数理解析研究所講究録1022 巻 1997 年 77-86

77.

$\mathrm{U}\mathrm{N}\mathrm{I}\mathrm{F}\mathrm{O}..\mathrm{R}\mathrm{M}$$\mathrm{p}\mathrm{E}\mathrm{R}\mathrm{F}\backslash \cdot \mathrm{E}\mathrm{C}\mathrm{T}\mathrm{N}\mathrm{E}\mathrm{S}\mathrm{s}$OF THE LIM.IT. SETS AND TRANSLATIONLENGTHS OF. KLEINIAN GROUPS
須川敏幸(京都大学・理学部)

TOSHIYUKI SUGAWA (KYOTO UNIVERSITY)

1. INTRODUCTION
In this note, we shall consider the uniform perfectness of the limit sets of Kleinian. As soon as the proof of uniform perfectness of the limit sets of Schottkygroups appeared in [1], this result was generalized to theof finitely generated
non-elementary Kleinian groups by Maskit, Marden, etc. (cf. [8] and [9]). Afterward,
Canary remarked in [3] that the same result holds for analytically finite Kleiniangroups.
Recently, J\"arvi and Vuorinen [4] proved the uniform
for finitely generated Kleinian groups in the higher dimensional case. It is noteworthy that their
proof does not rely on Ahlfors’ Finiteness Theorem.
In this note, we will present a more general condition for the uniform perfectness of
the limit set (Theorem 3.3 or Corollary 3.4). Our method also provides a bound foruniform perfectness by some geometric quantity. In practice, it is important to know
an explicit bound because the uniform perfectness connects with various quantity
relating to the geometry of the quotient surface (cf. Sugawa [10]). Indeed, we shallgive some applications in Sections 5 and 6. In Section 5, we state results relating
to the regularity in the sense of Dirichlet and the Hausdorff dimention of the limit
sets. Section 6 is devoted to an estimate of translation length by the(or themultiplier) of a loxodromic element of a Kleinian group. We shall conclude this noteby giving an example of those Kleinian groups whose limit sets are not uniformly
perfect.
$\mathrm{g}\mathrm{r}‘ \mathrm{o}\mathrm{u}\mathrm{p}\mathrm{s}$$\mathrm{c}a\mathrm{s}\mathrm{e}$$\mathrm{p}\mathrm{e}\mathrm{r}\dot{\mathrm{f}}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$2. RIEMANN ORBIPOLDS

A 1-dimensional complex orbifold is a Hausdorff topological space locally modeledon the quotient of an open subset of the.complex plane by an action of finitegroup consisting of biholomorphic.maps. By a Riemann orbifold (or, an orbifold,for short), we will mean a connected 1-dimensional complex orbifold. For a precise
deffiition and fundamental properties of orbifolds, we refer the reader to$\mathbb{C}$$\mathrm{M}\mathrm{c}\mathrm{M}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{n}’ \mathrm{S}$Key words and phmses. uniformly perfect, Kleinian group, translation lengh, Hausdorff
dimension.

78book [6]. Let $X=(R, \nu)$ be a hyperbolic orbifold, in other words, $X$ is a pairsuch that the set ofand a multiplicity map :of a Riemann surface
singularities (or, branch points) $b(X)=\{x\in R;\nu(x)>1\}$ is discrete, and has affom the upper half planebranched holomorphic normal covering map :. ThenH. We set
for any
onto such thatcan be regarded as a usual Riemann surface. If $N$ is the constant 1, then $X$
can be naturally indentified with the underlying Riemann surface . The coveringtransformation group $\Gamma=\Gamma_{X}=\{\gamma\in Aut(\mathbb{H});p\circ\gamma=p\}$ is called a Fuchsian
$R$

$\nu$

$Rarrow \mathbb{N}$$R$$\mathbb{H}arrow R$

$p$

$R$

$\mathbb{H}$$[mathring]_{X}=R\backslash b(R)$$\zeta\in$$\mathrm{o}\mathrm{r}\mathrm{d}_{\zeta p}=\nu(p(\zeta))$$X^{\mathrm{o}}$$R$distance function on $X$ naturally defined by the hyperbolic metric.
of non-trivial closed curves
denote the set of free homotopy classes
Let$X$
in , where a curve is said to be non-trivial if it is covered by an elementof infinite order, more precisely, there exists a lift via such that the initial pointis translated to the terminal one by , and said to be freely homotopic to anotherif both are covered by the same element of . (The free homotopy class precisely
corresponds to the conjugacy class of the element of which covers the curve andbe the set of free homotopy classes which are covered
has infinite order.) And letby hyperbolic elements of .
. By definition, if
andWe write ascovers a, then$\alpha$$[\alpha]$$C_{X}$$\gamma\in\Gamma$$\alpha$$\tilde{\alpha}$$p$$\gamma$$\Gamma$$\alpha’$$\Gamma$$C_{X}^{*}$$\Gamma$$\ell_{X}(\alpha)=\int_{\alpha}\rho_{X}$(2.1)$\gamma\in\Gamma$$l_{X}[ \alpha]=\inf_{\alpha’\in[\alpha}]^{\ell}\mathrm{x}(\alpha’)$$P_{x}[ \alpha 1=l=\mathrm{c}\mathrm{o}\gamma \mathrm{s}\mathrm{h}^{-1}(\frac{\mathrm{t}\mathrm{r}\gamma}{2})$where denotes the translation lengh. of.
of in
Now we set$l_{\gamma}$$\gamma$$L(X)= \inf_{[\alpha]\in C\mathrm{x}}\ell \mathrm{x}[\alpha]$and$\gamma$$\mathrm{S}\mathrm{L}(2, \mathbb{R})$and$\mathrm{t}\mathrm{r}\gamma$,is the trace of a representative
..

..

.’

$L^{*}(X)= \inf_{[\alpha]\in C*X}\ell_{X}[\alpha]$$\cdot\cdot$.

We call $X$ is modulated if $L(X)>0$ . We note here that if $X$ is a Riemann surface, whereis the injectivity radius of (see, the constant $L(R)$ is nothing but[10] .
Let $A_{2}(X)$ and $B_{2}(X)$ be the complex Banach spaces consisting of holomorphic.
quadratic differentials on $X$ with norms$2I_{R}$$R$$R$$I_{R}$$)$$\varphi$$|| \varphi||1=\int\int_{X}|\varphi|-arrow\int\int_{X}|\varphi(_{Z)}|dXdy$,,$|| \varphi||_{\infty}=\sup\rho_{\mathrm{x}}^{-}|2\varphi|=\sup\rho_{X}-2(_{Z})|\varphi(Z)|$respectively. The spaces $A_{2}(X)$ and $B_{2}(X)$ are canonically isomorphic to the spacesof integrable and bounded holomorphic automorphic forms
and$A_{2}(\mathbb{H}, \Gamma)$$B_{2}(\mathbb{H}, \Gamma)$

79

of weight-4 on for , respectively.
And we set $\kappa(X)=\sup$ {with}. For these spaces, theinclusion problem was first settled by Niebur and Sheingorn [7]. The following stro.ng
form is due to Matsuzaki [5].
$\Gamma$$\mathbb{H}$$||\varphi||_{\infty};\varphi\in A_{2}(X)$$||\varphi||\leq 1$

Theorem 2.1. The space $A_{2}(X)$ is (continuously) included by $B_{2}(X)$ if and only if$L^{*}(X)>0$ . Furthermore, there exist universal constants
and such that$r_{0}$$r_{1}$$\frac{1}{2\pi L^{*}([mathring]_{X})}\leq\kappa(X)\leq\max\{\frac{r_{0}}{L^{*}(X)}, r_{1}\}$.

Note that the inclusion map
induces an isometric isomorphism $A_{2}(X)arrow$and a bounded linear operator $B_{2}(X)arrow B_{2}\{X$ ) withby
$X^{\mathrm{o}}arrow X$$\circ$$A_{2}(X)\circ$$||\varphi||_{B_{2}([mathring]_{x})}\leq||\varphi||B_{2}(X)$the monotonicity of hyperbolic metrics:
In fact, the following holds.
Proposition 2.2.

$\rho_{X}\leq\rho_{X}\circ.\dot{\mathrm{T}}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{f}_{0}\mathrm{r}\mathrm{e}$$\kappa(X^{\circ})\leq\kappa(X)\leq 3\kappa([mathring]_{X})$Proof. It suffices to show that

, we have$\kappa(X^{\circ})\leq\kappa(X)$

.

.

. Let
be the mirror imageof $X$ , where
denotes the lower half plane. By the Bers-Greenberg isomorphism
theorem [2], the inclusion map :induces the natural biholomorphism:
between Teichm\"uller spaces of. Through the Bers
andembedding, we regard the Teichm\"uller spaces $T(X^{*})$ andare embedded in$B_{2}(X)$ and, respectively. Here, we note that the Fr\’echet derivative:$B_{2}(X)arrow B_{2}(X)\circ$ofat is nothing but the pull-back by the inclusion map :as quadratic differentials. The Ahlfors-Weill theorem and the Nehari-Kraus
theorem imply thatand $T(X^{*})\subset B_{2}(X)_{6}$ , thus
, whereopen
denotes theball of the Banach spacecentered at
and with radius as usual. Sincemaps to , Schwarz’s lemma yields
that. This is equivalent to the desired
assertion.
$\kappa(X)\leq 3\kappa(X^{\circ})$$X^{*}$$\mathbb{H}^{*}/\hat{\Gamma}$$\mathbb{H}^{*}$$x^{\mathrm{o}}*arrow X^{*}$$\overline{j}$$\overline{j}^{\#}$$T(X^{*})arrow T([mathring]_{X}^{*})$$[mathring]_{X}^{*}$$X^{*}$$T([mathring]_{X}^{*})$$B_{2}(X)\circ$$d_{0}\overline{j}\#$$\overline{j}^{\#}$

$0$

$j$

$j^{*}$$X^{\mathrm{o}}arrow X$$B_{2}([mathring]_{X})_{2}\subset T([mathring]_{X}^{*})$$B_{2}(X)_{6}$$(\overline{j}\#)-1(B_{2}([mathring]_{X})_{2})\subset$$B_{2}(*)_{r}$$0$$B_{2}(*)$$(\overline{j}^{\#})^{-1}$

$r$

$0$

$0$

$||d_{0}(\overline{j}\#)^{-1}||=||(d_{0\overline{j}^{\#})}-1|.|=||(j^{*})^{-1}||\leq 3$$\square$In this article, we will say that $X$ is of Lehner type if $L^{*}(X)>0$ . Generally,for (possibly disconnected) 1-dimensional complex orbifold $X$ , we define $L(X)=$
inf $L(X_{0})$ and $L^{*}(X)=$ inf, where the infima are taken over all connected$X$
componentsof , and we say that $X$ is modulated or of Lehner type if $L(X)>0$
or $L^{*}(X)>0$ , respectively.
A closed set $C$ in the Riemann sphere with $\neq C\geq 3$ is called uniformly perfect
if the complement
is modulated.
$L^{*}(x_{0})$$X_{0}$$\hat{\mathbb{C}}$$R=\hat{\mathbb{C}}\backslash C$

80
3. KLEINIAN GROUPS AND MAIN RESULTS
is a discrete
Let be a Kleinian group acting on the Riemann sphere , i.e.,
is not empty. We denote bywhose region of discontinuity
subgroup of. In the following, we always assume thatthe limit set of , i.e.,. Thenis known to be a perfectis non-elementary, in ohter words,
set. The quotient space $X(G)=\Omega(G)/G$ has a natural hyperbolic l-dimensional
complex orbifold structure with which the canonical projection : $\Omega(G)arrow X(G)$ isa covering map.
and be a connected component of
Let $X$ be a connected component of$\pi^{-1}(X)$ . If
is a holomorphic universal covering map, then clearly
is a holomorphic universal covering map for $X$ . Let $H=H_{\Omega}$ be the component. And
subgroup of corresponding to , i.e.,
denote by and the covering transformation groups of and , respectively. Then,
we have a natural exact sequence
$G$

$G$

$\hat{\mathbb{C}}$$\Omega(G)$$\mathrm{P}\mathrm{S}\mathrm{L}(2, \mathbb{C})$$G$$\Lambda(G)$$\Lambda(G)=\hat{\mathbb{C}}\backslash \Omega(G)$$G$$\neq\Lambda(G)\geq 3$$\Lambda(G)$$\pi$$\dot{X}(G)$$\Omega$$q:\mathbb{H}arrow\Omega$$p^{=\pi\circ}q$$G$.

$H=\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}_{c}(\Omega)=\{g\in G;g(\Omega)=\Omega\}$$\Omega$$\overline{\Gamma}$$\Gamma$$p$$q$$1arrow\Gammaarrow\overline{\Gamma}arrow\chi Harrow 1$of group homomorphisms, here can be described as
For each $h\in H$ , we denote by the translation length of$q\circ\gamma=\chi(\gamma)\circ q$$\chi$$l_{h}$(3.1)$l_{h}:= \inf_{z\in\Omega}d_{\Omega(}Z,$$h(z))$

$h$

in

:

$\Omega$

for all$\gamma\in\overline{\Gamma}$

.

.

Let

{ $h\in H;h$ has a fixed point in }{ $h\in H;h$ is the identity or elliptic and $l_{h}=0$ } and{ $h\in H;h$ corresponds to a puncture of $X=\Omega/H$ }{ $h\in H;h$ is parabolic and $l_{h}=0$ }.

$H_{e}=$$\Omega$$=$$H_{p}=$$=$We remark that there may exist an elliptic or parabolic element of $H$ with $l_{h}>0$ .
Such a parabolic element is sometimes called an accidental parabolic transformation.
as follows:
Now we define$h$$\lambda_{H},$$\lambda_{H}^{*}$$\lambda_{H}=\inf_{h\in H\backslash H_{\mathrm{e}}}l_{h}$

,

.

and

$\lambda_{H}^{*}=h\in H\backslash (\inf_{H_{p}H_{e}\cup)}l_{h}$Noting that an elliptic or parabolic element in a Fuchsian group always represents a
branch point or a puncture of the quotient surface, similarly we set

inf$\lambda_{\overline{\Gamma}}=$$\overline{\Gamma}\ni h$: of infinite order$l_{h}$

andinf$\lambda_{\Gamma}^{*}\sim=$$\overline{\Gamma}\ni h$$l_{h}$: hyperbolic

.

since
in the same manner, however it holds that
We can define
ordh.
for $H$ be defined by
has no isolated points. Let the number$N_{H}’=+\infty$,
as follows: If
is non-empty, then
And we define the number
otherwise we set $N_{H}’=N_{H}$ .
$\lambda_{\Gamma},$$\lambda_{\Gamma}=\lambda_{\Gamma}^{*}$$\lambda_{\Gamma}^{*}$$N_{H}$$N_{H}’$$N_{H}= \sup_{h\in H_{\mathrm{e}}}$$H_{p}$$\Lambda(G)$81
For these constants, the next result is fundamental for our present aim.

Theorem 3.1.
(3.2)
(3.3)

$\min\{\frac{1}{N_{H}}\lambda_{\Gamma}, \lambda_{H}\}\leq\lambda_{\tilde{\Gamma}}\leq\min\{\lambda_{\Gamma}, \lambda_{H}\}$,$\min\{\frac{1}{N_{H}’}\lambda_{\Gamma}, \lambda_{H}^{*}\}\leq\lambda\frac{*}{\Gamma}\leq\min\{\lambda_{\Gamma}, \lambda_{H}^{*}\}$.

andBy the relation (2.1), we obtain thatandand the similar
relations forandhold, hence the above theorem is equivalent to the following$\lambda_{\Gamma}=L(\Omega)$$\lambda_{\overline{\Gamma}}=L(X)$$L^{*}$$\lambda^{*}$

Theorem 3.2.
(3.4)$\min\{\frac{1}{N_{H}}L(\Omega), \lambda H\}\leq L(X)\leq\min\{L(\Omega), \lambda_{H}\}$(3.5)$\min\{\frac{1}{N_{H}}, L(\Omega), \lambda^{*}\}H\leq L^{*}(X)\leq\min\{L(\Omega), \lambda_{H}^{*}\}$

,

and

.

$N$ and $N’$ by
Now we define
and $\sup_{H}N_{H}’$ ,$H$
respectively, whereruns over all component subgroups of . Remark here thatthe constants
andare determined only by the conjugacy class of $H$
in , i.e., by the component $X=\Omega/H$ of $X(G)$ . Then, we immediately obtain thefollowing$\lambda(G),$ $\lambda^{*}(G),$$\inf_{H}\lambda_{H},$$\inf_{H}\lambda^{*}\sup_{HH}H’ N$$G$$\lambda_{H},$$\lambda_{H}^{*},$$N_{H}$$N_{H}’$$G$

Theorem 3.3. For any non-elementary Kleinian group$G$

it follows that$\min\{\frac{1}{N}L(\Omega(G)), \lambda(c)\}\leq L(X(G))\leq\min\{L(\Omega(G)), \lambda(G)\}$,$\min\{\frac{1}{N},L(\Omega(G)), \lambda^{*}(G)\}\leq L^{*}(X(G))\leq\min\{L(\Omega(G)), \lambda^{*}(G)\}$

and.

Corollary 3.4. Thus we have $L(\Omega(G))\geq L^{*}(X(G))$ . In particular, if $X(G)$ is of
Lehner type thenis uniformly perfect.
$\Lambda(G)$Remark. To guarantee that $\lambda(G)>0$ it is sufficient to assume that
, wheredenotes the injectivity radius ofat (oral communication with
Prof. Matsuzaki). But, this condition seems to be hard to check.
$\sup_{z\in\Omega(G)}\iota(z)<$$\infty$$\iota(z)$$\Omega(G)$$z$

In the case that is analytically finite, i.e., $X(G)$ consists of a finite number ofRiemann orbifolds of finite type, it is easily verified that $L^{*}(X(G))>0$ thus we havethe following collorary.
$G$

Corollary 3.5 (Canary [3]). For an analytically finite $non- elementar_{J}\iota$ Kleinian group, the limit set
is uniformly perfect.

$G$$\Lambda(G)$We should remark that by Ahlfors’ Finiteness Theorem this result produces thefinitely generated case.

82

4. PROOF OF THEO

$R\mathrm{E}\mathrm{M}3.1$

Under the same situation as the theorem, first we prove the next elementary
Lemma 4.1. For any element $h\in H$, we have

.

(4.1)

$\gamma\in x(h)\underline{\inf_{1}}l\gamma=\gamma\in x(\underline{\min_{h1)}}l_{\gamma}=l_{h}$, thenby definition. For any$z=q(\zeta)$ . Then, by the definition of metrics, we have

Proof. Let$\gamma\in\chi^{-1}(h)$$q\mathrm{o}\gamma=h\circ q$$\zeta\in \mathbb{H}$

we put.

$d_{\mathbb{H}}(\zeta,\gamma(\zeta))\geq d\Omega(q(\zeta), q(\gamma(\zeta)))=d_{\Omega(h(Z}Z,))\geq lh$. Now we prove the opposite inequality.
is arbitrary, it follows thatFirst, when $l_{h}=0$ , the covering transformation covers some loop in $X$ around a
singular point or a puncture. Now it is clear that this loop is covered by an elliptic
and $\chi(\gamma)=h$ . Next, weor parabolic element in , so in any case, we haveattaining the infimum in (3.1).
assume that $l_{h}>0$ , then there exists a point$q(\zeta)=z$,
a geodesic joining andthere
exists
point
with
then
Choose a$h(z)$ such that
. Let be a lift of via with initial point (, then
. We note here thatfor some
the terminal point of can be written as$\chi(\gamma)=h$ by definition. Therefore, we haveSince$l_{\gamma}\geq l_{h}$$\zeta$$h$$\overline{\Gamma}$$l_{\gamma}=0$$\gamma$$z\in\Omega$$\zeta\in \mathbb{H}$$z$$\alpha$$d_{\Omega}.(z, h(Z))= \int_{\alpha}\rho_{\Omega}$$\beta$$\alpha$$q$$\gamma\in\overline{\Gamma}$$\beta$$\gamma(\zeta.)$,$d_{\Omega}(z, h(Z))= \int_{\alpha}\rho_{\Omega}=\int_{\beta}\rho_{\mathbb{H}}\geq d_{\mathbb{H}}(\zeta,\gamma(\zeta))$

thus

some

$l_{h}\geq l_{\gamma}$. Combining this with the former part of the proof, we have$\gamma\in\chi^{-1}(h)$. Now the proof is completed.

$l_{h}=l_{\gamma}$for$\square$Now we shall prove the Theorem 3.1. First, we note that if. Therefore we conclude that$\gamma\in\overline{\Gamma}$

is elliptic, then$h=\chi(\gamma)\in H_{e}\backslash \{1\}$$\Gamma\cup\chi^{-}(1H\backslash H_{e})=x-1(1)\cup\chi^{-}1(H\backslash H_{e})\subset\overline{\Gamma}$,

and this immediately yields the right-hand side inequality in (3.3).
which may contribute to the infimum consists
The residual partof the non-ellipitic element such that $h=\chi(\gamma)$ is of finite order . Therefor$\chi^{-1}(1)=\Gamma$, so we have
, hence. By this observation,(3.3).
Noting that any parabolic
we are convinced the validity of the left-hand side ofelement of is mapped to a parabolic one in $H$ with translation length under thehomomorphism , we can show the inequality (3.2) in the same way as above.
$\chi^{-1}(H_{e}\backslash \{1\})\backslash \overline{\Gamma}_{e}$$n$$\gamma$$nl_{\gamma}=l_{\gamma^{n}}\geq\lambda_{\Gamma}$$\overline{\Gamma}$$\gamma^{n}\in$$l_{\gamma} \geq\frac{1}{n}\lambda_{\Gamma}\geq\frac{1}{N_{H}}\lambda_{\Gamma}$$0$$\chi$

83
5. SOME CONSEQUENCES
In this section, we shall exhibit several applications of our theorems. We denote by
the supremum of the moduli of annuli (round annuli, respectively), where the modulus of an annulus is defined here as the number
separating$m$ when this annulus is conformally equivalent to the round annulus$1<$$|z|<e^{m}\}$ and the round annulus means a bounded annulus with concentric circles, then we can define another constantas boundary. Further, if
by, wheredenotes the Euclidean.
to
For these constants, we know the following estimates.
$M_{\Omega(G)}$$(M_{[mathring]_{\Omega}(G)})$$\Lambda(G)$$\{z\in \mathbb{C},$$\infty\in\Lambda(G)$$\inf_{z\in\Omega(G)}\delta(z)\rho_{\Omega(}c)(Z)$$C_{\Omega(G)}$d.istance. f.r.om$\delta(z)$Theorem 5.1 (cf. [10]). For a non-elementary Kleinian group$L(\Omega(G))$ then we obtain that

$G$

$z$

$\Lambda(G)$, if we set $L=$,

$L \leq\frac{\pi^{2}}{M_{\Omega(G)}}\leq\min\{Le^{\iota}, \frac{1}{2}L2\coth 2(\frac{L}{2})\}$$\frac{1}{2}M_{\Omega(G)\Omega}-K\leq M^{\circ}(c)\leq M_{\Omega(G})$,

where $K$ is an absolute constant $\leq 1.7332\cdots$ . Moreover if
that
(5.1)

$\frac{\tanh L/2}{4}\leq C_{\Omega()}c\leq\frac{\sqrt{3}L}{\sqrt{\pi^{2}+4L^{2}}}$$\infty\in\Lambda(G)$, it is also valid.

In particular, $L(\Omega(G))>0$ if and only if.
Furthermore, Pommerenke has given a remarkable characterization of the uniform
perfectness in the words of capacity density.
$M_{\Omega(G)}<\infty$

Theorem 5.2 (Pommerenke [8]).
is uniformly perfect if and only if there
exists a constant $c\in(0,1]$ such thatfor any $a\in\Lambda(G)$$0<r\leq$, where Cap denotes the logarithmic cap.a city and $B(a, r)$anddiam
the closed disk centered at with radius .
$\Lambda(G)$$\mathrm{c}_{\mathrm{a}_{\mathrm{P}}}(\Lambda(G)\cap B(a, r))\geq cr$$(\Lambda(G))$

$a$

$r$

We note thatandcan be taken as the constant[10] . In particular, by virtue of Wiener’s criterion, we have the following$\mathrm{c}_{\mathrm{a}_{\mathrm{P}}}(B(a, r))=r$$2^{-7}e^{-M_{\Omega()}^{\circ}}G$

$c$

(see

$)$

Corollary 5.3. If $L(\Omega(G))>0$ the limit set$\Lambda(G)i\mathit{8}$regular in the sense of Dirichlet.

For general Kleinian groups, at least we can state the following
Corollary 5.4. Let
be a non-elementary Kleinian group. Any loxodromic orparabolic fixed point of is a regular point ofin the sense of Dirichlet.
$G$

$G$

$\Lambda(G)$84

In fact, if
is a fixed point of a loxodromic or parabolic element of , thenis contained by the limit set of some finitely generated non-elementary subgroupis uniformly perfect, by Theorem 5.2, we see thatof containing . Since, which implies(see, for example, Tsuji [11] p.104). Further, we should
the regularity ofatnote that the set of loxodromic fixed points of any non-elementary Kleinian group isdense in the limit set.
Another application of unifom perfectness is concerned with the Hausdorff dimension. The next result is essentially due to J\"arvi-Vuorinen [4].
$z_{0}$$G$$\gamma$$z_{0}$$G$$G_{0}$$\Lambda(G_{0})$$\gamma$$\varlimsup_{rarrow 0}\mathrm{c}_{\mathrm{a}}\mathrm{p}(\Lambda(G)\cap B(z_{0}, r))/r\geq\varlimsup_{rarrow 0}\mathrm{C}\mathrm{a}\mathrm{p}(\Lambda(G_{0})\cap B(z_{0}, r))/r>0$$\Lambda(G)$$z_{0}$

Theorem 5.5 (cf. [10]). The Hausdorff dimension
estimated from below as follows.

$\mathrm{H}-\dim(\Lambda(G))$$\mathrm{H}-\dim(\Lambda(G))\geq\frac{\log 2}{\log(2e^{M_{\Omega(G}})+1)0}(\geq\frac{\log 2}{M_{\Omega(G)}^{\mathrm{O}}+\log 3})$of$\Lambda(.G).c,an$be.

As an immediate consequence of this, we can see that any non-elementary Kleiniangroup has the limit set of posi.t.ive Hausdorff dimemsion.
6.

OF..TRANSLATION LENGTH BY THE TRACEAs an application of (5.1), we present here an estimate of the translation length.
of a loxodromic element of Kleinian group in the region of discontinuity by its$H$$L(\Omega(G))>0$
and
a component
Let be a non-elementary Kleinian group with.
subgroup of corresponding to a component of the region of discontinuity$H$
We set $C= \frac{1}{4}\tanh(L(\Omega(G))/2)>0$ . And let be a loxodromic element of ,, we may. By the M\"obius-invariance of
i.e.,
and
.
assume that has the form $h(z)=\lambda z$ , where. We denote by
In this time, we note thatthe principal branch. Let
of $\log z$ , i.e.,
is the branch ofdetermined by. If we denote by,
the distance from to. Letby (5.1), we haveforandbecausegeodesic
in
be a
arc
from towith minimal length and a lift of. We denote by
via the exponential map
the initial and terminal
point of . By construction, the difference $w_{1}-w_{0}$ is a logarithm of
we note. Then, we computehere$\mathrm{E}_{S\mathrm{T}\mathrm{I}}\mathrm{M}\mathrm{A}T\mathrm{E}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$

$G$

$G$

$\Omega(G)$$\Omega$$h$$\tau=\mathrm{t}\mathrm{r}^{2}(h)\in \mathbb{C}\backslash [0,4]$$l_{h}$$\mathrm{t}\mathrm{r}^{2}(h)$$\tau=(\sqrt{\lambda}+\sqrt{\lambda}^{-1})^{2}=\lambda+\lambda^{-1}+2$$h$$0,$ $\infty\in\Lambda(G)$${\rm Log} z$$\log$${\rm Log} z$$-\pi<\mathrm{I}\mathrm{m}{\rm Log} z\leq\pi$$\zeta={\rm Log}\lambda=\pm{\rm Log}\frac{\tau-2+\sqrt{(\tau-2)^{2}-4}}{2}$$\delta(z)\rho\Omega(Z)\geq C$$\Omega$$z_{0}$$\delta(z)$$z\in\Omega$$z$$\delta(z)\leq|z|$$\mathrm{O}\in\Lambda(G)$$h(z_{0})=\lambda z_{0}$$\mathrm{e}\mathrm{x}\mathrm{p}:\mathbb{C}arrow \mathbb{C}^{*}$$\beta$$w_{0},$ $w_{1}$$\beta$$\backslash \lambda,$$.\mathrm{s}\mathrm{o}$$|w_{1}-\cdot$ $w_{0}|\geq|{\rm Log}\lambda|$$d_{\Omega}(Z_{0}, h(z0))= \int_{\alpha}\rho_{\Omega}(z)|d_{Z|}\geq c\int\alpha\geq\frac{|dz|}{\delta(z)}c\int_{\alpha}\frac{|dz|}{|z|}$$=C \int_{\beta}|dw|\geq C|w_{1}-w\mathrm{o}|\geq C|{\rm Log}\lambda|$

Since$z_{0}$

was arbitrary, we have$l_{h}\geq C|{\rm Log}\lambda|$.

. Now we have proved the next$\Lambda(G)$$\alpha$$\alpha$85
Proposition 6.1. Let $H$ be a component subgroup of a $non- elementa\eta$ Kleiniangroup $G$ and suppose that $C= \frac{1}{4}\tanh(L(\Omega(G))/2)>0$ . Then, for any loxodromicelement $h\in H$, the tmnslation length can be estimated as$l_{h}$$l_{h}\geq C|{\rm Log}\lambda|$

where$\lambda$

is the multiplier of$h,$$i.e.,$,.

$\mathrm{t}\mathrm{r}^{2}(h)=\lambda+\lambda^{-1}+2$

Remark. In general, we camot estimate the translation length
above by the. This can be understood by the existence of accidental parabolic transformations.
$\mathrm{h}\mathrm{o}\mathrm{m}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$7. EXAMPLE
In this concluding section, we present a simple example of (infinitely generated)
Kleinian groups with non-uniformly perfect limit sets. Looking at the Theorem 3.3,
we can guess that if quotient orbifold has arbitrarily short geodesics which are liftedto non-closed curves in the region of discontinuity. In fact, such an example can begiven by infinitely generated Schottky groups as Pommerenke indicated in [8].

Letbe sequences tending to , and $r_{j}>0$ andwithbegiven so that all closed disks $A_{j}=B(a_{j}, r_{j}),$ $B_{j}=B(b_{j}, r_{j})$ are disjoint $(j=1,2, \ldots)$ .
We set, thenandare the isometric circles for and,respectively, thus
is an infinitely generated Schottky group with afundamental domain.
Now we set. Then, we can directly see that, henceis not uniformly perfect if.
$a_{j},$ $b_{j}\in \mathbb{C}$$\infty$$g_{j}(z)=b_{j}- \frac{\alpha_{\mathrm{j}}r^{2}}{z-a_{j}}$$A_{j}$$B_{j}$$\alpha_{j}\in \mathbb{C}$$|\alpha_{j}|=1$$g_{j}$$g_{j}^{-1}$$G=\langle g_{1}, g_{2}, \ldots\rangle$$\mathbb{C}\backslash \bigcup_{j}(A_{j}\cup B_{j})\subset\Omega(G)$$\tilde{r}_{j}=\mathrm{d}\mathrm{i}_{\mathrm{S}}\mathrm{t}(a_{j}, (\bigcup_{k\neq j}A_{k})\cup(\bigcup_{k}B_{k}))>r_{j}$$M_{[mathring]_{\Omega}(G)} \geq\sup_{j^{\tilde{r}_{j}}}/r_{j}$$\Lambda(G)$$\sup_{j^{\tilde{r}_{j}}}/r_{j}=\infty$

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DEPARTMENT OF MATHEMATICS, KYOTO UNIVERSITY, 606-01 SAKYO-KU, KYOTO CITY,
JAPAN-mail address: sugawa@kusm.kyoto-u.ac.jp$E$