2.1.1 Hyperbolic surfaces

We fix once and for all a closed oriented surface $\Sigma $ of genus $g\ge 2$ and denote by $\Gamma :=\pi _1(\Sigma )$ its fundamental group.

Definition 2.2 (Teichmüller Space)

The Teichmüller space of $\Sigma $ , denoted by $\mathcal {T}$ , is the space of equivalence classes of marked hyperbolic structures on $\Sigma $ . It can be realized as a connected component of the space

$$\begin{align*}\mathcal{T}\subset\mathrm{Hom}(\Gamma,\mathrm{PSL}_2(\mathbb{R}))/\mathrm{PSL}_2(\mathbb{R}), \end{align*}$$

where $\mathrm {PSL}_2(\mathbb {R})$ acts on the space of representations by conjugation.

2.1.2 Geodesic laminations

To study the geometry of hyperbolic surfaces, it is quite useful to look at the behavior of their geodesic laminations, which are 1-dimensional objects generalizing simple closed geodesics.

Definition 2.3 (Space of Geodesics)

The space of (unoriented) geodesics on $\mathbb {H}^2$ is naturally identified with the set of pairs of endpoints

$$\begin{align*}\mathcal{G}:=\{(x,y)\in\mathbb{RP}^1\times\mathbb{RP}^1\mid x\neq y\}/(x,y)\sim(y,x), \end{align*}$$

where $(x,y)$ corresponds to the line with endpoints $x,y$ .

For more details, we address the reader to Chapter I.4 of [Reference Canary, Epstein and Green10].

2.1.3 Currents and measured laminations

Both Teichmüller space and measured laminations can be seen inside the space of geodesic currents as introduced by Bonahon (see [Reference Bonahon4]). This framework is well-suited to study length functions thanks to the presence of a natural geometric intersection form, as we now explain.

2.1.4 Length functions

Every hyperbolic surface X has a (marked) length spectrum $\{L_X(\gamma )\}_{\gamma \in \Gamma -\{1\}}$ given by the lengths of its closed geodesics. Conveniently, Bonahon [Reference Bonahon4] proves that the length function $L_X(\bullet )$ extends continuously to geodesic currents as follows:

Definition 2.9 (Liouville Current)

The Liouville current $\mathcal {L}$ on $\mathcal {G}$ is the $\mathrm {PSL}_2(\mathbb {R})$ -invariant Borel measure on $\mathcal {G}$ defined by

$$\begin{align*}\mathcal{L}([a,b]\times[c,d]):=\beta^{\mathbb{R}}(a,b,c,d) \end{align*}$$

on boxes $[a,b]\times [c,d]$ with $[a,b]\cap [c,d]=\emptyset $ (these sets generate the Borel algebra of $\mathcal {G}$ ), where $\beta ^{\mathbb {R}}$ denotes the standard cross ratio of $\mathbb {RP}^1$ . The Liouville current has the property that

$$\begin{align*}L_X(\gamma)=i(\mathcal{L},\delta_\gamma) \end{align*}$$

for every $\gamma \in \Gamma $ (see [Reference Bonahon4]). Therefore, $i(\mathcal {L},\bullet )$ extends continuously the length function $L_X(\bullet )$ to the space of geodesic currents.

2.2.1 Boundary at infinity

In this model, the boundary at infinity $\partial \mathbb {H}^{2,1}$ of $\mathbb {H}^{2,1}$ identifies with the topological boundary of $\mathrm {PSL}_2(\mathbb {R})$ in $\mathbb {P}(M_2(\mathbb {R}))$ :

$$\begin{align*}\partial\mathrm{PSL}_2(\mathbb{R})=\{[X]\in\mathbb{P}(M_2(\mathbb{R}))\mid\;\mathrm{det}(X)=0\}. \end{align*}$$

Observe that $\partial \mathrm {PSL}_2(\mathbb {R})$ consists of rank one matrices and can be $\mathrm {PSL}_2(\mathbb {R})\times \mathrm {PSL}_2(\mathbb {R})$ -equivariantly identified with $\mathbb {RP}^1\times \mathbb {RP}^1$ via the map

$$\begin{align*}\begin{matrix} \partial\mathrm{PSL}_2(\mathbb{R})& \longrightarrow & \mathbb{RP}^1\times\mathbb{RP}^1\\ [X] & \longmapsto & ([\mathrm{Im}(X)],[\mathrm{Ker}(X)]).\\ \end{matrix} \end{align*}$$

2.2.2 Subspaces

Totally geodesic subspaces in anti-de Sitter $3$ -space $\mathbb {H}^{2,1}$ are of the form $\mathbb {P}(V)\cap \mathrm {PSL}_2(\mathbb {R})$ , where $V\subset M_2(\mathbb {R})$ is a linear subspace intersecting $\mathrm { SL}_2(\mathbb {R})$ . In particular, we have

• ○ timelike geodesics isometric to $\mathbb {R}/\pi \mathbb {Z}\Leftrightarrow V$ 2-plane of signature $(0,2)$ .

• ○ spacelike geodesics isometric to $\mathbb {R}\Leftrightarrow V$ 2-plane of signature $(1,1)$ .

• ○ spacelike planes isometric to $\mathbb {H}^2\Leftrightarrow V\ 3$ -plane of signature $(2,1)$ .

Let $\tilde {x}, \tilde {y}$ be representatives of two distinct points $x, y \in \mathbb {H}^{2,1}$ that satisfy $\langle \tilde {x}, \tilde {x} \rangle = \langle \tilde {y}, \tilde {y} \rangle = -1$ . Then, x and y are joined by

• ○ a spacelike geodesic if and only if $|\langle \tilde {x}, \tilde {y} \rangle |>1$ ;

• ○ a lightlike geodesic if and only if $|\langle \tilde {x}, \tilde {y} \rangle |=1$ ;

• ○ a timelike geodesic if and only if $|\langle \tilde {x}, \tilde {y} \rangle |<1$ .

The geodesic $\gamma (t)$ starting at $x\in \mathbb {H}^{2,1}$ with velocity $v\in T_x\mathbb {H}^{2,1}=x^\perp $ is represented by

$$\begin{align*}\tilde{\gamma}(t)=\bigg\{ \begin{array}{l l} \cosh(t)\tilde{x}+\sinh(t)v &\mathrm{if}\ \langle v,v\rangle=1,\\ \cos(t)\tilde{x}+\sin(t)v &\mathrm{if}\ \langle v,v\rangle=-1.\\ \end{array} \end{align*}$$

(Throughout the rest of the exposition, we often do not distinguish between $x \in \mathbb {H}^{2.1}$ and the choice of a representative $\tilde {x}$ satisfying $\langle \tilde {x}, \tilde {x} \rangle = -1$ .)

2.2.3 Acausal sets and pseudo-metrics

The last concept that we need is the one of acausality:

Definition 2.11 (Pseudo Metric)

On acausal subsets $S\subset \mathbb {H}^{2,1}$ we have a pseudo-metric $d_{\mathbb {H}^{2,1}}(\bullet ,\bullet )$ , defined as follows:

$$\begin{align*}\cosh(d_{\mathbb{H}^{2,1}}(x,y))=|\langle x,y\rangle|. \end{align*}$$

Notice that $d_{\mathbb {H}^{2,1}}$ does not satisfy the triangle inequality in general.

3.1.1 Boundary maps

Every Mess representation $\rho _{X,Y}$ comes with a natural equivariant boundary map

$$\begin{align*}\xi:\partial\Gamma \longrightarrow \partial\mathbb{H}^{2,1}. \end{align*}$$

It can be described explicitly as follows. Recall that $\partial \mathrm {PSL}_2(\mathbb {R})$ is naturally identified with $\mathbb {RP}^1\times \mathbb {RP}^1$ . Let $h_X,h_Y:\partial \Gamma \to \mathbb {RP}^1$ be the unique $\rho _X,\rho _Y$ -equivariant homeomorphism. The boundary map $\xi :\partial \Gamma \to \mathbb {RP}^1\times \mathbb {RP}^1$ is just $\xi =(h_X,h_Y)$ .

Its image $\xi (\partial \Gamma )=\Lambda _{X,Y}$ is the graph of the unique $(\rho _X-\rho _Y)$ -equivariant homeomorphism $h_{X,Y}:\mathbb {RP}^1\to \mathbb {RP}^1$ .

It is not difficult to check that $\Lambda _{X,Y}$ has the property that, for every $a,b,c\in \mathbb {RP}^1$ , the $3$ -space $\mathrm {Span}\{(a,h_{X,Y}(a)),(b,h_{X,Y}(b)),(c,h_{X,Y}(c))\}$ has signature $(2,1)$ . Let us assume without loss of generality that $a<b<c$ . As $h_{X,Y}$ is an orientation-preserving homeomorphism, we have $h_{X,Y}(a)<h_{X,Y}(b)<h_{X,Y}(c)$ . Hence, up to the action of $\mathrm {PSL}_2(\mathbb {R})\times \mathrm { PSL}_2(\mathbb {R})$ , we can assume that $(a,b,c)=(h_{X,Y}(a),h_{X,Y}(b),h_{X,Y}(c))=((0,0),(1,1),(\infty , \infty ))$ . Tracing back the identification with $\partial \mathrm {PSL}_2(\mathbb {R})$ , we see that

$$\begin{align*}{\tiny (0,0)=\left[ \begin{array}{c c} 0 &0\\ 1 &0\\ \end{array} \right],\; (1,1)=\left[ \begin{array}{c c} 1 &-1\\ 1 &-1\\ \end{array} \right],\; (\infty,\infty)=\left[ \begin{array}{c c} 0 &1\\ 0 &0\\ \end{array} \right] }. \end{align*}$$

Notice now that the span of these three elements equals the subspace of $2\times 2$ traceless matrices, which, by the expression of the bilinear pairing $\langle \bullet , \bullet \rangle $ from Section 2.2, coincides with the orthogonal of the identity. Being the identity matrix a spacelike vector with respect to $\langle \bullet , \bullet \rangle $ , the conclusion follows.

3.1.2 Domain of discontinuity

From the boundary curve $\Lambda _{X,Y}\subset \partial \mathbb {H}^{2,1}$ , one constructs a standard open domain:

$$\begin{align*}\Omega_{X,Y}:=\{y\in\mathbb{H}^{2,1}\mid\;[x,y]\ \text{spacelike}\ \forall x\in\Lambda_{X,Y}\}. \end{align*}$$

It can also be described as a connected component of

$$\begin{align*}\mathbb{H}^{2,1}-\bigcup_{x\in\Lambda_{X,Y}}{\{\langle x,\bullet\rangle=0\}} , \end{align*}$$

which is a properly convex subset of $\mathbb {P}(M_2(\mathbb {R}))$ whenever $X \neq Y$ , whose closure contains $\Lambda _{X,Y}$ . In particular, it contains a natural closed $\rho _{X,Y}(\Gamma )$ -invariant convex subset, namely the convex hull $\mathcal {CH}_{X,Y}$ of the limit set $\Lambda _{X,Y}$ .

As $\Omega _{X,Y}$ does not contain closed timelike geodesics, it has a well-defined timelike distance.

Definition 3.2 (Timelike Distance)

The timelike distance $\delta _{\mathbb {H}^{2,1}}(\bullet ,\bullet ) \in [0,\pi ]$ on $\Omega _{X,Y}$ is defined by setting

$$\begin{align*}\delta_{\mathbb{H}^{2,1}}(x,y) = 0 \end{align*}$$

if the segment $[x, y]$ is either lightlike or spacelike, and otherwise

$$\begin{align*}\delta_{\mathbb{H}^{2,1}}(x,y):= \int \sqrt{- \langle \dot{\alpha}(t), \dot{\alpha}(t) \rangle} dt , \end{align*}$$

where $\alpha = \alpha (t)$ denotes some lift to $\mathrm {SL}_2(\mathbb {R})$ of the parametrized timelike segment $[x,y] \subset \Omega _{X,Y} \subset \mathrm {PSL}_2(\mathbb {R})$ .

The group $\rho _{X,Y}(\Gamma )$ acts freely and properly discontinuosly on $\Omega _{X,Y}$ (see [Reference Mess17]). The quotient $M_{X,Y}:=\Omega _{X,Y}/\rho _{X,Y}(\Gamma )$ is the Mess manifold associated with $X,Y\in \mathcal {T}$ .

Let us mention the fact that $M_{X,Y}$ is a so-called globally hyperbolic maximal Cauchy compact (GHMC) anti-de Sitter $3$ -manifold. In particular, this means that $M_{X,Y}$ contains a closed spacelike surface S homeomorphic to $\Sigma $ , which intersects every inextensible timelike geodesic exactly once. From this property, it is not difficult to deduce that $M_{X,Y}$ is diffeomorphic to $\Sigma \times \mathbb {R}$ . Mess proves in [Reference Mess17] that, in fact, all GHMC manifolds M whose Cauchy surfaces are homeomorphic to $\Sigma $ have the form $M=M_{X,Y}$ for some $X,Y\in \mathcal {T}$ .

3.2.1 Pleated sets

Let $\rho _{X,Y}$ be a Mess representation with boundary map $\xi :\partial \Gamma \to \Lambda _{X,Y}$ .

Definition 3.3 (Geometric Realization)

Let $\lambda \in \mathcal {GL}$ be a lamination. The geometric realization of $\lambda $ for $\rho _{X,Y}$ is

$$\begin{align*}{\hat \lambda}:=\bigcup_{(a,b)\in\lambda}{[\xi(a),\xi(b)]}\subset\mathcal{CH}_{X,Y} , \end{align*}$$

where $(a,b)$ is the leaf of $\lambda $ with endpoints $a,b$ and $[\xi (a),\xi (b)]$ is the spacelike geodesic with endpoints $\xi (a),\xi (b)$ .

Definition 3.4 (Pleated Set)

Let $\lambda \in \mathcal {GL}_m$ be a maximal lamination. The pleated set associated with $\lambda $ and $\rho _{X,Y}$ is

$$\begin{align*}{\hat S}_\lambda:={\hat \lambda}\cup\bigcup_{\Delta(a,b,c)\subset\mathbb{H}^2-\lambda}{\Delta(\xi(a),\xi(b),\xi(c))}\subset\mathcal{CH}_{X,Y} , \end{align*}$$

where $\Delta (a,b,c)$ is the plaque of $\lambda $ with vertices $a,b,c$ and $\Delta (\xi (a),\xi (b),\xi (c))$ is the ideal spacelike triangle with endpoints $\xi (a),\xi (b),\xi (c)$ .

Incidentally, in combination with classical $3$ -dimensional topology, Proposition 3.5 has also the following topological corollary:

3.2.2 Bending locus

The pleated set ${\hat S}_\lambda $ is not necessarily bent along all the lines in ${\hat \lambda }$ .

We have the following:

3.2.3 1-Lipschitz developing map

Unfolding pleated sets along the bending locus naturally maps them to $\mathbb {H}^2$ . We formalize this heuristic picture as follows:

Developing maps have a couple of useful general properties which we now describe. First, they are isometric outside the bending locus.

Secondly, developing maps are contracting with respect to the natural path metric structure on pleated sets.

Definition 3.11 (Regular Path)

A (weakly) regular path is a map $\gamma :I=[a,b]\to \mathbb {H}^{2,1}$ that satisfies the following:

• ○ The path $\gamma $ is Lipschitz.

• ○ The tangent vector ${\dot \gamma }(t)$ is spacelike (or lightlike) for almost every $t\in I$ (at which $\dot \gamma $ is defined).

The length of a weakly regular path is

$$\begin{align*}L(\gamma):=\int_I{\sqrt{\langle{\dot \gamma}(t),{\dot \gamma}(t)\rangle} dt}. \end{align*}$$

The Lipschitz property implies that the length $L(\gamma )$ is always finite.

Lemma 3.12 (Claim 2 of Lemma 6.4 in [Reference Mazzoli and Viaggi16])

Let ${\hat S}\subset \mathbb {H}^{2,1}$ be an acausal subset and let $\gamma :I=[a,b]\to {\hat S}$ be a weakly regular path. Then,

$$\begin{align*}L(\gamma)=\lim_{\epsilon\to 0}\int_I{\frac{d_{\mathbb{H}^{2,1}}(\gamma(t),\gamma(t+\epsilon))}{\epsilon} \text{dt}}. \end{align*}$$

3.2.4 Pleated surfaces

The following result makes sure that every pleated set ${\hat S}_\lambda $ admits a natural 1-Lipschitz developing map:

We can finally define pleated surfaces:

Let us conclude this discussion by observing that pleated surfaces for a fixed Mess representation $\rho _{X,Y}$ have some useful compactness properties:

Lemma 3.17. Let $\rho _{X,Y}$ be the Mess representation with parameters $X,Y\in \mathcal {T}$ . Then, the space of intrinsic hyperbolic structures on the pleated sets

$$\begin{align*}\{Z_\lambda\}_{\lambda\in\mathcal{GL}_m} \end{align*}$$

is pre-compact in $\mathcal {T}$ .

Proof. Recall that $\rho _{X,Y}(\Gamma )$ acts cocompactly on $\mathcal {CH}_{X,Y}$ . Let $F\subset \mathcal {CH}_{X,Y}$ be a compact fundamental domain. For every maximal lamination $\lambda \in \mathcal {GL}_m$ with associated pleated set ${\hat S}_\lambda \subset \mathcal {CH}_{X,Y}$ , choose a basepoint $x_\lambda \in {\hat S}_\lambda \cap F$ . Let $f_\lambda :{\hat S}_\lambda \to \mathbb {H}^2$ be a $(\rho _{X,Y}-\rho _\lambda )$ -equivariant 1-Lipschitz developing map normalized so that $f_\lambda (x_\lambda )=o\in \mathbb {H}^2$ , a fixed basepoint. The equivariance and the 1-Lipschitz property tell us that

$$\begin{align*}d_{\mathbb{H}^2}(o,\rho_\lambda(\gamma)o)\le d_{\mathbb{H}^{2,1}}(x_\lambda,\rho_{X,Y}(\gamma)x_\lambda) \end{align*}$$

for every $\gamma \in \Gamma $ . Notice that the right-hand side is bounded from above by a uniform constant $K_\gamma $ independent of $\lambda $ , since $x_\lambda \in F$ is contained in a compact set and

$$\begin{align*}\cosh(d_{\mathbb{H}^{2,1}}(x_\lambda,\rho_{X,Y}(\gamma)x_\lambda))=|\langle x_\lambda,\rho_{X,Y}(\gamma)x_\lambda\rangle|. \end{align*}$$

Therefore, the set of representations $\{\rho _\lambda \}_{\lambda \in \mathcal {GL}_m}\subset \mathcal {T}\subset \mathrm {Hom}(\Gamma ,\mathrm {PSL}_2(\mathbb {R}))$ is pre-compact.

3.2.5 Convex core

An example of pleated surfaces is given by the two connected components of the boundary of the convex core $\partial \mathcal {CH}_{X,Y}=\partial ^+\mathcal {CH}_{X,Y}\cup \partial ^-\mathcal {CH}_{X,Y}$ . Each of them has the structure of a pleated set with bending loci $\lambda ^+$ and $\lambda ^-$ and intrinsic hyperbolic structures $Z_{\lambda ^+},Z_{\lambda ^-}\in \mathcal {T}$ . As we mentioned in the introduction, measuring the total turning angles along paths $\alpha :I\to \partial ^\pm \mathcal {CH}_{X,Y}$ equips the geodesic laminations $\lambda ^\pm $ with a transverse measure and hence identifies a pair of points $\mu ^\pm \in \mathcal {ML}$ . Mess proves that we have the following relations:

where $E_{\mu ^+}^l$ , $E_{\mu ^-}^l$ , $E_{\mu ^+}^r$ , $E_{\mu ^-}^r$ are the left and right earthquakes induced by the measured laminations $\mu ^+,\mu ^-$ . Heuristically speaking, an earthquake is the generalization to laminations of a twist deformation along a simple closed geodesic. Given a closed geodesic $\gamma $ on a hyperbolic surface X and a real parameter $\theta>0$ , we do the following operation. We lift $\gamma $ to a $\rho _X(\Gamma )$ -invariant family of pairwise disjoint geodesics $\lambda \subset \mathbb {H}^2$ . We cut $\mathbb {H}^2$ along $\lambda $ . We reglue all the ideal polygons $P\subset \mathbb {H}^2-\lambda $ by composing all the initial identifications $\ell \subset \partial P\to \ell '\subset \partial P'$ (left-to-right) with the isometry of $\ell '$ given by $t\mapsto t+\theta $ (the identification $\ell '=\mathbb {R}$ is determined by the boundary orientation). The result is still isometric to $\mathbb {H}^2$ , but the action of $\Gamma $ on it is the holonomy of a different hyperbolic structure, which, depending on the choices of orientations, is $E^l_{\theta \gamma }(X)$ or $E^r_{\theta \gamma }(X)$ .

We will describe more carefully the various elements that enter this picture in the next section, where we will prove a generalization of the result of Mess.

3.2.6 Initial and terminal singularities

We end this section by describing the initial and terminal singularities of $\Omega _{X,Y}$ , which are subsets of $\partial \Omega _{X,Y}$ dual to the boundary components of the convex core. Duality is understood in the sense of the duality induced by the quadratic form $\langle \bullet ,\bullet \rangle _{(2,2)}$ on $M_2(\mathbb {R})$ . Explicitly, we have

$$\begin{align*}\mathbb{P}(L)\leftrightarrow\mathbb{P}(L^\perp), \end{align*}$$

where $L^\perp \subset M_2(\mathbb {R})$ is the linear subspace orthogonal to L with respect to the quadratic form $\langle \bullet ,\bullet \rangle _{(2,2)}$ .

Define the following:

Let us start with the following observation:

Notice that, by Lemma 3.19, the initial and terminal singularities $\mathcal {S}^\pm $ are $\rho _{X,Y}(\Gamma )$ -invariant, acausal and contained in $\partial \Omega _{X,Y}$ . Benedetti and Guadagnini [Reference Benedetti and Guadagnini2] prove that they have the structure of an $\mathbb {R}$ -tree and relate them to the bending measured laminations $\mu ^\pm $ .

Benedetti and Guadagnini [Reference Benedetti and Guadagnini2] show the following:

For a proof, we refer to [Reference Benedetti and Guadagnini2] and [Reference Benedetti and Bonsante1].

4.1.1 Transverse cocycles

Shear-bend cocycles are a special case of transverse cocycles for $\lambda $ .

4.1.2 Measured laminations

Every measured lamination $\mu \in \mathcal {ML}_\lambda $ determines a natural transverse cocycle which, with a little abuse of notation, we will still denote by $\mu \in \mathcal {H}(\lambda ;\mathbb {R})$ . It is defined as follows. Let $P,P'$ be plaques of $\lambda $ . Let $\ell \subset \partial P, \ell ' \subset \partial P'$ be the (oriented) edges that separate $P,P'$ . Then,

$$\begin{align*}\mu(P,P'):=\mu([\ell,\ell']), \end{align*}$$

the measure, determined by $\mu $ , of the box $[\ell ,\ell ']\subset \mathcal {G}$ consisting of those geodesics separating $\ell $ and $\ell '$ .

4.1.3 Hyperbolic structures

Every hyperbolic structure X on $\Sigma $ also determines a transverse cocycle $\sigma _\lambda ^X\in \mathcal {H}(\lambda ;\mathbb {R})$ , the so-called shear cocycle of X. It is defined as follows. Let $P,P'$ be plaques of $\lambda $ . Let $\ell \subset \partial P,\ell '\subset \partial P'$ be the (oriented) edges that separate $P,P'$ . Denote by $x\in \ell ,x'\in \ell '$ the orthogonal projections of the opposite vertices in $P,P'$ .

Consider the partial foliation $\lambda _{PP'}$ of the region $[\ell ,\ell ']$ bounded by $\ell ,\ell '$ given by all the leaves that separate P from $P'$ . Note that $[\ell ,\ell ']-\lambda _{PP'}$ is a union of wedges – that is, regions bounded by a pair of leaves of $\lambda _{PP'}$ that are asymptotic in one or the other direction. Each of the wedges can be foliated by adding all the geodesics separating the boundary leaves and asymptotic to their common endpoint at infinity. Thus, we get a natural geodesic foliation of $[\ell ,\ell ']$ . The line field on $[\ell ,\ell ']$ which is orthogonal to this foliation is integrable and, following its leaves, it provides a natural isometric identification $\pi :\ell \to \ell '$ . Define

$$\begin{align*}\sigma_\lambda^X(P,P'):=d_{\ell'}(\pi(x),x') , \end{align*}$$

where $d_{\ell '}$ is the signed distance along $\ell '$ .

A straightforward computation in $\mathbb {H}^2$ shows the following:

(See, for example, ([Reference Mazzoli and Viaggi16], Lemma 4.11) for a proof of the second assertion.) Bonahon proves the following:

Theorem 4.3 (Theorems A and B of [Reference Bonahon5])

Let $\lambda $ be a maximal lamination. For every $X\in \mathcal {T}$ , the function $\sigma _\lambda ^X(\bullet ,\bullet )$ is a transverse cocycle. The map

$$\begin{align*}\begin{array}{c} \Phi:\mathcal{T}\to\mathcal{H}(\lambda;\mathbb{R})\\ X\to\sigma_\lambda^X\\ \end{array} \end{align*}$$

is a real analytic diffeomorphism onto its image, which coincides with the open convex cone

$$\begin{align*}\Phi(\mathcal{T})=\{\sigma\in\mathcal{H}(\lambda,\mathbb{R})|\;\omega_{\mathrm{Th}}(\sigma,\bullet)>0\ \mathrm{on}\ \mathcal{ML}_\lambda\} , \end{align*}$$

where $\omega _{\mathrm {Th}}(\bullet ,\bullet )$ is the Thurston’s symplectic form on $\mathcal {H}(\lambda ;\mathbb {R})$ .

The resulting set of coordinates for Teichmüller space are called shear coordinates relative to $\lambda $ .

The Thurston’s symplectic form $\omega _{\mathrm {Th}}(\bullet ,\bullet )$ is a natural symplectic form on the vector space $\mathcal {H}(\lambda ;\mathbb {R})$ . For our purposes, we do not need a precise definition of this object (we refer to Bonahon [Reference Bonahon5] for details), as we will only use the following property.

Theorem 4.4 (Theorem E of [Reference Bonahon5])

Let $\lambda $ be a maximal lamination. Let $\omega _{\mathrm {Th}}(\bullet ,\bullet )$ be the Thurston’s symplectic form on $\mathcal {H}(\lambda ;\mathbb {R})$ . Then, for every $\mu \in \mathcal {ML}_\lambda $ and $X\in \mathcal {T}$ , we have

$$\begin{align*}\omega_{\mathrm{Th}}(\sigma_\lambda^X,\mu)=L_X(\mu). \end{align*}$$

4.1.4 Continuity of cocycles

In order to talk about continuity properties of cocycles, we need to compare $\mathcal {H}(\lambda ';\mathbb {R})$ with $\mathcal {H}(\lambda ;\mathbb {R})$ for $\lambda '$ close to $\lambda $ . This can be done using the weights system $\mathcal {W}(\tau ;\mathbb {R})$ of a train track $\tau $ carrying $\lambda $ . As above, we will not need a precise definition of these objects but rather the following facts (see the proof of Lemma 13 in Bonahon [Reference Bonahon6] or Proposition 5.10 and Corollary 5.11 in [Reference Mazzoli and Viaggi16]).

• ○ $\tau $ determines an open set $U_\tau \subset \mathcal {GL}_m$ containing $\lambda $ .

• ○ $\mathcal {W}(\tau ;\mathbb {R})$ is a real vector space, and there is a canonical linear isomorphism $\mathcal {H}(\lambda ';\mathbb {R})\to \mathcal {W}(\tau ;\mathbb {R})$ for every $\lambda '\in U_\tau $ .

• ○ For every $\lambda _1,\lambda _2\in U_\tau $ , the following diagram commutes:

  

• ○ For every $X\in \mathcal {T}$ , the map $U_\tau \ni \lambda \mapsto \sigma ^X_\lambda \in \mathcal {W}(\tau ;\mathbb {R})$ is continuous.

4.2.1 Exponential and logarithm

The para-complex exponential function $\exp :\mathbb {B}\to \mathbb {B}$ is given by $\exp (z):=\sum _{k=0}^\infty \frac {z^k}{k!}$ . In terms of the classical exponential, we have $e^{x+\tau y}=e^x (\cosh (y)+\tau \sinh (y))$ . The para-complex exponential map is injective, but not surjective. Its image coincides with

$$\begin{align*}\mathbb{B}^+ :=\{x+\tau y\in\mathbb{B}\mid x> 0\ \text{and}\ \vert x+\tau y \vert^2>0\}. \end{align*}$$

The inverse of the exponential is the para-complex logarithm $\log :\mathbb {B}^+\to \mathbb {B}$ .

In coordinates $\mathbb {B}=\mathbb {R}\times \mathbb {R}$ , we have $\mathbb {B}^+=\{(\lambda ,\mu )\in \mathbb {R}\times \mathbb {R}|\;\lambda ,\mu>0\}$ . The exponential satisfies $\exp (\lambda \, e_l+\mu \, e_r)=\exp (\lambda ) \,e_l+\exp (\mu ) \, e_r$ , while the logarithm is $\log \left (\lambda \, e_l+\mu \, e_r\right )=\log (\lambda ) \,e_l+\log (\mu ) \,e_r$ .

4.2.2 Projective para-complex line

The boundary $\partial \mathbb {H}^{2,1}=\mathbb {RP}^1\times \mathbb {RP}^1$ can be identified with the para-complex projective line $\mathbb {BP}^1=(\mathbb {B}^2-\{0\})/\mathbb {B}^*$ via

$$\begin{align*}([u],[v])\in\mathbb{RP}^1\times\mathbb{RP}^1\mapsto\left[\frac{1+\tau}{2}u+\frac{1-\tau}{2}v\right]\in\mathbb{BP}^1 , \end{align*}$$

and $\mathrm {PSL}_2(\mathbb {R})\times \mathrm {PSL}_2(\mathbb {R})$ can be thought of as the para-complex projective linear transformations $\mathrm {PSL}_2(\mathbb {B})=\mathrm {SL}_2(\mathbb {B})/\{\pm 1, \pm \tau \}$ , via the isomorphism

$$\begin{align*}([A],[B])\in\mathrm{PSL}_2(\mathbb{R})\times\mathrm{PSL}_2(\mathbb{R})\mapsto\left[\frac{1+\tau}{2}A+\frac{1-\tau}{2}B\right]\in\mathrm{PSL}_2(\mathbb{B}). \end{align*}$$

The para-complex projective line $\mathbb {BP}^1$ is equipped with a natural para-complex cross ratio:

Definition 4.6 (Cross Ratio)

The para-complex cross ratio is defined by

$$\begin{align*}\beta^{\mathbb{B}}(z_1,z_2,z_3,z_4)=\frac{z_1-z_3}{z_1-z_4}\cdot\frac{z_2-z_4}{z_2-z_3}\in\mathbb{B} , \end{align*}$$

for any $4$ -tuple $(z_1,z_2,z_3,z_4) \in \mathbb {B}^4$ such that $z_1 - z_4, z_2 - z_3 \in \mathbb {B}^*$ .

The following is an elementary computation.

Lemma 4.7. For every $a,b,c,d\in \mathbb {BP}^1=\mathbb {RP}^1\times \mathbb {RP}^1$ such that $a - d, b - c \in \mathbb {B}^*$ , we have

$$\begin{align*}\beta^{\mathbb{B}}(a,b,c,d)=\frac{1+\tau}{2}\beta^{\mathbb{R}}(a_l,b_l,c_l,d_l)+\frac{1-\tau}{2}\beta^{\mathbb{R}}(a_r,b_r,c_r,d_r) , \end{align*}$$

where $x_l$ and $x_r$ denote the first and second components of $x \in \{a,b,c,d\} \subset \mathbb {RP}^1\times \mathbb {RP}^1$ , respectively.

4.3.1 Elementary shear

Let us start with an elementary shear-bend.

We define

$$\begin{align*}\sigma^{\mathbb{B}}(\Delta,\Delta'):=\log(-\beta^{\mathbb{B}}(\ell^+,\ell^-,u,u'))\in\mathbb{B}. \end{align*}$$

4.3.2 Maximal laminations with countably many leaves

We then consider the case of maximal laminations with countably many leaves.

These laminations always have the following structure: there is a canonical collection of simple sublaminations

$$\begin{align*}\lambda'=\lambda_1\sqcup\cdots\sqcup\lambda_n\subset\lambda, \end{align*}$$

where each $\lambda _j$ consists of the orbit of the axis of an element $\gamma _j\in \Gamma -\{1\}$ representing a simple closed curve. The complement $\lambda -\lambda '$ is made of isolated geodesics asymptotic to leaves of $\lambda '$ .

Let $\lambda \subset \mathbb {H}^2$ be a maximal lamination with countably many leaves and let $P,Q\subset \mathbb {H}^2-\lambda $ be a pair of distinct plaques. We denote by $\ell _1,\cdots ,\ell _m$ the leaves of $\lambda '$ that separate P from Q, oriented so that P and Q lie on the left and on the right, respectively, of each $\ell _j$ . For any $j \in \{1, \dots , m\}$ , select two plaques $R_j^P$ and $R_j^Q$ that lie on the left and on right of $\ell _j$ , respectively, and have an ideal vertex equal to one of the endpoints of $\ell _j$ (possibly distinct, compare with Lemma 4.2). Set $R_0^Q : = P$ and $R_{m+1}^P : = Q$ .

For every $j = 0, \dots , m$ , the plaques $R_{j}^Q$ and $R_{j+1}^P$ are separated by finitely many adjacent plaques

$$\begin{align*}T_{j, 0} : = R_j^Q, T_{j,1}, \dots, T_{j,n_j} , T_{j,n_j + 1} : = R_{j+1}^P , \end{align*}$$

indexed so that $T_{j,i}$ is adjacent to $T_{j,i-1}$ and $T_{j,i + 1}$ for any i. In this case, we set

$$\begin{align*}\sigma_\rho(R^Q_j,R^P_{j+1}) := \sum_{i = 0}^{n_j} \sigma^{\mathbb{B}}(T_{j,i}, T_{j, i+1}) , \quad j \in \{0, \dots , m\}. \end{align*}$$

Moreover, by Lemma 4.2, the elementary shear between $R_j^P$ and $R_j^Q$ can be computed as the logarithm of an explicit product of cross ratios, which depend only on the vertices of the plaques $R_j^P$ and $R_j^Q$ (compare also with Lemma 4.7). Finally, we set

$$\begin{align*}\sigma_\rho(P,Q) := \sum_{j = 0}^{m - 1} \left(\sigma^{\mathbb{B}}(R_j^Q,R_{j+1}^P) + \sigma^{\mathbb{B}}(R_{j+1}^P, R_{j+1}^Q)\right) + \sigma^{\mathbb{B}}(R_{m}^Q,Q) , \end{align*}$$

where $\rho = \rho _{X,Y}$ .

As observed in ([Reference Mazzoli and Viaggi16], Section 4.4), a simple cross ratio identity shows that a different choice of plaques $R_j^P,R_j^Q$ asymptotic from the left and from the right to the leaves $\ell _j\in \lambda '$ separating P from Q gives the same value for $\sigma _\rho (P,Q)$ . The fact that $\sigma _\rho (P,Q)$ is well-defined immediately implies that it also satisfies the properties of a transverse cocycle. Therefore, we have the following:

Furthermore, we have the following:

Proof. Proposition 6.7 in [Reference Mazzoli and Viaggi16] characterizes the intrinsic hyperbolic structure $Z_\lambda \in \mathcal {T}$ in terms of its shear coordinates. More precisely, one sees that the shear cocycle of $Z_\lambda $ can be reconstructed from a natural cross ratio $\beta _\xi $ associated to the $\rho _{X,Y}$ -equivariant limit map $\xi : \partial \Gamma \to \partial \mathbb {H}^{2,1}$ and defined purely in terms of the pseudo-Riemannian structure of $\mathbb {R}^{2,2}$ . An elementary computation shows that the cross ratios $\beta _\xi $ and $\beta ^{\mathbb {B}}$ are related by the identity

$$\begin{align*}| \beta^{\mathbb{B}}(\xi(x_1),\xi(x_2),\xi(x_3),\xi(x_4)) |^2 = \beta_\xi(x_1,x_2,x_3,x_4)^2 , \end{align*}$$

as we identify $\partial \mathbb {H}^{2,1}$ with $\mathbb {BP}^1$ . (Here, $| \cdot |^2$ denotes the natural pseudo-norm on $\mathbb {B}$ .) It then follows from the properties of the para-complex logarithm that the shear $\sigma _\xi $ , defined through the cross ratio $\beta _\xi $ , and the para-complex shear $\sigma ^{\mathbb {B}}$ satisfy

$$\begin{align*}\Re(\sigma^{\mathbb{B}}(P,Q)) = \sigma_\xi(P,Q) \end{align*}$$

for any pair of distinct plaques $P, Q$ of $\lambda $ , from which we deduce the desired statement.

4.3.3 General maximal laminations

Lastly, we describe the natural finite approximation process that defines the shear-bend cocycle in general, extending the previous case. Let $\lambda \subset \mathbb {H}^2$ be an arbitrary maximal lamination. As before, let $P,Q\subset \mathbb {H}^2-\lambda $ be a pair of plaques and let $\mathcal {P}_{PQ}$ be the set of plaques separating P from Q. Let

$$\begin{align*}\mathcal{P}=\{P_1,\cdots,P_m\}\subset\mathcal{P}_{PQ} \end{align*}$$

be a finite subset of plaques ordered from P to Q. Any two consecutive $P_j,P_{j+1}$ cobound a (possibly empty) region $U_j$ . We decompose its boundary as $\partial U_j=\ell _j\cup \ell _{j+1}$ with $\ell _j\subset \partial P_j$ and $\ell _{j+1}\subset \partial P_{j+1}$ . We add to the finite collection $\mathcal {P}$ of plaques the triangles

$$\begin{align*}\Delta(\ell_j^+,\ell_j^-,\ell_{j+1}^+),\Delta(\ell_j^-,\ell_{j+1}^-,\ell_{j+1}^+), \end{align*}$$

obtaining a chain of triangles $P=T_1,T_2,\cdots ,T_{3m-2},T_{3m-1}=Q$ .

We define

$$\begin{align*}\sigma_\rho(P,Q):=\sum_{j=1}^{3m-2}{\sigma^{\mathbb{B}}(T_j,T_{j+1})}. \end{align*}$$

We then carefully choose an exhaustion $\{\mathcal {P}_n\}_{n\in \mathbb {N}}$ of $\mathcal {P}_{PQ}$ by finite subsets and we set

$$\begin{align*}\sigma_\rho(P,Q):=\lim_{n\to\infty}{\sigma^{\mathbb{B}}_{\mathcal{P}_n}(P,Q)}. \end{align*}$$

The existence of the limit as well as the independence of the choices made to define it and the fact that the limit object is a $\mathbb {B}$ -transverse cocycle are proved in [Reference Mazzoli and Viaggi16].

The following is a summary of results in Sections 4, 5 and 6 of [Reference Mazzoli and Viaggi16].