The Fenchel–Nielsen twist in terms of the cross ratio coordinates

The Fenchel–Nielsen twist is a generalization of the Dehn twist. It is a deformation in the Teichmüller space defined by cutting a hyperbolic surface along a simple closed geodesic, rotating one side of the cut relative to the other, and attaching these sides. If the scale of the rotation is equal to the length of the simple closed geodesic, then the Fenchel–Nielsen twist is equal to the Dehn twist.

Thurston’s earthquake deformation is a generalization of the Fenchel–Nielsen twist. It is a deformation cutting a hyperbolic surface along a lamination instead of a simple closed geodesic ([Thu]). An earthquake deformation is one of the main actors of the two-dimensional hyperbolic geometry and used to solve the Nielsen realization problem by Kerckhoff ([Ker]). Moreover, the density of the earthquake flow in the moduli space is studied by Mirzakhani ([Mir]).

In the case of closed surfaces, for any earthquake deformation $E$, there exists a sequence of the Fenchel–Nielsen twists converging $E$. In the case of marked surfaces, instead of the Fenchel–Nielsen twists, for any earthquake deformation $E$, there exists a sequence of earthquake deformations along ideal arcs converging $E$. We calculated earthquake deformations along ideal arcs ([Asa] and [AIK]).

In this paper, we calculate the Fenchel–Nielsen twist in the enhanced Teichmüller space of a marked surface by the cross ratio coordinates. For a marked surface $(\Sigma,M)$ and a closed curve $C$ on $\Sigma\setminus M$, we assume that any connected component of $\Sigma\setminus C$ has at least one marked point. Then, we can embed the annulus $A_{1,1}$ with one point on each boundary component into $\Sigma$ so that $A_{1,1}$ includes $C$ as Figure 1.

We take an element $g$ of the enhanced Teichmüller space $\widehat{\mathcal{T}}(\Sigma)$ of $(\Sigma,M)$. Let the cross ratio coordinates of $g$ for the numbered ideal arcs in Figure 1 be $(X_{1},X_{2},X_{3},X_{4})$ and the length of the simple closed geodesic on $g$ corresponding to $C$ be $L(f_{2})$. We take a rotation parameter $t\in\mathbb{R}_{\geq 0}$.

Also, the Fenchel–Nielsen twist was calculated by the coordinates of triangle length or by the trace coordinates by Garden ([Gar]).

We start with the same settings as [AIK]. Let $\Sigma$ be an oriented compact topological surface which may have boundaries and $M$ be a finite subset in $\Sigma$. We assume that each boundary component has at least one point of $M$. We call $(\Sigma,M)$ a marked surface.

Let $M_{\circ}$ be the intersection of $M$ and the interior of $\Sigma$. We define $\Sigma^{\ast}:=\Sigma\setminus M_{\circ}$. An ideal arc in $(\Sigma,M)$ is an isotopy class of an arc in $\Sigma^{\ast}$ connecting the points of $M$ without self-intersection except for its endpoints. We require that an ideal arc not to be represented by a boundary segment. A nonempty maximal disjoint collection $\Delta$ of ideal arcs without a self-folded triangle is called an ideal triangulation of $\Sigma$. We assume that there is an ideal triangulation in $(\Sigma,M)$. A labeled triangulation is an ideal triangulation $\Delta$ equipped with a bijection $\ell:I\to\Delta$, where $n$ is the number of ideal arcs and $I:=\{1,\dots,n\}$ is a fixed index set.

We introduce the cross ratio coordinates into $\widehat{\mathcal{T}}(\Sigma)$. Let $\mathbb{H}$ be the upper half space in the complex plane and $\mathbb{S_{\infty}^{1}}=\mathbb{R}\cup\{\infty\}$ be the boundary of $\mathbb{H}$.

In order to define the Fenchel–Nielsen twist, we give another description of $\widehat{\mathcal{T}}(\Sigma)$. We take $[\Sigma^{h}_{0},f_{0}]\in\widehat{\mathcal{T}}(\Sigma)$ and denote the closure in $\mathbb{H}\cup\mathbb{S_{\infty}^{1}}$ of the lift $\widetilde{\Sigma^{h}_{0}}$ of $\Sigma^{h}_{0}$ as $\overline{\widetilde{\Sigma^{h}_{0}}}$. Let $\mathcal{F}_{\infty}(\Sigma)=\overline{\widetilde{\Sigma^{h}_{0}}}\cap\mathbb{S_{\infty}^{1}}$. We consider that $\mathcal{F}_{\infty}(\Sigma)$ has the same cyclic order as $\mathbb{S_{\infty}^{1}}$.

We consider the set

$$\left\{(\rho,\psi)\middle|\begin{array}[]{lll}\rho:\pi_{1}(\Sigma)\rightarrow\mathrm{PSL}(2,\mathbb{R})\text{ is a Fuchsian representation,}\\ \psi:\mathcal{F}_{\infty}(\Sigma)\rightarrow\mathbb{S_{\infty}^{1}}\text{ is }\rho(\pi_{1}(\Sigma))\text{-equivalent and cyclic-order-preserving.}\end{array}\right\}$$

and say that $(\rho_{i},\psi_{i})\ (i=1,2)$ are equivalent if there exists $\gamma\in\mathrm{PSL}(2,\mathbb{R})$ satisfying $\rho_{2}=\gamma\cdot\rho_{1}\cdot\gamma^{-1}$ and $\psi_{2}=\gamma\cdot\psi_{1}$. We denote the set of all of these equivalent classes as $\widehat{\mathcal{T}}_{\infty}(\Sigma)$.

Indeed, given $[\Sigma^{h},f]\in\widehat{\mathcal{T}}(\Sigma)$, the representation $\rho$ is the monodromy of the hyperbolic structure $h$, and $\psi$ is given by the continuous extension of the lift $\widetilde{f}:\widetilde{\Sigma}\to\widetilde{\Sigma^{h}}\subset\mathbb{H}$ to $\mathbb{S_{\infty}^{1}}$.

For a simple closed curve $C$ of $\Sigma\setminus M$ and $t\in\mathbb{R}_{\geq 0}$, we define the Fenchel–Nielsen twist $E_{tC}:\widehat{\mathcal{T}}_{\infty}(\Sigma)\rightarrow\widehat{\mathcal{T}}_{\infty}(\Sigma)$ as follows. We take $[\rho,\psi]\in\widehat{\mathcal{T}}_{\infty}(\Sigma)$ and let $\widetilde{\Sigma_{\rho}}$ be the universal cover of the hyperbolic surface $\Sigma_{\rho}:=\mathbb{H}/\rho(\pi_{1}(\Sigma))$. Let $\widetilde{C}$ be the union of all lifts of $C$ in $\widetilde{\Sigma_{\rho}}$. We call each connected component of $\widetilde{\Sigma_{\rho}}\setminus\widetilde{C}$ a gap. We fix a gap $S_{0}$. Let the length of $C$ in $\Sigma_{\rho}$ be $L(C)$. We endow each gap $S$ with a hyperbolic element $E_{S}\in\mathrm{PSL}(2,\mathbb{R})$ as follows.

• •

  $E_{S_{0}}$ is the identity.

• •

  For any gap $S_{1}$ adjacent to $S_{0}$, let $l_{1}$ be the intervening geodesic between these gaps. Let $E_{S_{1}}$ be the hyperbolic element whose axis is $l_{1}$, whose translation distance is $t\cdot L(C)$ and whose direction is to the left (i.e. the restriction $E_{S_{1}}|_{S_{1}}$ of $E_{S_{1}}$ moves $S_{1}$ to the left from $S_{0}$’s perspective).

• •

  For any gap $S_{2}$ adjacent to $S_{1}$ and other than $S_{0}$, let $l_{2}$ be the intervening geodesic between these gaps. Let $\overline{E_{S_{2}}}$ be the hyperbolic element whose axis is $l_{2}$, whose translation distance is $t\cdot L(C)$ and whose direction is to the left. We define $E_{S_{2}}:=E_{S_{1}}\circ\overline{E_{S_{2}}}$.

• •

  Inductively iterating, for any gap $S_{n}$ adjacent to $S_{n-1}$ and other than $S_{n-2}$, let $l_{n}$ be the intervening geodesic between these gaps. Let $\overline{E_{S_{n}}}$ be the hyperbolic element whose axis is $l_{n}$, whose translation distance is $t\cdot L(C)$ and whose direction is to the left. We define $E_{S_{n}}:=E_{S_{n-1}}\circ\overline{E_{S_{n}}}$.

We define $\widetilde{E_{tC}}:\widetilde{\Sigma_{\rho}}\setminus\widetilde{C}\to\mathbb{H}$ as $\widetilde{E_{tC}}:=\displaystyle\bigcup_{S\text{ : gap}}E_{S}|_{S}$ and continuously extend it to $\partial_{\infty}\widetilde{E_{tC}}:\partial_{\infty}\widetilde{\Sigma_{\rho}}\rightarrow\mathbb{S_{\infty}^{1}}$, where $\partial_{\infty}\widetilde{\Sigma_{\rho}}$ is the intersection of $\widetilde{\Sigma_{\rho}}$ and $\mathbb{S_{\infty}^{1}}$.

1. 1.

   We define a group-homomorphism $\rho^{\prime}:\pi_{1}(\Sigma)\to\mathrm{PSL}(2,\mathbb{R})$ by the condition $\partial_{\infty}\widetilde{E_{tC}}\circ\rho(\gamma)=\rho^{\prime}(\gamma)\circ\partial_{\infty}\widetilde{E_{tC}}$ on $\partial_{\infty}\widetilde{\Sigma_{\rho}}$ for any $\gamma\in\pi_{1}(\Sigma)$. Then, $\rho^{\prime}$ is a Fuchsian representation by [BKS, Proposition 3.2.].

2. 2.

   Define $\psi^{\prime}:\mathcal{F}_{\infty}(\Sigma)\to\mathbb{S_{\infty}^{1}}$ by $\psi^{\prime}:=\partial_{\infty}\widetilde{E_{tC}}\circ\psi$. Then, $\psi^{\prime}$ is $\rho^{\prime}$-equivariant and order-preserving. See [BB, BKS].

As above, we define

$$E_{tC}:\widehat{\mathcal{T}}_{\infty}(\Sigma)\rightarrow\widehat{\mathcal{T}}_{\infty}(\Sigma),\ [\rho,\psi]\mapsto[\rho^{\prime},\psi^{\prime}].$$

Moreover, $E_{tC}:\widehat{\mathcal{T}}(\Sigma)\rightarrow\widehat{\mathcal{T}}(\Sigma)$ is naturally induced by Proposition 2.4.