数理解析研究所講究録 1223 巻 2001 年 31-32

31Locally connected tree-like invariant continua under Kleinian groups

KATSUHIKO MATSUZAKI
お茶の水女子大学理学部数学科

松崎克彦

This note deals with invariant continua under Kleinian groups. Here, acon, and aKleiniantinuum is acompact connected subset of the Riemann spheregroup is adiscrete subgroup of M\"obius transformations of .
Let be acontinuum on . By definition, is locally connected at apoint $y\in L$
if, for any neighborhoodof there exists asmaller neighborhood $V\subset U$ such
$L\capV$thatis connected. We say that is locally connected if it is locally connected
at any point $y\in L$ . We say that acontinuum is tree-like if the complement $S^{2}-L$
is connected and if the interior of is empty. Locally connected, tree-like continua
are characterized by the following property. See [1, Section 10].
$S^{2}$$S^{2}$$L$$S^{2}$$L$$U$$y$ )

$L$

$L$

$L$

Proposition. Let
any points and in

be a locally connected, tree-like continuum. Then, for, there exists a unique arc $xy$ in that connects and .

$L\subset S^{2}$$x$

$y$

$L$

$L$

$x$

$y$

Apoint 4on alocally connected, tree-like continuum is called an endpoint if
there exists no arc Ain such that 4is an interior point of Awith respect to therelative topology on A. This is equivalent to saying that $L-\{\xi\}$ is connected.
Let be aKleinian group. Aloxodromic fixed point of is apoint that is fixedby aloxodromic element of . The limit set
for is the closure of the set of all
loxodromic fixed points of . We say thatis apoint of approximation (or aconical limit point) for if there exists asequence of elements
and distinct
points and onconverge to locallyconverge to andsuch thatuniformly for $z\in S^{2}-\{\xi\}$ . See [3, p.22]. Points of approximation belong to thelimit set. If all the points in the limit set
are points of approximation, then theKleinian group is convex cocompact ASchottky group is aconvex cocompact,
free Kleinian group.
Abikoff [1, Lemma 1] proved that any loxodromic fixed point of aKleinian groupwith the locally connected, tree-like limit set
is its endpoint. In this note,
we extend this result in the following form.
$L$$L$$\Gamma$$\Gamma$$\Gamma$$\Gamma$$\Lambda(\Gamma)$$\xi\in S^{2}$$\Gamma$$\Gamma$$x$$S^{2}$$y$$\gamma_{n}\in\Gamma$$\gamma_{n}(\xi)$$x$$\gamma_{n}(z)$$y$$\Lambda(\Gamma)$$\Gamma$$\Gamma$$\Lambda(\Gamma)$

Theorem. Let be a Kleinian group and a locally connected, tree-like continuum
that is invariant under . Then any point of approximation for is an endpoint$L$$\Gamma$$\Gamma$n-$\Gamma$

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Proof. Suppose that apoint 4of approximation for is not an endpoint of L. Then$\Gamma$there exists an arc Ain such that 4is in its interior. Let and be the endpointsof A. Since is apoint of approximation, there exists asequence of elements
and distinct points and onsuch thatconverge to andconverge
to for $i=1,2$ . The -invariance of implies thatlies inas well as belongs to .
Let be an open neighborhood of such that is not contained in the closureof and that $L\cap V$ is connected. For asufficiently large ,is contained$L\capV$in $V$ but
is not. Since
can be connected with in, we take anarc
there. Thenfor $i=1,2$ are distinct arcs inconnecting and. However, this contradicts the uniqueness of the arc in
as in the previous proposition.
$L$$z_{2}$$z_{1}$$\xi$$\gamma_{n}\in\Gamma$

$x$

$y$

$S^{2}$$\gamma_{n}(\xi)$

$y$

$x$

$L$

$\Gamma$$\gamma_{n}(z_{i})$$L$$\gamma_{n}(\overline{z_{i}\xi})=\overline{\gamma_{n}(z_{i})\gamma_{n}(\xi)}$

$L$

$y$

$V$

$x$

$y$

$V$

$n$

$\gamma_{n}(\xi)$$\gamma_{n}(z_{i})$$\overline{y\gamma_{n}(z_{i})}$$\gamma_{n}(z_{i})$$y$$\overline{y\gamma_{n}(z_{i})}\cup\overline{\gamma_{n}(z_{i})\gamma_{n}(\xi)}$$L$$\gamma_{n}(\xi)$

$y$

$L$

$\square$Corollary. Let be a convex cocompact Kleinian group and a locally connected,
tree-like continuum that is invariant under . Thenis connected.
Maskit [2] considered this problem for the case thatis the limit set for adegenerate Kleinian group and is aSchottky subgroup of . His arguments did
not involve any assumption on local connectivity for , however, acertain property
for seems to have been necessary to complete the proof. It is conjectured thatthe limit set for adegenerate Kleinian group is locally connected (cf. [1]), however,
only partial solutions have so far been obtained.
$\Gamma$$L$$\Gamma$$L-\Lambda(\Gamma)$

$L$

$G$

$G$

$\Gamma$$L$$L$

REFERENCES
1. W. Abikoff, Kleinian groups –geometrically finite and geometrically perverse, Geometryof group representations, Contemporary Math. 74, American Mathematical Society, 1988,
pp. 1-50.
2. B. Maskit, A remark on degenerate groups, Math. Scand. 36 (1975), 17-20.
3. B. Maskit, Kleinian groups, Springer, 1988.
To

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