Yoshida, Naoshi

Osaka City University Repository

Yoshida, N.
Osaka J. Math.
39 (2002), 89–96
A SEQUENCE IN THE CLASSICAL SCHOTTKY SPACE
NAOSHI YOSHIDA
(Received June 5, 2000)
1. Introduction
Let M be the topological group of all linear fractional transformations. Its multi-
plication is the composition of mappings and its topology is the uniform convergence
topology on the extended complex plane Ĉ.
Let be a positive integer. We denote the free group with basis {1 . . . } by
. The mapping from θ ∈ Hom( M) to (θ(1) . . . θ( )) ∈ M is bijective. We
give Hom( M) the topology such that this bijection is a homeomorphism. When
θ ∈ Hom( M) is a monomorphism, θ−1 is the inverse of the isomorphism θ whose
range is restricted to Im θ. For ϕ ∈ M and θ ∈ Hom( M), we define ϕθ ∈
Hom( M) to be (ϕθ)( ) = ϕ ◦ θ( ) ◦ ϕ−1 for every in . In this way, M acts
on Hom( M).
Let be a positive integer greater than one. Define the Schottky space S of rank
to be
S = {θ ∈ Hom( M) | Im θ is a Schottky group of rank }
S is M-invariant. The Schottky space of rank defined in Chuckrow [2] is S /M.
But the results of Chuckrow [2] which we use also hold for the Schottky space in
our sense. We denote by ∂S the boundary of S in Hom( M). An element of ∂S
is called a cusp if its image has parabolic transformations. The following results are
shown in Chuckrow [2]:
(1) S is open and connected in Hom( M) (Chuckrow [2, Lemma 5]).
(2) Every element of ∂S is a monomorphism and has an image without elliptic trans-
formations (Chuckrow [2, Theorem 4]).
(3) If θ ∈ ∂S is not a cusp, then Im θ does not act discontinuously on any open
subset of Ĉ (Chuckrow [2, Theorem 5]).
Define the classical Schottky space S0 of rank to be
S0 = {θ ∈ Hom( M) | Im θ is a classical Schottky group of rank }
Let S0 be the closure of S0 in Hom( M). If θ belongs to ∂S ∩ S0, then Im θ acts
2000 Mathematics Subject Classification. Primary 30F40; Secondary 32G15, 37F35.
90 N. YOSHIDA
discontinuously on some open subset of Ĉ (Marden [4, Proposition 3.1]). Thus every
element of ∂S ∩ S0 is a cusp.
For each loxodromic transformation , we denote the multiplier of by λ( )
(|λ( )| > 1). The main result of this paper is as follows:
Theorem. Let be an integer greater than one. If a sequence {θ }+∞
=1 in S0 con-
verges to θ in ∂S ∩S0 as tends to +∞, then for each parabolic transformation ϕ of
Im θ, λ(θ ◦θ−1(ϕ)) converges to 1 conically as tends to +∞. Namely, λ(θ ◦θ−1(ϕ))
converges to 1 and { |λ(θ ◦ θ−1(ϕ))− 1|
|λ(θ ◦ θ−1(ϕ))| − 1
}+∞
=1
is bounded.
Using McMullen [7, Theorem 7.3], we obtain the following:
Corollary. Let be an integer greater than one. If a sequence {θ }+∞
=1 in S0
converges to θ in ∂S ∩ S0 as tends to +∞, then
(1) Im θ converges to Im θ geometrically;
(2) the limit set of Im θ converges to the limit set of Im θ in the sense of Hausdorff
convergence;
(3) the Patterson-Sullivan measure of Im θ converges to the measure of Im θ weakly;
(4) the critical exponent of Im θ converges to the critical exponent of Im θ,
as tends to +∞.
In section 2, we will recall the definition of a Schottky group, and we will also
prove a lemma. In section 3, we will prove our theorem. In section 4, we will show
that S0 in our theorem cannot be replaced with S even if θ belongs to ∂S ∩ S0.
The author is deeply grateful to Professor Hiroki Sato for his valuable suggestions
and encouragement. Also he is gratitude to Professors Katsuhiko Matsuzaki and Yoshi-
hide Okumura. Thanks are due to the referees for their careful reading and valuable
suggestions.
2. Schottky Groups
Let be an integer greater than one. A subgroup of M is a Schottky group
of rank if there exist a set of generators 1 . . . of and 2 mutually disjoint
Jordan curves 1 −1 . . . − on Ĉ which satisfy the following conditions:
(1) 1 −1 . . . − bound a 2 -ply connected region .
(2) For each in {1 . . . }, maps onto − .
(3) For each in {1 . . . }, ( ) and are mutually disjoint.
A SEQUENCE IN THE CLASSICAL SCHOTTKY SPACE 91
In the above definition, if Jordan curves can be replaced with circles, then is called
a classical Schottky group of rank . A Schottky group of rank is free of rank ,
purely loxodromic and acts discontinuously on some open subset of Ĉ.
EXAMPLE (cf. McMullen [6, Theorem 3.1]). For each positive integer , let
1 . . . +1 be circles on Ĉ which bound an ( + 1)-ply connected region ( ≥
2). Suppose that 1 . . . +1 converge to circles 1 . . . +1 as tends to +∞,
respectively; 1 . . . +1 may be tangent but cannot intersect. Define θ , θ ∈
Hom( M) to be
θ ( ) = ρ +1 ◦ ρ θ( ) = ρ +1 ◦ ρ for every in {1 . . . }
respectively, where ρ and ρ are the reflections in and on Ĉ, respectively
( = 1 . . . + 1). It is shown that {θ }+∞
=1 is contained in S0 and converges to θ as
tends to +∞. If ϕ ∈ Im θ is parabolic, then there exist , ∈ {1 . . . +1} such that ϕ
and ρ ◦ ρ are conjugate in the group generated by ρ1 . . . ρ +1 (in this case, and
are tangent). Since the composite of two reflections in two mutually disjoint circles
is hyperbolic, λ(θ ◦ θ−1(ϕ)) is real for every . Therefore, λ(θ ◦ θ−1(ϕ)) converges
to 1 conically as tends to +∞: this is a special case of our theorem.
We notice the following:
Lemma 1 (Marden [4, Lemma 4.1]). Suppose that is a Schottky group and
that , and are three distinct limit points of . Fix a region as in the above
definition of a Schottky group. Then there exists one and only one ϕ ∈ such that ,
and belong to three distinct components of Ĉ− ϕ( ).
In order to prove our theorem, we will prove the following lemma.
Lemma 2. Let be a classical Schottky group. Suppose that and belong
to and have no common fixed points. Let , and be the repulsive fixed point of
, the attractive fixed point of and the attractive fixed point of , respectively. Then
there exist two closed disks and in Ĉ which have the following properties:
(1) and contain and , respectively and they do not intersect each other.
(2) ( ) contains and and it does not contain .
(3) contains at least one of ( ) and ( ).
Proof. Let be the rank of . Suppose that is a region as in the above defi-
nition of a Schottky group. Since is classical, we may assume that every component
of ∂ is a circle. Note that , and are limit points of . By Lemma 1, there ex-
ists ϕ ∈ such that , and belong to three distinct components of Ĉ−ϕ( ). Let
, and be components of Ĉ− ϕ( ) which contain , and , respectively. By
92 N. YOSHIDA
the definitions of and , we can show that ( ) contains Ĉ− and does not con-
tain . In particular, ( ) contains and . If the repulsive fixed point of does
not belong to (or ), then ( ) (or ( )) belongs to . Thus we can put =
and = .
3. Proof of Theorem
Choose a loxodromic transformation ψ of Im θ which does not fix the fixed point
of ϕ. We define ϕ = θ ◦ θ−1(ϕ) and ψ = θ ◦ θ−1(ψ) for each . Note that ϕ and
ψ have no common fixed points. Let and be the repulsive fixed point of ϕ
and the attractive fixed point of ϕ , respectively. We write for λ(ϕ ). Clearly,
converges to 1.
Choose an element γ of M such that γ ◦ ϕ ◦ γ−1( ) = /( + 1). Both γ( ) and
γ( ) converge to 0 as tends to +∞. We assume that is sufficiently large such that
neither γ( ) nor γ( ) is ∞. For each , define γ ∈M to be
γ ( ) = 1−
γ( )− γ( ) (γ( )− γ( ))
We write
γ ◦ ϕ ◦ γ−1( ) = +
+
( − = 1)
for each . Note that 6= 0 and that 2 converges to 1. Since γ( ) and γ( ) are the
solutions of the quadratic equation 2 − ( − ) − = 0,
(γ( )− γ( ))2 = (γ( ) + γ( ))2 − 4γ( )γ( ) = ( + )
2 − 4
2
Using ( + )2 = + −1 + 2, we have
(γ( )− γ( ))2 = ( − 1)
2
2
Since both and 2 converge to 1,(
1−
γ( )− γ( )
)2
=
2 −→ 1 ( −→ +∞)
Thus γ converges to γ, or some subsequence of {γ } converges to −γ, where
(−γ)( ) = −(γ( )). Considering fixed points and multipliers, we can show γ ◦ ϕ ◦
γ−1( ) = /( + ). Since γ ◦ϕ◦γ−1( ) = /( +1) and converges to 1, γ converges
to γ as tends to +∞.
A SEQUENCE IN THE CLASSICAL SCHOTTKY SPACE 93
Let σ ∈M map to 1/ . Define and to be
= σ ◦ γ ◦ ϕ ◦ γ−1 ◦ σ−1 and = σ ◦ γ ◦ ϕ ◦ γ−1 ◦ σ−1
respectively. Then ( ) = +1 and ( ) = +1. Note that 1/(1− ) is the repulsive
fixed point of . Define and to be
= σ ◦ γ ◦ ψ ◦ γ−1 ◦ σ−1 and = σ ◦ γ ◦ ψ ◦ γ−1 ◦ σ−1
respectively. Clearly, converges to as tends to +∞. Let and be the at-
tractive fixed points of and , respectively. Note that neither nor is ∞. By
Lemma 2, there exist two closed disks and in Ĉ for each which have the
following properties:
(1) and contain 1/(1− ) and , respectively and they do not intersect each
other.
(2) ( ) contains and and it does not contain ∞.
(3) contains at least one of (∞) and (1/(1− )).
From (2), both and are contained in C. We put
= { ∈ C | | − α | ≤ ρ }
We easily obtain
( ) = { ∈ C | | − ( α + 1)| ≤ ρ | |}
From ⊂ ( ), we deduce that
|α ( − 1) + 1| ≤ ρ (| | − 1)
Let be the ray which has α as its initial point and which passes through the center
(in the Euclidean sense) of . Suppose that crosses ∂ at ′ , ∂ at and ,
and (∂ ) at ′ ( lies between ′ and ). Under this condition,
| − | ≤ | ′ − ′ | = | ′ − α | − ρ ≤ |α ( − 1) + 1| + ρ | | − ρ
Using |α ( − 1) + 1| ≤ ρ (| | − 1), we have
| − | ≤ 2ρ (| | − 1)
We assume that is sufficiently large such that the following inequalities are satisfied:
| − | < | − (∞)|
4
| (∞)− (∞)| < | − (∞)|
4
94 N. YOSHIDA∣∣∣∣ (∞)− ( 11−
)∣∣∣∣ < | − (∞)|4
From these inequalities, we obtain
| − (∞)|
2
< | − (∞)|
| − (∞)|
2
<
∣∣∣∣ − ( 11−
)∣∣∣∣
Since contains and at least one of (∞) and (1/(1− )), and | − | is
the diameter (in the Euclidean sense) of ,
| − (∞)|
2
< | − |
Since | − | ≤ 2ρ (| | − 1),
| − (∞)| < 4ρ (| | − 1)
Using this inequality and |α ( − 1) + 1| ≤ ρ (| | − 1), we have
1 ≥ |α ( − 1) + 1|
ρ (| | − 1)
≥ |α |
ρ
| − 1|
| | − 1 −
1
ρ (| | − 1)
>
|α |
ρ
| − 1|
| | − 1 −
4
| − (∞)|
Since does not belong to ,
1 <
| − α |
ρ
≤ | |
ρ
+
|α |
ρ
<
4| |(| | − 1)
| − (∞)| +
|α |
ρ
Since | |(| | − 1) converges to 0 as tends to +∞, |α |/ρ is greater than 1/2 for
sufficiently large . Therefore,
| − 1|
| | − 1 <
ρ
|α |
(
1 +
4
| − (∞)|
)
< 2
(
1 +
4
| − (∞)|
)
for sufficiently large . This completes the proof.
4. Convergence of critical exponents
Let 3 be the unit ball model of three-dimensional hyperbolic space, and let ∂ 3
be the sphere at infinity of 3. M acts naturally on both of 3 and ∂ 3. A discrete
A SEQUENCE IN THE CLASSICAL SCHOTTKY SPACE 95
subgroup of M acts on 3 discontinuously. A discrete subgroup of M is called geo-
metrically finite if there exists a finite-sided fundamental polyhedron for its action on
3 and geometrically infinite otherwise. A Schottky group is geometrically finite.
Let be a discrete subgroup of M. Define the critical exponent δ( ) of to be
δ( ) = inf
{
α ≥ 0
∣∣∣∣ ∑
∈
exp(−αρ(o (o))) < +∞
}
where o = (0 0 0) and ρ(o (o)) is the hyperbolic distance between o and (o). Fur-
thermore, suppose that is geometrically finite. Then, there exists one and only one
Borel probability measure µ on ∂ 3 such that it is supported on the limit set of
and that for every in and every Borel subset of ∂ 3, the following equality
holds:
µ( ( )) =
∫
| ′( )|δ( ) µ( )
where | ′( )| is the linear distortion of at in the spherical metric on ∂ 3 (Sullivan
[8, Theorem 1]). We call this µ the Patterson-Sullivan measure of .
Let be an integer greater than one. For every θ in ∂S , Im θ is discrete (Marden
[4, Lemma 2.2]). Using McMullen [7, Theorem 7.3], we obtain the following:
Proposition. Suppose that a sequence {θ }+∞
=1 in S converges to a cusp θ as
tends to +∞ and that Im θ is geometrically finite. If for each parabolic transformation
ϕ of Im θ, λ(θ ◦ θ−1(ϕ)) converges to 1 conically as tends to +∞, then
(1) Im θ converges to Im θ geometrically;
(2) the limit set of Im θ converges to the limit set of Im θ in the sense of Hausdorff
convergence;
(3) the Patterson-Sullivan measure of Im θ converges to the measure of Im θ weakly;
(4) the critical exponent of Im θ converges to the critical exponent of Im θ,
as tends to +∞.
For every θ in ∂S ∩ S0, Im θ is geometrically finite (Jørgensen, Marden and
Maskit [3]). Hence from this we obtain the corollary stated in the introduction.
Finally, we will show that S0 in our theorem cannot be replaced with S . If θ ∈
∂S is not a cusp, then Im θ is geometrically infinite. Using Mostow rigidity, we can
prove this claim (see, for example, Matsuzaki and Taniguchi [5, Theorem 4.25]). If
a sequence {η }+∞
=1 in S converges to η and if Im η is geometrically infinite, then
δ(Im η ) converges to 2 as tends to +∞ (Bishop and Jones [1, Theorem 6.2]). It is
essentially proved in Chuckrow [2] that ∂S removed all cusps is dense in ∂S . Con-
sequently, by diagonal method, for each θ in ∂S , there exists a sequence {θ }+∞
=1 in
S such that θ converges to θ and δ(Im θ ) converges to 2 as tends to +∞. On
the other hand, if a discrete subgroup of M is geometrically finite and if the limit
96 N. YOSHIDA
set of does not coincide with Ĉ, then δ( ) is less than 2 (Sullivan [8, Theorem 1]).
Therefore, S0 in our theorem cannot be replaced with S even if θ belongs to ∂S ∩S0.
References
[1] C.J. Bishop and P.W. Jones: Hausdorff dimension and Kleinian groups, Acta Math. 179 (1997),
1–39.
[2] V. Chuckrow: On Schottky groups with applications to kleinian groups, Ann. of Math. 88
(1968), 47–61.
[3] T. Jørgensen, A. Marden and B. Maskit: The boundary of classical Schottky space, Duke Math.
J. 46 (1979), 441–446.
[4] A. Marden: Schottky groups and circles, Contributions to Analysis, Academic Press, New York
and London, 273–278, 1974.
[5] K. Matsuzaki and M. Taniguchi: Hyperbolic Manifolds and Kleinian Groups, Oxford Univ.
Press, Oxford, 1998.
[6] C.T. McMullen: Hausdorff dimension and conformal dynamics III: Computation of dimension,
Amer. J. Math. 120 (1998), 691–721.
[7] C.T. McMullen: Hausdorff dimension and conformal dynamics I: Strong convergence of
Kleinian groups, J. Diff. Geom. 51 (1999), 471–515.
[8] D. Sullivan: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite
Kleinian groups, Acta Math. 153 (1984), 259–277.
Graduate School of Science and Engineering
Shizuoka University
836 Ohya, Shizuoka 422–8529, Japan


A sequence in the classical Schottky space

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A sequence in the classical Schottky space

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