1 Introduction
The space
$\mathcal {T}$
of hyperbolic metrics on a closed orientable surface
$\Sigma $
of genus
$g\ge 2$
up to isotopy, known as Teichmüller space, is an object that appears ubiquitously as a space of parameters but also as a geometric object.
Comparing different hyperbolic metrics on
$\Sigma $
according to various measurements of distortion endows
$\mathcal {T}$
with a wealth of geometry. An example is the Lipschitz distortion, which corresponds to the so-called Thurston’s asymmetric metric. Thurston proves in [Reference Thurston21] that, given two hyperbolic metrics
$g_X,g_Y$
on
$\Sigma $
, we have
$$\begin{align*}\min_{f\ \text{homotopic to Id}}\{\mathrm{Lip}(f)|\;f:(\Sigma,g_X)\to(\Sigma,g_Y)\}=\sup_{\gamma\in\pi_1(\Sigma)-\{1\}}{\frac{L_Y(\gamma)}{L_X(\gamma)}}, \end{align*}$$
where
$L_X(\gamma ),L_Y(\gamma )$
are the lengths of the geodesic representatives of
$\gamma $
with respect to
$g_X,g_Y$
.
This phenomenon of expressing the measurement of distortion in terms of length spectra
$L_Z(\bullet )$
is not exclusive of the Thurston metric. For example, the Teichmüller and Weil-Petersson metrics on
$\mathcal {T}$
also have this property.
It is therefore important to understand how length functions behave on Teichmüller space. Often, this behavior is related to certain geometric structures on low-dimensional manifolds. A celebrated example is the relation between quasi-Fuchsian hyperbolic
$3$
-manifolds and Teichmüller geodesics discovered by Minsky [Reference Minsky18].
Following an analogy between quasi-Fuchsian
$3$
-manifolds and the so-called globally hyperbolic maximal Cauchy compact (GHMC) anti-de Sitter
$3$
-manifolds, in this article we bring together the following:
-
○
$3$
-dimensional anti-de Sitter geometry, -
○ Convexity of length functions along shear paths and earthquakes.
In particular, we use the global scale geometry of GHMC anti-de Sitter manifolds to give a proof of (strict) convexity of length functions. Using the same bridge, we also develop geometric bounds for the second variation on those functions along earthquakes. Our methods are inspired from ideas in
$3$
-dimensional hyperbolic geometry.
1.1 Anti-de Sitter geometry
Anti-de Sitter geometry in dimension 3 is the geometry of
$\mathbb {H}^{2,1}:=\mathrm {PSL}_2(\mathbb {R})$
endowed with its natural pseudo-Riemannian metric of signature
$(2,1)$
. The link between Teichmüller theory and anti-de Sitter
$3$
-manifolds comes from the basic fact that the group of symmetries of this space is
$$\begin{align*}\mathrm{Isom}_0(\mathrm{PSL}_2(\mathbb{R}))=\mathrm{PSL}_2(\mathbb{R})\times\mathrm{PSL}_2(\mathbb{R}), \end{align*}$$
where
$(A,B)\cdot X:=AXB^{-1}$
and, at the same time,
$\mathrm {PSL}_2(\mathbb {R})=\mathrm {Isom}^+(\mathbb {H}^2)$
. A vast literature explores various aspects of this relation starting with the seminal work of Mess [Reference Mess17] (for a survey on the topic and recent developments, see Bonsante and Seppi [Reference Bonsante and Seppi8]).
Mess representations
Let
$\Sigma $
be a closed orientable surface of genus
$g\ge 2$
that we fix once and for all. We denote by
$\Gamma :=\pi _1(\Sigma )$
its fundamental group.
We realize the Teichmüller space
$\mathcal {T}$
of hyperbolic metrics on
$\Sigma $
up to isotopy as a component of the representation space
$$\begin{align*}\mathcal{T}\subset\mathrm{Hom}(\Gamma,\mathrm{PSL}_2(\mathbb{R}))/\mathrm{PSL}_2(\mathbb{R}) \end{align*}$$
by associating to each hyperbolic structure X (the conjugacy class of) its holonomy representation
$\rho _X:\Gamma \to \mathrm {PSL}_2(\mathbb {R})$
.
For any
$X,Y\in \mathcal {T}$
, we can consider the representation
$$\begin{align*}\rho_{X,Y}=(\rho_X,\rho_Y):\Gamma\to\mathrm{PSL}_2(\mathbb{R})\times\mathrm{PSL}_2(\mathbb{R}). \end{align*}$$
We will refer to any such
$\rho _{X,Y}$
as a Mess representation. The group
$\rho _{X,Y}(\Gamma )$
acts on
$\mathbb {H}^{2,1}$
convex cocompactly, meaning that
-
○ there is an equivariant boundary map
whose image
$$ \begin{align*}\xi:\partial\Gamma\to\partial\mathrm{PSL}_2(\mathbb{R})=\mathbb{P}\{A\in M_2(\mathbb{R})\mid\operatorname{\mathrm{rank}}(A)=1\} ,\end{align*} $$
$\xi (\partial \Gamma )=\Lambda _{X,Y}$
has the property that, for every
$a,b,c\in \partial \Gamma $
, the subspace
$\mathbb {P}\{\mathrm {Span}\{\xi (a),\xi (b),\xi (c)\}\}\cap \mathrm {PSL}_2(\mathbb {R})$
is a spacelike plane – that is, it is isometric to
$\mathbb {H}^2$
;
-
○ there is a canonical
$\rho _{X,Y}(\Gamma )$
-invariant properly convex open subset
$\Omega _{X,Y}\subset \mathrm {PSL}_2(\mathbb {R})$
on which the action is properly discontinuous; and -
○ we have
$\partial \Omega _{X,Y}\cap \partial \mathrm {PSL}_2(\mathbb {R})=\Lambda _{X,Y}$
, and the group
$\rho _{X,Y}(\Gamma )$
acts cocompactly on the convex hull
$\mathcal {CH}_{X,Y}\subset \Omega _{X,Y}$
of
$\Lambda _{X,Y}$
.
In order to study the geometry of Mess representations, we will use laminations and pleated surfaces, as we introduced in [Reference Mazzoli and Viaggi16]. Let us briefly recall the construction.
Laminations and pleated surfaces
A geodesic lamination on a hyperbolic surface X is a
$\rho _X(\Gamma )$
-invariant closed subset
$\lambda \subset \mathbb {H}^2$
that can be decomposed as a disjoint union of complete geodesics, the leaves of the lamination. The connected components of
$\mathbb {H}^2-\lambda $
are ideal polygons, the plaques of the lamination. The lamination is called maximal if all the plaques are ideal triangles. Conveniently, the data of a geodesic lamination can be encoded, by recording the endpoints of the leaves, as a
$\Gamma $
-invariant closed subset of the space of geodesics
$$\begin{align*}\{(x,y)\in\partial\Gamma\times\partial\Gamma\mid\;x\neq y\}/(x,y)\sim(y,x). \end{align*}$$
The boundary map
$\xi :\partial \Gamma \to \Lambda _{X,Y}$
and the property of the curve
$\Lambda _{X,Y}$
allow us to associate with every maximal lamination
$\lambda $
a geometric realization
$$\begin{align*}{\hat \lambda}:=\bigcup_{(a,b)\in\lambda}{[\xi(a),\xi(b)]}\subset\mathcal{CH}_{X,Y} \end{align*}$$
and a pleated set
$$\begin{align*}{\hat S_\lambda}:={\hat \lambda}\cup\bigcup_{\Delta(a,b,c)\subset\mathbb{H}^2-\lambda}{\Delta(\xi(a),\xi(b),\xi(c))}\subset\mathcal{CH}_{X,Y}. \end{align*}$$
Here,
$[\xi (a),\xi (b)]$
denotes the spacelike geodesic with endpoints
$\xi (a),\xi (b)$
, whereas
$\Delta (\xi (a),\xi (b),\xi (c))$
is the ideal spacelike triangle contained in the spacelike plane
$\mathbb {P}(\mathrm { Span}\{\xi (a),\xi (b),\xi (c)\})\cap \mathrm {PSL}_2(\mathbb {R})$
with vertices
$\xi (a),\xi (b),\xi (c)$
.
For every element
$\gamma \in \Gamma -\{1\}$
with attracting and repelling fixed points
$\gamma ^+$
and
$\gamma ^- \in \partial \Gamma $
, respectively, the isometry
$\rho _{X,Y}(\gamma )$
acts by translations on the spacelike line
$\ell = [\xi (\gamma ^-), \xi (\gamma ^+)] \subset \mathcal {CH}_{X,Y}$
with minimal displacement
$$\begin{align*}L_{\rho_{X,Y}}(\gamma) = \frac{L_X(\gamma) + L_Y(\gamma)}{2}. \end{align*}$$
We refer to
$L_{\rho _{X,Y}}(\gamma )$
as the length of
$\gamma $
with respect to
$\rho _{X,Y}$
. As in the case of Fuchsian representations, the length function
$L_{\rho _{X,Y}}$
admits a natural extension to the space of measured laminations
$\mathcal {ML}$
, which is continuous with respect to its weak* topology. Given any maximal geodesic lamination
$\lambda $
, we set
$$\begin{align*}\mathcal{ML}_\lambda = \{ \mu \in \mathcal{ML} \mid \mathrm{support}(\mu) \subset \lambda \}. \end{align*}$$
Then we have the following structural result:
Theorem. (Theorems A, B and C of [Reference Mazzoli and Viaggi16])
Let
$\rho _{X,Y}$
be a Mess representation. Consider a maximal lamination
$\lambda \subset \Sigma $
and let
${\hat S_\lambda }\subset \mathcal {CH}_{X,Y}\subset \Omega _{X,Y}$
be the corresponding pleated set. Then,
-
1.
${\hat S}_\lambda $
is an acausal Lipschitz disk with boundary
$\Lambda _{X,Y}$
(i.e., for every pair of points
$x,y\in {\hat S}_\lambda $
, the geodesic
$[x,y]$
joining them is spacelike). In particular,
${\hat S}_\lambda $
has a pseudo-metric
$d_{\mathbb {H}^{2,1}}(x,y):=\mathrm {length}([x,y])$
. -
2.
${\hat S}_\lambda $
carries an intrinsic hyperbolic structure
$Z_\lambda \in \mathcal {T}$
, which satisfies
$L_{Z_\lambda }(\mu )=L_{\rho _{X,Y}}(\mu )$
for every
$\mu \in \mathcal {ML}_{\lambda }$
. -
3. Let
$\rho _\lambda $
be the holonomy representation of
$Z_\lambda \in \mathcal {T}$
. Then there exists a
$(\rho _{X,Y}-\rho _\lambda )$
-equivariant homeomorphism
${\hat f}:{\hat S}_\lambda \to \mathbb {H}^2$
which is 1-Lipschitz, in the sense that
$d_{\mathbb {H}^{2,1}}(x,y)\ge d_{\mathbb {H}^2}({\hat f}(x),{\hat f}(y))$
, and that is isometric on each leaf and plaque.
Furthermore, we have
$$\begin{align*}L_{Z_\lambda}(\gamma)\le L_{\rho_{X,Y}}(\gamma) \end{align*}$$
for every
$\gamma \in \Gamma -\{1\}$
, with strict inequality if and only if
$\gamma $
intersects transversely the bending locus of
$\hat {S}_\lambda $
.
Mess [Reference Mess17], inspired by work of Thurston (Chapter 8 of [Reference Thurston20]), observes that
$\partial ^\pm \mathcal {CH}_{X,Y}$
is the pleated set
${\hat S}_{\lambda ^\pm }$
of a lamination
$\lambda ^\pm $
and that measuring the total turning angle along paths
$\alpha :I\to \partial ^\pm \mathcal {CH}_{X,Y}$
endows
$\lambda ^\pm $
with a natural transverse measure, the bending measure
$\mu ^\pm $
, with
$\mathrm {support}(\mu ^\pm ) \subset \lambda ^\pm $
. Then he shows that the surfaces
$X,Y$
and
$Z_{\lambda ^+},Z_{\lambda ^-}$
are related by the following diagram:
where
$E_{\mu ^+}^l,E_{\mu ^-}^l,E_{\mu ^+}^r,E_{\mu ^-}^r$
are the left and right earthquakes induced by the measured laminations
$\mu ^+,\mu ^-$
.
Recall that by the work of Bonahon [Reference Bonahon5] and Thurston [Reference Thurston21], for every maximal geodesic lamination
$\lambda $
of
$\Sigma $
, the Teichmüller space
$\mathcal {T}$
can be realized as an open convex cone in a finite dimensional
$\mathbb {R}$
-vector space
$\mathcal {H}(\lambda ;\mathbb {R})$
via the so-called shear coordinates
$\sigma _\lambda :\mathcal {T}\to \mathcal {H}(\lambda ;\mathbb {R})$
. Generalizing Mess, we prove the following:
Theorem 1. Let
$\rho _{X,Y}$
be a Mess representation. Consider a maximal lamination
$\lambda \subset \Sigma $
and let
$S_\lambda ={\hat S_\lambda }/\rho _{X,Y}(\Gamma )$
be the corresponding pleated surface. Then, in shear coordinates
$\sigma _\lambda (\mathcal {T}) \subset \mathcal {H}(\lambda ;\mathbb {R})$
for
$\lambda $
, we have the following:
-
1. The intrinsic hyperbolic structure
$Z_\lambda $
of
$S_\lambda $
satisfies
$$\begin{align*}\sigma_\lambda(Z_\lambda)=\frac{\sigma_\lambda(X)+\sigma_\lambda(Y)}{2}. \end{align*}$$
-
2. The intrinsic bending cocycle
$\beta _\lambda $
of
$S_\lambda $
satisfies
$$\begin{align*}\beta_\lambda=\frac{\sigma_\lambda(X)-\sigma_\lambda(Y)}{2}. \end{align*}$$
Length functions
We now come to the main novelty of this article, namely, the anti-de Sitter perspective on length functions in Teichmüller theory.
Recall that, given any Mess representation
$\rho _{X,Y}$
and any nontrivial element
$\gamma \in \Gamma - \{1\}$
, the isometry
$\rho _{X,Y}(\gamma )$
acts by translation of
$L_{\rho _{X,Y}} = \frac {1}{2}(L_X(\gamma ) + L_Y(\gamma ))$
on its axis
$\ell \subset \mathcal {CH}_{X,Y}$
. In fact,
$\rho _{X,Y}(\gamma )$
preserves also the dual line
$\ell ^* \subset \mathbb {H}^{2,1} - \Omega _{X,Y}$
of
$\ell $
, which consists of all points in
$\mathbb {H}^{2,1}$
at timelike distance
$\frac {\pi }{2}$
from
$\ell $
. The element
$\rho _{X,Y}(\gamma )$
acts also on
$\ell ^*$
by translations and with minimal displacement
$$\begin{align*}\theta_{\rho_{X,Y}}(\gamma) = \frac{L_X(\gamma) - L_Y(\gamma)}{2}. \end{align*}$$
We prove the following:
Theorem 2. Let
$\rho _{X,Y}$
be a Mess representation and let
$\lambda $
be a maximal lamination. Denote by
$Z_\lambda \in \mathcal {T}$
the intrinsic hyperbolic structure on
${\hat S}_\lambda /\rho _{X,Y}(\Gamma )$
, where
${\hat S}_\lambda \subset \mathcal {CH}_{X,Y}$
is the pleated set associated with
$\lambda $
. Consider
$\gamma \in \Gamma -\{1\}$
a nontrivial element whose image
$\rho _{X,Y}(\gamma )$
has axis
$\ell \subset \mathcal {CH}_{X,Y}$
.
-
1. Let
$\delta $
be the maximal timelike distance of
$\ell $
from
${\hat S}_\lambda $
. Then,
$$\begin{align*}\cosh(L_{Z_\lambda}(\gamma))\le\cos(\delta)^2\cosh(L_{\rho_{X,Y}}(\gamma))+\sin(\delta)^2\cosh(\theta_{\rho_{X,Y}}(\gamma)). \end{align*}$$
-
2. Let
$\delta ^\pm $
be the maximal timelike distance of
$\ell $
from
$\lambda ^\pm $
. Then, where
$$\begin{align*}\cosh(i(\lambda^\pm,\gamma))\le\sin(\delta^\pm)^2\cosh(L_{\rho_{X,Y}}(\gamma))+\cos(\delta^\pm)^2\cosh(\theta_{\rho_{X,Y}}(\gamma)) , \end{align*}$$
$i(\bullet ,\bullet )$
denotes the geometric intersection form (see, for example, [Reference Bonahon4]).
When combined, the previous results (Theorem 1 and Theorem 2) give a purely anti-de Sitter proof of (strict) convexity of length functions in shear coordinates, recovering simultaneously results of Bestvina, Bromberg, Fujiwara and Souto [Reference Bestvina, Bromberg, Fujiwara and Souto3], and Théret [Reference Théret19]:
Theorem 3. Let
$\lambda \subset \Sigma $
be a maximal lamination. Given a measured lamination
$\mu \in \mathcal {ML}$
, denote by
$\mathscr {L}_\mu :\mathcal {T} \to (0,\infty )$
its associated length function,
$\mathscr {L}_\mu (X) : = L_X(\mu )$
. Then,
-
1. For any
$\gamma \in \Gamma -\{1\}$
nontrivial loop, the length function
$\mathscr {L}_\gamma $
is convex in shear coordinates for
$\lambda $
. Moreover, convexity is strict if
$\gamma $
intersects essentially every leaf of
$\lambda $
. -
2. For any measured lamination
$\mu \in \mathcal {ML}$
, the length function
$\mathscr {L}_\mu $
is convex in shear coordinates for
$\lambda $
. Furthermore, convexity is strict if the support of
$\mu $
intersects transversely each leaf of
$\lambda $
.
Note that (b) does not imply (a). In (a), the loop
$\gamma $
does not necessarily represent a simple curve.
In the case of earthquakes, the geometry of Mess
$3$
-manifolds allows us to get the following infinitesimal geometric bound. We should mention that these bounds can also be deduced from work of Kerckhoff [Reference Kerckhoff13] and Wolpert [Reference Wolpert23], respectively.
Theorem 4. Let
$\mu \in \mathcal {ML}$
be a measured lamination and let
$\epsilon _\mu :[a,b]\to \mathcal {T}$
be an earthquake path driven by
$\mu $
. Set
$\mathscr {L}_{\gamma }(t):=L_{\epsilon _\mu (t)}(\gamma )$
. Then, for every
$\gamma \in \Gamma -\{1\}$
, we have
$$\begin{align*}{\ddot {\mathscr{L}}_\gamma}\ge\frac{1}{\sinh(\mathscr{L}_\gamma)}\left|{\dot{\mathscr{L}}}_\gamma\right|\left(i(\gamma,\mu)-\left|{\dot{\mathscr{L}}}_\gamma\right|\right). \end{align*}$$
Let us point out that, by Kerckhoff’s formula for the first variation [Reference Kerckhoff13], we always have
$|{\dot {\mathscr {L}}}_\gamma |\le i(\gamma ,\mu )$
with strict inequality if
$\gamma $
intersects
$\mu $
essentially.
Anti-de Sitter proofs
We now briefly discuss the main new ideas and ingredients that go into the anti-de Sitter proofs.
Theorem 2
The idea is that, as we move a closed geodesic
$\gamma \subset M_{X,Y}$
orthogonally along timelike directions, its length shrinks. Heuristically speaking, every closed geodesic
$\gamma \subset M_{X,Y}$
is the core of an (immersed) anti-de Sitter annulus
$A_\gamma \subset M_{X,Y}$
whose intrinsic metric has the form
$\mathrm {d}s^2=-\mathrm {d}t^2+\sin (t)^2\mathrm {d}\ell ^2$
. Hence, the length of
$\gamma (s)=(0,s)$
(in
$(t,\ell )$
coordinates) contracts as we move it away from the core
$\{t=0\}$
along orthogonal timelike directions. In the proof of the theorem, we make some aspects of this picture precise. In particular, we understand how various avatars of
$A_\gamma $
intersect the pleated surfaces
${\hat S}_\lambda /\rho _{X,Y}(\Gamma )$
and
$\partial ^\pm \mathcal {CC}(M_{X,Y})=\partial ^\pm \mathcal {CH}_{X,Y}/\rho _{X,Y}(\Gamma )$
.
Theorem 3
(Strict) convexity is equivalent to the (strict) inequality in
$$\begin{align*}\mathscr{L}_\gamma\left( Z_\lambda \right)\le\frac{\mathscr{L}_\gamma(X)+\mathscr{L}_\gamma(Y)}{2} \end{align*}$$
for every
$X \neq Y \in \mathcal {T}$
, where
$\sigma _\lambda (Z_\lambda ) = \frac {\sigma _\lambda (X) + \sigma _\lambda (Y)}{2} \in \mathcal {H}(\lambda ;\mathbb {R})$
. We note that the right-hand side is equal to
$L_{\rho _{X,Y}}(\gamma )$
and, by Theorem 1, the left-hand side is equal to
$L_{Z_\lambda }(\gamma )$
, where
$Z_\lambda $
is the hyperbolic structure on the pleated surface
${\hat S}_\lambda /\rho _{X,Y}(\gamma )$
associated with
$\lambda $
and
$\rho _{X,Y}$
. The inequality is then a consequence of part (a) of Theorem 2. The inequality is not strict exactly when the maximal distance
$\delta $
between
$\ell $
and
$\hat {S}_\lambda $
vanishes or, in other words, when
$\ell \subset {\hat S}_\lambda $
. This is possible if and only if
$\gamma $
does not intersect transversely the bending locus.
The proof for laminations requires a significantly more refined argument based on the following heuristic principle: every time
$\ell $
passes at timelike distance
$\delta>0$
from
${\hat S}_\lambda $
it creates a gap of size
$\kappa>0$
between
$L_{Z_\lambda }(\mu )$
and
$L_{\rho _{X,Y}}(\mu )$
.
Theorem 4
The idea is to analyze the geometry of the representations
$\rho _t:=\rho _{Z_{-t},Z_t}$
as
$t\to 0$
, where
$Z_t=E^l_{t \mu }(Z)$
for some fixed
$Z \in \mathcal {T}$
. Notice that, by Theorem 1, the bending measure on
$\partial ^+\mathcal {CH}_{Z_{-t},Z_t}$
is
$\mu _t^+=t\mu $
and its associated hyperbolic structure is constant
$Z_t^+=Z$
. The main tool is again Theorem 2: combining the inequalities of part (a) and part (b) of Theorem 2, we have
$$\begin{align*}\cosh(t\cdot i(\mu,\gamma))-\cosh(\theta_{\rho_t}(\gamma))\le\cosh(L_{\rho_t}(\gamma))-\cosh(L_Z(\gamma)). \end{align*}$$
The conclusion follows from basic analysis, essentially the mean value theorem
$\cosh (x)-\cosh (y)=\sinh (\xi )(x-y)$
, where
$\xi \in [x,y]$
, and the fact that
$(f(-t)+f(t)-2f(0))/t^2\to {\ddot f}$
, which we apply to
$(L_{\rho _t}(\gamma )-L_Z(\gamma ))/t^2=(\mathscr {L}_\gamma (Z_{-t})+\mathscr {L}_\gamma (Z_t)-2\mathscr {L}_\gamma (Z)))/2t^2\to \ddot {\mathscr {L}}_\gamma /2$
.
Shear-bend parametrization
As an application of our computations on the intrinsic hyperbolic structure and intrinsic bending of a non-convex pleated surface, we also obtain a shear-bend parametrization of the space of Mess
$3$
-manifolds, in the spirit of Bonahon’s work [Reference Bonahon5]. Consider the space of Mess representations
$$\begin{align*}\mathcal{MR}:=\mathcal{T}\times\mathcal{T}\subset\mathrm{Hom}(\Gamma,\mathrm{PSL}_2(\mathbb{B}))/\mathrm{PSL}_2(\mathbb{B}), \end{align*}$$
where
$\mathbb {B}:=\mathbb {R}[\tau ]/(\tau ^2-1)=\mathbb {R}\oplus \tau \mathbb {R}$
denotes the ring of para-complex numbers. Let
$\mathcal {H}(\lambda ;\mathbb {B})$
be the finite dimensional
$\mathbb {B}$
-module of transverse cocycles for
$\lambda $
with values in
$\mathbb {B}$
, as introduced in Bonahon [Reference Bonahon5]. Notice that there are natural identifications
$\mathrm {PSL}_2(\mathbb {B})=\mathrm { PSL}_2(\mathbb {R})\times \mathrm {PSL}_2(\mathbb {R})$
and
$\mathcal {H}(\lambda ;\mathbb {B})=\mathcal {H}(\lambda ;\mathbb {R})\oplus \tau \, \mathcal {H}(\lambda ;\mathbb {R})$
. We have the following:
Theorem 5. Let
$\lambda \subset \Sigma $
be a maximal lamination. Then,
-
1. The map
which associates to
$$\begin{align*}\begin{matrix} \Phi: & \mathcal{MR} & \longrightarrow & \mathcal{H}(\lambda;\mathbb{B})\\ & \rho = \rho_{X,Y} & \longmapsto & \sigma_\lambda(Z_\lambda)+\tau \, \beta_\lambda , \end{matrix} \end{align*}$$
$\rho $
the shear-bend cocycle
$\sigma _\rho $
of the unique pleated surface
$S_\lambda ={\hat S}_\lambda /\rho (\Gamma )$
determined by
$\lambda $
, is an analytic para-complex embedding.
-
2. If
$\omega ^{\mathbb {B}}_{\mathrm {Th}}(\bullet ,\bullet )$
denotes the (
$\mathbb {B}$
-valued) Thurston’s symplectic form on
$\mathcal {H}(\lambda ;\mathbb {B})$
, then for every measured lamination
$$\begin{align*}\omega_{\mathrm{Th}}^{\mathbb{B}}(\sigma_\rho,\mu)=L_\rho(\mu)+\tau \, \theta_\rho(\mu) \end{align*}$$
$\mu \in \mathcal {ML}_\lambda $
and every
$\rho = \rho _{X,Y} \in \mathcal {MR}$
.
-
3. The image of the embedding is given by
Here,
$$ \begin{align*} \Phi(\mathcal{MR}) &=\left\{\sigma+\tau\beta\in\mathcal{H}(\lambda;\mathbb{B})\bigg|\;\sigma+\beta,\sigma-\beta\in\mathcal{T}\subset\mathcal{H}(\lambda;\mathbb{R})\right\}\\ &=\left\{\sigma+\tau\beta\in\mathcal{H}(\lambda;\mathbb{B})\bigg|\;|\omega_{\mathrm{Th}}^{\mathbb{B}}(\sigma+\tau\beta,\bullet)|_{\mathbb{B}}^2>0\ \mathrm{on}\ \mathcal{ML}_\lambda\right\}. \end{align*} $$
$|x+\tau y|_{\mathbb {B}}^2=x^2-y^2$
is the para-complex norm.
-
4. The pull-back of
$\omega ^{\mathbb {B}}_{\mathrm {Th}}$
to
$\mathcal {MR}=\mathcal {T}\times \mathcal {T}$
coincides with where
$$\begin{align*}\Phi^*\omega_{\mathrm{Th}}=c\cdot(\omega_{\mathrm{WP}}\oplus \omega_{\mathrm{WP}} + \tau \, \omega_{\mathrm{WP}}\oplus (- \omega_{\mathrm{WP}})) , \end{align*}$$
$\omega _{\mathrm {WP}}(\bullet ,\bullet )$
is the Weil-Petersson symplectic form.
Notice that the para-complex structure and the symplectic structure appearing in items (1) and (4), respectively, are part of the natural para-hyperKähler structure defined on the deformation space of GHMC anti-de Sitter
$3$
-manifolds (see Mazzoli, Seppi and Tamburelli [Reference Mazzoli, Seppi and Tamburelli15] for details).
It is worth mentioning that Mausburger and Scarinci [Reference Mausburger and Scarinci14] described a similar set of coordinates for GHMC spacetimes of dimension
$3$
that admit a cusped Cauchy surface, which they call generalised shear coordinates. Although the present work deals with the case of a closed orientable surface
$\Sigma $
of genus larger than
$1$
and any maximal geodesic lamination
$\lambda $
, in [Reference Mausburger and Scarinci14], the authors consider constant curvature (not necessary equal to
$-1$
) GHMC spacetimes homeomorphic to
$S \times \mathbb {R}$
, where S is a finite type surface with at least one puncture and
$\lambda $
is an ideal triangulation.
Structure of the article
The paper is organized as follows:
-
○ In Section 2, we recall some basic facts in Teichmüller theory and anti-de Sitter
$3$
-dimensional geometry. -
○ In Section 3, we introduce Mess representations and pleated surfaces and recall some of their properties.
-
○ In Section 4, we compute the intrinsic shear-bend cocycles of pleated surfaces and prove Theorems 1 and 5.
-
○ In Section 5, we study the behavior of length functions for Mess representations and prove Theorem 2.
-
○ In Section 6, we discuss the purely anti-de Sitter proofs of Theorems 3 and 4.
2 Teichmüller and anti-de Sitter space
In this section, we recall the amount of basic Teichmüller theory and anti-de Sitter
$3$
-dimensional geometry that we will need in the next sections.
2.1 Teichmüller theory
We start with hyperbolic surfaces and (measured) geodesic laminations.
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 2.1 (Hyperbolic Structures)
A marked hyperbolic structure on
$\Sigma $
is a homeomorphism between
$\Sigma $
and a quotient
$\mathbb {H}^2/\rho _X(\Gamma )$
of the hyperbolic plane
$\mathbb {H}^2$
by the image of a faithful and discrete representation
$\rho _X:\Gamma \to \mathrm { PSL}_2(\mathbb {R})$
, the holonomy of the structure. Two marked hyperbolic structures
$X,X'$
on
$\Sigma $
are equivalent if their holonomies
$\rho _X,\rho _{X'}$
are conjugate.
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
$x,y$
.
Definition 2.4 (Geodesic Lamination)
Let
$X=\mathbb {H}^2/\rho _X(\Gamma )$
be a hyperbolic surface. A geodesic lamination on X is a
$\rho _X(\Gamma )$
-invariant closed subset
$\lambda \subset \mathbb {H}^2$
, which can be expressed as a disjoint union of complete geodesics, the leaves of the lamination. The complementary regions
$\mathbb {H}^2-\lambda $
are ideal polygons (with possibly infinitely many sides) and are called the plaques of
$\lambda $
.
The geodesic lamination
$\lambda $
is maximal if all its plaques are ideal triangles. A geodesic lamination on X is completely determined by the endpoints on
$\mathbb {RP}^1$
of the leaves which form a closed
$\rho _X(\Gamma )$
-invariant subset of
$\mathcal {G}$
. We denote by
$\mathcal {GL}$
the space of geodesic laminations and by
$\mathcal {GL}_m$
the subspace consisting of maximal ones.
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.
Definition 2.5 (Geodesic Current)
Let
$X=\mathbb {H}^2/\rho _X(\Gamma )$
be a hyperbolic surface. A geodesic current on X is a
$\rho _X(\Gamma )$
-invariant locally finite Borel measure on
$\mathcal {G}$
. We denote by
$\mathcal {C}$
the space of geodesic currents.
Definition 2.6 (Closed Geodesics)
A basic example of geodesic current is the one associated to a free homotopy class of a loop
$\gamma \in \Gamma -\{1\}$
. It is defined as
$\delta _\gamma :=\sum _{[\alpha ]\in \Gamma /\langle \gamma \rangle }{\delta _{\ell _\alpha }}$
, where
$\ell _\alpha $
is the axis of
$\rho _X(\alpha )$
and
$\delta _\ell $
is the Dirac mass on the point
$\ell \in \mathcal {G}$
.
Definition 2.7 (Geometric Intersection)
On
$\mathcal {C}$
, there is a natural intersection form
$i(\bullet ,\bullet )$
, defined as follows. Let
$\alpha ,\beta \in \mathcal {C}$
be geodesic currents. Consider the space of intersecting geodesics
$\mathcal {I}:=\{(\ell ,\ell ')\in \mathcal {G}\times \mathcal {G}|\ell \cap \ell '\neq \emptyset \}$
. The group
$\rho _X(\Gamma )$
acts properly discontinuously and freely on
$\mathcal {I}$
. By invariance, the measure
$\alpha \times \beta $
on
$\mathcal {I}$
descends to a Borel measure on
$\mathcal {I}/\rho _X(\Gamma )$
. Define
$i(\alpha ,\beta ):=(\alpha \times \beta )(\mathcal {I}/\rho _X(\Gamma ))$
. A crucial property of the geometric intersection form
$i(\alpha ,\beta )$
is that it is continuous in
$\alpha ,\beta $
.
Definition 2.8 (Measured Lamination)
Let
$X=\mathbb {H}^2/\rho _X(\Gamma )$
be a hyperbolic surface. A measured lamination on X is a geodesic current
$\mu \in \mathcal {C}$
with
$i(\mu ,\mu )=0$
. We denote by
$\mathcal {ML}$
the space of measured laminations.
The support of a measured lamination
$\mathrm {support}(\mu )$
is a geodesic lamination (see [Reference Bonahon4]). We denote by
$\mathcal {ML}_\lambda :=\{\mu \in \mathcal {ML}|\;\mathrm { support}(\mu )\subset \lambda \}$
the space of measured laminations whose support is contained in
$\lambda $
.
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 The
$\mathrm {PSL}_2(\mathbb {R})$
model of
$\mathbb {H}^{2,1}$
The second central object that we discuss is the anti-de Sitter
$3$
-space
$\mathbb {H}^{2,1}$
. We will mostly work in its linear and projective models, which we now describe. For more details on the material we present here, we refer the reader to [Reference Bonsante and Seppi8].
The group
$\mathrm {SL}_2(\mathbb {R})$
sits inside the vector space of
$2\times 2$
matrices with real entries
$M_2(\mathbb {R})$
as the hyperboloid of vectors of norm
$-1$
for the quadratic form
$\langle \bullet ,\bullet \rangle $
of signature
$(2,2)$
given by
$$\begin{align*}2 \, \langle X,Y\rangle:=\mathrm{det}(X)+\mathrm{det}(Y)-\mathrm{det}(X+Y)=-\mathrm{tr}(XY^\star), \end{align*}$$
where
${\tiny \left [ \begin {array}{c c} a &b\\ c &d\\ \end {array} \right ]^\star := \left [ \begin {array}{c c} d &-b\\ -c &a\\ \end {array} \right ]}$
.
Note that, for every
$X\in \mathrm {SL}_2(\mathbb {R})$
, the restriction of the quadratic form to
$T_X\mathrm {SL}_2(\mathbb {R})=X^\perp $
has signature
$(2,1)$
and hence induces a
$(2,1)$
-pseudo-Riemannian metric on
$\mathrm { SL}_2(\mathbb {R})$
(experts will have recognized the Killing form of
$\mathrm {SL}_2(\mathbb {R})$
). The group
$\mathrm {SL}_2(\mathbb {R})\times \mathrm {SL}_2(\mathbb {R})$
acts on
$M_2(\mathbb {R})$
by left and right multiplications as
$(A,B)\cdot X:=AXB^{-1}$
, and the action is isometric with respect to
$\langle \bullet ,\bullet \rangle $
. Passing to the projectivization
$\mathrm {PSL}_2(\mathbb {R})\subset \mathbb {P}(M_2(\mathbb {R}))$
, we obtain the projective model of anti-de Sitter
$3$
-space
$\mathbb {H}^{2,1}$
.
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 2.10 (Acausal Set)
A subset
$S\subset \mathbb {H}^{2,1}\cup \partial \mathbb {H}^{2,1}$
is acausal if, for every
$x,y\in S$
, the geodesic
$[x,y]$
is spacelike.
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 Mess representations and pleated surfaces
The goal of the section is to describe Mess representations and the geometry of their pleated surfaces. In particular, at the end of the section, we discuss the structure of the boundary of the convex core associated with a Mess representation.
3.1 Mess representations
First of all, we introduce the following class:
Definition 3.1 (Mess Representation)
Let
$X,Y\in \mathcal {T}$
be hyperbolic structures. The Mess representation with parameters
$X,Y$
is
$$\begin{align*}\rho_{X,Y}:=(\rho_X,\rho_Y):\Gamma\to\mathrm{PSL}_2(\mathbb{R})\times\mathrm{PSL}_2(\mathbb{R}), \end{align*}$$
where
$\rho _X,\rho _Y$
are the holonomy representations of
$X,Y$
.
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 Laminations and pleated surfaces
Mess representations are examples of maximal representations in
$\mathrm {PSL}_2(\mathbb {R})\times \mathrm {PSL}_2(\mathbb {R})=\mathrm {PSO}_0(2,2)$
, as introduced in Burger, Iozzi and Wienhard [Reference Burger, Iozzi and Wienhard9]. In fact, by a celebrated result of Goldman [Reference Goldman12], every maximal representation in
$\mathrm {PSL}_2(\mathbb {R})\times \mathrm {PSL}_2(\mathbb {R})$
is a Mess representation.
As a consequence, we can apply the results of [Reference Mazzoli and Viaggi16] to our setting. In this section, we recall the pleated surface construction from [Reference Mazzoli and Viaggi16] and describe some geometric properties of these objects.
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
$\xi (a),\xi (b),\xi (c)$
.
Proposition 3.5 (Proposition 3.7 of [Reference Mazzoli and Viaggi16])
The pleated set
$\widehat {S}_\lambda \subset \mathcal {CH}_{X,Y}$
is a
$\rho _{X,Y}(\Gamma )$
-invariant topological Lipschitz acausal subsurface.
Incidentally, in combination with classical
$3$
-dimensional topology, Proposition 3.5 has also the following topological corollary:
Corollary 3.6. Let
$\rho _{X,Y}$
be a Mess representation with parameters
$X,Y\in \mathcal {T}$
. Identify the Mess manifold
$M_{X,Y}:=\Omega _{X,Y}/\rho _{X,Y}(\Gamma )$
with
$\Sigma \times \mathbb {R}$
and let
$\alpha \subset \Sigma $
be an essential multicurve. Then, the geodesic realization of
$\alpha $
in
$M_{X,Y}$
is isotopic to
$\alpha \subset \Sigma \times \{0\}$
.
Proof. Let
$\lambda _\alpha $
be a maximal lamination obtained from
$\alpha $
by adding finitely many geodesics spiraling around the curves in
$\alpha $
. By Proposition 3.5, there exists an embedded
$\pi _1$
-injective (Lipschitz) surface
$S_{\alpha }={\hat S}_{\lambda _\alpha }/\rho _{X,Y}(\Gamma )\subset M_{X,Y}$
containing the geodesic realization of the curves in
$\alpha $
. By Proposition 3.1 and Corollary 3.2 of [Reference Waldhausen22], such a surface, being embedded and
$\pi _1$
-injective, is isotopic to
$\Sigma \times \{0\}$
.
3.2.2 Bending locus
The pleated set
${\hat S}_\lambda $
is not necessarily bent along all the lines in
${\hat \lambda }$
.
Definition 3.7 (Bending Locus)
Let
$\rho _{X,Y}$
be a Mess representation. Consider
$\lambda $
a maximal lamination with geometric realization
${\hat \lambda }$
and denote by
$\widehat {S}_\lambda $
the corresponding pleated set. A point
$x\in \ell \subset {\hat \lambda }$
is in the bending locus of
$\widehat {S}_\lambda $
if there is no (necessarily spacelike) geodesic segment k entirely contained in
$\widehat {S}_\lambda $
and such that
$\mathrm { int}(k)\cap \ell =x$
.
We have the following:
Proposition 3.8 (Proposition 3.11 of [Reference Mazzoli and Viaggi16])
The bending locus is a sublamination of
${\hat \lambda }$
, and its complement in
$\widehat {S}_\lambda $
is a union of 2-dimensional totally geodesic spacelike regions.
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:
Definition 3.9 (Developing Map)
Let
$\rho _{X,Y}$
be a Mess representation. Let
${\hat S}_\lambda \subset \mathcal {CH}_{X,Y}$
be the pleated set associated with a maximal lamination
$\lambda $
. A 1-Lipschitz developing map is a homeomorphism
$f:{\hat S}_\lambda \to \mathbb {H}^2$
with the following properties:
-
1. It is isometric on every plaque and sends every leaf of
${\hat \lambda }$
onto a complete geodesic. -
2. It is 1-Lipschitz with respect to the intrinsic pseudo-metric on
${\hat S}_\lambda $
and the hyperbolic metric on
$\mathbb {H}^2$
.
Developing maps have a couple of useful general properties which we now describe. First, they are isometric outside the bending locus.
Lemma 3.10 (Lemma 6.2 of [Reference Mazzoli and Viaggi16])
Let
$\rho _{X,Y}$
be a Mess representation and let
${\hat S}_\lambda $
be the pleated set associated to a maximal lamination
$\lambda $
. Then, every 1-Lipschitz developing map
$f:{\hat S}_\lambda \to \mathbb {H}^2$
is isometric on the complement of the bending locus of
${\hat S}_\lambda $
.
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,
$$\begin{align*}L(\gamma)=\lim_{\epsilon\to 0}\int_I{\frac{d_{\mathbb{H}^{2,1}}(\gamma(t),\gamma(t+\epsilon))}{\epsilon} \text{dt}}. \end{align*}$$
Lemma 3.13 (Lemma 6.4 of [Reference Mazzoli and Viaggi16])
Let
$\rho _{X,Y}$
be a Mess representation and let
${\hat S}_\lambda $
be the pleated set associated to a maximal lamination
$\lambda $
. Then, every 1-Lipschitz developing map
$f:{\hat S}_\lambda \to \mathbb {H}^2$
sends weakly regular paths
$\gamma :I\to {\hat S}_\lambda $
to Lipschitz (hence rectifiable) paths
$f\gamma :I\to \mathbb {H}^2$
of smaller length
$L(\gamma )\ge L(f\gamma )$
.
3.2.4 Pleated surfaces
The following result makes sure that every pleated set
${\hat S}_\lambda $
admits a natural 1-Lipschitz developing map:
Proposition 3.14 (Proposition 6.6 in [Reference Mazzoli and Viaggi16])
Let
$\rho _{X,Y}$
be a Mess representation. For every maximal lamination
$\lambda \in \mathcal {GL}_\lambda $
, there is
-
○ an intrinsic hyperbolic structure
$Z_\lambda \in \mathcal {T}$
; -
○ a
$(\rho _{X,Y}-\rho _\lambda )$
-equivariant 1-Lipschitz developing map
$f:{\hat S}_\lambda \to \mathbb {H}^2$
where
$\rho _\lambda $
is the holonomy of
$Z_\lambda $
.
We can finally define pleated surfaces:
Definition 3.15 (Pleated Surface)
Let
$\rho _{X,Y}$
be a Mess representation. The pleated surface associated with the maximal lamination
$\lambda \in \mathcal {GL}$
consists of the following data:
-
1. The pleated set
${\hat S}_\lambda $
. -
2. The intrinsic hyperbolic holonomy
$\rho _\lambda :\Gamma \to \mathrm {PSL}_2(\mathbb {R})$
of
$Z_\lambda $
. -
3. A
$(\rho _{X,Y}-\rho _\lambda )$
-equivariant 1-Lipschitz developing map
$f:{\hat S}_\lambda \to \mathbb {H}^2$
.
Remark 3.16. It is worth mentioning that there exist examples of distinct Fuchsian structures
$\rho _\lambda , \rho _\lambda ' \in \mathcal {T}$
and equivariant homeomorphisms
$f_\lambda , f_\lambda ' : \hat {S}_\lambda \to \mathbb {H}^2$
associated with the same pleated set
$\hat {S}_\lambda $
that satisfy item (1) from Definition 3.9.
To see this, consider a maximal lamination
$\lambda $
whose support admits a non-atomic measured lamination
$\mu \neq 0$
and let
$Z_\lambda = \mathbb {H}^2/\rho _\lambda (\Gamma )$
and
$f_\lambda : \hat {S}_\lambda \to \mathbb {H}^2$
be as in Definition 3.9. Then, the left earthquake along
$\mu $
satisfies
$E^l_\mu (Z_\lambda ) \neq Z_\lambda $
, and its associated
$((E^l_\mu )_*(\rho _\lambda ) - \rho _\lambda )$
-equivariant map
$\mathcal {E}^l_\mu : \mathbb {H}^2 \to \mathbb {H}^2$
is continuous, is isometric on the complementary regions of the lift of
$\mathrm {support}(\mu ) \subset \lambda $
, and sends leaves into complete geodesics. It follows that
$\mathcal {E}^l_\mu \circ f_\lambda : \hat {S}_\lambda \to \mathbb {H}^2$
satisfies item (1) of Definition 3.9. This shows in particular that the requirement (2) in Definition 3.9 is necessary if we want to have uniqueness of the intrinsic hyperbolic structure and of the developing map of
$\hat {S}_\lambda $
(up to the action of
$\mathrm {Diff}_0(\Sigma )$
).
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:
Definition 3.18 (Initial and Terminal Singularities)
The sets
$\mathcal {S}^\pm $
of dual points of supporting planes of
$\partial ^\pm \mathcal {CH}_{X,Y}$
are the initial and terminal singularities.
Let us start with the following observation:
Lemma 3.19. Let
$H=P(V)\cap \mathbb {H}^{2,1}$
be a supporting plane of
$\partial ^\pm \mathcal {CH}_{X,Y}$
. Then,
-
○ H is spacelike and defines a dual point
$P(V^\perp )\in \mathbb {H}^{2,1}$
. Let
$w\in V^\perp $
be a unit timelike vector pointing outside
$\mathcal {CH}_{X,Y}$
. -
○ For every
$x\in H\cap \mathcal {CH}_{X,Y}$
, the timelike geodesic
$\gamma (t)=\cos (t)x-\sin (t)w$
, with
$t\in [0,\pi /2)$
, is contained in
$\Omega _{X,Y}$
, while
$w=\gamma (\pi /2)\in \partial \Omega _{X,Y}$
.
Any two distinct supporting planes
$H_1,H_2$
of
$\partial ^+\mathcal {CH}_{X,Y}$
intersect in a spacelike geodesic
$H_1\cap H_2$
. If
$w_1,w_2$
are the dual points of
$H_1,H_2$
, then
$[w_1,w_2]$
is spacelike.
Proof. The first point: Recall that
$\partial \mathbb {H}^{2,1}=\mathbb {RP}^1\times \mathbb {RP}^1$
and that
$\Lambda _{X,Y}$
is the graph of an orientation preserving homeomorphism
$h_{X,Y}:\mathbb {RP}^1\to \mathbb {RP}^1$
. If H is a supporting hyperplane for
$\mathcal {CH}_{X,Y}$
, then
$\partial H$
does not intersect
$\Lambda _{X,Y}$
transversely. The fact that H must be spacelike follows from the following observations. The boundary of a lightlike plane has the form
$(\{s\}\times \mathbb {RP}^1) \cup (\mathbb {RP}^1\times \{t\})$
. The boundary of a totally geodesic plane in
$\mathbb {H}^{2,1}$
with signature
$(1,1)$
is the graph of an orientation reversing linear transformation
$\mathbb {RP}^1\to \mathbb {RP}^1$
. In the former case, either
$(s,t) \in \Lambda _{X,Y}$
, in which case H does not intersect
$\partial ^+\mathcal {CH}_{X,Y}$
, or
$\partial H$
intersects
$\Lambda _{X,Y}$
transversely. In the latter case, namely if H has signature
$(1,1)$
,
$\partial H$
has always transverse intersection with
$\Lambda _{X,Y}$
.
The second point: Recall that
$\Omega _{X,Y}$
is the set of points that can be connected to every point in
$\Lambda _{X,Y}$
by a spacelike geodesic. A point
$x\in \mathbb {H}^{2,1}$
and a point
$p\in \partial \mathbb {H}^{2,1}$
are connected by a spacelike geodesic if and only if
$\langle x,p\rangle \neq 0$
. Let us show that
$\gamma (t)\in \Omega _{X,Y}$
for every
$t\in [0,\pi /2)$
. In order to do so, lift
$\Lambda _{X,Y}$
continuously to
$M_2(\mathbb {R})$
. As
$x\in \Omega _{X,Y}$
, we have
$\langle x,p\rangle \neq 0$
for every
$p\in \Lambda _{X,Y}$
and, by continuity, we can assume that it is negative for every
$p\in \Lambda _{X,Y}$
. As H is a supporting hyperplane and w is a timelike vector orthogonal to H and pointing outside
$\mathcal {CH}_{X,Y}$
, we have
$\langle p,w\rangle \ge 0$
for every
$p\in \Lambda _{X,Y}$
. Therefore,
$\langle \gamma (t),p\rangle =\cos (t)\langle x,p\rangle -\sin (t)\langle w,p\rangle <0$
for every
$p\in \Lambda _{X,Y}$
and
$t<\pi /2$
. In order to conclude, it is enough to observe that
$w=\gamma (\pi /2)\not \in \Omega _{X,Y}$
, as
$\langle w,p\rangle =0$
for every
$p\in \partial H\cap \Lambda _{X,Y}\neq \emptyset $
.
For the last part, notice that
$H_1\cap H_2$
is either empty or a spacelike geodesic. Suppose that
$H_1\cap H_2=\emptyset $
. Then,
$\mathbb {H}^{2,1}-(H_1\cup H_2)$
consists of two connected components – one of them containing
$\mathcal {CH}_{X,Y}$
. As
$H_1,H_2$
lie on opposite sides of
$\mathcal {CH}_{X,Y}$
in such component, they cannot be supporting hyperplanes for the same boundary component of
$\partial \mathcal {CH}_{X,Y}$
. This is a contradiction.
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 $
.
Definition 3.20 (
$\mathbb {R}$
-tree)
An
$\mathbb {R}$
-tree is a geodesic metric space
$(\mathcal {S},d_{\mathcal {S}}(\bullet ,\bullet ))$
such that between two points
$x,y\in \mathcal {S}$
there is a unique (up to reparametrization) injective path
$\alpha :[0,1]\to \mathcal {S}$
with
$\alpha (0)=x,\alpha (1)=y$
.
Benedetti and Guadagnini [Reference Benedetti and Guadagnini2] show the following:
Proposition 3.21. Let
$\rho _{X,Y}$
be a Mess representation and let
$\mathcal {S}^\pm \subset \partial \Omega _{X,Y}$
be the initial and terminal singularities. Then,
-
○
$\mathcal {S}^\pm $
is
$\rho _{X,Y}(\Gamma )$
-invariant, acausal and path-connected by regular paths. In particular, it has an intrinsic path metric where
$$\begin{align*}d_{\mathcal{S}^\pm}(x,y)=L(\alpha) , \end{align*}$$
$\alpha :[0,1]\to \mathcal {S}^\pm $
is a regular path joining x to y.
-
○ For every pair of points
$w,w'\in \mathcal {S}^\pm $
, there is a unique continuous injective path connecting them inside
$\mathcal {S}^\pm $
. -
○ For every
$\gamma \in \Gamma -\{1\}$
, the minimal displacement coincides with
$$\begin{align*}\min_{x\in\mathcal{S}^\pm}\{d_{\mathcal{S}^\pm}(x,\rho_{X,Y}(\gamma)x)\} \end{align*}$$
$i(\gamma ,\mu ^\pm )$
and it is realized by some point
$x\in \mathcal {S}^\pm $
.
Here,
$\mu ^\pm \in \mathcal {ML}$
is the bending lamination of
$\partial ^\pm \mathcal {CH}_{X,Y}$
, and
$i(\bullet ,\bullet )$
is the geometric intersection form.
For a proof, we refer to [Reference Benedetti and Guadagnini2] and [Reference Benedetti and Bonsante1].
4 A generalization of a result of Mess
The goal of the section is to define the shear-bend cocycles of pleated surfaces and prove Theorem 1.
We begin by recalling the Thurston-Bonahon shear parametrization of Teichmüller space (as discussed by Bonahon in [Reference Bonahon5]), which we will generalize to the space of Mess representations in Theorem 5, at the end of the section.
4.1 Shear coordinates
We refer to Bonahon [Reference Bonahon5] for more details on the material presented in this section.
4.1.1 Transverse cocycles
Shear-bend cocycles are a special case of transverse cocycles for
$\lambda $
.
Definition 4.1 (Transverse Cocycle)
Let
$\mathbb {A}$
be an Abelian group and let
$\lambda \subset \mathbb {H}^2$
be a maximal lamination. An
$\mathbb {A}$
-transverse cocycle for
$\lambda $
is a function
$\sigma (\bullet ,\bullet )$
of pairs of plaques satisfying the following properties:
-
○ Invariance:
$\sigma (\gamma P,\gamma Q)=\sigma (P,Q)$
for every
$\gamma \in \Gamma $
and plaques
$P,Q$
; -
○ Symmetry:
$\sigma (P,Q)=\sigma (Q,P)$
for every plaques
$P,Q$
; -
○ Additivity:
$\sigma (P,R)=\sigma (P,Q)+\sigma (Q,R)$
for every plaques
$P,Q,R$
such that R separates P from Q.
The space of
$\mathbb {A}$
-transverse cocycles is denoted by
$\mathcal {H}(\lambda ;\mathbb {A})$
. It has a natural structure of group, which is isomorphic to
$\mathbb {A}^{-3\chi (\Sigma )}$
whenever
$\mathbb {A}$
has no
$2$
-torsion (see Bonahon [Reference Bonahon5]).
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:
Lemma 4.2. Let
$\beta ^{\mathbb {R}}$
be the cross ratio on
$\mathbb {RP}^1$
. We have the following:
-
○ If
$P,P'$
are adjacent triangles and
$\ell =\ell '$
, then where
$$\begin{align*}\sigma_\lambda^X(P,P')= \log ( - \beta^{\mathbb{R}}(\ell^+,\ell^-,v,v')), \end{align*}$$
$v, v'$
are the ideal vertices of
$P, P'$
opposite to
$\ell =\ell '$
, respectively, and
$\ell $
is oriented so that P and
$P'$
lie on its left and on its right, respectively.
-
○ If
$P,P'$
lie on opposite sides of a leaf
$\ell \subset \lambda $
and each shares an ideal vertex with
$\ell $
(possibly distinct), then where
$$\begin{align*}\sigma_\lambda^X(P,P')= \log( \beta^{\mathbb{R}}(\ell^+,u,v,\ell^-) \, \beta^{\mathbb{R}}(\ell^-,\ell^+,u',u) \, \beta^{\mathbb{R}}(\ell^-,u',v',\ell^+) ) , \end{align*}$$
$u, u'$
are the ideal vertices of the sides
$e\subset P,e'\subset P'$
that are not endpoints of
$\ell $
and that separate the plaques
$P,P'$
;
$v, v'$
are the vertices of
$P, P'$
opposite to
$e, e'$
; and
$\ell $
is oriented so that P and
$P'$
lie on its left and on its right, respectively.
(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 Para-complex numbers
In order to define the shear-bend cocycle of pleated surfaces, it is convenient to exploit the natural para-complex cross ratio on the boundary of
$\mathbb {H}^{2,1}$
(see Section 2 of Danciger [Reference Danciger11]).
Definition 4.5 (Para-complex Numbers)
The ring of para-complex numbers is
$\mathbb {B}:=\mathbb {R}[\tau ]/(\tau ^2-1)$
. Similarly to the case of complex numbers, every element
$z=x+\tau y$
has
-
○ a conjugate
${\bar z}:=x-\tau y$
; -
○ a pseudo-norm
$\vert z\vert ^2:=z\bar {z}=x^2-y^2\in \mathbb {R}$
.
However,
$\mathbb {B}$
has also non trivial zero-divisors: an element
$z\in \mathbb {B}$
is invertible if and only if
$\vert z\vert ^2\neq 0$
, in which case
$z^{-1}=\bar {z}/\vert z\vert ^2$
. We denote by
$\mathbb {B}^*$
the set of invertible elements of
$\mathbb {B}$
.
It is convenient to decompose
$\mathbb {B}$
as
$\mathbb {R}\times \mathbb {R}$
. Consider
$$\begin{align*}e_l:=\frac{1+\tau}{2},e_r:=\frac{1-\tau}{2}. \end{align*}$$
The elements
$e_l,e_r$
are idempotent
$e_j^2=e_j$
, orthogonal
$e_l \, e_r = 0$
and conjugate
${\bar e_l}=e_r$
. This implies that the map
$(\lambda ,\mu )\in \mathbb {R}\times \mathbb {R}\mapsto \lambda e_l+\mu e_r\in \mathbb {B}$
is a ring isomorphism. In these coordinates, the conjugate of an element is
$\overline {\lambda e_l+\mu e_r}=\mu e_l+\lambda e_r$
, and its norm is
$\vert \lambda e_l+\mu e_r\vert =\lambda \mu $
.
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 Shear-bend cocycle
We now recall the natural shear-bend cocycle and its geometric interpretation, as given in Sections 4 and 5 of [Reference Mazzoli and Viaggi16].
Let
$\rho = \rho _{X,Y}$
be a Mess representation with limit curve
$\Lambda _{X,Y}$
.
4.3.1 Elementary shear
Let us start with an elementary shear-bend.
Lemma 4.8. Let
$\Delta =(u,\ell ^-,\ell ^+),\Delta '=(u',\ell ^+,\ell ^-)\subset \mathbb {H}^{2,1}$
be ideal triangles sharing a common edge
$\ell =[\ell ^-,\ell ^+]$
and with vertices on
$\Lambda _{X,Y}$
ordered as
$u<\ell ^-<u'<\ell ^+$
. Then,
$- \beta ^{\mathbb {B}}(\ell ^+,\ell ^-,u,u')\in \mathbb {B}^+$
.
Proof. Recall that
$\Lambda _{X,Y}$
is the graph of the unique
$(\rho _X-\rho _Y)$
-equivariant homeomorphism
$h_{X,Y}:\mathbb {RP}^1\to \mathbb {RP}^1$
. For a point
$p\in \mathbb {RP}^1\times \mathbb {RP}^1$
, denote by
$p_l,p_r$
its left and right components, respectively. Then, we have
$u_a<\ell ^-_a<u^{\prime }_a<\ell ^+_a$
on
$\mathbb {RP}^1$
for
$a=l,r$
. The conclusion follows from Lemma 4.7.
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:
Definition 4.9 (Intrinsic Shear-Bend I)
Let
$\rho = \rho _{X,Y}$
be a Mess representation. Let
$\lambda $
be a maximal lamination with countably many leaves. The cocycle
$\sigma _\rho (\bullet ,\bullet )\in \mathcal {H}(\lambda ;\mathbb {B})$
is the intrinsic shear-bend cocycle of the pleated set
${\hat S}_\lambda $
.
Furthermore, we have the following:
Proposition 4.10 (Proposition 6.7 in [Reference Mazzoli and Viaggi16])
Let
$\rho = \rho _{X,Y}$
be a Mess representation. Let
$\lambda $
be a maximal lamination with countably many leaves. Then,
$\Re \sigma _\rho = (\sigma _\rho +{\bar \sigma }_\rho )/2\in \mathcal {H}(\lambda ;\mathbb {R})$
is the shear cocycle of the intrinsic hyperbolic structure
$Z_\lambda \in \mathcal {T}$
of the pleated set
${\hat S}_\lambda $
.
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].
Theorem 4.11 (Theorem B of [Reference Mazzoli and Viaggi16])
Let
$\rho = \rho _{X,Y}$
be a Mess representation. For every maximal geodesic lamination
$\lambda \in \mathcal {GL}$
, the finite approximation process converges and defines a
$\mathbb {B}$
-transverse cocycle
$\sigma _\rho \in \mathcal {H}(\lambda ;\mathbb {B})$
.
Definition 4.12 (Intrinsic Shear-Bend II)
Let
$\rho = \rho _{X,Y}$
be a Mess representation. Let
$\lambda $
be a maximal lamination. The cocycle
$\sigma _\rho \in \mathcal {H}(\lambda ;\mathbb {B})$
provided by Theorem 4.11 is the intrinsic shear-bend cocycle of the pleated set
${\hat S}_\lambda $
.
The following is a summary of results in Sections 4, 5 and 6 of [Reference Mazzoli and Viaggi16].
Proposition 4.13. We have the following properties:
-
1. If
$\lambda $
has countably many leaves, the definitions I and II coincide. -
2.
$(\sigma _\rho +{\bar \sigma }_\rho )/2$
is the shear cocycle of the intrinsic hyperbolic structure
$Z_\lambda \in \mathcal {T}$
. -
3. The map
$\lambda \in \mathcal {GL}_m\mapsto \sigma _\rho \in \mathcal {W}(\tau ;\mathbb {B})$
is continuous with respect to the Hausdorff topology on
$\mathcal {GL}_m$
. Here,
$\mathcal {W}(\tau ;\mathbb {R})$
is the weight space of a train track
$\tau $
carrying
$\lambda $
.
4.4 Gauss map
In order to prove Theorem 1, we study the Gauss map of the pleated set
${\hat S}_\lambda $
, which we now describe. To this purpose, let us begin with some general observations.
The group
$\mathrm {PSL}_2(\mathbb {R})\times \mathrm {PSL}_2(\mathbb {R})$
acts transitively on oriented timelike geodesics. The stabilizer of
$\gamma (t)=\cos (t)I+\sin (t)J\in \mathrm {PSO}(2)$
, where
$J={\tiny \left (\begin {array}{c c} 0 &-1\\ 1 &0\\ \end {array}\right )}$
, is equal to
$\mathrm {PSO}(2)\times \mathrm {PSO}(2)$
.
Therefore, the space of oriented timelike geodesics is
$\mathrm {PSL}_2(\mathbb {R})\times \mathrm {PSL}_2(\mathbb {R})$
-equivariantly identified with
$$\begin{align*}(\mathrm{PSL}_2(\mathbb{R})/\mathrm{PSO}(2))\times(\mathrm{PSL}_2(\mathbb{R})/\mathrm{PSO}(2)) \cong \mathbb{H}^2 \times \mathbb{H}^2 \end{align*}$$
(that is, the symmetric space of
$\mathrm {PSL}_2(\mathbb {R})\times \mathrm {PSL}_2(\mathbb {R})$
). We identify
$\mathbb {RP}^1$
with
$\mathbb {P}\{A\in M_2(\mathbb {R})|\;\mathrm {rk}(A)=1\}/\mathrm {PSO}(2)$
and
$\mathbb {H}^2$
with
$\mathrm {PSL}_2(\mathbb {R})/\mathrm {PSO}(2)$
.
Lemma 4.14. Let
$H\subset \mathbb {H}^{2,1}$
be a (oriented) spacelike plane. Consider the map
$g=(g_l,g_r):H\to \mathbb {H}^2\times \mathbb {H}^2$
, where
$g(x)$
is the future-pointing timelike geodesic orthogonal to H at x. Then,
$g_j$
is isometric and extends continuously to the map
$g_j:\partial H\subset \mathbb {RP}^1\times \mathbb {RP}^1\to \mathbb {RP}^1$
, sending
$g_j(a_l,a_r)=a_j$
for
$j=l,r$
.
Proof. By equivariance, it is enough to check the claim for a specific hyperplane
$H\subset \mathbb {H}^{2,1}=\mathrm {PSL}_2(\mathbb {R})$
. We choose H to be the dual plane of I – that is,
$H=\mathbb {P}\{M\in \mathrm { SL}_2(\mathbb {R})|\;\mathrm {tr}(M)=0\}$
. As above, let
$\gamma =\mathrm {PSO}(2)$
.
Notice that
$J=H\cap \gamma $
and that H,
$\gamma $
intersect orthogonally at J. It follows that
$g(J)=\gamma =([I],[I])$
. As the diagonal group of
$\mathrm {PSL}_2(\mathbb {R})\times \mathrm {PSL}_2(\mathbb {R})$
preserves H and acts transitively on it, by equivariance we have
$g_j(AJ)=[A]$
. In other words, the components
$g_j$
are the restrictions of the standard projections
$\pi :\mathrm {PSL}_2(\mathbb {R})\to \mathrm {PSL}_2(\mathbb {R})/\mathrm { PSO}(2)$
to H. Also observe that, as
$\gamma $
is orthogonal to H at J, the differential
$\mathrm {d}\pi _J$
is isometric. Thus, by equivariance,
$\mathrm { d}\pi $
is isometric everywhere.
The boundary of H is
$\partial H=\mathbb {P}(\{M\in M_2(\mathbb {R})|\;\mathrm {tr}(M)=0,\mathrm {rk}(M)=1\})$
. Notice that, by Hamilton-Cayley, every
$M\in M_2(\mathbb {R})$
satisfies
$M^2-\mathrm {tr}(M)M+\mathrm { det}(M)=0$
. Therefore, if
$M\in \partial H$
, then
$M^2=0\Longleftrightarrow \mathrm {Im}(M)=\mathrm {Ker}(M)$
. The map
$g_j(AJ)=[A]$
extends continuously to a map
$\partial H\to \mathbb {RP}^1$
sending
$g_j(\mathrm {Im}(M),\mathrm {Ker}(M))=[\mathrm {Im}(M)]=[\mathrm {Ker}(M)]$
.
Let
$\rho _{X,Y}$
be a Mess representation with limit curve
$\Lambda _{X,Y}\subset \mathbb {RP}^1\times \mathbb {RP}^1$
.
Lemma 4.15. Consider two ideal spacelike adjacent triangles
$\Delta =\Delta (a,b,c)$
and
$\Delta '=(c,b,a')$
sharing a common edge
$[b,c]$
and with vertices ordered as
$a<b<a'<c$
along
$\Lambda _{X,Y}$
. Let
$g=(g_l,g_r):\mathrm { int}(\Delta )\cup \mathrm {int}(\Delta ')\to \mathbb {H}^2\times \mathbb {H}^2$
be the map sending x to the future pointing timelike normal
$g(x)\in \mathbb {H}^2\times \mathbb {H}^2$
. Then,
$$ \begin{align*} \sigma(\Delta,\Delta')=\frac{\sigma_{\mathbb{H}^2}(g_l(\Delta),g_l(\Delta'))+\sigma_{\mathbb{H}^2}(g_r(\Delta),g_r(\Delta'))}{2}, \\ \beta(\Delta,\Delta')=\frac{\sigma_{\mathbb{H}^2}(g_l(\Delta),g_l(\Delta'))-\sigma_{\mathbb{H}^2}(g_r(\Delta),g_r(\Delta'))}{2}, \end{align*} $$
where
$\sigma _{\mathbb {H}^2}(\Delta _1,\Delta _2)$
denotes the hyperbolic elementary shear of the adjacent ideal triangles
$\Delta _1,\Delta _2\subset \mathbb {H}^2$
.
Proof. Identify
$\mathbb {BP}^1$
with
$\mathbb {RP}^1\times \mathbb {RP}^1$
. By Lemma 4.14, the left and right projections of
$\Delta ,\Delta '$
are the ideal triangles
$g_j(\Delta )=\Delta (a_j,b_j,c_j),g_j(\Delta ')=\Delta (c_j,b_j,a^{\prime }_j)$
, where
$j=l,r$
, respectively. Notice that we have
$a_j<b_j<a_j'<c_j$
on
$\mathbb {RP}^1$
because the set
$\Delta \cup \Delta '$
is acausal. In particular,
$$ \begin{align*}\sigma_{\mathbb{H}^2}(g_j(\Delta),g_j(\Delta')) = \log(-\beta^{\mathbb{R}}(b_j,c_j,a_j,a_j'))\end{align*} $$
by Lemma 4.2. Recall that
$\sigma (\Delta ,\Delta '),\beta (\Delta ,\Delta ')$
are the real and para-complex parts of
$\sigma ^{\mathbb {B}}(\Delta ,\Delta ')=\sigma ^{\mathbb {B}}(a,b,c,d)$
and that, by definition,
$\sigma ^{\mathbb {B}}(b,c,a,a')=\log (-\beta ^{\mathbb {B}}(b,c,a,a'))$
. The conclusion follows from Lemma 4.7.
We are ready to prove Theorem 1.
4.5 The proof of Theorem 1
Let
$\rho _{X,Y}$
be a Mess representation.
Consider the pleated set
${\hat S}_\lambda $
associated with the maximal lamination
$\lambda $
. Every point
$x\in {\hat S}_\lambda -{\hat \lambda }$
lies in a plaque and, therefore, has a well-defined future-pointing, timelike, unit normal direction
$g(x)$
. The map
$g=(g_l,g_r):{\hat S}_\lambda -{\hat \lambda }\to \mathbb {H}^2\times \mathbb {H}^2$
is the Gauss map of the pleated set
${\hat S}_\lambda $
. By Lemma 4.14, it is
$\rho _{X,Y}$
-equivariant and isometric on each plaque.
Proof of Theorem 1
We split the proof into two cases.
Maximal laminations with countably many leaves. Let
$P,Q$
be distinct plaques. By definition and by Lemma 4.2, it is enough to consider the case where
$P,Q$
are either adjacent or asymptotic to the same leaf. The claim then follows from the computations of Lemmas 4.14 and 4.15.
General maximal laminations. The general case follows from the density of finite leaved maximal laminations in
$\mathcal {GL}_m$
and from the continuity properties of cocycles, as given in Theorem 4.13.
4.6 Shear-bend parametrization
The proof of Theorem 5 is a combination of Theorem 1 and some properties of the classical shear coordinates
$\Phi :\mathcal {T}\to \mathcal {H}(\lambda ;\mathbb {R})$
.
Proof of Theorem 5
We have
$$\begin{align*}\mathcal{H}(\lambda;\mathbb{B})=\frac{1+\tau}{2}\mathcal{H}(\lambda;\mathbb{R})\oplus\frac{1-\tau}{2}\mathcal{H}(\lambda;\mathbb{R}) \end{align*}$$
as
$\mathbb {B}$
-modules.
Part (1). Recall that
$\sigma ^{\mathbb {B}}_\lambda =\sigma +\tau \beta $
and that, by Theorem 1, we have
$\sigma =(\sigma _\lambda ^X+\sigma _\lambda ^Y)/2$
and
$\beta =(\sigma _\lambda ^X-\sigma _\lambda ^Y)/2$
. Therefore, in terms of the above splitting, the shear-bend map decomposes as
$$\begin{align*}\Psi:\rho_{X,Y}\mapsto\sigma^{\mathbb{B}}_\lambda=\frac{1+\tau}{2}\sigma_\lambda^X\oplus\frac{1-\tau}{2}\sigma_\lambda^Y. \end{align*}$$
The single components
$\Phi (X),\Phi (Y)=\sigma _\lambda ^X,\sigma _\lambda ^Y$
are analytic by Theorem 4.3. Injectivity also follows from the injectivity in the same theorem, since every pair of Mess representations
$\rho = \rho _{X,Y}, \rho ' = \rho _{X',X'}$
satisfies
$$ \begin{align*} \sigma_\rho^{\mathbb{B}}=\sigma_{\rho'}^{\mathbb{B}}\Longleftrightarrow\sigma_\lambda^X=\sigma_\lambda^{X'}\text{ and }\sigma_\lambda^Y=\sigma_\lambda^{Y'}. \end{align*} $$
It remains to be checked that the map respects the para-complex structures of
$\mathcal {T}\times \mathcal {T}$
and
$\mathcal {H}(\lambda ;\mathbb {B})$
. The para-complex structure
$\mathbb {J}$
acts on
$T_X\mathcal {T}\oplus T_Y\mathcal {T}$
simply as
$\mathbb {J}(u,v)=(u,-v)$
and acts on
$\mathcal {H}(\lambda ;\mathbb {B})$
as the multiplication by
$\tau $
. Denoting by
$\Phi :\mathcal {T}\to \mathcal {H}(\lambda ;\mathbb {R})$
the classical shear coordinates, we observe
$$ \begin{align*} \mathrm{d}\Psi(\mathbb{J}(u,v)) &=\mathrm{d}\Psi(u,-v)=\frac{1+\tau}{2}\mathrm{d}\Phi(u)\oplus\frac{1-\tau}{2}(-\mathrm{d}\Phi(v))\\ &=\tau\left(\frac{1+\tau}{2}\mathrm{d}\Phi(u)\oplus\frac{1-\tau}{2}\mathrm{d}\Phi(v)\right)=\tau \, \mathrm{d}\Psi(u,v). \end{align*} $$
Part (2). The Thurston’s symplectic form on
$\mathcal {H}(\lambda ;\mathbb {B})$
splits as
$$\begin{align*}\omega_{\mathrm{Th}}^{\mathbb{B}}=\frac{1+\tau}{2} \, \omega_{\mathrm{Th}}^{\mathbb{R}}\oplus\frac{1-\tau}{2} \, \omega_{\mathrm{Th}}^{\mathbb{R}}, \end{align*}$$
with respect to the above decomposition. Thus, by Theorem 4.4, we have
$$ \begin{align*} \omega_{\mathrm{Th}}^{\mathbb{B}}(\sigma_\rho^{\mathbb{B}},\mu) &=\frac{1+\tau}{2}\omega_{\mathrm{Th}}^{\mathbb{R}}(\sigma_\lambda^X,\mu)+\frac{1-\tau}{2}\omega_{\mathrm{Th}}^{\mathbb{R}}(\sigma_\lambda^Y,\mu)\\ &=\frac{1+\tau}{2}L_X(\mu)+\frac{1-\tau}{2}L_Y(\mu)\\ &=\frac{L_X(\mu)+L_Y(\mu)}{2}+\tau\frac{L_X(\mu)-L_Y(\mu)}{2} \end{align*} $$
for every
$\mu \in \mathcal {ML}_\lambda $
. We will see in the next section that
$L_\rho =(L_X+L_Y)/2$
and
$\theta _\rho =(L_X-L_Y)/2$
for any
$\rho = \rho _{X,Y}$
.
Part (3). By part (1), the image of
$\Psi $
is
$$\begin{align*}\left\{\sigma+\tau\beta\in\mathcal{H}(\lambda;\mathbb{B})\bigg|\;\sigma+\beta,\sigma-\beta\in\mathcal{T}\subset\mathcal{H}(\lambda;\mathbb{R})\right\}. \end{align*}$$
By Theorem 4.3, we have
$$\begin{align*}\mathcal{T}=\{\sigma\in\mathcal{H}(\lambda,\mathbb{R})|\;\omega_{\mathrm{Th}}(\sigma,\bullet)>0\ \mathrm{on}\ \mathcal{ML}_\lambda\}. \end{align*}$$
Thus,
$$ \begin{align*} &\sigma+\tau\beta\in\Psi(\mathcal{T}\times\mathcal{T})\Leftrightarrow\omega_{\mathrm{Th}}(\sigma\pm\beta,\mu)>0\\ &\Leftrightarrow\omega_{\mathrm{Th}}(\sigma,\mu)^2-\omega_{\mathrm{Th}}(\beta,\mu)^2=\left|\omega_{\mathrm{Th}}^{\mathbb{B}}(\sigma+\tau\beta,\mu)\right|{}^2_{\mathbb{B}}>0 \end{align*} $$
for every
$\mu \in \mathcal {ML}_\lambda $
.
Part (4). By work of Bonahon and Sözen [Reference Bonahon and Sözen7], we have that
$\Phi ^*\omega _{\mathrm {Th}}=c\cdot \omega _{\mathrm {WP}}$
. The conclusion then follows from the fact that
$\Psi $
splits as
$\frac {1+\tau }{2}\Phi \oplus \frac {1-\tau }{2}\Phi $
and
$\omega _{\mathrm {Th}}^{\mathbb {B}}$
splits as
$\frac {1+\tau }{2}\omega _{\mathrm {Th}}^{\mathbb {R}}\oplus \frac {1-\tau }{2}\omega _{\mathrm {Th}}^{\mathbb {R}}$
.
5 Length functions in anti-de Sitter
$3$
-manifolds
In this section, we study the anti-de Sitter length functions associated with Mess representations and prove Theorem 2.
5.1 Moving endpoints orthogonally
Let us start with some estimates in
$\mathbb {H}^{2,1}$
on how the length of a spacelike segment changes if we move its endpoints orthogonally in timelike directions. The following is an elementary computation:
Lemma 5.1. Let
$[x,y]$
be a spacelike segment. Let
$v\in T_x\mathbb {H}^{2,1},w\in T_y\mathbb {H}^{2,1}$
be unit timelike vectors orthogonal to
$[x,y]$
. Consider
$p=\cos (t)x+\sin (t)v$
and
$q=\cos (t)y+\sin (t)w$
. Then,
-
1.
$[v,w]$
lies on the dual geodesic of
$[x,y]$
. In particular, it is spacelike. -
2. We have
$$\begin{align*}-\langle p,q\rangle=\cos(t)^2\cosh(d_{\mathbb{H}^{2,1}}(x,y))+\sin(t)^2\cosh(d_{\mathbb{H}^{2,1}}(v,w)). \end{align*}$$
As
$-\langle p,q\rangle>1$
,
$[p,q]$
is spacelike and
$\cosh (d_{\mathbb {H}^{2,1}}(p,q))=-\langle p,q\rangle $
.
In order to manipulate better some inequalities, later we will use several times the following estimates on hyperbolic trigonometric functions.
Lemma 5.2. We have the following:
-
1. For every
$a_0> 0$
and
$b_0 \in (0,\frac {\pi }{2})$
, there exists
$\kappa>0$
such that for all
$$\begin{align*}\cos(b)^2\cosh(a)+\sin(b)^2\cosh(a-a_0)\le\cosh(a-\kappa) \end{align*}$$
$a \in [a_0,+\infty )$
and
$b \in [b_0,\pi /2]$
.
-
2. For every
$a_0> 0$
, there exists
$c_0 \in (0,1)$
such that for all
$$\begin{align*}c \, \cosh(a)\ge\cosh(a-\eta(c)) \end{align*}$$
$a \ge a_0$
and
$c\in [c_0,1]$
, where
$\eta (c) := \mathrm {arccosh}(1/c)$
.
Proof. A straightforward computation shows that, for every
$u>0$
, the function
$x \mapsto \cosh (x-u)/\cosh (x)$
is strictly decreasing over
$\mathbb {R}$
. In particular, for every
$x \geq x_0> 0$
, we have
$$\begin{align*}e^{- u} < \frac{\cosh(x - u)}{\cosh(x)} \leq \frac{\cosh(x_0 - u)}{\cosh(x_0)}. \end{align*}$$
Inequality (1). We first rewrite the desired statement as
$$\begin{align*}\cos(b)^2+\sin(b)^2\frac{\cosh(a-a_0)}{\cosh(a)}\le\frac{\cosh(a-\kappa)}{\cosh(a)}. \end{align*}$$
As
$x \mapsto \cosh (x-a_0)/\cosh (x)$
is decreasing and
$b \geq b_0 \in (0,\frac {\pi }{2}]$
, we have
$$ \begin{align*} \cos(b)^2+\sin(b)^2\frac{\cosh(a-a_0)}{\cosh(a)} & \le \cos(b)^2+\sin(b)^2\frac{1}{\cosh(a_0)} \\ & = 1 - \sin(b)^2 \left(1-\frac{1}{\cosh(a_0)}\right) \\ & \leq 1 - \sin(b_0)^2 \left(1-\frac{1}{\cosh(a_0)}\right) \\ & = \cos(b_0)^2+\sin(b_0)^2\frac{1}{\cosh(a_0)} <1. \end{align*} $$
Since
$\cosh (a - \kappa )/\cosh (a)> e^{-\kappa }$
, it is enough to choose
$\kappa>0$
so that
$\cos (b_0)^2+\sin (b_0)^2/\cosh (a_0)<e^{-\kappa }$
.
Inequality (2). We write
$c = 1/\cosh (\delta )$
for some
$\delta \geq 0$
. For every
$\delta \in [0,a_0]$
and for every
$a \geq a_0$
, we have
$$ \begin{align*} \frac{\cosh(a - \delta)}{c} & = \cosh(\delta) \cosh(a - \delta) \\ & \leq \cosh(\delta) \cosh(a - \delta) + \sinh(\delta) \sinh(a - \delta) \\ & = \cosh(a). \end{align*} $$
Hence, the assertion follows if we set
$c_0 : = 1/\cosh (a_0)$
.
5.2 Length and pleated surfaces
We now introduce loxodromic transformations of
$\mathbb {H}^{2,1}$
and the length functions associated to Mess representations.
Definition 5.3 (Loxodromic)
An isometry
$\gamma =(A,B)\in \mathrm {PSL}_2(\mathbb {R})\times \mathrm {PSL}_2(\mathbb {R})$
is loxodromic if
$A,B$
are both loxodromic transformations of
$\mathrm {PSL}_2(\mathbb {R})$
. A loxodromic transformation
$\gamma $
preserves two disjoint (dual) lines
$$\begin{align*}\ell=[(x_A^+,x_B^+),(x_A^-,x_B^-)],\ell^*=[(x_A^+,x_B^-),(x_A^-,x_B^+)]\subset\mathbb{H}^{2,1}, \end{align*}$$
where
$x_A^\pm ,x_B^\pm $
are the attracting and repelling fixed points of
$A,B$
on
$\mathbb {RP}^1$
, and it acts on them by translations by
$$\begin{align*}L(\gamma)=\frac{L(A)+L(B)}{2}\;\text{and }\;\theta(\gamma)=\frac{\left|L(A)-L(B)\right|}{2} , \end{align*}$$
respectively, where
$L(A),L(B)$
are the translation lengths of
$A,B$
. The quantities
$L(\gamma )$
and
$\theta (\gamma )$
are the translation length and torsion of
$\gamma $
.
Notice that if
$\rho _{X,Y}$
is a Mess representation, then for every
$\gamma \in \Gamma -\{1\}$
, the transformation
$\rho _{X,Y}(\gamma )=(\rho _X(\gamma ),\rho _Y(\gamma ))$
is loxodromic because
$\rho _X,\rho _Y$
are holonomies of hyperbolic structures. Furthermore, as
$\Lambda _{X,Y}\subset \mathbb {RP}^1\times \mathbb {RP}^1$
is the graph of the unique
$(\rho _X-\rho _Y)$
-equivariant homeomorphism
$h_{X,Y}:\mathbb {RP}^1\to \mathbb {RP}^1$
, we see that the axis
$\ell _\gamma $
of
$\rho _{X,Y}(\gamma )$
, having the endpoints on
$\Lambda _{X,Y}$
, is contained in
$\mathcal {CH}_{X,Y}$
.
We are now ready to prove the first part of Theorem 2.
Proposition 5.4. Let
$\rho = \rho _{X,Y}$
a Mess representation. Consider
$\gamma \in \Gamma -\{1\}$
a nontrivial element and denote by
$\ell \subset \mathcal {CH}_{X,Y}$
the axis of
$\rho (\gamma )$
. Let
$\lambda \subset \Sigma $
be a maximal lamination and let
$Z_\lambda \in \mathcal {T}$
be the intrinsic hyperbolic structure on
${\hat S}_\lambda /\rho (\Gamma )$
, where
${\hat S}_\lambda \subset \mathcal {CH}_{X,Y}$
is the pleated set associated with
$\lambda $
. Let
$\delta $
denote the maximal timelike distance of
$\ell $
from
${\hat S}_\lambda $
. Then,
$$\begin{align*}\cosh(L_{Z_\lambda}(\gamma))\le\cos(\delta)^2\cosh(L_\rho(\gamma))+\sin(\delta)^2\cosh(\theta_\rho(\gamma)). \end{align*}$$
Proof. Let
$x\in \ell ,y\in {\hat S}_\lambda $
be points that realize the maximal timelike distance
$\delta $
. Notice that the timelike segment
$[x,y]$
is orthogonal to
$\ell $
at x. Denote by
$v\in T_x\mathbb {H}^{2,1}$
the unit timelike vector tangent to
$[x,y]$
that points towards y. We can write
$y=\cos (\delta )x+\sin (\delta )v$
.
We now apply Lemma 5.1 to the spacelike segment
$[x,\rho _{X,Y}(\gamma )x]\subset \ell $
and the timelike unit tangent vectors
$v,\rho _{X,Y}(\gamma )v$
. We have
$$ \begin{align*} &\cosh\left(d_{\mathbb{H}^{2,1}}(y,\rho_{X,Y}(\gamma)y)\right)\\ &=\cos(\delta)^2\cosh\left(d_{\mathbb{H}^{2,1}}(x,\rho_{X,Y}(\gamma)x)\right)+\sin(\delta)^2\cosh\left(d_{\mathbb{H}^{2,1}}(v,\rho_{X,Y}(\gamma)v)\right). \end{align*} $$
Notice that
$d_{\mathbb {H}^{2,1}}(x,\rho _{X,Y}(\gamma )x)=L_\rho (\gamma )$
and
$d_{\mathbb {H}^{2,1}}(v,\rho _{X,Y}(\gamma )v)=\theta _\rho (\gamma )$
.
The conclusion then follows from Proposition 3.14, which tells us that the intrinsic hyperbolic distance between
$y,\rho _{X,Y}(\gamma )y$
on
${\hat S}_\lambda $
is smaller than
$d_{\mathbb {H}^{2,1}}(y,\rho _{X,Y}(\gamma )y)$
, and from the fact that
$L_Z(\gamma )$
coincides with the minimal displacement of
$\rho _{X,Y}(\gamma )$
with respect to the hyperbolic metric on
${\hat S}_\lambda $
.
5.3 Intersection and pleated surfaces
We then prove the second part of Theorem 2.
Proposition 5.5. Let
$\rho = \rho _{X,Y}$
be a Mess representation. Let
$\gamma \in \Gamma -\{1\}$
be a nontrivial element and denote by
$\ell \subset \mathcal {CH}_{X,Y}$
the axis of
$\rho (\gamma )$
. Let
$\delta ^\pm $
be the maximal timelike distance of
$\ell $
from
$\lambda ^\pm $
. Then,
$$\begin{align*}\cosh(i(\lambda^\pm,\gamma))\le\sin(\delta^\pm)^2\cosh(L_\rho(\gamma))+\cos(\delta^\pm)^2\cosh(\theta_\rho(\gamma)). \end{align*}$$
Proof. Let
$[x,x^\pm ]$
be a timelike segment that realizes the maximal timelike distance
$\delta ^\pm $
, with
$x\in \ell ,x^\pm \in \ell ^\pm \subset \lambda ^\pm $
. Notice that
$[x,x^\pm ]$
is orthogonal to both
$\ell ,\ell ^\pm $
. Let
$v\in T_x\mathbb {H}^{2,1},v^\pm \in \mathcal {T}_{x^\pm }\mathbb {H}^{2,1}$
be the unit speed timelike vectors tangent to the geodesic
$[x,x^\pm ]$
at the endpoints.
Claim 1. We have
$$\begin{align*}\cosh(d_{\mathbb{H}^{2,1}}(v^\pm,\rho_{X,Y}(\gamma)v^\pm))=\sin(\delta^\pm)^2\cosh(L_\rho(\gamma))+\cos(\delta^\pm)^2 \cosh(\theta_\rho(\gamma)). \end{align*}$$
Proof of the claim
Note that
$$\begin{align*}v^\pm=-\cos(\pi/2-\delta^\pm)x+\sin(\pi/2-\delta^\pm)v \end{align*}$$
and that v and
$\rho _{X,Y}(\gamma )v$
are both orthogonal to the segment
$[x,\rho _{X,Y}(\gamma )x]\subset \ell $
. The claim follows from Lemma 5.1.
Claim 2. Let
$v,v',v"\in \mathbb {H}^{2,1}$
be dual to the pairwise distinct supporting planes
$H,H',H"$
of
$\partial ^+\mathcal {CH}_{X,Y}$
.
-
1. If
$v,v',v"$
all lie on a common minimizing path inside
$\mathcal {S}^+$
, then
$H\cap H'\cap H"$
is either empty or equal to a line. The latter happens if and only if
$v, v',v"$
lie on a spacelike geodesic segment of
$\mathbb {H}^{2,1}$
. -
2. If
$v,v',v"$
all lie on a common minimizing path inside
$\mathcal {S}^+$
and
$v < v' < v"$
, then the reverse triangle inequality holds:
$$\begin{align*}d_{\mathbb{H}^{2,1}}(v,v")\ge d_{\mathbb{H}^{2,1}}(v,v')+d_{\mathbb{H}^{2,1}}(v',v"). \end{align*}$$
Proof of the claim
Consider the faces
$F,F',F"=H,H',H"\cap \partial ^+\mathcal {CH}_{X,Y}$
. As
$\mathcal {S}^+$
is an
$\mathbb {R}$
-tree, there are two possibilities: either one of the faces separates the other two on
$\partial ^+\mathcal {CH}_{X,Y}$
, or there is a unique face
$G\subset \partial ^+\mathcal {CH}_{X,Y}$
different from
$H, H', H"$
that separates every pair of them. The first case corresponds to the configuration where the dual points
$v,v',v"$
lie on a minimizing path inside
$\mathcal {S}^+$
. The second case corresponds to the configuration where
$v,v',v"$
are the vertices of a tripod in
$\mathcal {S}^+$
with center w, the dual point of G. Let us consider the first case. In addition, let us assume that
$v < v' < v"$
without loss of generality. Then, either the lines
$H\cap H'$
,
$H'\cap H"$
, and
$H \cap H"$
coincide, or
$F'$
separates
$H\cap H'$
from
$H'\cap H"$
in
$H'$
. Hence, the triple intersection
$H\cap H'\cap H"$
is either empty or equal to a line.
The second part of the claim follows from Lemma 6.3.5 of [Reference Benedetti and Bonsante1].
Claim 3. Let
$v,w\in \mathcal {S}^+$
be distinct points. Then,
$$\begin{align*}d_{\mathcal{S}^+}[v,w]\le d_{\mathbb{H}^{2,1}}(v,w). \end{align*}$$
Proof of the claim
Let
$\alpha :I=[0,1]\to \mathcal {S}^+$
be an injective weakly regular path joining v and w. By Lemma 3.12, we have
$$\begin{align*}L=\int_I{\vert{\dot \alpha}(t)\vert \text{dt}}=\lim_{\epsilon\to 0}\int_{[0,1-\varepsilon]}{\frac{d_{\mathbb{H}^{2,1}}(\alpha(t),\alpha(t+\epsilon))}{\epsilon} \text{dt}}. \end{align*}$$
If
$\epsilon <\epsilon _0$
, then
$$\begin{align*}\left|\int_I{\frac{d_{\mathbb{H}^{2,1}}(\alpha(t),\alpha(t+\epsilon))}{\epsilon} \text{dt}}-L\right|<\delta. \end{align*}$$
Choose
$\epsilon =1/2^k$
. For convenience, we take dyadic approximations of the integral with Riemann sums:
$$\begin{align*}\int_I{\frac{d_{\mathbb{H}^{2,1}}(\alpha(t),\alpha(t+1/2^k))}{1/2^k}\text{dt}}=\lim_{n\to\infty}{\sum_{p=0}^{2^n-2^{n-k}}{\frac{d_{\mathbb{H}^{2,1}}(\alpha(p/2^n),\alpha(p/2^n+1/2^k))}{1/2^k}}\cdot\frac{1}{2^n}}. \end{align*}$$
We reorganize the sum as
$$\begin{align} \frac{2^k}{2^n} \cdot \sum_{p=0}^{2^n - 2^{n-k}}& d_{\mathbb{H}^{2,1}}(\alpha(\tfrac{p}{2^n}),\alpha(\tfrac{p}{2^n}+\tfrac{1}{2^k})) = \nonumber\\ &=\frac{2^k}{2^n} \cdot \sum_{j=0}^{2^{n-k} - 1}\left(\sum_{q=0}^{2^k-2}{d_{\mathbb{H}^{2,1}}(\alpha(\tfrac{j}{2^n}+\tfrac{q}{2^k}),\alpha(\tfrac{j}{2^n}+\tfrac{q+1}{2^k}))}\right) \nonumber\\ &\qquad + \frac{2^k}{2^n} \, d_{\mathbb{H}^{2,1}}(\alpha(1 - \tfrac{1}{2^k}), \alpha(1)) \nonumber\\ &\leq\frac{2^k}{2^n} \cdot \sum_{j=0}^{2^{n-k} - 1} d_{\mathbb{H}^{2,1}}(\alpha(\tfrac{j}{2^n}), \alpha(\tfrac{j}{2^n} + 1 - \tfrac{1}{2^k})) \end{align}$$
$$\begin{align} &\quad\qquad\ \kern1pt\quad+ \frac{2^k}{2^n}\, d_{\mathbb{H}^{2,1}}(\alpha(1 - \tfrac{1}{2^k}), \alpha(1)) \nonumber\\&\qquad\ \,\le\frac{2^k}{2^n}\cdot\sum_{j=0}^{2^{n-k} - 1}{d_{\mathbb{H}^{2,1}}\left(\alpha(0),\alpha(1)\right)}=d_{\mathbb{H}^{2,1}}\left(v,w\right). \end{align}$$
The assertion follows by taking the limits first as
$n \to \infty $
, and then as
$k \to \infty $
.
We have
$$\begin{align} &\!\!\!\!\cosh(i(\lambda,\gamma)) \le\cosh(d_{\mathcal{S}^+}(v,\rho_{X,Y}(\gamma) v))\qquad\qquad\qquad\end{align}$$
$$\begin{align} &\!\!\!\!\le\cosh(d_{\mathbb{H}^{2,1}}(v,\rho_{X,Y}(\gamma) v)) \qquad\quad\qquad\end{align}$$
$$\begin{align} &\quad\qquad=\sin(\delta^\pm)^2\cosh(L_\rho(\gamma))+\cos(\delta^\pm)^2\cosh(\theta_\rho(\gamma)). \end{align}$$
6 Length functions in Teichmüller space
In this section, we carry out an anti-de Sitter analysis of length functions on Teichmüller space, on both global and infinitesimal scales, and prove Theorems 3 and 4.
6.1 Orthogonal projection to a line
We begin with some explicit computations on the orthogonal projection
$\pi :\mathbb {H}^{2,1}\to \ell $
to a spacelike geodesic.
Lemma 6.1. Let
$y,\ell $
be a point and a spacelike line in
$\mathbb {H}^{2,1}$
such that the rays
$[y,\ell ^\pm ]$
are spacelike. Then,
$$\begin{align*}m_{y,\ell}=\min_{x\in\ell}\{-\langle y,x\rangle\})=\sqrt{\frac{2\langle y,\ell^+\rangle\langle y,\ell^-\rangle}{-\langle \ell^+,\ell^-\rangle}} , \end{align*}$$
and it is realized at the unique point
$$\begin{align*}x=\frac{1}{\sqrt{-2\langle\ell^+,\ell^-\rangle}}\left(\sqrt{\frac{\langle y,\ell^-\rangle}{\langle y,\ell^+\rangle}}\ell^++\sqrt{\frac{\langle y,\ell^+\rangle}{\langle y,\ell^-\rangle}}\ell^-\right)\in\ell \end{align*}$$
such that
$[y,x]$
is orthogonal to
$\ell $
.
Proof. Write
$\ell (t)=(e^t\ell ^++e^{-t}\ell ^-)/\sqrt {-2\langle \ell ^+,\ell ^-\rangle }$
and consider the function
$f(t)=-\langle \ell (t),y\rangle $
. As
$[y,\ell ^+],[y,\ell ^-]$
are spacelike, we have
$f(t)\to \infty $
as
$|t|\to \infty $
. Hence,
$f(t)$
has a minimum which is a critical point. The unique critical point of the function is at
$e^{2t}=\langle y,\ell ^-\rangle /\langle y,\ell ^+\rangle $
. The conclusion follows by elementary computations.
6.2 Convexity of length functions
We now describe the purely anti-de Sitter proof of (strict) convexity of length functions on Teichmüller space
$\mathcal {T}$
in shear coordinates for an arbitrary maximal lamination
$\lambda \subset \Sigma $
.
We prove separately the two parts of Theorem 3.
Proposition 6.2. Let
$\lambda \subset \Sigma $
be a maximal lamination. Let
$\gamma \in \Gamma -\{1\}$
be a nontrivial loop. The length function
$\mathscr {L}_\gamma :\mathcal {T}\subset \mathcal {H}(\lambda ;\mathbb {R})\to (0,\infty )$
is convex. Moreover, convexity is strict if
$\gamma $
intersects essentially every leaf of
$\lambda $
.
Proof. Recall that a function
$L:U\subset \mathbb {R}^n\to \mathbb {R}$
defined on an open convex subset
$U\subset \mathbb {R}^n$
is (strictly) convex if and only if for every
$x,y\in U$
, we have a (strict) inequality
$$\begin{align*}L\left(\frac{x+y}{2}\right)\le\frac{L(x)+L(y)}{2}. \end{align*}$$
Consider
$X,Y\in \mathcal {T}$
and let
$\rho = \rho _{X,Y}$
be the corresponding Mess representation. Let
${\hat S}_\lambda \subset \mathcal {CH}_{X,Y}$
be the pleated set associated with
$\lambda $
and let
$\rho _\lambda :\Gamma \to \mathrm {PSL}_2(\mathbb {R})$
be the holonomy of the intrinsic hyperbolic structure
$Z_\lambda \in \mathcal {T}$
on
${\hat S}_\lambda /\rho _{X,Y}(\Gamma )$
. By Theorem 1, we have
$\sigma _\lambda (Z_\lambda )=(\sigma _\lambda (X)+\sigma _\lambda (Y))/2$
in
$\mathcal {H}(\lambda ;\mathbb {R})$
. By Theorem 2, we have
$$\begin{align*}\cosh(L_{Z_\lambda}(\gamma))\le\cos(\delta)^2\cosh(L_\rho(\gamma))+\sin(\delta)^2\cosh(\theta_\rho(\gamma)) , \end{align*}$$
where
$\delta $
is the maximal timelike distance of the axis
$\ell $
of
$\rho _{X,Y}(\gamma )$
from the pleated set
${\hat S}_\lambda $
. Notice that
$\delta>0$
unless
$\ell \subset {\hat S}_\lambda $
, in which case
$\ell $
does not intersect transversely the bending locus. Also observe that, unless
$\rho $
is Fuchsian, which happens precisely when
$X=Y$
, the bending locus cannot be empty. Therefore, if
$X,Y$
are distinct and the support of
$\gamma \in \mathcal {C}$
intersects essentially every leaf of
$\lambda $
, we have
$\delta>0$
and
$$\begin{align*}\cos(\delta)^2\cosh(L_\rho(\gamma))+\sin(\delta)^2\cosh(\theta_\rho(\gamma))<\cosh(L_\rho(\gamma)) \end{align*}$$
as
$L_\rho (\gamma )<\theta _\rho (\gamma )$
. Since
$\cosh (\bullet )$
is strictly increasing on
$(0,\infty )$
, we conclude
$L_Z(\gamma )<L_\rho (\gamma )$
.
Proposition 6.3. Let
$\lambda \subset \Sigma $
be a maximal lamination and let
$\mu \in \mathcal {ML}$
be a measured lamination. Then, the length function
$\mathscr {L}_\gamma :\mathcal {T}\subset \mathcal {H}(\lambda ;\mathbb {R})\to (0,\infty )$
is convex. Furthermore, convexity is strict if the support of
$\mu $
intersects transversely each leaf of
$\lambda $
.
Proof. We immediately deduce convexity by Proposition 6.2, the density of weighted simple curves in
$\mathcal {ML}$
and the
$\mathcal {C}^\infty $
-convergence of length functions
$a_n \,\mathscr {L}_{\gamma _n}\to \mathscr {L}_\mu $
if
$a_n \gamma _n\to \mu $
in
$\mathcal {ML}$
.
We now discuss strict convexity.
Consider
$X,Y\in \mathcal {T}$
and the Mess representation
$\rho :=\rho _{X,Y}$
. Let
${\hat S}_\lambda \subset \mathcal {CH}_{X,Y}$
be the pleated set associated with
$\lambda $
. Let
$\mu \in \mathcal {ML}$
be a measured lamination whose support contains a leaf
$\ell $
that intersects the bending locus of
${\hat S}_\lambda $
(which is nonempty, unless the representation is Fuchsian).
Because
$\ell $
intersects the bending locus, its geometric realization
${\hat \ell }$
is not contained on
${\hat S}_\lambda $
. Let
$x\in {\hat \ell }$
and
$y\in {\hat S_\lambda }$
be points that realize the maximal timelike distance
$\delta =\max \{\delta _{\mathbb {H}^{2,1}}(z,t)|\;z\in {\hat S}_\lambda ,t\in {\hat \ell }\}>0$
.
Let
$K:=I\times J$
denote the neighborhood of
$\ell $
in the space of geodesics
$\mathcal {G}$
consisting of those lines with one endpoint in I and another endpoint in J.
Recall that, by Lemma 6.1, we have
$$\begin{align*}m_{z,\ell}:=\min_{t\in\hat{\ell}}\{-\langle z,t\rangle\})=\sqrt{\frac{2\langle z,\hat{\ell}^+\rangle\langle z,\hat{\ell}^-\rangle}{-\langle \hat{\ell}^+,\hat{\ell}^-\rangle}} \end{align*}$$
and that the minimum is realized at a point
$\pi (z)\in \hat {\ell }$
, the orthogonal projection of z to
$\hat {\ell }$
, described explicitly by
$$\begin{align*}\pi(z)=\frac{1}{\sqrt{-2\langle\hat{\ell}^+,\hat{\ell}^-\rangle}}\left(\sqrt{\frac{\langle z,\hat{\ell}^-\rangle}{\langle z,\hat{\ell}^+\rangle}}\hat{\ell}^+ + \sqrt{\frac{\langle z, \hat{\ell}^+\rangle}{\langle z,\hat{\ell}^-\rangle}}\hat{\ell}^-\right). \end{align*}$$
As y is connected to
$\hat {\ell }$
by a timelike segment of length
$\delta $
orthogonal to
$\hat {\ell }$
, we have
$m_{y,\ell }=\cos (\delta )$
. As
$x\in \hat {\ell }$
, we have
$m_{x,\ell }=1$
. By continuity of the above expressions, we have the following.
Claim 1. For every
$\epsilon> 0$
, there exist a neighborhood
$K(\epsilon )=I(\epsilon ) \times J(\epsilon )$
of
$\ell \in \mathcal {G}$
and a neighborhood
$U(\epsilon )$
of x in
$\mathbb {H}^{2,1}$
with the following properties:
-
1.
$m_{y,\ell '}\in (\cos (2\delta ),\cos (\delta /2))$
for every
$\ell '\in K(\epsilon )$
. In particular, y is connected to every
$\hat {\ell }'$
by a timelike segment of length at least
$\delta /2$
and, hence,
$\delta _{\mathbb {H}^{2,1}}(y,\hat {\ell }')\ge \delta /2$
; -
2. For every
$\ell '\in K(\epsilon )$
,
$\hat {\ell }'$
intersects
$U(\epsilon )$
; -
3. If
$\ell _1,\ell _2\in K(\epsilon )$
are so that
$\hat {\ell }_1\cup \hat {\ell }_2$
is acausal and
$z_j\in \hat {\ell }_j\cap U(\epsilon )$
, then
$d_{\mathbb {H}^{2,1}}(z_1,z_2)<\epsilon $
; -
4.
$m_{z,\ell '}\in (\cos (\epsilon ),\cosh (\epsilon ))$
for every
$\ell '\in K(\epsilon )$
and
$z\in U(\epsilon )$
; -
5. For every
$z,w\in U(\epsilon )$
and
$\ell '\in K(\epsilon )$
, we have
$d_{\mathbb {H}^{2,1}}(\pi '(z),\pi '(w))<\epsilon $
where
$\pi '$
is the orthogonal projection onto
$\hat {\ell }'$
.
Let
$K(\epsilon )$
and
$U(\epsilon )$
be the neighborhoods provided by the claim. As
$\ell $
lies in the support of
$\mu $
, we have
$m(\epsilon ):=\mu (K(\epsilon ))> 0$
.
We approximate
$\mu $
in
$\mathcal {ML}$
with a sequence of weighted simple closed curves
$a_n\gamma _n$
. By convergence of
$a_n\gamma _n$
to
$\mu $
, we have
$a_n m_{n}(\epsilon ) := a_n\gamma _n(K(\epsilon ))\to m(\epsilon )$
. Notice that
$m_n = m_{n}(\epsilon )$
is the number of distinct leaves of the geometric realization
${\hat \gamma }_n$
contained in
$K(\epsilon )$
. Let
$\ell _n$
be one of those leaves.
Claim 2. There exists a constant
$M> 1$
that depends only on the representation
$\rho $
such that, for any
$\epsilon> 0$
,
$n \in \mathbb {N}$
, and
$\ell _n$
as above, we can find elements
$$\begin{align*}\alpha_{n,1},\cdots,\alpha_{n,m_{n}}\in\Gamma \end{align*}$$
and corresponding points
$$\begin{align*}z_{n,0} < z_{n,1}<\cdots<z_{n,m_{n} - 1} < z_{n,m_n} = \rho(\gamma_n) z_{n,0} \end{align*}$$
on
${\hat \ell }_n$
with the following properties:
-
1.
$\alpha _{n,m_{n}}\cdots \alpha _{n,1}=\gamma _n$
; -
2.
$z_{n,0}, \rho (\alpha _{n,j}\cdots \alpha _{n,1})^{-1} z_{n,j} \in U(\epsilon /M)$
and for every
$$ \begin{align*}d_{\mathbb{H}^{2,1}}(z_{n,j},\rho(\alpha_{n,j}\cdots\alpha_{n,1})z_{n,0})<\epsilon/M < \epsilon\end{align*} $$
$j \in \{1, \dots , m_n\}$
;
-
3. The axis of
$\alpha _{n,j}$
lies inside
$\alpha _{n,j-1}\cdots \alpha _{n,1}(K(\epsilon ))$
.
Proof of the claim
Let us start by applying Claim 1 to an arbitrary
$\epsilon '> 0$
, and denote by
$$\begin{align*}\ell_n=\ell_{n,0},\cdots,\ell_{n,m_n - 1} \end{align*}$$
the
$m_n$
translates of
$\ell _n$
contained in
$K(\epsilon ')$
. We will later determine sufficient conditions on
$\epsilon '$
(in terms of
$\epsilon $
) that guarantee the desired properties.
By Claim 1 part (ii), for every
$j \in \{0, \dots , m_n - 1\}$
, there exists some point
$w_{n,j} \in \hat {\ell }_{n,j} \cap U(\epsilon ')$
. Since the leaves
$\ell _{n,j}$
are in the same
$\Gamma $
-orbit, we can find elements
$\beta _{n,j} \in \Gamma $
such that
$z_{n,j}:=\rho (\beta _{n,j})\, w_{n,j}$
belongs to the spacelike geodesic segment
$[w_{n,0}, \rho (\gamma _n) w_{n,0}] \subset \hat {\ell }_n$
. Notice that
$\beta _{n,0} = \mathrm {id} \in \Gamma $
and
$z_{n,0} = w_{n,0}$
. We also set
$\beta _{n,m_n} := \gamma _n$
,
$z_{n,m_n} : = \rho (\gamma _n) w_{n,0}$
. Up to reindexing the leaves
$\ell _{n,j}$
, we can assume that the points
$z_{n,j}$
appear in linear order along
$[w_{n,0}, \rho (\gamma _n) w_{n,0}]$
; that is,
$$\begin{align*}w_{n,0} = z_{n,0} < z_{n,1} < \dots < z_{n,m_n-1} < z_{n,m_n} = \rho(\gamma_n) w_{n,0}. \end{align*}$$
For any
$j \in \{1, \dots , m_n\}$
, we then define
$\alpha _{j,n} : = \beta _{n,j} \beta _{n,j-1}^{-1}$
.
Property (i). The identity
$\alpha _{n,m_n} \cdots \alpha _{n,1} = \beta _{n,m_n} = \gamma _n$
follows directly from our construction.
Property (ii). Notice that
$$\begin{align*}w_{n,0} \in U(\epsilon') \cap \hat{\ell}_{n,0}, \; w_{n,j} = \rho(\beta_{n,j})^{-1} z_{n,j} = \rho(\alpha_{n,j} \cdots \alpha_{n,1})^{-1} z_{n,j} \in U(\epsilon') \cap \hat{\ell}_{n,j}. \end{align*}$$
Because the leaves
$\ell _{n,0}$
and
$\ell _{n,j}$
are lifts of a common simple closed curve in S, the union of their geometric realizations
$\hat {\ell }_{n,0} \cup \hat {\ell }_{n,j}$
is acausal. In particular, it follows from Claim 1 part (iii) that
$$\begin{align*}d_{\mathbb{H}^{2,1}}(\rho(\alpha_j \cdots \alpha_1) z_{n,0},z_{n,j}) = d_{\mathbb{H}^{2,1}}(w_{n,0}, \rho(\alpha_{n,j} \cdots \alpha_{n,1})^{-1} z_{n,j}) < \epsilon'. \end{align*}$$
Property (iii). We deduce the last property from the stability of quasi-geodesics inside
${\hat S}_n$
, the pleated set associated with the lamination
$\lambda _n$
consisting of the closed geodesic
$\gamma _n$
suitably completed to a maximal lamination of
$\Sigma $
by adding finitely many leaves spiraling around
$\gamma _n$
.
Let us explain how. For any
$j \in \{1,\dots , m_n\}$
, consider the concatenation of the translates
$$\begin{align*}g_{n,j} = \bigcup_{k\in\mathbb{Z}} {\rho(\alpha_{n,j})^k\left([z_{n,j-1},z_{n,j}]_{{\hat S}_n}\cup[z_{n,j},\rho(\alpha_{n,j}) z_{n,j -1}]_{{\hat S}_n}\right)} , \end{align*}$$
where
$[a,b]_{\hat {S}_n}$
denotes the length-minimizing path inside
$\hat {S}_n$
between
$a, b \in \hat {S}_n$
. Notice that
$$ \begin{align}d_{\hat{S}_n}(z_{n,j},\rho(\alpha_{n,j}) z_{n,j -1}) & = d_{\hat{S}_n}(\rho(\beta_{n,j})^{-1} z_{n,j}, \rho(\beta_{n,j-1}^{-1}) \, z_{n,j -1}) \nonumber\\& = d_{\hat{S}_n}(w_{n,j}, w_{n,j-1}) \nonumber\\& \leq d_{\mathbb{H}^{2,1}}(w_{n,j}, w_{n,j-1}) \\& < \epsilon' , \nonumber\end{align} $$
where in the last step we applied Claim 1 part (iii) to
$w_{n,j - 1} \in U(\epsilon ') \cap \hat {\ell }_{n,j - 1}$
,
$w_{n,j} \in U(\epsilon ') \cap \hat {\ell }_{n,j}$
. By basic hyperbolic geometry,
$g_{n,j}$
is a uniform quasi-geodesic on
${\hat S}_n$
with respect to the intrinsic hyperbolic metric, with quasi-geodesic constants that are
$O(\epsilon ')$
-close to
$1$
(see, for example, Section I.4.2 of [Reference Canary, Epstein and Green10]). Hence, the invariant axis of
$\rho (\alpha _{n,j})$
on
${\hat S}_n$
lies in the
$O(\epsilon ')$
-neighborhood of
$g_{n,j}$
with respect to the hyperbolic metric. In particular, such endpoints are close to the endpoints of
$(\alpha _{n,j -1} \cdots \alpha _{n,1})\ell _n$
on the Gromov boundary
$\partial _\infty {\hat S}_n$
.
Let
$\phi _n:\partial _\infty {\hat S}_n\to \partial \Gamma $
be the unique
$\Gamma $
-equivariant homeomorphism. By Lemma 3.17, the hyperbolic structures
${\hat S}_n/\rho (\Gamma )$
lie in a compact subspace of Teichmüller space
$\mathcal {T}$
. Thus, as the boundary maps
$\phi _n$
depend continuously on
$S_n$
, they are uniformly equicontinuous. In particular, it is not restrictive to assume that the function
$O(\epsilon ')$
is independent of the hyperbolic structure of
$S_n$
, the leaf
$\ell _n$
and the weighted simple closed curves
$a_n \gamma _n$
. It follows that, if
$\epsilon '$
is small enough (or, equivalently, if
$M> 1$
is sufficiently large and
$\epsilon ' : = \epsilon / M$
), the endpoints of the axis of
$\alpha _j$
are contained in
$K(\epsilon )$
for every
$j \leq m_n$
.
Let
$\alpha _j = \alpha _{n,j} \in \Gamma $
and
$z_j = z_{n,j} \in \hat {\ell }_n$
be the elements provided by Claim 3 and define
$x_j:=\rho (\alpha _j\cdots \alpha _1)x$
and
$y_j=\rho (\alpha _j\cdots \alpha _1)y$
, where
$x_0 : = x \in \hat {\ell }$
and
$y_0 : = y \in \hat {S}_\lambda $
maximize the timelike distance between
$\hat {\ell }$
and
$\hat {S}_\lambda $
.
For any
$j \in \{0,\dots , m_n - 1\}$
, let
$\delta _j:=\delta _{\mathbb {H}^{2,1}}(y_{j},\hat {\ell }_{\alpha _{j+1}})$
be the timelike distance of
$y_j \in {\hat S}_\lambda $
from
$\hat {\ell }_{\alpha _{j+1}}$
, the axis of
$\rho (\alpha _{j+1})$
. By Property (iii) of Claim 2, we have
$(\alpha _{j}\cdots \alpha _1)^{-1}\ell _{\alpha _{j+1}}\in K(\epsilon )$
. Hence, Claim 1 part (i) implies that
$$ \begin{align} \delta_j=\delta_{\mathbb{H}^{2,1}}(y_{j},\hat{\ell}_{\alpha_{j+1}})=\delta_{\mathbb{H}^{2,1}}(y,\rho(\alpha_{j}\cdots\alpha_1)^{-1}\hat{\ell}_{\alpha_{j+1}})>\delta/2. \end{align} $$
Claim 3. There exists
$\kappa> 0$
, depending only on
$X, Y \in \mathcal {T}$
and on
$\delta> 0$
, such that for every
${j \in \{0,\dots , m_n - 1\}}$
,
$$ \begin{align*} \cosh(d_{\mathbb{H}^{2,1}}(y_{j},y_{j+1}))=\cos(\delta_j)^2\cosh(L_\rho(\alpha_{j+1}))+\sin(\delta_{j})^2\cosh(\theta_\rho(\alpha_{j+1}))\\ \le\cosh(L_\rho(\alpha_{j+1})-\kappa). \end{align*} $$
Proof of the claim
Let
$\pi _{j+1}(p)\in \hat {\ell }_{\alpha _{j+1}}$
be the unique point such that the segment
$[p,\pi _{j+1}(p)]$
is orthogonal to
$\hat {\ell }_{\alpha _{j+1}}$
. Observe that
$\pi _{j+1}(y_{j+1}) = \rho (\alpha _{j+1})\pi _{j+1}(y_j)$
. Lemma 5.1 applied to the spacelike segment
$[\pi _{j+1}(y_j),\pi _{j+1}(y_{j+1})]$
of length
$L_\rho (\alpha _{j+1})$
and the orthogonal timelike segments
$$ \begin{align*}[y_j,\pi_{j+1}(y_j)], \quad \rho(\alpha_{j+1})[y_j,\pi_{j+1}(y_j)] = [y_{j+1},\rho(\alpha_{j+1})\pi_{j+1}(y_j)]\end{align*} $$
implies the first identity of our statement. To deduce the upper bound, we argue as follows. By relation (4),
$\delta _j> \delta /2$
for every
$j \in \{0, \dots , m_n - 1\}$
and
$$\begin{align*}L_\rho(\alpha_{j + 1}) - \theta_\rho(\alpha_{j + 1}) = L_Y(\alpha_j) \geq \min\{\mathrm{sys}(X), \mathrm{sys}(Y)\} =: r> 0 , \end{align*}$$
where
$\rho = \rho _{X,Y}$
and
$\mathrm {sys}(Z)> 0$
denotes the systole of the hyperbolic structure
$Z \in \mathcal {T}$
. In particular, it follows that
$$ \begin{align*} \cosh(d_{\mathbb{H}^{2,1}}(y_{j},y_{j+1})) =\cos(\delta_j)^2\cosh(L_\rho(\alpha_{j+1}))+\sin(\delta_{j})^2\cosh(\theta_\rho(\alpha_{j+1}))\\ \le \cos(\delta_j)^2\cosh(L_\rho(\alpha_{j+1}))+\sin(\delta_{j})^2\cosh(L_\rho(\alpha_{j+1}) - r). \end{align*} $$
However, we have
$$\begin{align*}L_\rho(\alpha_{j+1}) = \frac{1}{2} (L_X(\alpha_{j + 1}) + L_Y(\alpha_{j + 1})) \geq r \end{align*}$$
for every
$j \in \{0, \dots , m_n - 1\}$
. The conclusion now follows from Lemma 5.2 part (1) applied to
$a_0 = r$
,
$b_0 = \delta /2$
,
$a = L_\rho (\alpha _{j+1})$
and
$b = \delta _j$
. Notice in particular that the resulting
$\kappa $
depends only on r and
$\delta $
, as desired.
Claim 4. There exist constants
$C , \epsilon _0> 0$
that depend only on the representation
$\rho $
such that for every
$\epsilon \in (0,\epsilon _0)$
and for every
$j \in \{0, \dots , m_n - 1\}$
as in Claim 2, we have
$$\begin{align*}L_\rho(\alpha_{j + 1})- d_{\mathbb{H}^{2,1}}(z_{j},z_{j+1})\le C \epsilon. \end{align*}$$
Proof of the claim
As a first step, we show that
$$ \begin{align*}|d_{\mathbb{H}^{2,1}}(\rho(\alpha_{j+1}) z_j, z_j) - d_{\mathbb{H}^{2,1}}(z_{j+1}, z_j)|\leq d_{\mathbb{H}^{2,1}}(\rho(\alpha_{j+1}) z_j, z_{j+1}) .\end{align*} $$
To see this, notice that
$z_{j}$
,
$z_{j+1}$
and
$\rho (\alpha _{j+1}) z_{j}$
lie on a common spacelike plane, since
$z_{j}, z_{j+1} \in \hat {\ell }_n$
and
$\rho (\alpha _{j+1}) z_{j}$
belongs to
$\rho (\alpha _{j+1}) \hat {\ell }_n$
, which are entirely contained in
$\hat {S}_n$
. Therefore, the inequality is a reformulation of the standard triangle inequality. Since both points
$\rho (\alpha _{j+1}) z_{j}, z_{j+1}$
lie inside
$\rho (\alpha _{j+1} \cdots \alpha _1) U(\epsilon /M)$
, it follows from Claim 2 part (ii) and Claim 1 that
$$ \begin{align*}|d_{\mathbb{H}^{2,1}}(\rho(\alpha_{j+1}) z_{j}, z_{j}) - d_{\mathbb{H}^{2,1}}(z_{j+1}, z_{j})| < \epsilon/M. \end{align*} $$
We now prove that
$d_{\mathbb {H}^{2,1}}(\rho (\alpha _{j+1}) z_{j}, z_{j})> L_\rho (\alpha _{j+1}) - O(\epsilon )$
. Consider the orthogonal projection
$\pi _{j+1}$
onto
$\hat {\ell }_{\alpha _{j+1}}$
. By Claim 2 part (ii) and 1 part (iv), the quantity
$$\begin{align*}D_{j}=\min_{t\in\hat{\ell}_{\alpha_{j+1}}}\{-\langle z_{j},t\rangle\}=\sqrt{\frac{2\langle z_{j},\hat{\ell}_{\alpha_{j+1}}^+\rangle\langle z_{j},\hat{\ell}_{\alpha_{j+1}}^-\rangle}{-\langle\hat{\ell}_{\alpha_{j+1}}^+,\hat{\ell}_{\alpha_{j+1}}^-\rangle}}=-\langle z_{j},\pi_{j+1}(z_{j})\rangle \end{align*}$$
is contained in the interval
$(\cos (\epsilon /M),\cosh (\epsilon /M))$
.
If
$D_j>1$
, then the segment
$[z_{j},\pi _{j+1}(z_{j})]$
is spacelike. Write
$D_j=\cosh (d_j)$
and
$z_{j}=\cosh (d_j)\,\pi _{j+1}(z_{j})+\sinh (d_j)\,v_j$
with
$v_j$
orthogonal to
$\hat {\ell }_{\alpha _{j+1}}$
at
$\pi _{j+1}(z_{j})$
. We have
$$ \begin{align*} &\cosh(d_{\mathbb{H}^{2,1}}(z_{j},\rho(\alpha_{j+1})z_{j}))=-\langle z_{j},\rho(\alpha_{j+1})z_{j}\rangle\\ &=\cosh(d_j)^2\cosh(L_\rho(\alpha_{j+1}))-\sinh(d_j)^2\cosh(\theta_\rho(\alpha_{j+1})). \end{align*} $$
Hence, as
$L_\rho (\alpha _{j+1})>\theta _\rho (\alpha _{j+1})$
, we get
$$ \begin{align*}\cosh(d_{\mathbb{H}^{2,1}}(z_{j},\rho(\alpha_{j+1})z_{j}))>\cosh(L_\rho(\alpha_{j+1})).\end{align*} $$
If
$D_j<1$
, then the segment
$[z_{j},\pi _{j+1}(z_j)]$
is timelike. Write
$D_j=\cos (d_j)$
and
$z_{j}=\cos (d_j)\pi _{j+1}(z_{j})+\sin (d_j)v_j$
with
$v_j$
orthogonal to
$\hat {\ell }_{\alpha _{j+1}}$
at
$\pi _{j+1}(z_{j})$
. We have
$$ \begin{align*} &\cosh(d_{\mathbb{H}^{2,1}}(z_{j},\rho(\alpha_{j+1})z_{j}))=-\langle z_{j},\rho(\alpha_{j+1})z_{j}\rangle\\ &=\cos(d_j)^2\cosh(L_\rho(\alpha_{j+1}))+\sin(d_j)^2\cosh(\theta_\rho(\alpha_{j+1})). \end{align*} $$
Thus,
$$\begin{align*}\cosh(d_{\mathbb{H}^{2,1}}(z_{j},\rho(\alpha_{j+1})z_{j}))>\cos(d_j)^2 \cosh(L_\rho(\alpha_{j+1})). \end{align*}$$
Now let
$r> 0$
be the systole of the representation
$\rho $
. By part (2) of Lemma 5.2 applied to
$a_0 = r$
, there exists a constant
$c_0 \in (0,1)$
such that
$$\begin{align*}\cos(d_j)^2\cosh(L_\rho(\alpha_{j+1})) \geq \cosh(L_\rho(\alpha_{j+1}) - \eta(\cos(d_j)^2)) , \end{align*}$$
for every j that satisfies
$\cos (d_j)^2 \geq c_0$
, where
$\eta (c) : = \mathrm {arccosh}(1/c)$
. Since
$d_j \leq \epsilon /M$
, there exists a
$\epsilon _0$
such that for any
$\epsilon \leq \epsilon _0$
, the condition
$\cos (d_j)^2 \geq c_0$
holds for any j. A simple estimate shows that
$\eta (c) \leq \frac {3}{c} \sqrt {1 - c}$
for every
$c \in (0,1)$
, implying that
$$ \begin{align*} L_\rho(\alpha_{j+1}) - \eta(\cos(d_j)^2) & \geq L_\rho(\alpha_{j+1}) - \frac{3 \sin(d_j)}{\cos(d_j)^2} \\ & \geq L_\rho(\alpha_{j+1}) - \frac{3 \sin(\epsilon/M)}{\cos(\epsilon/M)^2} \\ & \geq L_\rho(\alpha_{j+1}) - C\epsilon , \end{align*} $$
for some constant
$C> 0$
depending only on
$\epsilon _0, M> 0$
. Finally, combining the estimates obtained above, we conclude that
$$ \begin{align*} \cosh(d_{\mathbb{H}^{2,1}}(z_{j},\rho(\alpha_{j+1})z_{j}))&> \cos(d_j)^2 \cosh(L_\rho(\alpha_{j+1})) \\ & \geq \cosh(L_\rho(\alpha_{j+1}) - C\epsilon) \end{align*} $$
for every
$j \in \{0,\dots , m_n - 1\}$
, which concludes the proof of the assertion.
Let
$\kappa , \epsilon _0, C> 0$
be the constants provided by Claims 3 and 4 and choose
$\epsilon = \frac {1}{2} \min \{\epsilon _0, \kappa /C\}$
. Then for every
$n \in \mathbb {N}$
, we have
$$ \begin{align} \!L_Z(\gamma_n) &\le d_{{\hat S}}(y,\gamma_ny) \qquad\nonumber\\ &\le\sum_j{d_{{\hat S}}(y_{j},y_{j+1})}\qquad\end{align} $$
$$ \begin{align} &\ \,\qquad\le\sum_j{d_{\mathbb{H}^{2,1}}(y_{j},y_{j+1})} \end{align} $$
$$\begin{align} &\!\le\sum_j{(L_\rho(\alpha_{j+1})-\kappa)} \qquad\qquad\qquad\end{align}$$
$$\begin{align} &\le\sum_j{(d_{\mathbb{H}^{2,1}}(z_{j},z_{j+1})+ C \epsilon - \kappa)}\qquad \\ &\leq L_\rho(\gamma_n) - \frac{m_n \kappa}{2} .\qquad \end{align}$$
Multiplying by
$a_n$
and taking the limit as
$n \to \infty $
, we deduce the desired assertion.
6.3 Second variation along earthquakes
In the case of earthquakes, we make quantitative estimates and compute the second variation of length functions as given in Theorem 4.
Proof of Theorem 4
Let
$Z_t:=\epsilon _\mu (t)$
and consider the Mess representation
$\rho _t:=\rho _{Z_{-t},Z_t}$
with parameters
$Z_{-t},Z_t\in \mathcal {T}$
. Notice that, by Theorem 1, we have
$\mu _t^+=t\mu $
and
$Z_{\lambda _t^+}=Z$
is constant. For convenience, we introduce
$L_t:=L_{\rho _t}(\gamma )$
and
$\theta _t:=\theta _{\rho _t}(\gamma )$
.
By Propositions 5.4 and 5.5, we have
$$\begin{align*}\cosh(\mathscr{L}_\gamma(Z))\le\cos(\delta_t^\pm)^2\cosh(L_t)+\sin(\delta_t^\pm)^2\cosh(\theta_t) \end{align*}$$
and
$$\begin{align*}\cosh(i(\mu_t^\pm,\gamma))\le\sin(\delta_t^\pm)^2\cosh(L_t)+\cos(\delta_t^\pm)^2\cosh(\theta_t). \end{align*}$$
Summing the inequalities, we get
$$\begin{align*}\cosh(t\cdot i(\mu,\gamma))-\cosh(\theta_t)\le\cosh(L_t)-\cosh(\mathscr{L}_\gamma(Z)). \end{align*}$$
By the mean value theorem, we can write
$$\begin{align*}\cosh(t\cdot i(\mu,\gamma))-\cosh(\theta_t)=\sinh(\xi_t)\left(t\cdot i(\mu,\gamma)-\left|\theta_t\right|\right), \end{align*}$$
where
$\xi _t\in [\left |\theta _t\right |,t\cdot i(\mu ,\gamma )]$
, and
$$\begin{align*}\cosh(L_t)-\cosh(\mathscr{L}_\gamma(Z))=\sinh(\zeta_t)\left(L_t-\mathscr{L}_\gamma(Z)\right), \end{align*}$$
where
$\zeta _t\in [\mathscr {L}_\gamma (Z),L_t]$
.
We now divide both right- and left-hand sides by
$t^2$
as follows:
$$\begin{align*}\frac{\sinh(\xi_t)}{t}\left(i(\mu,\gamma)-\frac{\left|\theta_t\right|}{t}\right)\le\sinh(\zeta_t)\frac{L_t-\mathscr{L}_\gamma(Z)}{t^2}. \end{align*}$$
We observe that as
$t\to 0$
, the terms converge to the following:
In the left-hand side,
-
○
$\left |\theta _t\right |/t=\left |\mathscr {L}_\gamma (t)-\mathscr {L}_\gamma (-t)\right |/2t\to {\dot {\mathscr {L}}}_\gamma $
. -
○
$\sinh (\xi _t)/t\ge \sinh (\left |\theta _t\right |)/t$
as
$\xi _t\ge \left |\theta _t\right |$
. -
○
$\sinh (\left |\theta _t\right |)/t\to \cosh (\theta _0){\dot \theta }_0={\dot {\mathscr {L}}}_\gamma $
.
In the right-hand side,
-
○
$\sinh (\zeta _t)\to \sinh (\mathscr {L}_\gamma (Z))$
as
$L_t=(\mathscr {L}_\gamma (t)+\mathscr {L}_\gamma (-t))/2\to \mathscr {L}_\gamma (Z)$
. -
○
$(L_t-\mathscr {L}_\gamma (Z))/t^2=(\mathscr {L}_\gamma (t)+\mathscr {L}_\gamma (-t)-2\mathscr {L}_\gamma (Z))/2t^2\to {\ddot {\mathscr {L}}}_\gamma /2$
.
The conclusion follows.
Let us conclude by recalling that an exact formula for the first variation of length functions along earthquakes has been computed by Kerckhoff [Reference Kerckhoff13].
Theorem 6.4 (Kerckhoff [Reference Kerckhoff13])
Let
$\mu \in \mathcal {ML}$
be a measured lamination. Let
$\epsilon _\mu :[a,b]\to \mathcal {T}$
be an earthquake path driven by
$\mu $
. Let
$\gamma \in \Gamma -\{1\}$
be a nontrivial loop. Set
$\mathscr {L}_\gamma (t):=L_{\epsilon _\mu (t)}(\gamma )$
. We have
$$\begin{align*}{\dot{\mathscr{L}}}_\gamma(t)=\int_{\mathcal{I}/\rho_{\epsilon_\mu(t)}(\Gamma)}{\cos(\theta_t)\;\text{d}\mu\times\delta_\gamma} , \end{align*}$$
where
$\mathcal {I}$
is the space of intersecting geodesics of
$\mathbb {H}^2$
and
$\theta _t(\ell ,\ell ')$
is the angle of intersection between (the
$\epsilon _\mu (t)$
-geodesic realizations of) the leaves
$\ell ,\ell '$
.
In particular, we always have
$|{\dot {\mathscr {L}}}_\gamma |\le i(\gamma ,\mu )$
with strict inequality if
$\gamma $
intersects the support of
$\mu $
essentially.
Acknowledgements
We thank Sara Maloni, Beatrice Pozzetti and Andrea Seppi for useful discussions and generous feedback on the article, and the anonymous referee, whose comments helped us to improve our exposition. Filippo and Gabriele would like to thank also their previous affiliations, University of Virginia and Heidelberg University, respectively, where the present work was developed.
Competing interest
None.
Funding statement
Gabriele and Filippo gratefully acknowledge the support of the DFG 427903332 and of Sara Maloni’s U.S. National Science Foundation grant DMS-1848346 (NSF CAREER), respectively.
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