数理解析研究所講究録
第 1660 巻 2009 年 128-138

128

Complexified Penner’s coordinates and its applications
Toshihiro Nakanishi (Shimane University)*1

Penner’s -lengths

1.1

A coordinate-system for Teichm\"uller space$\lambda$

Let $D=\{z\in \mathbb{C}: |z|<1\}$ be the unit disk, a model of hyperbolic plane and$SU(1,1)= \{(\frac{a}{b}\frac{b}{a}):a,$ $b\in \mathbb{C},$ $|a|^{2}-|b|^{2}=1\}$.

Then $PSU(1,1)$ is the group of orientation preserving hyperbolic motions of D.

Let $G=G_{g,n}$ be the punctured surface group of type $(g, n)$ , where $2g-2+n>0$ :$G=\langle a_{1},$$b_{1},$$\ldots,$$a_{g},$$b_{g},$$d_{1},$$\ldots,$$d_{n}:( \prod_{k=1}^{g}a_{k}b_{k}a_{k}^{-1}b_{k}^{-1})d_{1}\cdots d_{n}=1\rangle$.

A point of the Teichmuller space
is a class of faithful Fuchsian represen$G$
tations of into $PSU(1,1)$ which have finite covolume. We denote points in by, where is a Fuchsian group and $m$ :marked groups
is an isomorphism.
Elements
incorresponding toare parabolic. Choose
a horocycle
invariant undersuch that action ofonis the translation oflength one. Then the identification ofwithgives the following
statement.
$\mathcal{T}=\mathcal{T}_{g,n}$$\mathcal{T}$$D_{1},\ldots,$$H_{k}$$Garrow\Gamma$$\Gamma$$\Gamma_{m}$$D_{n}$$d_{1},\ldots,$$\Gamma_{m}\in \mathcal{T}$$D_{k}$$D_{k}$$\Gamma_{m}$$\mathcal{T}_{g,n}$$d_{n}$$H_{k}$$(\Gamma_{m}, H_{1}, \ldots, H_{n})$Is naturally embedded in the decorated Teichm\"uller space$\tilde{\mathcal{T}}_{g,n}$.

Therefore, by restricting them to this embedded subspace, Penner’s -length coorgive also global coordinates for the Teichm\"uller space
dinates for.
$\lambda$$\tilde{\mathcal{T}}_{g,n}$$\mathcal{T}_{g,n}$’A joint work with M. N\"a\"at\"anen. The author is grateful to Professor Robert Penner for helpful
dlscussions. He thanks Professor Michihiko FUjii for organizing a series of workshops on hyperbolic
geometry and its related topics.

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1.2

Distance between horocycles

Let be a point of the unit circle. A horvcycle at is a Euclidean circle intangent at to the ‘unit circle. The point is called the base point of .
Let
andbe horocycles based at different points and and the hyperbolic line between
and . Define
$h$

$p$

$p$

$D$

$p$

$h$

$p$

$h_{1}$$h_{2}$$p_{1}$$p_{1}$$p_{2}$$\gamma$$p_{2}$$\lambda=e^{\delta/2}$,(1)

where is the signed length of the portion of the geodesic intercepted between the$\delta>0$ if
two horocycles
andandare disjoint and $\delta<0$ otherwise. In
this way we can assign a positive number to the pair.
$\delta$$\gamma$$h_{1}$$h_{2},$$h_{1}$$h_{2}$$(h_{1}, h_{2})$$\lambda$

1.3$\lambda$-length of an ideal arc

Let be the oriented closed surface of genus$P=\{p_{1}, \ldots,p_{n}\}$ a set ofpoints.
An ideal arc of $(S, P)$ is a path joining two points andin $S-P$ . The ideal
arc is simple if $c\cap(S-P)$ is a simple arc.
, then there exists an orientation preserving homeomorphism
Let
$S$

$n$

$g,$

$c$

$p_{i}$$p_{j}$$c$$\Gamma_{m}\in \mathcal{T}_{g,n}$$f:S-Parrow D/\Gamma$

inducing $m$ . Let be the geodesic representative in the homotopy class of $f(c)$ for
the Poincar\’e metric of the punctured surface. By the identification of with, the horocycles at the endpoints of defines the -length.
$\gamma$$D/\Gamma$$(\Gamma_{m}, H_{1}, \ldots, H_{n})$Let$\Delta=\{c_{1}, c_{2}, \ldots, c_{q}\},$$\Gamma_{m}$$\lambda$$\gamma$$q=6g-6+3n$ , be an ideal triangulation of $(S, P)$ . Then

Theorem 1 (Penner [1])$\lambda_{\Delta}=\prod_{i=1}^{q}\lambda(c_{i}):\mathcal{T}_{g_{1}n}arrow(\mathbb{R}_{+})^{q}$is an embedding.

$\lambda(c, \Gamma_{m})$130

The image ofis a real algebraic variety determined bypolynomials. A
component ofis called a triangle in . The image ofis a real
algebraic variety determined by zero loci of algebraic equations, whereis easily obtained by triangles abutting on the kth puncture .
$\lambda_{\Delta}$$n$$S- \bigcup_{j=1}^{q}c_{j}$$\Delta$$\lambda_{\Delta}$$n$$D_{1},\ldots,$$D_{n}$$D_{k}$$p_{k}$$D_{k}( \lambda_{1}, \ldots, \lambda_{q})=\sum_{i=1}^{N}\frac{\lambda(e_{i})}{\lambda(a_{i})\lambda(b_{i})}-1$

1.4

.

(2)

The Ptolemy identityLet
be an ideal triangulation of $(S, P)$ . Let $e\in\Delta$ andandbe triangles being on the different sides of . It is possible that $T_{1}=T_{2}$ . Liftto a quadrangle
in D. Then is a diagonal of . Let
be the other diagonal and project to an ideal arc in. Then$\Delta=\{c_{1}, c_{2}, \ldots, c_{q}\}$$T_{1}$$T_{2}$$e$$T_{1}\cup e\cup T_{2}$$Q=\tilde{T}_{1_{\sim}}\cup\tilde{e}\cup\tilde{T}_{2}$

$f$

$Q$

$\tilde{e}$$\tilde{f}$$T_{1}\cup e\cup T_{2}$$f$$\Delta’=(\Delta-\{e\})\cup\{f\}$is another ideal triangulation of $(S, P)$ . We say thatmentary move on$e$$\Delta’$arises from$\Delta$by the ele-

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be the sides of andbe the sides of . Suppose that$d\in$ A
are opposite sides of . Let
be the projections of
The following theorems are proved in Penner’s paper:Let$(\tilde{a},\tilde{b},\tilde{e})$$\tilde{T}_{1}$$Q$$\tilde{c}$$(\tilde{c})\tilde{d},\tilde{e})$

$a,$

$b,$

$\tilde{T}_{2}$$\cap\Delta’$$c,$$\tilde{a}$and.

$\tilde{a},\tilde{b},\tilde{c},\tilde{d}$Theorem 2 (the Ptolemy identity, Penner [1])The -lengths function satisfy the identity$\lambda$(3)$\lambda(a)\lambda(c)+\lambda(b)\lambda(d)=\lambda(e)\lambda(f)$This theorem describes the coordinate-change between$\lambda_{\Delta}(\mathcal{T})$and$\lambda_{\Delta’}(\mathcal{T})$:$\lambda_{\Delta’}\circ\lambda_{\Delta}^{-1}(..., \lambda(a), \lambda(b), \lambda(c), \lambda(d), \lambda(e), \cdots)$$=( \cdots, \lambda(a), \lambda(b), \lambda(c), \lambda(d), \frac{\lambda(a)\lambda(c)+\lambda(b)\lambda(d)}{\lambda(e)}, \cdots)$

Theorem 3 (Penner [1]) For arbitrary ideal triangulations
there enists a finite sequence of ideal triangulations$\Delta=\Delta_{0},$where each$\Delta_{i}$arisesfrom$\Delta_{i-1}$$\Delta_{1},$$\cdots,$$\Delta_{m}=\Delta’$,

by an elementary move.

$\Delta$and$\Delta’$(4)of $(S, P)$ ,132

Using this theorem it can be shown that coordinate change between
coordinates associated with two ideal triangulations is a bi-rational map:

Theorem 4 If$\Delta$and$\Delta$‘$\lambda$-lengthare ideal triangulations of , then the coordinate change$F$$id\downarrow \mathcal{T}\mathcal{T}arrow^{arrow\lambda_{A’}\lambda_{A}}\lambda_{\Delta},(\mathcal{T})\subset(\mathbb{R}_{+}^{+})^{q}\lambda_{\Delta}(\mathcal{T})\subset(\mathbb{R})^{q}I^{\lambda_{A’}}\circ\lambda^{\frac{}{A}1}$

extends to a mtional transformation of$\mathbb{R}^{q}$Letdenote the mapping class group of $(S, P)$ . Eachon the Teichm\"uller space . The theorem above yields$\mathcal{M}C=\mathcal{M}C_{g,n}$$\varphi\in \mathcal{M}C$

acts$\mathcal{T}$Theorem 5 The comespondence$\phi\mapsto\phi_{*}=\lambda_{\varphi^{-1}(\Delta)}0\lambda_{\Delta}^{-1}$

gives an isomo rphism of2$SL(2, \mathbb{C})$$\mathcal{M}C$

to a group of mtional transformations.

-representation space of a punctured sur-face groupLetbe the space of classes of faithful representations $[m]$ of the punctured
surface group intosuch that $m(d_{i})$ is parabolic with tr $m(d_{i})=-2$ for. The Teichm\"uller space$i=1,2,$is a subspace of.
Our purpose is to give a $coordinatearrow system$ for
whose restriction tocoincides with Penner’s -lengths coordinate-system.
$\mathcal{R}=\mathcal{R}_{g,n}$$G$$\ldots,$$SL(2, \mathbb{C})$$n$$\mathcal{T}_{g,n}$$\mathcal{R}_{g,n}$$\mathcal{R}_{g,n}$$\mathcal{T}_{g,n}$$\lambda$

2.1

Parabolic elements ofDefine$\mathcal{P}=$If$P_{1}${ $P\in SL(2,$$\mathbb{C})$$SL(2, \mathbb{C})$: $P$ is parabolic with tr $P=-2$ }.

and $P_{2}\in P$ do not commute, then the square root of $-P_{1}l*$ in$Q= \pm\frac{1}{\sqrt{2-trP_{1}P_{2}}}(I-P_{1}P_{2})$,$SL(2, \mathbb{C})$$\langle$

5)is unique up to sign and satisfies$P_{2}=Q^{-1}P_{1}Q$

For the rest of this paper, the diagram$P_{1}arrow^{Q}P_{2}$

.

(6)

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will mean that $Q^{2}=-P_{1}P_{2}$ .
Cycles of parabolic elements$P_{n+1}=P_{1}\in \mathcal{P}$ .
Let
Suppose that no consecutive elements
andcommute. Let
be a square root of $-P_{i}P_{i+1},$ $(i=1,2, .., n)$ . Then, since$P_{i+1}=Q_{i}^{-1}P_{i}Q_{i},$commutes with ,$P_{1},$$P_{n},$$\ldots,$$P_{i}$$P_{i+1}$$Q_{i}$$Q_{1}Q_{2}\cdots Q_{n}$$P_{1}$

or $-2$ .

$trQ_{1}Q_{2}\cdots Q_{n}=+2$

Definition

is a $(+)$ -system or a $(-)$ -system according to if tr $Q_{1}Q_{2}\cdots Q_{n}=+2$$(Q_{1}, Q_{2}, \ldots, Q_{n})$

or-2.

2.2

(7)

A trace identity of Ptolemy typeLet

$P_{1},$$Q_{1},$$Q_{2},$$P_{2},$$P_{3}$$Q_{3},$

and$Q_{4},$$P_{4}$$Q_{5},$. Suppose that

$Q_{6},$$Q_{5}’,$

and$P_{i}$

do not commute unless $i=j$ . Chooseso that$P_{j}$$Q_{6}’\in SL(2, \mathbb{C})$$Q_{1}^{2}=-P_{1}P_{2}$,$Q_{2}^{2}=-P_{2}P_{3}$$Q_{4}^{2}=-P_{4}P_{1}$,$Q_{6}^{2}=-P_{3}P_{1}$$(Q_{5}’)^{2}=-P_{1}P_{3}$,,,$(Q_{6}’)^{2}=-P_{4}P_{2}$$Q_{3}=-P_{3}P_{4}$$Q_{6}=-P_{2}P_{4}$

,

where
$Q_{5}’=P_{1}Q_{5}P_{1}^{-1}$

Theorem 6 If $(Q_{1}, Q_{2}, Q_{5}),$,$(Q_{5}’, Q_{3}, Q_{4})$$Q_{6}’=P_{4}Q_{6}P_{4}^{-1}$

and

,
,

$(Q_{1}, Q_{6}, Q_{4})$tr $Q_{5}trQ_{6}=$ tr $Q_{1}trQ_{3}+$ tr$Q_{2}trQ_{4}$.

are $(-)- systems$, then
(8)

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3

Complexified -length

3.1

Definition of -length$\lambda$$\lambda$

A point of is represented by a marked group. Let
be the set of parabolic
elements in $[m(d_{1})]\cup\cdots\cup[m(d_{n})]$ , where $[m(d_{i})]$ is the conjugacy class of $m(d_{i})$ .
$\mathcal{R}$$\Gamma_{m}$$\mathcal{P}_{+}(\Gamma)$

Let be an ideal arc in $(S, P)$ . Then for eachdefines two parabolic, see the following figure. We define the -length of withelements
ofrespect toby$\lambda(c, \Gamma_{m})=trQ$ ,(9)where is a square root of. The -length is defined up to sign.
$c$$\Gamma_{m}\in \mathcal{R},$$P_{1},$$P_{2}$$c$$\lambda$$\mathcal{P}_{+}(\Gamma)$$c$$\Gamma_{m}$

$Q$

3.2

$\lambda$$-P_{1}P_{2}$$\lambda$-length coordinates for$\mathcal{R}_{g,n}$Let
be an ideal triangulation of $(S, P)$ . Let be a triangle in .
inherites the orientation of the surface . Label the sides of $T$ by
in order.
Then those sides determine matrices
whose traces give -lengths of.
and for$T$$\Delta=(c_{1}, c_{2}, \ldots, c_{q})$$T$$\Delta$$S$$Q_{a},$$c$$Q_{b},$$a,$$Q_{c}$

$c$

$a,$

$b$

$\Gamma_{m}$Lemma 1 It is possible to choose branches of -length functions
so thatis a $(-)$ -system for each triangle in .
$\lambda$$\lambda(c_{q})$$b,$$\lambda$$\lambda(c_{1}),$$T$$(Q_{a}, Q_{b}, Q_{c})$$\lambda(c_{2}),$$\ldots$,$\Delta$With the choice of branches of -lengths as depicted in the lemma, we obtain$\lambda$

Theorem 7 For each ideal triangulation$\Delta_{f}$$\lambda_{\Delta}=\prod_{i=1}^{q}\lambda(\alpha):\mathcal{R}_{g_{i}n}arrow(\mathbb{C}^{*})^{q}$is an embedding. The image is contained in an algebraic variety.

3.3

Rational representation of the mapping class group

As in the case of$\mathcal{T}$, the Ptolemy identity (8) yields

Theorem 8 The mapping class groupform ations.

$\mathcal{M}C$

acts on$\mathcal{R}$as a group of mtional tmns-

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4

Invariant holomorphic two-formLet$p=4g-2$ , be triangles in an ideal triangulation of a once-punctured
surface. Let the sequence of sides
agree with the positive orientationofof , then the 2-form$T_{1},\ldots,$$T_{p},$$b_{i},$$a_{i},$$c_{i}$$T_{i}$$T_{i}$(10)$\sum_{i=1}^{p}(d\log\lambda(a_{i})\wedge d\lambda(b_{i})+d\log\lambda(b_{i})\wedge d\log\lambda(c_{i})+d\log\lambda(c_{i})\wedge d\log\lambda(a_{i}))$is invariant under the mapping class groupthe corresponding result in [2].

$\mathcal{M}C$. The proof is similar to the one of5

A characterization of the rational map induced
by a mapping class

5.1

Example: Once punctured torus

The Teichm\"uller space
defined by$\mathcal{T}_{1,1}$of once punctured tori is represented as the subspace of$(\mathbb{R}_{+})^{3}$$x^{2}+y^{2}+z^{2}=xyz$ ,(11)where $y,$ are -length functions related to an (essentially unique) triangulationof the once punctured torus (or $y,$ are trace functions, tr , tr $AB$ , with $\{A, B\}$the canonical generator-system of
)
$x,$

$z$

$\lambda$

$x,$

$z$

$tr_{A}$$B$$G_{1,1}.$

The mapping class group$\mathcal{M}C_{1,1}$

has generators
and$\sigma(x,y, z)=(x, z,\frac{x^{2}+z^{2}}{y})$$\tau(x, y, z)=(\frac{x^{2}+y^{2}}{z}, y, x)$with relations$(\tau 0\sigma)^{3}=1$,$(\sigma 0\tau 0\sigma)^{2}=1$

,

.

Since
, the group of rational transformations generated byacts onand preserves the equation (11) and $(x, y, z)=(3,3,3)$ gives integer solutions of$\mathcal{M}C_{1,1}$$\mathcal{T}_{1_{t}1}$$\sigma$$\tau$(11).

Theorem 9 (Markoff) All positive integer solutions of (11) are in the oribit of(3, 3, 3) under the action of.
$\mathcal{M}C_{1,1}$

The viewpoint of understanding the Markoff transformations as mapping classes
actiong on
is given in Penner’s paper [1].
$\mathcal{T}_{1,1}$

With -length coordinates, the Teichm\"uller spaceis determined by algebraic equations and the group of rational transformations induced by the mappingclass groupkeep this space. So we can pursue analogies of the above result.
$\lambda$$\mathcal{T}_{g,n}$$\mathcal{M}C_{g,n}$

$n$

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Example: twice punctured torus

5.2

Let
be the ideal triangulation of the twice punctured torus as depicted in thefollowing figure.
$\Delta$$twi\infty$punctured torus

Consider the -lengths$\lambda$$\lambda_{a},$$\lambda_{b},$$\lambda_{c},$$\lambda_{d},$$\lambda_{e}$. The Teichm\"uller spaceassociated with . Then it holds thatspace
) is represented by the -lengths as the space$\Delta$$\lambda_{e}=\lambda_{f}$$\mathcal{T}_{1_{2}2}$(or the$\lambda$$\mathcal{R}_{1,2}$$\frac{\lambda_{e}}{\lambda_{a}\lambda_{b}}+\frac{\lambda_{a}}{\lambda_{b}\lambda_{e}}+\frac{\lambda_{b}}{\lambda_{a}\lambda_{e}}+\frac{\lambda_{c}}{\lambda_{d}\lambda_{e}}+\frac{\lambda_{d}}{\lambda_{c}\lambda_{e}}+\frac{\lambda_{\epsilon}}{\lambda_{c}\lambda_{d}}=1$

or$\lambda_{c}\lambda_{d}(\lambda_{a}^{2}+\lambda_{b}^{2}+\lambda_{e}^{2})+\lambda_{a}\lambda_{b}(\lambda_{c}^{2}+\lambda_{d}^{2}+\lambda_{e}^{2})=\lambda_{a}\lambda_{b}\lambda_{c}\lambda_{d}\lambda_{e}$The mapping class group
generators

.

(12)

(as a group of rational transformations) has$\mathcal{M}C_{1,2}$$\omega_{1*}(\lambda_{a},\lambda_{b},\lambda_{c},\lambda_{d},\lambda_{e})\omega_{2*}(\lambda_{a},\lambda_{b},\lambda_{c},\lambda_{d},\lambda_{e})$$==$$( \lambda_{d},\lambda_{b},\lambda_{c},\frac{\lambda_{e}^{2}}{\lambda_{c},\lambda}(\lambda_{d},\lambda_{a},\lambda_{b},\frac{+\lambda_{d}^{2}\prime f_{a}\lambda_{c}’+\lambda_{b}\lambda_{c}\lambda_{e})}{\lambda_{e}})$$\omega_{3*}(\lambda_{a}, \lambda_{b}, \lambda_{c}, \lambda_{d}, \lambda_{e})$$=$$( \lambda_{a}, \frac{\lambda_{b}^{2}+\lambda_{e}^{2}}{\lambda_{c}}, \lambda_{b}, \lambda_{d}, \lambda_{e})$,with relations$\omega_{2*}^{2}\omega_{1*}\omega_{2*}^{2}=\omega_{3*}$$(\omega_{1*}\omega_{2*})^{3}=1$,$\omega_{1*}\omega_{3*}=\omega_{3*}\omega_{1*}$$(\omega_{3*}\omega_{2*})^{3}=1$

The point $p=(6,6,6,6,6)$ gives integer solutions of (12). An analogous result to
the Markoff equation holds:

Theorem 10 The orbit(12), but not all of its integer solutions.
$\{\varphi_{*}(6,6,6,6,6) :\varphi\in \mathcal{M}C_{1,2}\}$, gives integer solutions of

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5.3

Diophantine equations

We consider a once punctured surface.

Lemma 2 Letbe the -length coordinate-system for
associatedto an ideal triangulation.
where $q=6g-3$ Then the -length of asimple ideal arc is expressed by a rational function of the form$\lambda$$(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{q})$$\mathcal{R}_{g,1}$$(c_{1}, c_{2}, \ldots, c_{q})_{f}$$\lambda$$c$$\frac{P(\lambda_{1},\lambda_{2}.’.\cdot\cdot.\cdot,\lambda_{q})}{\lambda_{1}^{m_{1}}\lambda_{2}^{m_{2}}\lambda_{q}^{m_{q}}}$where$P(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{q})$

(13)

,

is a homogeneous polynomial of degree$d=1+m_{1}+m_{2}+\cdots+m_{q}$ ,with positive integer coefficients andand in $S-P$ for $i=1,2,$$c_{i}$$\ldots,$For
letentries offollowing observation.
$\varphi\in \mathcal{M}C_{g,1}$$\varphi_{*}$$m_{i}$

is the geometric intersection number of

$c$

$q$

denote the rational transformation induced by . Thenare of the form as in (13). This fact leads us to the$\varphi$$\varphi_{*}(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{q})$

Let
(14)

$D(\lambda_{1}, \ldots, \lambda_{q})=0$

be the algebraic equation which determines
the rational transformationinduced by$\mathcal{T}_{g,1}$

in the -length coordinates. Then
preserves. $Morearrow$$\lambda$$D(\lambda_{1}, \ldots, \lambda_{q})$$\varphi\in \mathcal{M}C_{g,1}$$\varphi_{*}$

over, if$(\lambda, \lambda, \ldots, \lambda)$

gives integer solutions of (14), then so doesof$\varphi_{*}(\lambda, \lambda, \ldots, \lambda)$.

We remark that it is not true in general that all integer solutions are in the orbit.
under$\mathcal{M}C$$(\lambda, \lambda, \ldots, \lambda)$

3-manifolds which fiber over the circle

6

Let
. Let
be a manifold which fibers over the circle and whosemonodoromy is . If
denotes the action of on the fundamental group $G=$of the surfaceof type $(g, n)$ , then the fundamental group of
has thepresentation(15): $\varphi_{*}(g)=tgt^{-1}$ for all$\varphi\in \mathcal{M}C_{g_{\gamma}n}$$M_{\varphi}$$\varphi$$G_{g,n}$$\varphi_{*}$$\varphi$$S$$M_{\varphi}$$\tilde{G}=\langle G,$

If$m:\tilde{G}arrow SL(2, \mathbb{C})$$g\in G\rangle$$t$

is a faithful representation of$\tilde{G}$, then for all $g\in G$$(\varphi_{*}\circ m)(g)=m(t)m(g)m(t)^{-1}$

Henoe the class $[m]$ is a fixed point of$\varphi_{*}$.

for its action on$\mathcal{R}_{g,n}$

.

138

The -length coordinates ofrepresent
as a rational function. Hence thefixed point $[m]$ corresponds to a solution of the algebraic equation$\lambda$$\mathcal{R}_{g,n}$$\varphi_{*}$$\varphi_{*}(\lambda_{1}, \ldots, \lambda_{q})=(\lambda_{1}, \ldots, \lambda_{q})$

.

(16)

If
is reducible, then one of the solutions of (16) gives a faithful and discrete
representation $m$ of . We can find the M\"obius transformation $m(t)$ easily, because$m(t)$ sends the fixed point of $m(g)$ to that offor each parabolic element$g\in G$ . In this way hyperbolization of
can be done. However, to carry this
hyperbolization program into effect, we need efficient discreteness criteria.
$\varphi$$G$$m(\varphi_{*}(g))$$M_{\varphi}$

References[1] Penner, R. C., The decorated Teichm\"uller space of punctured surfaces, Commun. Math. Phys. 113 (1987), 299-339.

[2] Penner, R. C., Weil-Petersson volumes. J. Differential Geom. 35 (1992), 559-

608.

[3] T. Nakanishi and M. NaEt\"anen, Complexification of lambda length as parameter
representation space of punctured surface groups, J. London Math.
forSoc., 70 (2004), 383-404.
$SL(2, \mathbb{C})$[4] T. Nakanishi, A trace identity for parabolic elements ofJ., 30 (2007) 1-18.

$SL(2, \mathbb{C})$, Kodai Math.