Holomorphic Families of QuasiFuchsian Groups
K. Astala
1
M. Zinsmeister
Introduction
Let Γ ⊂ Möb(C) be a discrete group of Möbius transformations
g(z) =
az + b
cz + d
of the extended complex plane. We denote by L(Γ) its limit set, i.e. the set of
accumulation points of orbits of Γ, and assume that L(Γ) 6= C.
A holomorphic variation of the coefficients associated to the elements g ∈ Γ
gives rise to a holomorphic deformation Γλ of Γ: We assume that for each g ∈ Γ
we are given functions ag , bg , cg and dg holomorphic in λ ∈ ∆ = {z ∈ C : |z| <
1}, such that if
ag (λ)z + bg (λ)
gλ (z) =
,
cg (λ)z + dg (λ)
then we have g0 = g and gλ hλ = (gh)λ . Suppose moreover that the groups
Γλ = {gλ : g ∈ Γ} remain discrete; then by a theorem of Jørgensen [J] they
are all isomorphic to Γ. The above holomorphic deformation of a Möbius group
induces also a so called holomorphic motion of the limit set [Su2].
In case Γ is Fuchsian, i.e. Γ preserves a disk or a halfplane D, and the
limit set L(Γ) = ∂D, it was noticed already by Poincare that for small nonFuchsian deformations of Γ the limit set L(Γλ ) remains a Jordan curve but now
it is nowhere smooth. Much later, in fact, by almost a hundred years, Bowen
([Bo], see also [Su1]) proved that if Γ is in addition finitely generated then the
Hausdorff dimension of the limit set dimH (L(Γλ )) > 1. Finally, extending the
method of Bowen, Ruelle [R] showed that for a holomorphic deformation of a
finitely generated group Γ,
λ −→ dimH (L(Γλ ))
is real analytic when Γ is uniformly expanding, i.e. it does not contain parabolic
elements.
Ruelle’s approach to real analyticity, being based on the spectral properties
of the so called transfer operator, depends strongly on the assumption that the
groups are finitely generated. It is therefore of interest to know if the result
1
holds for general groups of Möbius transformations. The purpose of this paper
is to show that the answer to the general question is negative: There are large
families of Fuchsian groups, containing no parabolics or elliptic elements, for
which one can construct holomorphic deformations Γλ , λ ∈ ∆, such that the
Hausdorff dimension of the limit set does not depend real analytically on the
parameter λ.
As usually, we call a group of Möbius transformations quasiFuchsian if the
limit set L(Γ) is a Jordan curve, its complement consists of two quasiconformal
disks Ω1 and Ω2 and no element of Γ interchanges these two components. Then
both Ω1 and Ω2 are conformally equivalent to the open unit disk ∆ and a choice
of the conformal mappings gives a conjugancy of Γ to a pair of Fuchsian groups
G1 and G2 . Furthermore, taking the quotients ∆/Gi we have surfaces which are
quasiconformally equivalent to a (fixed) surface S and thus Γ represents a pair
of points in the Teichmüller space T (S). Conversely, given a Riemann surface
S without nondegenerate boundary arcs, any pair of points of T (S) can be
represented in this manner by a quasiFuchsian group Γ. Hence we can identify
T (S)×T (S) with a subspace of all quasiFuchsian groups; for this and additional
material see [Be].
The surfaces S we shall use in constructing the counterexamples are all planar domains, namely the Denjoy-Carleson domains with a totally disconnected
boundary. Recall that a Denjoy-domain Ω is a subdomain of C which is of the
form Ω = C\E, where E is a compact subset of the real line. In addition, Ω is
Carleson if, writing |.| for the Lebesque measure on R, we have
|E ∩ (x − t, x + t)| ≥ Ct
for a positive constant C and for all x ∈ E, 0 < t < diam(E). Such domains with
a totally disconnected boundary are easy to build, a Cantor set with variable
ratios will do, and clearly if Ω = ∆/G for a Fuchsian group G, then the group
is of the first kind and uniformly loxodromic.
Theorem 1 Let Ω be any Carleson-Denjoy domain with a totally disconnected
boundary. Then there is a holomorphic family of quasiFuchsian groups Γλ ∈
T (Ω) × T (Ω), λ ∈ ∆, such that λ −→ dimH (L(Γλ )) is not real analytic in λ.
In the above theorem the groups Γλ will be conjugates of the group Γ = Γ0 under
a holomorphic family of quasiconformal mappings Fλ : C → C. It follows that
dimH (L(Γλ )) depends continuously on λ. However, the precise or best possible
degree of smoothness remains open.
To prove that the Hausdorff dimension does not vary real analytically we
construct the deformations Γλ so that the dimension of the limit set is not
constant but it is ≡ 1 for |λ| small. We see hence that not even does Bowen’s
estimate dimH (L(Γλ )) > 1 hold for infinitely generated groups. In fact, one
obtains a new proof for the result [AZ1] that there are quasiFuchsian groups Γ
which are not Fuchsian but still the limit set L(Γ) is a rectifiable curve.
2
2
Group constructions
For the proof of Theorem 1, let us first consider Schottky groups Γ generated by
reflections. We are then given a family D of disks Di , 1 ≤ i ≤ n, with pairwise
disjoint closures; the group Γ= ΓD is the group generated by the products
sj o sk , where sj denotes the reflection across ∂Dj . The limit set L(ΓD ) of such
a group is a totally disconnected Cantor set.
Small variations of the disks Dj still keep Γ a Schottky group but general
deformations can degenerate it in many different ways.
We call circle-chain a
S
family D0 of disjoint Euclidean disks Di such that i Di divides the plane into
two components, while for each j, the complement of
[
Di
i6=j
is connected. Clearly the products of reflections in the ∂Di still generate a
Möbius group Γ0 , but now it is quasiFuchsian.
Lemma 2 Let Dt , t > 0, be families of n disjoint disks and let Γt be the
corresponding Schottky groups. If D0 is a circle-chain with Dt → D0 , i.e. if for
individual disks the Hausdorff distance
dH (Dt,i , D0,i ) −→ 0
as t → 0, then
lim inf dimH (L(Γt )) ≥ dimH (L(Γ0 )).
t→0
Proof: Denote by δ(t) the Hausdorff dimension of L(Γt ). Since the Möbius
groups Γt are finitely generated, it follows from [Su1] that there exists a unique
δ(t)-conformal measure, i.e. a probability measure µt supported on L(Γt ) such
that
Z
δ(t)
µt (g(E)) =
|g 0 | dµt
(1)
E
for all g∈Γt , E borelian. Furthermore, L(Γt ) does not support any δ -conformal
measures for δ < δ(t).
When taking the limit of the families Dt as t → 0, we have that L(Γt )
approaches L(Γ0 ) in the Hausdorff metric and the groups Γt converge to Γ0 in
the sense of uniform convergence of the generators. Therefore we can take a
weak limit of the δ(t)-conformal measures µt and obtain a measure µ on L(Γ0 )
that satisfies (1) with the exponent
δ = lim inf δ(t).
t→0
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It follows that
2
δ(0) ≤ lim inf δ(t).
t→0
As the next step towards Theorem 1 we recall Rubel and Ryff’s construction
of the covering group of a Denjoy domain Ω = C\E [RR]. They find first a
conformal mapping f from the upper-half plane onto a domain of the type
U = ∆\
∞
[
Dj
j=0
where the Dj ’s are pairwise disjoint disks with ∂Dj orthogonal to ∂∆; under the
mapping f the arcs ∂Dj ∩ ∆ correspond to the segments of R complementary
to E. The universal covering of Ω is then obtained by extending f −1 to the
whole disk by successive applications of the Schwarz reflection principle.
For simplicity we will assume that one of the disks, say D0 , is the upper
half plane. As above, let sj denote the reflection in ∂Dj ; then F=U ∪ s0 (U) is
a fundamental domain for Ω and Ω=∆/Γ, where Γ is the group generated by
s0 o sj , j ∈ N. In fact, since ∂Dj is the isometric circle for s0 o sj , F is the
Dirichlet fundamental domain of Γ centered at 0.
In the case of Carleson-Denjoy domains, the lengths of the closed geodesics
of Ω = C\E have a positive lower bound and thus, if the disks Dj are from now
on ordered by decreasing radii rj , we have
rk ≤ Cdist(Dj , Dk ),
j ≤ k,
(2)
where dist stands for Euclidean distance and C depends only on the Carleson
constant of E.
We then consider the configuration D consisting of the three disks Dj ,
1 ≤ j ≤ 3, together with their mirror images under s0 . If we suitably translate
them, so that the configuration remains symmetric with respect to the real axis,
we obtain a circle-chain whose corresponding Schottky group Γ0 is quasiFuchsian
but not Fuchsian (This is where we need at least 6 disks).
Results of Bowen and Sullivan ([Bo],[Su1]) assert that, since Γ0 is finitely
generated, one has
dimH (L(Γ0 )) > 1.
We can hence apply Lemma 2; the disks in D can be translated to a configuration
D∗ , symmetric with respect to R, such that the closures of the disks in D∗ are
pairwise disjoint and the induced Schottky group has a limit set of dimension
strictly greater than 1.
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Next, for ζ ∈ ∂∆ and ε > 0 denote by Cε (ζ) the union of two ”cones”,
the convex hull of B(0, ε) ∪ {ζ} and its image reflected across ∂∆. Then the
complement of
[
Cε (ζ)
(3)
S
ζ∈
(D∩∂∆)
D∈D
consists of 6 quadrilaterals. By (2), if ε is small enough these quadrilaterals
contain all the remaining Dj and s0 (Dj ), j ≥ 4. Let Q be one of the quadrilaterals connecting, say, the disks Di and Dk of D and choose from the closures
of the corresponding disks of D∗ the points zi and zk minimizing the distance
between the disks. We construct a new quadrilateral by taking the image of
Q under the similarity TQ which maps the corresponding vertices of Q to zi
and zk . The family D∗ can then be completed to a full chain with the disks
Dj∗ = TQ (Dj ), Dj ⊂ Q.
Let us then consider a diffeomorphism
F : C\
S
S
Dj ∪ s0 (Dj ) −→ C\
Dj∗ ∪ s0 (Dj∗ )
j≥1
j≥1
having a smooth homeomorphic extension to the closures, which is equal to
TQ in each quadrilateral Q and which commutes with s0 . By the action of
the group Γ, F extends to the whole plane as a quasiconformal mapping.
Moreover, it conjugates Γ to a quasiFuchsian group with limit set a Jordan
curve of Hausdorff dimension > 1. Denote by µF the dilatation of F defined
as
∂F
µF =
.
∂F
Finally, we make use of the Carleson measures. For a general domain G this
space, denoted by CM (G), consists of those planar measures ν for which
ν(B(ζ, R)) ≤ CR
(4)
for all ζ ∈ ∂G and all 0 < R < diam(G).
Lemma 3
|µF (z)|2 (1 − |z|2 )−1 dxdy ∈ CM (∆).
Proof: We estimate the integral (4) only inside the unit disk (the exterior
estimate is proved similarly) and denote by ν the restriction of the measure
|µF (z)|2 (1 − |z|2 )−1 dxdy to ∆. By the work of Carleson, see [G, Theorem
II.3.9], the claim is equivalent to proving the inequality
ZZ
Z
|g|dν ≤ C
|g|ds
(5)
∆
∂∆
5
for all g ∈ H 1 (∆), the usual Hardy space.
Since µF is Γ-invariant,
ZZ
XZZ
|g|dν =
|(g o γ)γ 0 |dν.
∆
F
γ∈Γ
On the other hand, as µF vanishes in the quadrilaterals Qj complementary to
(3) and as D is finite, we see that for any ζ ∈ ∂F and 0 < R < 2,
ZZ
dxdy
≤ CR.
ν(F ∩ B(ζ, R)) ≤
2
B(ζ,R)∩(F \∪Qj ) 1 − |z|
But, by (2) F is a Lavrentiev domain, i.e. a bilipschitz image of the unit disk. In
such domains Carleson’s theorem remains true [Z; 1.Lemma,Section 3]. In other
words, as dν F is a Carleson measure for F,
Z
ZZ
|h|ds,
|h|dν ≤ C
F
∂F
for all h ∈ H 1 (F), the L1 -closure on ∂F of holomorphic polynomials. In particular,
ZZ
Z
Z
0
0
|(g o γ)γ |dν ≤ C
|(g o γ)γ |ds = C
|g|ds.
F
∂F
γ(∂F )
Regrouping all the inequalities we obtain
ZZ
Z
|g|dν ≤ C
|g|ds,
∪γ(∂F )
∆
and we can invoke a theorem of Fernandez [F]: the Carleson-Denjoy domains
are exactly those Denjoy domains for which arclength on
[
γ(∂F)
γ∈Γ
is a Carleson measure for the unit disk. This gives (5) and hence proves the
lemma.2
Conclusion: We have constructed for the Carleson-Denjoy domain Ω a
Fuchsian representation Ω = ∆/Γ and for the group Γ a quasifuchsian conjugate
with limit set of dimension > 1 such that µF , the dilatation of the conjugating
mapping, satisfies the Carleson measure condition of Lemma 3. Let us now
denote by Fλ the mapping with dilatation λµF , λ ∈ ∆. Then the group Γλ =
Fλ oΓoFλ−1 is a group of Möbius transformations which depends holomorphically
on λ ([Be]). By a result of Semmes [Se] (c.f. also [AZ2]), Lemma 3 implies that
the limit set L(Γλ ) = Fλ (∂∆) is rectifiable if |λ| is small. This shows that the
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Hausdorff dimension is not real-analytic in λ and thus completes the proof of
Theorem 1 .
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The University of Helsinki and The University of Bordeaux
[email protected]
[email protected]
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