数理解析研究所講究録
1118 巻 1999 年 1-12

1Some topics of geodesics on tori
Ryuji ABE (阿部隆次)

1

Introduction

In [A1] and [A2] we constructed explicit conformal embeddings from once punctured tori
to closed tori corresponding to real lattices. Our principal aim of the construction ofthese mappings was to know explicit relations between the Teichm\"uller spaces of the once
punctured torus and the closed torus. In this note we consider these conformal embeddings

from another point of view and state some related topics with respect to geodesics on a

once punctured torus.
Our aim of this note is to show the following three points:
1. a number theoretical background of the conformal embedding from a once punctured
torus to a closed torus (\S 3),
2.

$\mathrm{M}\mathrm{c}\mathrm{S}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{e}’ \mathrm{s}$identity for the lengths of simple closed geodesics on a once punctured

torus (\S 4, \S 5),

3. an alternative proof of$\mathrm{M}\mathrm{c}\mathrm{S}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{e}’ \mathrm{s}$identity by using binary trees (\S 6).

Relations among the cusp and closed geodesics on a once punctured torus are very

important in 1 and 2. The relation $X^{2}+Y^{2}+Z^{2}=XYZ$ plays an important role in 1
and 3. This is also used in order to represent the Teichm\"uller space of the once punctured

torus (\S 2).

2

A model of$\mathcal{T}_{1,1}$

Let $M$ be a once punctured torus with a complete finite-area hyperbolic structure and let$G$

be the fundamental group of M.

the commutator
an element

$A$

in

$[a_{0}, b_{0}]$is a free group of two generators$a_{0},$$b_{0}$

such thatis peripheral. Throughout this note we use the convention that$\mathrm{P}\mathrm{S}\mathrm{L}(2, \mathrm{R})$

if

$G$

represents the M\"obius transformation induced by $A,$ $i.e.$ ,$A=\in \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathrm{R})$then $A(z)= \frac{az+b}{cz+d}$ .

2The group of isometries of the upper half-plane$\mathrm{H}$can be identified with

Then any complete finite-area hyperbolic structure on $M$ is given by$\rho:Garrow \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathrm{R})$

Let

$C$

is a discrete faithful representation such that$\mathrm{P}\mathrm{S}\mathrm{L}(2, \mathrm{R})$$\mathrm{H}/p(G)$$\rho([a_{0}, b_{0}])$.

, whereis parabolic.

be the set of isotopy classes of simple closed curves on $M$ , where simple curves are

the hyperbolic
curves which contain no self intersections. We can associat to eachof the unique closed geodesic in the isotopy class. Moreover, each homotopy
lengthclass of closed curves on $M$ corresponds to an equivalent class in , where $h\in G$ are$\gamma\in C$$e$$|\gamma|$$G$equivalent if is conjugate to
$g$

$h$

or

$h^{-1}$element $g\in G$ , then we can verify tr. If the homotopy class$\gamma$$g,$

is represented by a group, where tr denotes the trace of a$( \rho(g))=2\cosh\frac{1}{2}|\gamma|$

matrix.
for $A,$A Fuchsian groupare hyperbolic and tr $[B^{-1}, A^{-1}]=-2$ , where$\Gamma=\langle A, B\rangle$$B\in \mathrm{P}\mathrm{S}\mathrm{L}(2, \mathrm{R})$is called a Fricke group if $A,$$\Gamma=\langle A, B\rangle$$B$

is the free group generated by. We can verify that the image of the discrete faithful representation which
determines a complete finite-area hyperbolic structure on a once punctured torus is a
Fricke group. Conversely, we consider a once punctured torus which is uniformized by a
$A$

and

$B$

$\rho$

Fricke group$\Gamma$and take a normalized form for the presentation of$\Gamma$(see [A1], [A2]). By

AB, the above description of theandusing the quantities
Fricke group is characterized by $X^{2}+\mathrm{Y}^{2}+Z^{2}=X\mathrm{Y}Z$ and $X,$ $Z>2$ . Moreover, we$X=\mathrm{t}\mathrm{r}A,$ $Y=\mathrm{t}\mathrm{r}B$$Z=\mathrm{t}\mathrm{r}$$Y,$obtain the following theorem (see [W]).

Theorem 1 (Fricke, Keen) The Teichm\"ulier space
the sublocus of $X^{2}+Y^{2}+Z^{2}=X\mathrm{Y}Z$ with $X,$

$Y,$$\mathcal{T}_{1,1}$$Z>2$ .

In this note we denote a point in the Teichm\"uller spaceby a triplet (X,$Y,$ $Z$

Example 1 Letof the once punctured torus is$\mathcal{T}_{1,1}$of the once punctured torus).

$A=$ and $B=$ . Thention of (3, 3, 3) and is a subgroup of the modular group$\Gamma=\langle A, B\rangle$$\mathrm{P}\mathrm{S}\mathrm{L}(2, \mathbb{Z})$is a representa-of index 6.

The once punctured torus (3, 3, 3) plays an important role when we give a geometricalinterpretation of a classical result of the number theory (see \S 3).

3

Markoff forms and geodesics

We begin by recalling the classical theory. We define a quadratic form with integral

coefficients and irrational roots$\xi,$$\eta$

as follows:$Q(x, y)=ax^{2}+bxy+cy^{2}=a(x-\xi y)(x-\eta y),$$d=b^{2}-4ac>0$ ,3and define$m(Q)= \inf_{)(x,y\neq(0,0)}\frac{|Q(x,y)|}{\sqrt{d}}$.

If $(m_{1}, m_{2}, m_{3})$ is given by the positive solutions of $m_{1}^{2}+m_{2}^{2}+m_{3}^{2}=3m_{1}m_{2}m_{3}$ , we call$(m_{1}, m_{2}, m_{3})$a Markoff triple. Note that if we set $m_{1}=X/3,$$m_{2}=\mathrm{Y}/3$

the relation $X^{2}+Y^{2}+Z^{2}=XYZ$ in \S 2 is obtained. Suppose thatand $m_{3}=Z/3$ ,$i=1,2,3$ are$m_{i},$

integers, it is easily verified that $m_{i}\in\tilde{M}=\{1,2,5,13,29, \cdots\}$ . A number included in$\tilde{M}$is called a Markoff number.

We can construct a quadratic form corresponding to a Markoff triple, which is calleda Markoff form. The Markoff form corresponding towith $m_{1}\geq m_{2}\geq m_{3}$ is$(m_{1}, m_{2}, m_{3})$defined by $Q_{m_{1}}(x, y)=ax^{2}+bxy+cy^{2}$ such that$a=m_{1},$where$u=\mathrm{t}\mathrm{h}\mathrm{e}$$b=3m_{1}-2u,$

least positive integer of$c= \frac{1+u^{2}}{m_{1}}-3u$$\pm\frac{km_{1}-m_{2}}{m_{3}}$

(mod

,

$3m_{1}$), $k=0,1,2,$$\cdots$.

We show the first five cases:
if (1, 1, 1) then $Q_{1}(x, y)=x^{2}-xy-y^{2}$ and$m(Q_{1})= \frac{1}{\sqrt{5}}$,,if (2, 1, 1) then $Q_{2}(x, y)=2x^{2}-4xy-2y2$ and$m(Q_{2})= \frac{1}{2\sqrt{2}}$if (5, 2, 1) then $Q_{5}(x, y)=5x^{2}-11xy-5y2$ and$m(Q_{5})= \frac{5}{\sqrt{221}}$,if (13, 5, 1) then $Q_{13}(x, y)=13x^{2}-29xy-13y^{2}$ and$m(Q_{13})= \frac{13}{\sqrt{1517}}$,

if (29, 5, 2) then $Q_{29}(x, y)=29x^{2}-63xy-31y^{2}$ and$m(Q_{29})= \frac{29}{\sqrt{7565}}$.

We recall a relation between Markoff form and Diophantine approximation. Consider
the problem of finding the following$\nu(\theta)$

for a given irrational number . Let$\theta$

$p$

and

$q$

be

positive integers,$\nu(\theta)=\inf\{c||\theta-\frac{p}{q}|<\frac{c}{q^{2}}$for infinitely many

A. Markoff proved that there is a discrete set of values$\nu(\theta)>1/3$

then$\nu(\theta)=\nu_{i}$for some , where$i$$\nu_{i}$

$q\}$

.

decreasing to 1/3 such that if$\nu_{i}=1/\sqrt{9-\frac{4}{i^{2}}}$and is a Markoff number.
$i$

For the first five cases we can confirm $m(Q_{i})=\nu_{i}$ for $i=1,2,5,13,29$ .

4

Let

$\xi,$$\eta$

corresponds to the. The form
and . We can assume that $0<-\eta<1<\xi$
are given by continued fractions as follows:

be irrational roots of a Markoff form

whose endpoints are
geodesic on
without loss of generality, then$\mathrm{H}$

$\xi$

$Q$

$Q$

$\eta$$\xi \mathrm{a}\mathrm{n}\mathrm{d}-\eta$$\xi=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+}}\ldots=\{a_{0}, a_{1}, a2, \cdots\}$where$a_{i},$$i\in \mathbb{Z}$and,$- \eta=\frac{1}{a_{-1}+\frac{1}{a_{-2}+}}\ldots=\{0, a_{-1,-2}a, \cdots\}$are positive integers.

Markoff forms are characterized by the geodesics which remain below the horizontal line ${\rm Im} z=3/2$ . The projections of these geodesics to the

Theorem 2 (Cohn)$(a)$once punctured torus (3, 3, 3) are closed.
$(b)$

The sequence $\{-\eta, \xi\}=\{\cdots, a_{-2,-1,0,1,2}aaaa, \cdots\}$ is a periodic sequence consisting of integer pairs $(1, 1)$ and $(2, 2)$ .

A geodesic on$\mathrm{H}$is transferred to the extended plane$1-J(z)=\wp’(u)^{2}=4\wp(u)^{3}+1$

for$\mathrm{G}$by using the relation$z\in \mathrm{H},$$u\in$C.

(3.1)

The relation (3.1) is obtained by considering an abelianization of actions of$A$

and $B$

introduced in Example 1 (see [C], [A1]). Then it follows from Theorem 2 (b) that a closed
geodesic on the once punctured torus (3, 3, 3) corresponds to a direction in C. Indeed,
the integer pairs $(1, 1)$ and $(2, 2)$ means the following transformations, respectively,$z \text{ト}arrow 1+\frac{1}{1+\frac{1}{z}}=\frac{2z+1}{z+1}=B^{-1}(Z)$and$z \text{ト}arrow 2+\frac{1}{2+\frac{1}{z}}=\frac{5z+2}{2_{Z}+1}=B^{-}1A^{-}1B-1(z)$.

The assertion of Theorem 2 (a) was generalized to the case of Fricke groups by A. L.

Schmidt [S]. Generalizing the relation (3.1), we construct conformal embeddings fromclosed tori corresponding to real lattices to once punctured tori (see [A1], [A2]).
$e\mathrm{d}$4Simple geodesics on a hyperbolic surface

In the previous section we recalled that Markoff forms are characterized by the closed
geodesics on the once punctured torus (3, 3, 3) which is not near to the cusp. Here we
consider simple geodesics on a general hyperbolic surface.
Let $M$ be a hyperbolic surface, $i.e.,$ $M$ can be identified with the quotient spaceare projected toof the upper half-plane
under a Fuchsian group . Geodesics on$\mathrm{H}/G$$\mathrm{H}$geodesics on $M$ .

$G$$\mathrm{H}$5

Definition 1 A geodesic on $M$ is said to be complete if it is not strictly contained in any
other geodesic, $i.e.$ , it is either closed and smooth, or open and of infinite length in both
directions.

Theorem 3

Let $M$ be a hyperbolic surface with cusps. Then there exist
neighborhoods of cusps which do not have intersection with any simple geodesic on $M$$(\mathrm{J}\phi \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{S}\mathrm{e}\mathrm{n})$except for ones having both ends going up cusps.
It follows immediately from Theorem 3 that the hyperbolic surface $M$ with cusps is
not covered by its simple geodesics. Then we have the problem: what is the density ofthe set of points on simple geodesics? J. S. Birman and C. Series gave an answer for thisproblem in [BS]. For $k\geq 0$ we define $G_{k}=\{\gamma|\gamma$ is a complete geodesic on $M$ whichhave at most$k$transversal self-intersections}.

Theorem 4 (Birman-Series) For eack $k\geq 0,$ $S_{k}=\{p\in M|p\in\gamma\in G_{k}\}$ is nowhere
dense and Hausdorff dimension one.

Note thatand that$S_{0}\subset S_{1}\subset S_{2}\subset\cdots$$\bigcup_{k=1}^{\infty}S_{k}$is a dense subset of $M$ . Let$\ovalbox{\tt\small REJECT}$a closed torus which can be identified with the quotient space of the extended plane
under a lattice group. We can define the set$S_{0}=\mathrm{T}$case$S_{k}$

for$\mathrm{T}$be$\mathrm{G}$in the same way as for $M$ . Then

which presents a striking contrast to Theorem 4. For simplicity, we consider the$\mathrm{T}=\mathrm{C}/\langle 1, i\rangle$. A geodesic on$\ovalbox{\tt\small REJECT}$

is the projection of a line on C. Let be a line on$l$is always complete and simple, and is closed if and only if
C. The projection of on
the slope of is rational. Thus we have.
Let $M$ be a hyperbolic surface with cusps and without boundary.
$l$$\ovalbox{\tt\small REJECT}$$l$$S_{0}=\ovalbox{\tt\small REJECT}$Definition 2 We call a portion of the surface isometric to$\{z|{\rm Im} z\geq 1\}/\langle P\rangle,$$P=$

It follows from Theorem 3 that there exists a cusp region so that if an end of a simple

geodesic enters this region then it will never leave, that is, the end goes up the cusp. We
call this region a small cusp region.

Definition 3 A geodesic has an end up the cusp if that end is contained in a small cusp
region on $M$ . A geodesic is a cusp geodesic if it has both its ends up the cusp.

6

be a small cusp region. Define $E=\{x\in \mathcal{H}|x$ is on a complete simple
and thatis homeomorphic togeodesic}. Then we can obtain thatmeans the boundary
set {countably many isolated points}, whereLet$\mathcal{H}\subset M$$E$$(\partial \mathcal{H}\cap E)\mathrm{x}\mathbb{R}^{+}$$\partial \mathcal{H}$$\cup$$\partial \mathcal{H}\cap E=\mathrm{C}\mathrm{a}\mathrm{n}\mathrm{t}_{\mathrm{o}\mathrm{r}}$be aand the definition of isolated point is as follows: lettopological space and $X\subset Y$ then $x\in X$ is isolated if there is an open set $U,$ $x\in U$ suchof the small cusp region$Y$$\mathcal{H}$that $U\cap X=\{x\}$ . Using same notations, we introduce another definition: is a boundary
point if is the limit of points in $X$ and there is a connected open set contained in thesuch that is in the closure of . We call a maximal component ofcomplement
$x$

$U$

$x$

$\partial \mathcal{H}\backslash E$

$U$

$x$

$Y\backslash X$

a gap.

, the simple geodesic on which
is another end of , we call this the other end.
If$x\in\partial \mathcal{H}\cap E$$\gamma_{x}$$x$lies has an end up the cusp. There$\gamma_{x}$Theorem 5$(\mathrm{M}\mathrm{c}\mathrm{S}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{e})$Let$M,\mathcal{H},$$E$

be as above and$x\in\partial \mathcal{H}\cap E$

.

$(a)$

The point$x$

is isolated if and only if the other end also goes up a cusp.

$(b)$

The point$x$

is a boundary point if and only if the other end spirals to a closed simple

geodesic.
$(c)$which satisfy the following conEach gap is bounded by two points
on a cusp geodesic. The point lies either on a geodesic
ditions: The pointwith an end spiraling to a closed simple geodesic or on a geodesic with its other end$y,$$z\in\partial \mathcal{H}\cap E$$z$$ylie\mathit{8}$up some other cusp. If the surface has only one cusp, only the first case occurs.
By using Theorem 4 and 5 we can prove the following striking identity for the lengthsof simple closed geodesics on a once punctured torus.

Theorem 6$(\mathrm{M}\mathrm{c}\mathrm{S}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{e})$Let $M$ be$a$once punctured torus then$\sum_{\gamma}\frac{1}{1+e^{|\gamma|}}=\frac{1}{2}$where the sum is over all simple closed geodesics

5$\gamma$,on $M$ .

Outline of a proof of Theorem 6

First we consider a general surface: let $M$ be a hyperbolic surface with cusps. Take acusp of $M$ and denotes its small cusp region. We define in the same way as in \S 4.
be on a cusp geodesic . From Theorem 5, is an isolat point andLet$E$$\mathcal{H}$$x\in\partial \mathcal{H}\cap E$$\gamma$$x$$e\mathrm{d}$7Fig. 5.1

is in the closure of two gaps. Suppose that one gap is bounded bybounded by$x,$(see Fig. 5.1).

$z$$\alpha$. The cusp geodesic(resp.

$\beta$$\gamma$

$x,$

$y$

and the other isis contained in a unique embedded pair of pants) is a closed simple geodesic in Theorem 5 (c) to which the otherend of the geodesic through$y$(resp.

$z$) spirals. We need not consider the second case of

Theorem 5 (c), because our final aim is to prove the assertion for a once punctured torus.
The pants can be decomposed into four quadrilaterals $A,$ $B,$
obtain quadrilaterals $A’$ and $B’$ in$\mathrm{H}$which are congruent to

$A$

$C$

and

and

$B$

$D$

. Moreover, we, respectively (seeFig. 5.2). The broken lines in Fig. 5.2 represent geodesics ends of whose projections on$M$spiral to the simple closed geodesics$\alpha$and$\beta$. The points$x’,$ $y’,$$z’$

are congruent towhich bound two gaps. Then we call the shaded parts in Fig. 5.2 gap regions.
Now we can calculate the ratio of the area of the gap region to the area of the small cusp

$x,$

$y,$

$z$

region:$\frac{2(1-\tanh\frac{1}{2}|\alpha|)\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\frac{1}{2}|\beta|+2(1-\tanh\frac{1}{2}|\beta|)\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\frac{1}{2}|\alpha|}{2\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\frac{1}{2}|\alpha|+2\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{h}\frac{1}{2}|\beta|}$$=$$\frac{(1-\tanh\frac{1}{2}|\alpha|)\cosh\frac{1}{2}|\alpha|+(1-\tanh\frac{1}{2}|\beta|)\cosh\frac{1}{2}|\beta|}{\cosh\frac{1}{2}|\alpha|+\cosh\frac{1}{2}|\beta|}=\frac{2}{1+e^{\frac{1}{2}(|||\beta}\alpha+|)}$.

On the one hand, since these arguments can be applied to all cusp geodesics in the8$\frac{|\alpha|}{2}$$\frac{|\beta|}{2}$$\frac{|\alpha|}{2}$$\frac{|\beta|}{2}$Fig. 5.2

9small cusp region$\mathcal{H}$, we obtain$\sum_{\alpha,\beta}\frac{2}{1+e^{(|\alpha||\beta|}+)}\cross$(the area of small cusp region)($=$$\mathrm{t}\mathrm{h}\mathrm{e}$total area of gap region).

On the other hand, Theorem 4 means that (the area of small cusp region) is equal to (thetotal area of gap region), so the above equation is changed into$\sum_{\alpha,\beta}\frac{1}{1+e^{\frac{1}{2}(|||\beta}\alpha+|)}=\frac{1}{2}$.

If $M$ is a once punctured torus, the simple geodesics$\alpha$and$\beta$coincide. Therefore we

obtain the identity:$\sum_{\alpha}\frac{1}{1+e^{|\alpha|}}=\frac{1}{2}$6.

Another approach to Theorem 6

There exists an alternative proof of Theorem 6 by using binary trees (see [B]). In thisse ction we rewrite Theorem 6 by using binary trees and state a part of the proof in orderto show the advantage of using them.

Let$\Sigma$be a binary tree, $i.e.$ , a simplicial tree all of whose vertices have degree 3.

Suppose that$\Sigma$

is embedded in the plane. We call the closure of a connected componentof the complement of$\Sigma$a complementary region. Let$\Omega$be the set of complementary

reglons.

We recall that $M$ is a once punctured torus with a complete finite-area hyperbolic

structure and denotes the set of isotopy classes of simple closed curves on $M$ . There isa natural bijective correspondences between and . Let
be a free basis for the
$C$

$C$

$\Omega$$\{a, b\}$fundamental group $G=\pi_{1}(M)$ . We consider simple closed curves inclasses are represented by the group elements

faithful representation
where$\rho$$a,$$b$, ab and

whose homotopy. There exists a discretewith $\rho(a)=A,$ $p(b)=B,$ $\rho(ab)=AB$ and $p(a^{-1}b)=A^{-1}B$

is a Fricke group. We set $X=$ tr $A,$ $Y=$ tr $B,$ $Z=$ tr $AB$ and. Then $X,$ $Z,$ $W$ are elements of and are visualized as in Fig. 6.1.

$\Gamma=\langle A, B\rangle$$W=\mathrm{t}\mathrm{r}A^{-1}B$$a^{-1}b$

$C$

$Y,$

$\Omega$$Z$

Fig. 6.1

10
is a Fricke group, we get the relation $X^{2}+Y^{2}+Z^{2}=XYz$ . We call this the. Moreover, from the formulavertex relation because is the intersection of $X,$

Since$\langle A, B\rangle$$Y,$ $Z\in\Omega$$v_{1}$

tr

$A$

tr

, we obtain the relation $XY=Z+W$ , which is called theand . A binary tree each of whose
is the edge connecting$B=\mathrm{t}\mathrm{r}AB+\mathrm{t}\mathrm{r}A^{-1}B$edge relation because$e$$v_{2}$$v_{1}$

vertex (resp. edge) satisfies the vertex (resp. edge) relation is called a Markoff binary
tree.

Note that in Fig. 6.1 the relation $W^{2}+X^{2}+\mathrm{Y}^{2}=WXY$ is obtained from $X^{2}+$$Y^{2}+Z^{2}=XYZ$ and $X\mathrm{Y}=Z+W$ . Generally, if there exists a single vertex satisfying

the vertex relation, by using the edge relation inductively, we can construct the Markoff
binary tree including the vertex.

Example 2 A part of the Markoff binary tree constructed from (3, 3, 3) is as in Fig. 6.2.

Fig. 6.2
Let$\Sigma$be the markoff binary tree constructed from (X,$Y,$ $Z$)$\in \mathcal{T}_{1,1}$

and letset of complementary regions associat with . If we takeis the simple closed geodesic on the once punctured torus corresponding to$e\mathrm{d}$$\Sigma$$\Omega$$\Omega\ni\omega=2\mathrm{c}o\mathrm{s}\mathrm{h}\frac{1}{2}|\gamma_{\omega}|$$\omega$$\gamma_{\omega}$$h( \omega)=\frac{1}{1+e^{1}\gamma_{\omega}1}$where $h(x)= \frac{1}{2}(1-\sqrt{1-\frac{4}{x^{2}}})$ .

By using this notation, the identity in Theorem 6 can be represented as follows:$\sum_{\zeta v\in\Omega}h(\omega)=\frac{1}{2}$Fig. 6.3

Fig. 6.4.

Fig. 6.5

be the, where, then11

Here we only show$\Sigma_{\omega\in\Omega}h(\omega)\leq\frac{1}{2}$. We think of a directed edge. Define

the case of Fig. 6.3. Immediately we get $\psi(e)arrow+\psi(-earrow)=\frac{Z}{XY}+\frac{W}{XY}=1$

edge relation. In Fig. 6.4 it is obtained thatin
, which means the$\psi(e_{1}^{arrow)}+\psi(e_{2}^{arrow)}+\psi(e_{3}^{arrow)}=\frac{Z}{X\mathrm{Y}}+\frac{X}{YZ}+\frac{\mathrm{Y}}{XZ}=1$

which means the vertex relation. We consider the case of Fig. 6.5, from$\psi(e_{3}^{arrow)}=1$

Let

$C$

and

$\psi(e_{4}^{arrow})+\psi(e_{5}^{arrow)}+\psi(-e_{1}^{arrow})=1$$\psi(e)arrow=\frac{Z}{X\mathrm{Y}}$, we have,$\psi(e_{1}^{arrow)}+\psi(e_{2}^{arrow)}+$$\psi(e_{2}^{arrow)}+\psi(e_{3}^{arrow)}+\psi(e_{4}^{arrow)}+\psi(e^{arrow}5)=1$.

be the set of all directed edges which touch and are directed to a subtree (see Fig.

6.6). By using the above calculations inductively, we obtain

Fig. 6.6. In this figure$C$$\Sigma_{e_{i\in c\psi(}^{arrow}}e_{i}^{\prec}$) $=1$ .

is the set of edges with direction.

We go back to the case of Fig. 6.3. From direct calculations we can verify the following
inequality:$h(X)+h(Y)\leq h(X+Y)\leq\psi(e)\neg$

where$h(X+\mathrm{Y})=h(\sqrt{\frac{1}{X^{2}}+\frac{1}{Y^{2}}})$.

Consider the case of Fig. 6.4, there exist the following three inequalities:$h(X)+h(Y)\leq\psi(e_{1}^{arrow)}, h(Y)+h(Z)\leq\psi(e_{2}^{arrow)}$

and $h(Z)+h(X)\leq\psi(e_{3}^{arrow)}$ .

Summing each side of inequalities, respectively, we obtain $h(X)+h(Y)+h(Z) \leq\frac{1}{2}.$ By
using this argument inductively,
is proved.
$\Sigma_{\omega\in\Omega}h(\omega)\leq\frac{1}{2}$12References[A1]

R. Abe, On correspondences between once punctured tori and closed tori:
Fricke groups and real lattices, to appear in Tokyo J. of Math.

[A2]

R. Abe, Abelianization of Fricke groups and rhombic lattices, Preprint,

1998.
[BS]

J. S. Birman and C. Series, Geodesics with bounded intersection number on
surfaces are sparsely distributed, Topology 24 (1985), 217-225.

[B]

B. H. Bowditch, A proof of McShane’S identity via Markoff triples, Bull.

London Math. Soc. 28 (1996), 73-78.
[C]

H. Cohn, Representation of Markoff’s binary quadratic forms by geodesics
on a perforated torus, Acta. Arith. 18 (1971), 125-136.

[J]

T.

$\mathrm{J}\oint \mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{n}$, Simple geodesics on Riemann surfaces, Proc. A. M. S. 86(1982), 120-122.
[M]

G. McShane, Simple geodesics and a serie8 constant over

$TeiChm\ddot{u}lle\Gamma$

space,

Invent. Math. 132 (1998), 607-632.
[S|

A. L. Schmidt, Minimum of quadratic forms with respect to Fuchsiangroups. , J. reine angrew. Math. 286/287 (1976), 341-468.
$I$

[W]

S. Wolpert, On the K\"aler form of the moduli space of once punctured tori,
Comm. Math. Helv. 58 (1983), 246-256.

$\mathrm{E}$-mail address: ryu2@math.keio.ac.jp