数理解析研究所講究録 1270 巻 2002 年 88-92

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Sufficient conditions for Teichmiiller
modular groups to be of the second kind
Ege Fujikawa
Department of Mathematics, Tokyo Institute of Technology
藤川英華 (東京工業大学大学院理工学研究科数学専攻)

1Introduction
We consider the action of the reduced Teichmiiller modular groupfor ahyperbolic Riemann surface , which is agroup of automorphisms of
the reduced Teichmiiller space $T\#(R)$ . If is of analytically finite type, thereduced Teichmiiller modular group is nothing but the ordinary Teichmiiller
modular group Mod(7?), and it is well known that Mod(7?) acts properlydoes
discontinuously on $T(R)$ . However, if is of infinite type,not act properly discontinuously on $T\#(R)$ , in general. On the basis of thisfact, in [1], we have introduced new notions, the limit set and the region of
discontinuity for aTeichmiiller modular group as an analogy to the theoryof the Kleinian groups acting on the Riemann sphere.
$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$

$R$

$R$

$R$

$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$Definition 1We say that apoint in $T\#(R)$ is alimit point for asubgroupofif there exist apoint $q\in T\#(R)$ and asequenceof$\lim_{narrow\infty}d_{T}(\chi_{n}(q),p)=0$. The set of the limit
distinct elements of such that
$p$

$G$

$\{\chi_{n}\}$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$$G$. The complement
points is called the limit set of $G$ , and denoted by$T\#(R)-\Lambda(G)$ of the limit set is denoted by, and called the region of
discontinuity of $G$ .
$\Lambda(G)$$\Omega(G)$Similarly, for asubgroup of the ordinary modular group Mod(ff), we canof analyticallyin $T(R)$ . For aRiemann surface
anddefine. On the other hand, forfinite type, we haveaRiemann surface whose Fuchsian model is of the second kind, we always. This is the reason why we consider the reduced
have, not the ordinary modular group Mod(7?), for amodular group$R$ of infinite type
Riemann surface$G$$\Lambda(G)$$\Omega(G)$$\Lambda(\mathrm{M}\mathrm{o}\mathrm{d}(R))=\Lambda(\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R))=\emptyset$$R$$\Omega(\mathrm{M}\mathrm{o}\mathrm{d}(R))=\emptyset$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$

$R$

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Definition 2 For a subgroup G of, we say that G is of the firstkind if, and otherwise of the second kind.
$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$$\Omega(G)=\emptyset$

this note, we focus our attention on normal covering surfaces of Riemannsurfaces, and consider sufficient conditions for Teichm\"uller modular groupsto be of second kind.
$\ln$

2

Sufficient conditions

Throughout this note, we assume that a Riemann surface
has the nonabelian fundamental group. In [1] and [2], we have shown sufficient conditionsof Riemann surfaces forto be of the first kind or of the secondkind, which are stated below.
$R$

$R$

$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$Definition 3 For a given $M>0$ , we say that a point of belongs to a
subsetof if there exists a non-trivial simple closed curve containing
such that the hyperbolic length of is less than $M$ .
$p$$R_{M}$$R$$R$$c_{p}$$p$$c_{p}$

Definition 4 We say thatsatisfies the lower bound condition if thereexists ansuch thatconsists only of cusp neighborhoods. Further
we say that satisfifies the upper bound condition if there exist aconstant$M>0$ and a connected componentofsuch that ahomeomorphism
oftothat is induced by the inclusion map ofintoissurjective.
$R$$\epsilon>0$$R_{\epsilon}$$R$$R_{M}^{*}$$\pi_{1}(R_{M}^{*})$$\pi_{1}(R)$$R_{M}$$R$$R_{M}^{*}$

Theorem 1 ([1]) If R does not satisfy the lower bound condition, thenis of the first kind$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$

Theorem 2 ([1]) If R satisfies the lower and upper bound conditions, then$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$is of the second kind.

In connection with these results, we have stated the following conjecturein [1].

Conjecture
satisfifies the lower bound condition, thenis of thesecond kind. That is, considering Theorem 1, we conjecture thatis of the second kind if and only if satisfies the lower bound condition.
$1\mathrm{f}R$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$$R$, we have proved a partial solution of this conjecture, giving a weakercondition than the upper bound condition.
$\ln[2]$

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Theorem 3 ([2]) Let R be a Riemann surface that satisfies the following twoconditions:

1. R satisfies the lower bound condition.

2. There exists a constant $M>0$ such that, for any connected componentof the complement of , is either simply or doubly connectedconsists of finitely many connected components.
and$V$$R_{M}$$V$$R-\overline{V}$Then$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$is of the second kind.

Remark 1The upper and lower bound conditions and the conditions inTheorem 3are quasiconformally invariant. Then these conditions are regarded as conditions for the Teichmiiller space.

3Normal covering surfaces
be anormal covering surface of aRiemann
Throughout this section, letsurface which is not auniversal cover. First, we have given asufficient
to be of the second kind.
condition for$\tilde{R}$$R$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(\tilde{R})$Proposition 1([1]) If $R$ is of analytically finite type, thenupper and lower bound conditions. Thus, by Theorem 2,second kind.

$\tilde{R}$satisfies theis of the$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(\tilde{R})$Asimple example of this proposition is as follows.
Example 1 Let$\tilde{R}=\mathrm{C}-\{n|n\in \mathrm{Z}\}$.

is a normal
is of the second kind. Indeed,of $\mathrm{C}-\{0,1\}$ , which is an analytically finite Riemann surface.

Then$\tilde{R}$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(\tilde{R})$$cover\dot{\tau}ng$

surface

In connection with Proposition 1, we have the following.

Proposition 2([3]) If
satisfies the lower and upper bound conditions,then also satisfies these conditions.
$\tilde{R}$$R$

Remark 2In [3], we have considered aweakened upper bound condition.
The same proof can be applied also for Riemann surfaces that satisfy the(generalized) upper bound condition in Definition 4.

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Rom Theorem 2 and Proposition 2, we have a condition of anormal
covering surface forto be of the second kind.
$\tilde{R}$Proposition 3 If
and$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(\tilde{R})$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$$\tilde{R}$satisfies the lower and upper bound conditions, thenare of the second kind.

$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$

If does not satisfy the upper bound condition, then Proposition3is
not necessarily true.
$\tilde{R}$

Example 2 Let$\tilde{R}=\mathrm{C}-\bigcup_{n=1}^{\infty}\bigcup_{m\in \mathrm{Z}}\{\frac{m}{n}\pm n^{2}\sqrt{-1}\}$,, where $f(z)=z+1$ . Although does not satisfy the upper
andbound condition, satisfies the conditions in Theorem 3. Thenisof the second kind. On the other hand, satisfies the upper bound condition,
but does not satisfy the lower bound condition. Then, from Theorem 1,
is of the fifirst kind.
$R=\tilde{R}/\langle f\rangle$$\tilde{R}$$\cdot\tilde{R}$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(\tilde{R})$$R$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$

Thatfor both$\tilde{R}$satisfifies the upper bound condition is not anecessary condition
andto be of the second kind.

$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(\tilde{R})$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$

Example 3 Let$\tilde{R}=\mathrm{C}-\cup^{\infty}\cup n=1m\in \mathrm{Z}\mathrm{t}\frac{m}{n}+(2n+1)\sqrt{-1}\}$,, where $f(z)=z+1$ . Although does not satisfy the upperandbound condition, satisfies the conditions in Theorem 3. Thenis$R$of the second kind. Further , satisfies the lower and upper bound conditions.
Then, by Theorem 2,
is also of the second kind.
$R=\tilde{R}/\langle f\rangle$$\tilde{R}$$\tilde{R}$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(\tilde{R})$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$In the last of this note, we give afollowing problem.

Problem If$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(\tilde{R})$is of the fifirst kind, then so is$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$.

does not satisfy the lower bound condition, thenIt is clear that, ifneither does . Hence, from Theorem 1, if
does not satisfy the lower
bound condition, thenandare of the first kind. Further,
if the conjecture, which is stated in the previous section, is true, then theproblem is solved affirmatively$\tilde{R}$$R$$\tilde{R}$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(\tilde{R})$$\mathrm{M}\mathrm{o}\mathrm{d}^{\#}(R)$92
References[1] E. Fujikawa, Limit sets and regions of discontinuity of Teichm\"ullerular groups, preprint.

$\mathrm{m}\mathrm{o}\mathrm{d}-$[2] E. Fujikawa, Teichmiiller modular groups with non-empty regions ofdiscontinuity, preprint.
[3] E. Fujikawa, The order of conformal automorphisms of Riemann surfacesof infinite type, preprint.