal W^{uu}$ and $\mathcal W^{ss}$ of $\mathcal {QT}_g$ : given $q \in \mathcal {QT}_g$ , let $\mathcal W^{uu}(q)$ denote those $q'$ with the same (Whitehead equivalence class of) imaginary measured foliation, while $\mathcal W^{ss}(q)$ denotes those $q'$ with the same real foliation. In period coordinates, these foliations simply look like changing the real and imaginary parts of periods, respectively. This identification also gives a hint as to the meaning of the superscripts: deformations in $\mathcal W^{uu}$ come from either scaling the measure or are exponentially expanded under the Teichmüller geodesic flow, while deformations in $\mathcal W^{ss}$ that do not come from scaling are exponentially contracted.

At a global level, (3) implies that we can identify each leaf of $\mathcal W^{uu}$ and each leaf of $\mathcal W^{ss}$ with open subsets in $\mathcal {MF}_g$ . However, the intersection of these leaves with a stratum $\mathcal {Q}$ is not described just in terms of the topological type of the foliation (in part due to complications arising from taking Whitehead equivalence classes). A global description of $\mathcal W^{uu} \cap \mathcal {Q}$ is a consequence of the main theorem of our previous paper (see [Reference Calderon and FarreCF24, Corollary 2.6]).

4.4 The orthogeodesic foliation

Let $X\in \mathcal {T}_g$ and let $\lambda $ be a geodesic lamination. The complement of $\lambda $ in X is a (possibly disconnected) hyperbolic surface whose metric completion has totally geodesic (possibly noncompact) boundary. For each such complementary component Y, there is a piecewise geodesic $1$ -complex ${\mathsf {Sp}}$ called the spine consisting of points that are closest to two or more components of $\partial Y$ in its universal cover. Away from ${\mathsf {Sp}}$ , there is a unique nearest point in $\partial Y$ .

The fibers of the projection map $Y\setminus {\mathsf {Sp}} \to \partial Y$ form a foliation of $Y\setminus {\mathsf {Sp}}$ whose leaves extend continuously across ${\mathsf {Sp}}$ to a piecewise geodesic singular foliation $\mathcal {O}_{\partial Y}(Y)$ called the orthogeodesic foliation. This singular foliation has n-pronged singularities at the vertices of the spine of valence n, and every endpoint of every leaf of $\mathcal {O}_{\partial Y}(Y)$ meets $\partial Y$ orthogonally.

The orthogeodesic foliation $\mathcal {O}_{\partial Y}(Y)$ is equipped with a transverse measure: the measure assigned to a small enough transversal is Lebesgue after projection to $\partial Y$ , and this assignment is invariant by isotopy transverse to $\mathcal {O}_{\partial Y}(Y)$ . This construction produces a (nonsmooth) singular foliation on $X\setminus \lambda $ , which extends continuously across the leaves of $\lambda $ and defines a singular measured foliation $\mathcal {O}_{\lambda }(X)$ on S (after smoothing out the leaves in a neighborhood of ${\mathsf {Sp}}$ ); see [Reference Calderon and FarreCF24, Section 5] for more details. An essential feature of ${\mathcal {O}_\lambda }(X)$ is that for a geodesic segment of a leaf of $\lambda $ , the length of that segment is computed by the transverse measure to ${\mathcal {O}_\lambda }(X)$ , and this is preserved under isotopy transverse to ${\mathcal {O}_\lambda }(X)$ .

We have therefore produced a map

$$\begin{align*}\mathcal{O}_\lambda: \mathcal{T}_g \to \mathcal{MF}_g\end{align*}$$

recording the measure equivalence class of $\mathcal {O}_\lambda (X)$ . Suppose now that $\lambda $ admits a transverse measure of full support, and define $\mathcal {MF}(\lambda )$ to be the subset of measured foliations that bind together with $\lambda $ . In [Reference Calderon and FarreCF24], the authors proved the following:

Combining Theorem 4.3 with the Theorem of Gardiner and Masur, we obtain a $\operatorname {\mathrm {Mod}}_g$ -equivariant bijection

$$\begin{align*}\mathcal{O}: \mathcal{PT}_g \to \mathcal{QT}_g\end{align*}$$

mapping the unit length locus to the unit area locus, such that the imaginary and real foliations of $\mathcal {O}(X,\lambda )$ are measure equivalent to $\lambda $ and $\mathcal {O}_\lambda (X)$ , respectively. One can also give an explicit construction of $\mathcal {O}(X,\lambda )$ by “inflating” $\lambda $ and “deflating” the complementary subsurfaces to $\lambda $ ; this viewpoint makes it clear that the real foliation of $\mathcal {O}(X,\lambda )$ is isotopic to $\mathcal {O}_\lambda (X)$ , not just Whitehead-equivalent [Reference Calderon and FarreCF24, Proposition 5.10].

For use in the sequel, we record here an elementary geometric estimate that says that leaves of ${\mathcal {O}_\lambda }(X)$ are nearly $C^1$ in the spikes of $\lambda $ . It can be proven via elementary Euclidean geometry in the plane.

4.5 Arc systems

From now on we tacitly take the metric completion of $X\setminus \lambda $ , so each component is a finite area hyperbolic surface with totally geodesic boundary. For each component Y of $X\setminus \lambda $ and for each compact edge e of ${\mathsf {Sp}}$ , there is a dual arc $\alpha $ properly isotopic to any nonsingular leaf of $\mathcal {O}_{\partial Y}(Y)$ meeting e. It is not difficult to see that the union $\underline{\alpha }(X,\lambda )$ of all arcs obtained this way cuts $X\setminus \lambda $ into disks, that is, is a filling arc system on $X\setminus \lambda $ . To each arc $\alpha \in \underline{\alpha }(X,\lambda )$ we record a weight $c_\alpha := \mathcal {O}_{\partial Y}(Y)(e)$ , where $e\subset {\mathsf {Sp}}$ and $\alpha $ are dual. That is, $c_\alpha $ is the length of the projection of e to $\lambda $ . We remark that if the weight of $\underline{\alpha}$ is small, its shortest (orthogeodesic) representative may not necessarily be a leaf of $\mathcal{O}_\lambda(X)$ . This is related to the observation in Reference LuoLuo07 that “radius invariants” can be negative even when “radius coordinates” are positive.

It turns out that the formal sum

$$\begin{align*}\underline{A}(X,\lambda) := \sum_{\alpha \in \underline{\alpha}(X,\lambda)} c_\alpha \alpha\end{align*}$$

completely determines the geometry of $X\setminus \lambda $ in the following sense.

The arc complex $\mathscr {A}(Y)$ of a surface with boundary Y is the simplicial flag complex whose vertices are essential proper isotopy classes of embedded arcs, and where two vertices are joined if they have disjoint representatives. The weighted filling arc complex of Y is either the cone

$$\begin{align*}\mathscr{A}(Y) \times \mathbb{R}_{\ge 0} / (\mathscr{A}(Y) \times \{0\})\end{align*}$$

when Y is simply connected, or an $\mathbb {R}_{>0}$ -bundle over the subspace of the arc complex corresponding to filling arc systems, when Y is not. The weighted filling arc complex $|\mathscr {A}_{\text {fill}}(S\setminus \lambda )|_{\mathbb {R}}$ is the product over all complementary subsurfaces to $\lambda $ . Generalizing work of [Reference DoDo08, Reference LuoLuo07, Reference UshijimaUsh99] for when $\lambda $ is a multi-curve, in [Reference Calderon and FarreCF24, §6] we proved that the map

$$\begin{align*}\underline{A} : \mathcal{T}(S\setminus \lambda) \to |\mathscr{A}_{\text{fill}}(S\setminus\lambda)|_{\mathbb{R}}\end{align*}$$

is a piecewise analytic mapping class group equivariant homeomorphism.

This induces a map $\mathcal {T}_g \to |\mathscr {A}_{\text {fill}}(S\setminus \lambda )|_{\mathbb {R}}$ obtained by cutting $X\in \mathcal {T}$ along the geodesic realization of $\lambda $ and recording the geometric weighted filling arc complex; abusing notation, this map is also denoted by $\underline{A}$ . In general, this map is not surjective, but rather maps onto a subspace $\mathscr B(S\setminus \lambda )\subset |\mathscr {A}_{\text {fill}}(S\setminus \lambda )|_{\mathbb {R}}$ cut out by equations that are linear on each cell (see §7.3 of [Reference Calderon and FarreCF24]).

4.6 Hexagons, centers, and basepoints

Since the arc system $\underline{\alpha } = \underline{\alpha }(X,\lambda )$ is filling, the complement of $\lambda \cup \underline{\alpha }$ in X consists of topological disks. Generically, that is, if every vertex of ${\mathsf {Sp}}$ is trivalent, then when $\underline{\alpha}$ is realized orthogeodesically, each a component H of the metric completion of $X \setminus (\lambda \cup \underline{\alpha })$ is a partially ideal (degenerate) right angled hexagon where we consider spikes of H as sides of zero length. More generally, if H contains a vertex of ${\mathsf {Sp}}$ of valence n, then H is a partially ideal (degenerate) right angled $2n$ -gon. It is a consequence of Gauss–Bonnet that there is a uniform bound (depending only on genus) on the total valence of the vertices of ${\mathsf {Sp}}$ .

Abusing terminology, we will often refer to any component of $X\setminus (\lambda \cup \underline{\alpha })$ a as hexagon, regardless of its number of sides. Every vertex v of ${\mathsf {Sp}}$ is corresponds to a unique hexagon $H_v$ , and v is equidistant to each of the geodesic boundary components of $H_v$ corresponding to leaves of $\lambda $ . In other words, v is the center of a hyperbolic circle contained in $X\setminus \lambda $ tangent to $\lambda $ ; these points of tangency are also the first intersection points with $\lambda $ of the n-pronged singularity of $\mathcal {O}_\lambda (X)$ emanating from v. Note that this incircle is not necessarily contained in $H_v$ .

More generally, to any triple of pairwise disjoint complete geodesics in $\mathbb {H}^2$ , none of which separates the others, there is a unique inscribed circle. The center of the triple is the (hyperbolic) center of this circle; equivalently, it is the unique point which is equidistant from all three geodesics. The basepoints of the triple are the points of tangency of this circle with the geodesics. Equivalently, they are the closest-point projections of the center of the triple to the constituent geodesics.

The following observation gives a universal lower bound on distance from any center to its corresponding geodesics, that is, the radius of the incircle.

If, in addition, we have a lower bound on the injectivity radius of the hyperbolic structure X, we can give an upper bound on this radius.

Lemma 4.7 necessarily requires a bound on the thickness of X. For example, consider an ideal triangulation of a once-punctured surface. Opening the cusp into a boundary component, this determines a maximal orthogeodesic arc system on the bordered surface. The centers of each triangle all have definite injectivity radius, but the boundary is far away from the thick part of the surface. See Figure 1.

Figure 1

No uniform bound on radii over the entirety of moduli space.

4.7 Shear-shape cocycles

Let $\lambda \in {\mathcal {GL}^{cr}}(S)$ , and let $\underline{\alpha } = \cup _i \alpha _i$ be a filling arc system on $S\setminus \lambda $ . Following §7 of [Reference Calderon and FarreCF24], we define shear-shape cocycles and discuss the structure of the space of these objects.

A shear-shape cocyle $\sigma $ for the pair $(\lambda , \underline{\alpha })$ is an element of a certain cohomology theory. We need the notion of an orientation of $\lambda \cup \underline{\alpha }$ , which is a continuous orientation of $\lambda $ and a continuous co-orientation of $\underline{\alpha }$ that agrees with the orientation of $\lambda $ at the endpoints of all of the arcs. While $\lambda \cup \underline{\alpha }$ is not usually orientable, there is always a two-to-one orientation cover $\widehat {\lambda }\cup \widehat {\underline{\alpha }} \to \lambda \cup \underline{\alpha }$ .

Let $N_{\underline{\alpha }}$ be a snug neighborhood of $\lambda \cup \underline{\alpha }$ , that is, an open neighborhood of $\lambda \cup \underline{\alpha }$ on S satisfying that the complement of $\lambda \cup \underline{\alpha }$ on S and the complement of $N_{\underline{\alpha }}$ on S have the same topological type. Then the orientation cover $\widehat {\lambda }\cup \widehat {\underline{\alpha }} \to \lambda \cup \underline{\alpha }$ extends to a two-to-one cover $\widehat {N_{\underline{\alpha }}} \to N_{\underline{\alpha }}$ .

A shear-shape cocycle $\sigma $ for $(\lambda ,\underline{\alpha })$ is then a relative class $\sigma \in H^1(\widehat {N_{\underline{\alpha }}},\partial \widehat {N_{\underline{\alpha }}};\mathbb {R})^-$ that is anti-invariant with respect to the covering involution. We also require that $\sigma $ evaluates positively on standard transversals, which are choices of relative cycles $t_i$ crossing the arc $\widehat {\alpha }_i$ of $\widehat {\underline{\alpha }}$ (and no others) which are disjoint from $\widehat \lambda $ , given the orientation which agrees with the co-orientation of $\widehat {\alpha }_i$ . From positivity on standard transversals and anti-invariance of $\sigma $ under the covering involution, we recover a (positively) weighted filling arc system

$$\begin{align*}\underline{A}(\sigma) = \sum_{\alpha_i \in \underline{\alpha}} \sigma(t_i) \alpha_i \in |\mathscr{A}_{\text{fill}}(S\setminus\lambda)|_{\mathbb{R}}.\end{align*}$$

The space of shear-shape cocycles with arc system $\underline{\alpha }$ is denoted by $\mathcal {SH}(\lambda ;\underline{\alpha })$ ; it is routine to check that the definition does not depend on the choice of snug neighborhood $N_{\underline{\alpha }}$ [Reference Calderon and FarreCF24, Lemma 7.7].

In more concrete terms, a shear-shape cocycle $\sigma \in \mathcal {SH}(\lambda ;\underline{\alpha })$ defines a weighted filling arc system $\underline{A}(\sigma )$ as above and a function on the set of unoriented arcs that are transverse to $\lambda $ and disjoint from $\underline{\alpha }$ that satisfies:

1. (SH0) (support): If k does not intersect $\lambda $ then $\sigma (k)=0$ .

2. (SH1) (transverse invariance): If k and $k'$ are isotopic through arcs transverse to $\lambda $ and disjoint from $\underline{\alpha }$ , then $\sigma (k) = \sigma (k')$ .

3. (SH2) (finite additivity): If $k = k_1 \cup k_2$ where $k_i$ have disjoint interiors, then $\sigma (k) = \sigma (k_1) + \sigma (k_2)$ .

4. (SH3) ( $\underline{A}$ -compatibility): Suppose that k is isotopic rel endpoints and transverse to $\lambda $ to some arc which may be written as $t_i \cup \ell $ , where $t_i$ is a standard transversal and $\ell $ is disjoint from $\underline{\alpha }$ . Then the loop $k \cup t_i \cup \ell $ encircles a unique point p of $\lambda \cap \underline{\alpha }$ , and

   $$\begin{align*}\sigma(k) = \sigma(\ell) + \varepsilon c_{\alpha_i}\end{align*}$$
   where $\varepsilon $ denotes the winding number of $k \cup t_i \cup \ell $ about p (where the loop is oriented such that the edges are traversed k then $t_i$ then $\ell $ ).

The axioms (SH0) through (SH3) determine uniquely a shear-shape cocyle [Reference Calderon and FarreCF24, Proposition 7.14].

In §9 of [Reference Calderon and FarreCF24], the authors constructed “train track coordinates” for $\mathcal {SH}(\lambda , \underline{\alpha })$ ; the point is that a shear-shape cocyle is completely determined by the weights it deposits on a suitable train track. Compare §7 below, in which we construct geometric train track neighborhoods of laminations on hyperbolic surfaces and recall the standard terminology for train tracks.

The space of all shear-shape cocycles $\mathcal {SH}(\lambda )$ is glued together from spaces $\mathcal {SH}(\lambda ;\underline{\alpha })$ , where $\underline{\alpha }$ ranges over all filling arc systems of $S\setminus \lambda $ , and the gluings are encoded by the combinatorics of $\mathscr {A}_{\text {fill}}(S\setminus \lambda )$ . Then $\mathcal {SH}(\lambda )$ has the structure of an affine $\mathcal {H}(\lambda )$ bundle over $\mathscr {B}(S \setminus \lambda )$ , where $\mathcal {H}(\lambda )$ is Bonahon’s space [Reference BonahonBon97a, Reference BonahonBon97b] of transverse cocycles for $\lambda $ [Reference Calderon and FarreCF24, Theorem 8.1].Footnote ³

4.8 The shear-shape cocycle of a hyperbolic metric

Let $X\in \mathcal {T}_g$ and let $\lambda $ be a chain recurrent geodesic lamination. Let $\underline{A}(X,\lambda )\in \mathscr {B}(S \setminus \lambda )$ be the weighted filling arc system with underlying geometric arc system $\underline{\alpha }$ recording the geometry of $X\setminus \lambda $ as in §4.5. Following Bonahon [Reference BonahonBon96], we defined in [Reference Calderon and FarreCF24, §13] the geometric shear-shape cocycle $\sigma _\lambda (X) \in \mathcal {SH}(\lambda ;\underline{\alpha })$ of the hyperbolic metric X.

Using the axioms for shear-shape cocycles, it is enough to compute the value of $\sigma _\lambda (X)$ on small transversal arcs k to $\lambda $ that are disjoint from $\underline{\alpha }$ .Footnote ⁴ Briefly, a small enough transverse arc k has endpoints in hexagons $H_u$ and $H_v$ in the universal cover. These hexagons give basepoints to their boundary geodesics, and the orthogeodesic foliation defines an isometry between these two basepointed geodesics. The quantity $\sigma _\lambda (X)(k)$ is a signed distance between these two basepoints.

There is a bi-linear pairing

$$\begin{align*}\omega_{\mathcal{SH}}: \mathcal{SH}(\lambda)\times \mathcal H(\lambda) \to \mathbb{R}\end{align*}$$

that computes hyperbolic length: for every transverse measure $\mu $ with support contained in $\lambda $ , we have

(4) $$ \begin{align} \ell_X(\mu) = \omega_{\mathcal{SH}}(\sigma_\lambda(X), \mu)>0. \end{align} $$

Thus the image of $\sigma _\lambda $ is contained in the locus $\mathcal {SH}^+(\lambda )$ satisfying the positivity conditions (4). For $\lambda $ measurable, $\sigma _\lambda $ is a homeomorphism onto $\mathcal {SH}^+(\lambda )$ [Reference Calderon and FarreCF24, Theorem 12.1].

5.1 Examples

We now demonstrate that Corollary 5.2 is sharp with the following example, which shows that short curves of $(X, \lambda )$ may not be taken to short curves of $\mathcal {O}(X, \lambda )$ .

Example 5.4 (Short curve to long curve)

For $t>0$ , define an arc system on the 4-holed sphere as in Figure 2; as discussed in Section 4.5, this specifies a hyperbolic metric with boundary components of length $(1,1, 2t, 2t)$ . Glue together the boundary components of the same length (with any twisting) to yield a hyperbolic structure $X_t$ on a genus two surface equipped with two simple closed curves $\gamma _1$ and $\gamma _2$ such that

$$\begin{align*}\ell_{X_t}(\gamma_1) = 1 \text{ and } \ell_{X_t}(\gamma_2) = 2t.\end{align*}$$

Define $\underline{\gamma _t} = \gamma _1 + 2t \gamma _2$ ; then the quadratic differential $q_t := \mathcal {O}( X_t, \underline{\gamma _t})$ is the union of two tori along a slit of length t. Compare Figure 2.

Figure 2

Short curves on corresponding surfaces. While $\gamma _2$ is hyperbolically short on $X_t$ , it is not conformally short on $q_t$ . The shaded expanding annulus demonstrates that the separating curve $\gamma _3$ is conformally short on $q_t$ .

Now as $t \to 0$ , we see that the moduli of the two pieces remain bounded as they always lie on the same horocycle in $\mathcal M_{1,1}$ (corresponding to different choices of twisting). In particular, $\gamma _2$ is always the core curve of an embedded cylinder of height t and width $2t$ , that is, modulus 1/2.

However, we can find a large expanding annulus (shaded in Figure 2) homotopic to the curve $\gamma _3$ that separates the two tori, and hence the extremal length of $\gamma _3$ must go to $0$ as $t \to 0$ .

We now observe that the converse to Proposition 5.1 also does not hold: there are families of thick pairs $(X, \lambda )$ whose corresponding differentials exit the cusp of $\mathcal {Q}^1\mathcal {M}_g$ . Compare with [Reference WrightWri22, Remark 5.10].

Example 5.5 (Thick to thin)

Fix a surface of any genus and consider a pants decomposition P one of whose curves separates off a genus one subsurface $\Sigma $ . Let $\gamma $ denote the curve of P contained inside $\Sigma $ . Using Fenchel-Nielsen coordinates, set all curves of P to have length 1. this in particular ensures the existence of an orthogeodesic arc $\alpha $ of weight running between the two sides of $\gamma $ . Glue together the curves of $P \setminus \gamma $ with arbitrary twists, and glue together the two boundaries corresponding to $\gamma $ such that the endpoints of $\alpha $ match up; call the resulting hyperbolic structure X and resulting curve $\delta $ . By construction, we have that $\mathcal {O}_P(X)$ contains a foliated annulus of weight with core curve $\delta $ . See Figure 3.

Figure 3

Thick hyperbolic surfaces mapping to thin flat surfaces. As the coefficient of $\gamma $ gets smaller, it deposits less mass on $\delta $ and so there is a flat cylinder of larger modulus.

Define the weighted multicurve

$$\begin{align*}\underline{P}_t := \sum_{P \setminus \gamma} \gamma' + t \gamma.\end{align*}$$

Then the quadratic differentials $q_t := \mathcal {O}(X, \underline{P}_t)$ each have area in $=(3g-4, 3g-3]$ and each contain a vertical cylinder with core curve $\delta $ whose circumference is $i(\delta , \underline{P}_t) = t$ and whose height is ; compare Figure 3.

The extremal length of $\delta $ on $q_t$ is bounded above by the reciprocal of the modulus of an embedded cylinder with core curve $\delta $ , hence by $\frac {t}{1/2} = 2t$ . Since this tends to $0$ with t and extremal length provides upper bounds for hyperbolic length in the same conformal class, we see that $q_t$ leaves the end of $\mathcal {Q}^1\mathcal {M}_g$ .

6.1 Close triples have close centers

We consider two triples of geodesics satisfying that no geodesic in one triple separates the others in the same triple. If the configurations are close on a ball of large radius around a center of one of the triples, then the center of the other triple is very close to that of the first.

Lemma 6.1. Fix $r\ge \log (\sqrt 3)$ , let $e\ge 0$ , and suppose $\zeta <1$ . Let $g_1$ , $g_2$ , and $g_3$ be pairwise disjoint complete geodesics in $\mathbb {H}^2$ , none of which separates the others, with distance at most r from their center, u. Suppose $h_i$ are complete geodesics that are $\zeta $ -Hausdorff close to $g_i$ in the ball of radius $\log (1/\zeta )-e$ about u, for $i =1, 2, 3$ . Then the center v of $(h_1,h_2,h_3)$ satisfies

$$\begin{align*}d(u,v) \le O(\zeta^2),\end{align*}$$

where the implicit constant depends only on r and e.

In addition to the above estimate on the distance between centers of Hausdorff-close triples, we also want to be able to compare the projections of these centers to each of the geodesics in the triple. We begin with a general estimate on the distance between the projections of a point to Hausdorff-close geodesics.

Lemma 6.2. Fix $r\ge 0$ , let g be a complete geodesic in $\mathbb {H}^2$ , and suppose $u \in \mathbb {H}^2$ is distance at most r from g. Suppose that h is a complete geodesic that is $\zeta $ -Hausdorff-close on the ball of radius $b\log (1/\zeta )$ about u, where $b\in (0,1]$ and $\zeta <1$ . Then if $\pi _g$ and $\pi _h$ are the closest-point projection maps,

$$\begin{align*}d(\pi_g(u), \pi_h(u)) = O(\zeta^{1+b})\end{align*}$$

where the implicit constant depends only on r.

Proof. We refer to Figure 4 throughout the proof.

Figure 4

Projections to Hausdorff-close geodesics.

Let $\rho _g(u)$ denote the intersection of the line containing u and $\pi _h(u)$ with g, and let $\rho _h(u)$ be the intersection of the line containing u and $\pi _g(u)$ with h. Gromov hyperbolicity of $\mathbb {H}^2$ implies that g and h fellow travel at scale $\zeta $ on segments of length at least $b\log (1/\zeta ) - r- C$ to the right and to the left of $\pi _g(u)$ , where C is some universal constant. Negative curvature then bounds

$$\begin{align*}d(\rho_h(u),\pi_g(u)) \le \zeta e^{-(b\log(1/\zeta)-r-C)} = O(\zeta^{1+b}).\end{align*}$$

A symmetric argument gives the same bound $d(\rho _g(u),\pi _h(u)) =O(\zeta ^{1+b})$ .

Because $\pi _g$ is a closest point projection, we know that

$$\begin{align*}d(u, \pi_g(u)) \le d(u, \rho_g(u)) \le d(u, \pi_h(u)) + O(\zeta^{1+b})\end{align*}$$

where the last inequality is the triangle inequality. Similarly, we have

$$\begin{align*}d(u, \pi_h(u)) \le d(u, \rho_h(u)) \le d(u, \pi_g(u)) + O(\zeta^{1+b}).\end{align*}$$

Thus, the right triangle with vertices $\pi _g(u), \rho _g(u)$ and u is very nearly isosceles, with two sides whose lengths differ by $O(\zeta ^{1+b})$ . The hyperbolic law of cosines then gives

$$\begin{align*}\cosh(d(u,\pi_g(u))+O(\zeta^{1+b}))=\cosh(d(u,\rho_g(u))) =\cosh(d(u,\pi_g(u))\cosh(d(\rho_g(u), \pi_g(u))).\end{align*}$$

Rearranging the terms and expanding, this implies

$$\begin{align*}\frac{\cosh(d(u,\pi_g(u))+O(\zeta^{1+b}))}{\cosh(d(u,\pi_g(u)))} = 1+ O(\zeta^{2+2b}) = 1+ O(d(\rho_g(u), \pi_g(u))).\end{align*}$$

Thus $d(\rho _g(u),\pi _g(u))=O(\zeta ^{1+b})$ , with implicit constant depending only on r.

An application of the triangle inequality proves the lemma.

In particular, we highlight the following special case, which can be proven by taking $b = d(u,g) / \log (1/\zeta )$ and observing that if h fellow-travels g in a larger ball then the estimates in the Lemma get better.

Corollary 6.3. Let $r, g, u$ be as above and let B be any hyperbolic ball centered at u and meeting g. Then if h is any geodesic that has Hausdorff distance at most $\zeta $ from g in B, we have

$$\begin{align*}d(\pi_g(u), \pi_h(u)) = O(\zeta)\end{align*}$$

where the implicit constant depends only on the cutoff r.

Corollary 6.4. Let all notation be as in Lemma 6.1. If $p_i \in g_i$ and $q_i \in h_i$ are the basepoints for these geodesic triples, then we have

$$\begin{align*}d(p_i, q_i) = O(\zeta^{2})\end{align*}$$

where the implicit constant depends only on r and e.

Proof. Fix an $i \in \{1, 2, 3\}$ and let $\pi _g$ denote the closest-point projection of $\mathbb {H}^2$ to $g_i$ . Since the closest-point projection map is a contraction, we know that

$$\begin{align*}d(p_i, \pi_g(v)) = d(\pi_g(u), \pi_g(v)) \le d(u, v) = O(\zeta^{2})\end{align*}$$

where the last estimate follows by Lemma 6.1. On the other hand, Lemma 6.2 implies that

$$\begin{align*}d(\pi_g(v), q_i) = d(\pi_g(v), \pi_h(v)) = O(\zeta^{2}).\end{align*}$$

These two estimates plus the triangle inequality prove the desired statement.

6.2 Equidistant configurations

Valence n vertices of ${\mathsf {Sp}}$ occur when an n-tuple of geodesics are equidistant from a point, but these configurations are highly unstable as one deforms in the Hausdorff topology. Here we prove an estimate on nearly equidistant tuples that will be important in Section 10 to compare arc systems on nearby laminations.

The boundary of a regular ideal n-gon is $0$ -equidistant, as is any collection of disjoint geodesics equidistant from a given point.

The following lemma states that if two pairs of $4$ -tuples are close to one another on a ball of large radius but different pairs of opposite geodesics are closer in one $4$ -tuple than the other, then both configurations are nearly equidistant. Compare Corollary 10.10 and Figure 12 below. As discussed in Example 4.5, the weighted filling arc system of a configuration of $4$ pairwise disjoint geodesics in the plane, none of which separates the others, is homeomorphic to $\mathbb {R}$ .

Using Lemma 6.2, this of course implies that the basepoints of all of the triples of geodesics in G differ by $\zeta ^2$ , and the same for H. In particular, combined with Corollary 6.4, this gives us a bound of $O(\zeta ^2)$ for the diameter of the collection of all of the basepoints on the coresponding geodesics of G and H.

Part II. Train tracks and cellulations

In this part of the paper, we discuss a number of geometric-combinatorial constructions that we will make frequent use of in the later parts. For measured geodesic laminations on hyperbolic surfaces, these are (augmented) “equilateral train tracks,” while for quadratic differentials these are specific families of cellulations by saddle connections. The main point of this part is to show that these constructions are dual to each other and thereby develop a technical framework in which to state quantified versions of our continuity results.

7.1 Uniform train tracks

The following is a variant of Thurston’s construction of a train track neighborhood of a geodesic lamination on a hyperbolic surface ([Reference ThurstonThu82, §8.9] and [Reference ThurstonThu98, §4]).

A train track $\tau $ constructed as in Construction 7.1 from parameters X, $\lambda $ , and $\delta $ is called a uniform geometric train track. Any uniform geometric train track is implicitly equipped with the data of its collapse map $\pi : \mathcal {N}_\delta (\lambda )\to \tau $ .

We adopt the following standard definitions and notation:

• ○ The branches of $\tau $ are its edges, and the switches of $\tau $ are its vertices (with their $C^1$ structure).

• ○ A train track is generic or trivalent if all switches are trivalent.

• ○ The ties of $\tau $ or $\mathcal {N}_\delta (\lambda )$ are connected components of restrictions of the leaves of ${\mathcal {O}_\lambda }(X)$ to the neighborhood. All ties are rectifiable compact arcs and meet $\lambda $ at right angles.

• ○ A train path $\gamma $ is a $C^1$ edge-path in $\tau $ .

• ○ The width of $\tau $ or $\mathcal {N}_{\delta }(\lambda )$ is the supremum of the length of its ties.

• ○ Spikes of $S \setminus \lambda $ (or $S \setminus \tau $ ) are regions bounded by two segments of $\lambda $ or of $\tau $ which share an ideal endpoint (or tangential direction at a switch), together with an arc in the complement.

The spikes of $\tau $ often correspond to pairs of asymptotic geodesics in $\lambda $ , but may also arise when there is a leaf of $\mathcal{O}_\lambda(X)$ with length smaller than $2\delta $ . In §7.4, we discuss the notion of “proto-spikes,” which allows us to treat these two geometrically similar cases in a uniform manner.

7.2 Length along train tracks

Geometric train tracks inherit a length function from $\lambda $ . More precisely, we can identify each branch of $\tau $ with an interval as follows: given two points $p_1$ and $p_2$ in the same branch, the distance from $p_1$ to $p_2$ is the length of any segment of a leaf of $\lambda $ running along the branch between the ties $\pi ^{-1}(t_1)$ and $\pi ^{-1}(t_2)$ , and this is well-defined because transport along the orthogeodesic foliation preserves length along the lamination.

As an example, let us compute the lengths of the sides of triangles of $S \setminus \tau $ when the base lamination is maximal. Consider first an ideal hyperbolic triangle, equipped with the orthogeodesic foliation relative to its boundary. The foliation has a unique trivalent singular leaf, whose intersections with the boundary are the basepoints. Elementary hyperbolic geometry allows us to compute that the distance $D_\delta $ along the boundary between any basepoint and either of the leaves of the orthogeodesic foliation of length $2\delta $ is given by

(5) $$ \begin{align} D_\delta := \log\left(\frac{\coth(\delta)}2\right) = \log(1/\delta)-\log (2)+O(\delta^2). \end{align} $$

Note in particular that this is a constant depending only on $\delta $ .

Now suppose $\lambda $ is maximal and $\tau = \tau (X, \lambda , \delta )$ is a uniform geometric train track. Then each side of each triangle of $S \setminus \tau $ defines a train path running between spikes of $\tau $ , and by the above discussion this path must have length (as measured along $\lambda $ ) exactly $2D_\delta $ . Moreover, the midpoint of each of these train paths is the image of a basepoint of a complementary triangle under the collapse map.

We emphasize that when the base lamination is not maximal, the side lengths of $S \setminus \tau $ are not necessarily all equal (even if $\tau $ is maximal).

7.3 Equilateral geometric train tracks

The following construction of a geometric train track neighborhood reflects bettter the singular flat geometry of $\mathcal {O}(X,\lambda )$ when $\lambda $ is not maximal. Our main desideratum is that the distance between the tips of the spikes of $X \setminus \tau $ and the corresponding basepoints should be constant over all spikes. We therefore build neighborhoods with this property in mind.

Recall from (5) that $D_\delta \approx \log (1/2\delta )$ is the distance between any basepoint of an ideal triangle and the leaves of the orthogeodesic foliation of length $2\delta $ , as measured along the triangle’s boundary.

Construction 7.3 (Equilateral train track neighborhood)

Let $\delta <\log (\sqrt 3)$ be given and fix a crowned hyperbolic surface Y. The set of basepoints of the orthogeodesic foliation $\mathcal {O}_{\partial Y}(Y)$ is a finite set in $\partial Y$ , and we let $I \subset \partial Y$ denote those points that are at most $D_\delta $ away from a basepoint, as measured along $\partial Y$ . Set

$$\begin{align*}\text{Thick}_\delta(Y) := \{ y \in Y \mid \text{ the leaf of } \mathcal{O}_{\partial Y}(Y) \text{ through } y \text{ meets } I \}\end{align*}$$

and set Thin $_\delta (Y)$ to be its complement. Then we define the $\delta $ –equilateral neighborhood of the boundary as

$$\begin{align*}{\mathcal{E}}_{\delta}(\partial Y) := \mathcal{N}_{\delta}(I) \cup \text{Thin}_\delta(Y)\end{align*}$$

where $\mathcal {N}_{\delta }(I)$ is the uniform $\delta $ -neighborhood of I. See Figure 5.

Figure 5

Construction of the equilateral neighborhood (shaded) of the boundary of a crowned hyperbolic surface. The dark shaded regions are the $\delta $ -thin parts of Y; observe that they do not necessarily have to contain the orthogeodesic arc in the homotopy class of leaves. The shaded blue regions are both proto-spikes.

Given a hyperbolic surface X and a geodesic lamination $\lambda $ , we define a $\delta $ -equilateral neighborhood ${\mathcal {E}}_{\delta }(\lambda )$ of $\lambda $ in X by taking the union of the equilateral neighborhoods for each component of $X \setminus \lambda .$

We observe that the boundary of each component of $\operatorname {\mathrm {Thin}}_\delta (Y)$ decomposes into two geodesic segments of $\lambda $ , together with either one or two leaves of $\mathcal {O}_{\partial Y}(Y)$ .

One can also define the decomposition of Y into Thick $_\delta (Y)$ and Thin $_\delta (Y)$ as follows. Contracting along the leaves of $\mathcal {O}_{\partial Y}(Y)$ defines a deformation retract r of Y onto its spine ${\mathsf {Sp}}(Y)$ , and the lengths of projections of edges to $\partial Y$ allows us to identify each of ${\mathsf {Sp}}(Y)$ with an interval, hence defining a metric. Letting J denote the set of points of ${\mathsf {Sp}}(Y)$ at most $D_\delta $ away from a vertex with respect to this metric, we have that

$$\begin{align*}\text{Thick}_\delta(Y) = r^{-1}(J).\end{align*}$$

From this discussion, it becomes apparent that the complement of J in ${\mathsf {Sp}}(Y)$ is a union of intervals, hence each component of Thin $_\delta (Y)$ is a band foliated by parallel leaves of $\mathcal {O}_{\partial Y}(Y)$ .

Any equilateral neighborhood $\mathcal {E}_\delta (\lambda )$ is foliated by (segments of) leaves of ${\mathcal {O}_\lambda }(X)$ ; if the collapse map

$$\begin{align*}\pi:\mathcal{E}_\delta(\lambda) \to \mathcal{E}_\delta(\lambda)/\sim\end{align*}$$

is a homotopy equivalence, then we say that

$$\begin{align*}\tau( X, \lambda , \delta) := \mathcal{E}_\delta(\lambda)/\sim\end{align*}$$

is an equilateral train track and $\mathcal {E}_\delta (\lambda )$ is an equilateral train track neighborhood. Note that the length along $\lambda $ endows the branches of $\tau $ with a length function, as in Section 7.1.

For the rest of the paper, the expression $\tau (X, \lambda , \delta )$ will always refer to an equilateral train track (and not a uniform geometric one).

7.4 Geometrically (in)visible arcs

Fix an equilateral neighborhood ${\mathcal {E}_\delta (\lambda )} \subset X$ as in Construction 7.3. Suppose that $\mathcal {E}_\delta (\lambda )$ has no closed leaves so that the leaf space $\tau = \tau (X,\lambda ,\delta )$ is an equilateral train track. As noted above, the components of Thin $_\delta (X \setminus \lambda )$ are spikes and bands foliated by parallel leaves of $\mathcal {O}_{\lambda }(X)$ . Leaves of the compact components (bands) are necessarily parallel to some arc of $\underline{\alpha } = \underline{\alpha }(X, \lambda )$ . However, such an arc may not be completely contained in $ \mathcal {E}_\delta (\lambda )$ , which may occur when one side of an arc is “sharper” than the other; compare Figure 5.

The spikes of $\tau $ correspond either to pairs of asymptotic geodesics or invisible arcs. In either case, we call the region of $X \setminus \lambda $ that is foliated by leaves of ${\mathcal {O}_\lambda }(X)$ parallel to the switch leaf that lie on the same side of the orthogeodesic realization of that leaf (if one exists) a proto-spike at scale $\delta $ . Note that so long as $\delta $ is taken small enough, the orthogeodesic representatives of leaves of invisible arcs are always leaves of ${\mathcal{O}_\lambda}(X)$ . See Figure 5.

Let $\underline{\alpha }_\bullet = \underline{\alpha }_\bullet (X,\lambda ,\delta )$ denote the visible arc system and ${\mathcal {E}}= \mathcal {E}_\delta (\lambda )$ . For each arc of $\underline{\alpha}_\bullet$ , pick the leaf of ${\mathcal{O}_\lambda}(X)\setminus \lambda$ in the “middle” of the isotopy class, i.e., such that the distances (as measured along $\lambda$ ) between the leaf and the basepoints on either side are equal. Each such leaf intersects ${\mathcal {E}}$ in two orthogeodesic segments, and so its image under the collapse map is an arc with two well-defined endpoints on the equilateral train track $\tau $ . We may therefore consider the augmented equilateral train track

$$\begin{align*}\tau\cup \underline{\alpha}_\bullet = (\mathcal{E} \cup \underline{\alpha}_\bullet)/ \sim\end{align*}$$

which can be thought of as a train track with arcs attached at right angles. In Section 9.1 below, we show how to smooth the incidences of $\alpha _\bullet $ onto $\tau $ to obtain a train track $\tau _{\underline{\alpha }}$ extending $\tau $ .

7.5 Equilateral and uniform neighborhoods

The two Constructions 7.1 and 7.3 of geometric train track neighborhoods are related as follows. This estimate will allow us to make geometric statements about (the more useful) equilateral train tracks using (the simpler) uniform ones.

Proposition 7.7 (Comparing equilateral and uniform)

There are universal constants $\delta _{7.7}<\log (\sqrt 3)$ and $w_{7.7}\ge 1$ such that for all $\delta \le \delta _{7.7}$ ,

$$\begin{align*}\mathcal{N}_{w_{7.7}^{-1}\delta}(\lambda) \subset \mathcal{E}_{\delta}(\lambda)\subset \mathcal{N}_{w_{7.7}\delta}(\lambda).\end{align*}$$

The proof of this statement follows from bounding the following geometric quantity.

We note that if Thin $_\delta (Y)$ is nonempty, then $\Delta _V(\delta )$ will always be achieved by the corresponding boundary leaf. This quantity allows us to compare the width of an equilateral neighborhood with that of a uniform one; Proposition 7.7 is therefore an immediate consequence of the following lemma.

Lemma 7.9. There are universal constants $\delta _{7.7}<\log (\sqrt 3)$ and $w_{7.7}\ge 1$ such that for any crowned hyperbolic surface Y and any proto-spike V at scale $\delta \le \delta _{7.7}$ , we have

$$\begin{align*}w_{7.7}^{-1}\delta \le{\Delta_V(\delta)} \le w_{7.7} \delta.\end{align*}$$

It is not difficult to see that if $\delta $ is small enough, then any leaf of $\mathcal {O}_{\partial Y}(Y)$ with distance along $\partial Y$ at most $\log (2+\sqrt 3)$ from a leaf of length $2\delta $ has length at most $\log (3)$ . Let $\delta _{7.7}$ be as such.

Assume that V is a protospike at scale $\delta \le \delta _{7.7} $ . In particular, if V is bounded by an arc $\alpha $ , then $\ell (\alpha )< \log (3)$ . Using [Reference Calderon and FarreCF24, Lemma 6.6], we know that for any such protospike, there is a leaf of the orthogeodesic foliation of length $\log 3$ in V. Note that V inherits basepoints from the hexagon containing it.

Proof. Suppose first that V is a spike. If the distance $E_V$ between the leaf of length $\log 3$ and the center of the hexagon H containing it was $\log 2 $ or larger, then the center of H would be infinitely far away, which is absurd; see Figure 6.

Figure 6

Left: The angle $\varphi $ between the leaf of $\mathcal {O}_{\partial Y}(Y)$ of length $\log (3)$ in V and the spine of Y. Right: The distances between leaves in a spike.

Now suppose that V is bounded by an arc $\alpha $ ; by assumption, $\ell (\alpha )\le 2\delta <\log 3$ . A computation shows that the larger angle $\varphi $ made by the leaf of the orthogeodesic foliation of length $\log 3$ in V and the edge of the spine that it meets ranges between $\pi /2$ and $2\pi /3$ ; these values of $\varphi $ correspond, respectively, to the cases $\ell (\alpha )= \log 3$ and $\ell (\alpha ) =0$ , that is, that V is a spike. Consider the complete geodesic in $\mathbb {H}^2$ that extends the relevant edge of the spine, and the projection of its ideal endpoints to a geodesic comprising the spike V (see Figure 6). Then $E_V$ is bounded above by the distance between these points and the leaf of length $\log 3$ . Another explicit computation shows that this upper bound varies monotonically between $\log (2)$ when $\varphi = 2\pi /3$ and $\log (2+\sqrt 3)$ at the other extreme where $\varphi = \pi /2$ . This completes the proof.

Proof of Lemma 7.9

First we prove the assertion for spikes in a component Y of $X\setminus \lambda $ .

The leaf through the basepoints of V has length at least $\log (3)$ . By definition of $D_\delta $ , the distance in the spike V along its boundary between the leaf of length $2\delta $ and the leaf of length $\log (3)$ is exactly $D_\delta $ . By definition of $\Delta _V(\delta )$ , the distance in V along its boundary between the basepoint on V and the leaf of length $2\Delta _V(\delta )$ is also $D_\delta $ . Thus, $\delta \le \Delta _V(\delta )$ and the distance between the leaf of length $2\delta $ and the leaf of length $2\Delta _V(\delta )$ is equal to $E_V$ . Since $\delta \le \delta _{7.7}$ and $E_V\le \log (2+\sqrt 3)$ by Lemma 7.10, we have $2\Delta _V(\delta )\le \log (3)$ . Compare Figure 6.

Now the lengths of leaves of the orthogeodesic foliation of length at most $\log (3)$ in a spike are comparable to the lengths of the horocyclic segments with the same endpoints by a universal multiplicative factor C. Because horocycles grow exponentially in spikes, this implies that $\Delta _V(\delta )$ cannot be larger than $C\delta e^{E_V}$ . Applying Lemma 7.10 yields the conclusion for spikes.

The proof for proto-spikes bounded by an arc $\alpha $ of length at most $\log 3$ is similar. This time, however, we have to consider the possibility that Thin $_\delta (Y)\cap V$ is empty but that there is a component Z of Thin $_\delta (Y)$ contained in the proto-spike $V'$ that meets V along their shared arc (as in Figure 5). Then $E_V$ still computes the distance along the boundary of $V\cup V'$ between the leaf of the orthogeodesic foliation contained in V of length $2\delta $ and the boundary component of Z facing V. Again since $\delta \le \delta _{7.7}$ , we have $2\Delta _V(\delta )\le \log (3)$ .

Now we use the fact that for every leaf of the orthogeodesic foliation in $V \cup V'$ there is a hypercycle with the same endpoints on $\partial Y$ . The lengths of a hypercycles parallel to $\alpha $ distance d and $d+E_v$ away are $\ell (\alpha )\cosh (d)$ and $\ell (\alpha )\cosh (d+E_V)$ , respectively. The ratio of these lengths does not exceed $2\cosh (E_V)$ .

For leaves of $\mathcal {O}_{\partial Y}(Y)$ with length at most $\log (3)$ , there is a constant $C'$ such that the ratio of the length of a leaf of $\mathcal {O}_{\partial Y}(Y)$ in V and the length of the corresponding hypercyclic arc with the same endpoints is bounded above by $C'$ and below by $1/C'$ . The existence of such a $C'$ follows from the fact that as $d \to \infty $ , proto-spikes converge to spikes and hypercycles to horocycles, and hence we can apply the bounds of the previous paragraph.

In particular, comparing the leaves of $\mathcal {O}_{\partial Y}(Y)$ of length $2\delta $ and $2\Delta _V(\delta )$ to their corresponding hypercycles, we obtain

$$\begin{align*}\frac{\delta}{C^{\prime2}2\cosh(E_V)}\le \Delta_V(\delta) \le C^{\prime2}2\cosh(E_V)\delta.\end{align*}$$

Appealing to Lemma 7.10 completes the proof.

7.6 Geometry of equilateral train tracks

The width of a geometric train track neighborhood shrinks linearly in the defining parameter, where the linear rate is uniform for all surfaces in a compact part of the moduli space.

Proof. We will first produce a bound on the length of a tie of $\mathcal {N}_\delta (\lambda )$ .

Let t be a tie of $\mathcal {N}_\delta (\lambda )$ . As the intersection $t\cap \lambda $ has $1$ -dimensional Lebesgue measure zero (Lemma 4.2 item (3)), it suffices to bound the sum of the lengths of each component of $t\setminus \lambda $ . For each proto-spike V, we consider the components of $t\setminus \lambda $ corresponding to V, that is, $t\cap V$ . The longest such component has length at most $2\delta $ . There is a smooth train path that follows the proto-spike in forward time, and recurrences to t correspond to shrinking components of $t\cap V$ . Subsequent recurrences to t define homotopically nontrivial loops in S, and therefore have length at least s.

As in the proof of Lemma 7.9, the lengths of leaves of the orthogeodesic foliation decay at rate $Ce^{-d}$ , where d is the distance along the spike and C is a universal constant. Thus the total length of $V \cap t$ is at most

$$\begin{align*}\sum_{r = 0}^\infty 2C \delta e^{-rs}.\end{align*}$$

There are at most $6|\chi (S)|$ proto-spikes of $\mathcal {E}_{\delta }(\lambda )$ , so the total length of a tie t of $\mathcal {N}_\delta (\lambda )$ is at most $6|\chi (S)|$ times the bound above. Invoking Proposition 7.7 now yields a bound on the length of a tie of $\mathcal {E}_\delta (\lambda )$ .

Thus, if the width is smaller than length of the systole, then $\mathcal {E}_\delta (\lambda )$ is a train track neighborhood of $\lambda $ .

8.1 Cellulations from train tracks

Let $X\in \mathcal {T}_g$ , let $\lambda \in \mathcal {ML}_g$ , let $\delta $ be small enough that $\mathcal {E}_\delta (\lambda )\subset X$ is an equilateral train track neighborhood whose leaf space $\tau = \tau (X,\lambda ,\delta )$ is trivalent, let $\underline{\alpha }_\bullet =\underline{\alpha }_{\bullet }(X,\lambda ,\delta )$ be the visible arc system, and let $\tau \cup \underline{\alpha }_\bullet $ be the augmented track, considered as a 1-complex with some tangential data.

Since $\tau \cup \underline{\alpha }_\bullet $ is filling (Lemma 7.6), its dual complex is a cellulation $\mathsf T$ . Its vertices correspond to the complementary components of $S \setminus (\tau \cup \underline{\alpha }_\bullet )$ , which are in turn in bijection with hexagons of $S \setminus (\lambda \cup \underline{\alpha })$ and hence both the vertices of the spine and the zeros of $\mathcal {O}(X, \lambda )$ . Some of its edges of $\mathsf T$ are dual to (segments of) branches of $\tau $ , while others are dual to visible arcs.

See [Reference AgolAgo11, Reference GuéritaudGué16, Reference Landry, Minsky and TaylorLMT23] for the genesis of this terminology.

For $z\in \mathbb {C}$ we recall from [Reference Calderon and FarreCF24] the definition of the following function:

$$\begin{align*}[z]_+:= \left\{ \begin{array}{rl} z & \arg(z) \in [0, \pi) \\ -z & \arg(z) \in [\pi, 2\pi) \end{array}\right.\end{align*}$$

The point of this function is that while periods of a quadratic differential are only defined up to $\pm 1$ , the function $[z]_+$ distinguishes a choice of square root for each saddle connection.

Proposition 8.2 (Geometric cellulation dual to train track)

The edges of the cellulation $\mathsf T$ of $\mathcal {O}(X,\lambda )$ dual to $\tau \cup \underline{\alpha }_\bullet $ are realized by veering saddle connections. Moreover, for each nonhorizontal edge e of $\mathsf T$ , we have

$$\begin{align*}[\operatorname{\mathrm{hol}}_{\mathcal{O}(X,\lambda)}(e)]_+ = \sigma_\lambda(X)(t_e) +i\lambda(t_e)\end{align*}$$

where $t_e$ is a tie crossing the segment of the branch of $\tau $ dual to e, and for each horizontal edge, we have that $[\operatorname {\mathrm {hol}}_{\mathcal {O}(X,\lambda )}(e)]_+$ is the weight of the arc of $\underline{\alpha }$ dual to e.

That the quantity $\sigma _\lambda (X)(t_e)$ is well-defined can be checked using the axioms for shear-shape cocycles (see §4.7 or [Reference Calderon and FarreCF24, §7]; see also the discussion on “admissible routes” in [Reference Calderon and FarreCF24, §14.5]).

Proof. Let b be an edge of $\tau \cup \underline{\alpha }_{\bullet }$ with dual edge e joining components of $X\setminus \tau \cup \underline{\alpha }_\bullet $ that have centers v and w. If b is an arc of $\underline{\alpha }_\bullet $ , then then e is the dual edge of the spine of $\mathcal {O}_\lambda (X)$ , which is realized as a horizontal saddle connection on $\mathcal {O}(X,\lambda )$ , whose length is exactly the weight of the arc corresponding to b.

Otherwise, b comes from a branch of $\tau $ . Choose a nonswitch tie $t_e$ of $\mathcal {E}_{\delta }(\lambda )|_b$ . By construction of the equilateral neighborhood $\mathcal {E}_\delta (\lambda )$ , since $t_e$ terminates in the component of $X\setminus \tau \cup \underline{\alpha }_{\bullet }$ containing v, the distance along $\lambda $ between $t_e$ and the corresponding basepoint $p_v$ of v is strictly less than $D_\delta $ (by our choice of representatives for $\underline{\alpha}_{\bullet}$ ). Furthermore, the same is true for some basepoint $p_w$ of w. Using the definition of the equilateral neighborhood $\mathcal {E}_\delta (\lambda )$ , the distance between $t_e$ along $\lambda $ (in the universal cover) and the center of any other component of $X\setminus (\lambda \cup \underline{\alpha })$ meeting $t_e$ is strictly greater than $D_\delta $ . See Figure 7.

Figure 7

Distances along a lamination from a tie to basepoints of incident components of $X\setminus \tau \cup \underline{\alpha }{}_{\bullet }$ . The data of a geometric train track encodes a cellulation by veering saddle connections. The highlighted orange rectangle is singularity-free.

This implies that there is a band of isotopic leaves of ${\mathcal {O}_\lambda }(X)$ running from $p_v$ to $p_w$ about $t_e$ such that any leaf of $\lambda $ meeting all of these leaves has length $2 D_\delta $ . There is a map from $\mathcal E_\delta (\lambda )$ to $\mathcal {O}(X,\lambda )$ , obtained by integrating the transverse measure on ties coming from (the measure on) $\lambda $ , which is locally isometric along the leaves of $\lambda $ . Compare with [Reference Calderon and FarreCF24, Proposition 5.10]. Under this map, the band of isotopic leaves about $t_e$ becomes an embedded rectangle in (the universal cover of) $\mathcal {O}(X, \lambda )$ of length $2D_\delta $ , height $\lambda (t_e)$ , and containing the singularities corresponding to v and w on its boundaries. In particular, there is a nonsingular geodesic segment connecting the corresponding singularities.

Since the horizontal measure in this rectangle is given by intersection with ${\mathcal {O}_\lambda }(X)$ (equivalently the hyperbolic length measured along $\lambda $ ) and the vertical measure is given by intersection with $\lambda $ , the period of this saddle is as claimed. The sign of the real part of the period of e dual to b is consistent with the sign of $\sigma _\lambda (X)(t_e)$ ; see, for example, [Reference Calderon and FarreCF24, Thoerem 13.13].

The proof of the Proposition actually shows the following:

8.2 Train tracks from cellulations

Given a cellulation $\mathsf T$ of q by saddle connections, we would like to produce a $C^1$ structure at the vertices of the dual graph making it a train track that carries the imaginary folliation of q. For the following class of cellulations, such a choice can be made unambiguously.

This class of cellulations arises naturally. Indeed, the dual cellulations considered above are of this form.

Proof. Each polygon of the dual cellulation is dual to a vertex of $\tau \cup \underline{\alpha }_\bullet $ . The edges of the cellulation either cross an arc of $\underline{\alpha }_{\bullet }$ , in which case they are horizontal, or cross $\tau $ , in which case they are not. The statement now follows by observing that every vertex of $\tau \cup \underline{\alpha }_\bullet $ has at most four tangential directions: two coming from the tangential structure of $\tau $ and up to two coming from incidences of $\underline{\alpha }_{\bullet }$ with $\tau $ , each of which has a unique arc emanating from it. Compare Figure 8.

Figure 8

Simply horizontally convex cellulations and their dual train tracks. The tangential data of the edge dual to the saddle connection s depends on the slope of s. Dashed red arcs denote arcs dual to horizontal saddle connections.

We can construct a train track out of a (simply) horizontally convex cellulation as follows. The following is a generalization of [Reference MirzakhaniMir08, §4.4] and of [Reference Calderon and FarreCF24, Construction 10.4].