Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 18, 1993, 23–29

A PLY INEQUALITY FOR KLEINIAN GROUPS

Carsten Lunde Petersen

Århus Universitet, Matematisk institut DK-8000 Århus C, Denmark; clp@mi.aau.dk

Abstract. We prove an inequality for the multipliers of loxodromic elements in certain Kleinian groups. The inequality is a generalization of Bers inequality (see L. Bers [B, Theorem 3] and C. McMullen [McM, Theorem 6.4]). It is also an analogue of an inequality by Ch. Pommerenke, G. M. Levin and J.-C. Yoccoz for multipliers of repelling periodic orbits of rational maps [Po], [L], [Y] and [Pe].

1. Introduction

Let Γ be a Kleinian group. Suppose Ω_{0} is a simply connected domain of discontinuity for Γ and ϕ: Ω_{0} → D is a Riemann map. Denote the stabilizer of Ω_{0} by Γ_{0} and denote by Γ̂_{0} the Fuchsian group conjugate to Γ_{0} by ϕ. We will call Γ̂_{0} a flat model of Γ on Ω_{0} .

Let γ ∈ Γ_{0} and γ̂ ∈ Γ̂_{0} be conjugate, i.e. γ̂ = ϕ ∘ γ ∘ ϕ^{−1} . If γ is not elliptic, the fixed point(s) of γ is (are) accessible boundary points of Ω_{0} . In fact: (1) If γ̂ is hyperbolic, it stabilizes a unique hyperbolic geodesic α̂. The axis α of γ in Ω_{0} given by α := ϕ^{−1}(α̂) defines accesses to the fixed point(s) of γ; (2) If γ̂ is parabolic, γ is parabolic and the preimage of the line segment between 0 and the fixed point for γ̂ defines an access to the fixed point of γ.

The main theorem of this paper is Theorem C. Theorems A and B are corollaries of Theorem C. They have been singled out to exhibit more clearly the contents of Theorem C. The layout of the paper is to formulate the theorems in the order that their prerequisites are introduced and to leave the proofs to the end.

Acnowledgement. The author would like to thank B. Maskit and B. Branner for their suggestions and support. This paper was written during the author’s stay at I.H.E.S., and the author would also like to thank I.H.E.S. for its hospitality and “Rejse Legat for Matematikere” for financial support during his stay.

1991 Mathematics Subject Classification: Primary 30F40; Secondary 30D05, 30C35.

2. Theorems

Let γ ∈ Γ be loxodromic. Conjugating by a Möbius transform if necessary, we can assume that γ(z) = λz, with |λ| > 1, so that the fixed points are 0 and ∞. Suppose there exist k simply connected domains of discontinuity for Γ, Ω_{1}, …, Ω_{k}, with γ in the intersection of their stabilizers. Then the two fixed points 0 and ∞ are common boundary points of the k domains. Let γ̂_{1}, …, γ̂_{k} be the corresponding elements of the flat models of Γ on Ω_{1}, …, Ω_{k}. Let further ϱ_{1}, …, ϱ_{k} be the multipliers of the repelling fixed points for γ̂_{1}, …, γ̂_{k}; then:

Theorem A. There exists a logarithm L of λ such that

|L| ≤ \frac{2 · sin(θ)}{∑_{j=1}^{k}(1/ log ϱ_{j})},

where θ is the angle between L and 2πi.

For quasi-Fuchsian groups this is the Bers inequality.

Let us next relax the assumptions on γ and introduce the combinatorial rotation number. Suppose there exist a simply connected domain of discontinuity Ω_{0} and q ∈ N such that γ^{q} ∈ Γ_{0} . We will suppose q to be minimal with this property. Let γ̃ be the conjugate ϕ ∘ γ^{q} ∘ ϕ^{−1} . Let α_{0} be the axis of γ^{q} in Ω_{0} . The orbit of α_{0} under γ consists of q disjoint arcs between 0 and ∞. We define a cyclic order on these arcs in the following way. Let C(r) be the circle with center 0 and radius r. For each arc mark its first intersection with C(r) starting from 0. These intersections have a clockwise cyclic order on C(r), and we assign the same order to the arcs. It is easy to see, using the Jordan curve theorem, that the cyclic order of the arcs does not depend on r. Let α_{0}^{0}, …, α_{0}^{q−1} be the q arcs in the orbit of α_{0} under γ labelled counter clockwise. Since the cyclic order does not depend on r, there exists a p ∈ {0, …, q − 1}, (p, q) = 1 such that

γ(α_{0}^{j}) = α_{0}^{(j+p) mod q}.

Definition. The number p/q is called the combinatorial rotation number for γ.

An easy application of the Jordan curve theorem shows that if γ^{q′} (with q′ minimal) is in the stabiliser of another simply connected domain of discontinuity Ω_{1} , then q = q′, and Ω_{1} gives rise to the same combinatorial rotation number p/q. Thus the combinatorial rotation number is well defined.

The motivation for introducing the combinatorial rotation number comes from the theory of iteration of polynomials and, more generally, rational maps of the Riemann sphere. More precisely, it comes from the Yoccoz estimate of the size of limbs of the Mandelbrot set (see [Y]).

Theorem B. Let γ(z) = λz be loxodromic. Suppose γ has combinatorial rotation number p/q and γ^{q} is in the stabilizer of Ω_{0} . Let γ̃ be the conjugate of γ^{q} in a flat model of Γ on Ω_{0} and let ϱ be the eigenvalue of the repelling fixed point for γ̃. Then there exists a logarithm L of λ such that

|L − \frac{p}{q}2πi| ≤ \frac{2 sin(θ)}{q^{2}} · log(ϱ),

where θ is the angle between L − (p/q)2πi and 2πi.

We next introduce the logarithmic density to improve our inequality. This improvement is motivated by Levin ([L]). Let Area(W, η) denote the area with respect to the conformal metric η := |dz|/|z| of some Borel subset W of C. Suppose 0 is a boundary point of the Borel subset U ⊂ C. Let r > 0 be given and suppose the following limit exists:

B := lim_{δ→0} \frac{Area ((U ∩ A(δ, r)), η)}{Area (A(δ, r), η)},

where A(δ, r) = {z ∈ C | δ < |z| < r}. Then B ∈ [0, 1] does not depend on r, and we say that the set U has logarithmic density B at 0. Note that if λU = U for some complex number λ, |λ| > 1, then U has a logarithmic density at 0.

Let γ, Ω_{0} and p/q be as in Theorem B. Suppose further that there exist simply connected domains of discontinuity Ω_{1}, …, Ω_{k} with γ^{q} in their stabilizers and that the orbits of Ω_{0}, …, Ω_{k} under γ are disjoint. Let γ̃_{0}, …, γ̃_{k} be the elements of the flat model of Γ on Ω_{0}, …, Ω_{k} corresponding to γ^{q}. Let further ϱ_{1}, …, ϱ_{k} be the multipliers of the repelling fixed points for γ̃_{0}, …, γ̃_{k}; then:

Theorem C. There exists a logarithm L of λ such that

|L − \frac{p}{q}2πi| ≤ B · \frac{2 sin(θ)}{q^{2} · ∑_{j=1}^{k}(1/ log ϱ_{j})},

where θ is the angle between L−(p/q)2πi and 2πi and B ∈ [0, 1] is the logarithmic density of the set ∪_{j=0}^{k} ∪_{l=0}^{q−1} γ^{l}(Ω_{j}) at 0.

Complement to Theorems A, B and C. Geometrically the inequality in Theorem B indicates that L is contained in the disc D(r + (p/q)2πi, r) and thus

λ ∈ exp (D(r + i2π \frac{p}{q}, r)),

where r = log ϱ/q^{2} and D(r + i2π(p/q), r) is the disc with center r + i2π(p/q) and radius r. A similar interpretation is valid in Theorems A and C.

Figure 1.

The groups with which this paper is concerned do exist. B. Maskit [M1] has used combination theorems to construct Kleinian groups with an element γ of prescribed combinatorial rotation number and a prescribed number of cycles of simply connected domains of discontinuity stabilized by the appropriate iterate of γ.

To get an idea of the capabilities of Theorems A, B and C, let us consider a family {Γ_{ε}} of Kleinian groups which are deformations of some Kleinian group, say Γ_{0} . Suppose each Γ_{ε} has a simply connected domain of discontinuity, Ω_{ε,0} , with stabilizer Γ_{ε,0} (for instance Γ_{ε} could be a function group, in which case Ω_{ε,0} would be invariant). Suppose in addition the Γ_{ε,0} have a common flat model Γ̂_{0} on Ω_{ε,0} . Then Theorem A gives uniform estimates for the multipliers of the loxodromic elements of Γ_{ε,0} , depending only on the flat model Γ̂_{0} .

We also have an immediate consequence of Theorem B. Let Γ be a Kleinian group with a simply connected domain of discontinuity Ω. Further let ψ_{t}: \overline{C} → \overline{C} be a family of quasiconformal deformations with Γ-invariant Beltrami differentials supported in the orbit of Ω under Γ. Let ψ̃_{t}: Γ → PSL(2, C) be the corresponding family of Kleinian group monomorphisms, i.e. ψ̃_{t}(γ) = ψ_{t} ∘ γ ∘ ψ_{t}^{−1} for γ ∈ Γ. Suppose that the conjugate (γ̃)_{t} of ψ̃_{t}(γ^{q}) in the flat model of ψ̃_{t}(Γ) on ψ_{t}(Ω) tends to a parabolic element when t tends to 0. Then it follows from Theorem B that the multipliers of ψ̃_{t}(γ) tend to exp(±p/q · 2πi). Thus, if the coefficients of ψ̃_{t}(γ) converge to the coefficients of some element γ_{0} , i.e. the sequence algebraically converges to γ_{0} for t tending to 0 and p/q ≠ 0/1, then γ_{0} is an elliptic element.

It was pointed out to the author by Maskit that this in conjunction with Chuckrow’s theorem leads to the corollary below. For completeness we also state Chuckrow’s theorem.

Chuckrow’s theorem. Let Γ be a non elementary, finitely generated Kleinian group, and let {ψ_{m}} be a sequence of type-preserving isomorphisms of Γ into PSL(2, C). Suppose that ψ_{m}(Γ) is Kleinian for each m, and that ψ_{m} algebraically converges to ψ: Γ → Γ̃. Then ψ is an isomorphism.

The reader will find a proof of Chuckrow’s theorem in [M2, p. 97].

Corollary. Suppose {ψ̃_{t}: Γ → PSL(2, C)}_{t∈I} is a family of group monomorphisms as described above and p/q ≠ 0/1. Then the family cannot have an algebraic limit when t tends to 0.

Proof. Suppose this is false and that ψ̃ is an algebraic limit of ψ̃_{t} when t tends to 0. Then it follows from Chuckrow’s theorem that ψ̃ is a group isomorphism. On the other hand, we have shown above that the image by ψ̃ of the loxodromic element γ (thus of infinite order), is an elliptic element of (finite) order q. This is a contradiction.

3. Proof of Theorems A, B and C

We need a few basic tools: the modulus of a torus with a non-trivial homotopy class of Jordan curves and a Grötzsch inequality. Let T be a torus isomorphic to C/G, where G := L · Z ⊕ 2πi · Z, Re(L) > 0. Furthermore let Π: C → T denote the corresponding universal covering. A flat metric on T is a conformal metric, which makes Π a local isometry, when C is equipped with a flat metric (proportional to the euclidean). Suppose U is a Borel subset of T and let ϱ be a flat metric on T. Define Area(U) := mes(U, ϱ); then the quotient Area(U)/ Area(T) does not depend on the choice of a flat metric ϱ on T, and we call it the relative flat area of U.

Suppose κ: [0, 1] → T, κ(0) = κ(1) is a non-trivial (i.e. not homotopic to a constant) Jordan curve in the torus T. Let [κ] denote the homotopy class of κ in T. A metric ϱ on T is called admissible if it is conformal Borel metric and l_{ϱ}(κ′) ≥ 1 for all κ′ ∈ [κ]. The modulus for the pair (T, κ) is a conformal invariant defined by mod(T, κ) = inf { mes(T, ϱ) | ϱ admidssible }. The modulus is evidently a conformal invariant. Let us remind the reader that the modulus of an annulus is defined in the same way.

There exists p, q ∈ Z, p and q relatively prime, such that any lift κ̃ of κ to Π satisfies κ̃(1) = κ̃(0) + q · L − p · 2πi. The number σ := q · L − p · 2πi does not depend on the choice of generator L, whereas the number p generally does. We call σ the associated segment of κ or rather of [κ], as it is a homotopy invariant. Changing the orientation of κ if necessary, we can suppose that q ≥ 0. If q = 0, then p = 1. Further, if q > 0, then a change of L to L + 2πi leads to a change of the number p to p + q. Thus for a suitable choice of L we have p ∈ {0, …, q − 1}. With this normalization we call p/q the combinatorial rotation number of [κ].

Grötszch inequality (Annuli in a torus). Let T be a torus and let κ be a non-trivial Jordan curve in T with combinatorial rotation number p/q, 0 < q and associated segment σ. Let {A_{j}}_{j∈J} be any family of disjoint annuli homotopic to κ in T. Denote by B the relative flat area of the family, B := ∑_{j∈J} Area(A_{j})/ Area(T). Then:

∑_{j∈J} mod(A_{j}) ≤ B · mod(T, κ) = B · \frac{2π · sin(θ)}{q · |σ|},

where θ is the oriented angle from σ to 2πi. Furthermore, equality is attained if and only if each of the sets Π^{−1}(A_{j}) is a straight strip in C parallel to σ.

Proof. Let ϱ be the flat metric on T corresponding to 1/|σ| on C. Then ϱ is admissible for both T and each A_{j}. Furthermore, the standard length-area argument shows that mod(T, κ) = mes(T, ϱ). We get

∑_{j∈J} mod(A_{j}) ≤ ∑_{j∈J} mes(A_{j}, ϱ) = B · mes(T, ϱ) = B · \frac{2π · sin(θ)}{q · |σ|}.

Equality is attained if and only if the restriction of ϱ to each A_{j} is the extremal (or flat) metric on A_{j} (see e.g. [L-H, p. 33]), that is, the last condition of the theorem is satisfied.

Proof of Theorem C. Let T be the torus generated by the action of γ on C^{∗}. Furthermore, let Π̂: C^{∗} → T denote the corresponding projection map. Then T is isomorphic to C/G, where G = {n · L + m · 2πi | n, m ∈ Z, exp(L) = λ}. For j = 0, …, k let α_{j} be the axis of γ^{q} in Ω_{j}. Define κ_{j} as the projection of α_{j} by Π̂. Then the κ_{j} are disjoint Jordan curves and thus homotopic in T. Further, the combinatorial rotation number of the κ_{j} equals the combinatorial rotation number p/q for the α_{j}. Let L be the choice of a logarithm of λ, such that the associated segment of the κ_{j} is q · L − p · 2πi. For each j = 0, …, k let A_{j} := Π̂(Ω_{j}), let ϕ_{j}: Ω_{j} → D be a Riemann map and let γ̃_{j} be the conjugate of γ^{q} by ϕ_{j}. Then Ã_{j} := D/γ̃_{j} is an annulus of modulus π/ log(ϱ_{j}), where ϱ_{j} is the multiplier of the repelling fixed point for γ̃_{j}. The map ϕ_{j} induces a biholomorphic map Φ_{j}: A_{j} → Ã_{j}. We have:

a) A_{j} is an annulus homotopic to κ_{j} in T with mod(A_{j}) = π/ log(ϱ_{j}). The preimage of A_{j} by Π̂ equals the orbit of Ω_{j} under γ; thus

b) the A_{j} are disjoint. Furthermore, the conformal metric 1/|z| descends by Π̂ as a flat metric on T, and thus

c) the logarithmic density B of ∪_{j=1}^{k} ∪_{l=0}^{q−1} γ^{l}(Ω_{j}) at 0 equals the relative flat area of ∪_{j=1}^{k}A_{j} in T. Using a), b) and c) in the Grötzsch inequality we get

∑_{j=0}^{k} π/ log(ϱ_{j}) = ∑_{j=0}^{k} mod(A_{j}) ≤ B · \frac{2π · sin(θ)}{q|q · L − p · 2πi|},

where θ is the angle between L and 2πi. This is equivalent to the inequality in Theorem C.

A PLY inequality for Kleinian groups 29

Theorem A is obtained from Theorem C by using B ≤ 1 and p/q = 0/1. Theorem B is similarly obtained from Theorem C by using B ≤ 1 and k = 0.

References

[B] Bers, L.: On boundaries of Teichmüller spaces and on Kleinian groups I. - Ann. of Math. 91, 1970, 570–600.

[L] Levin, G.M.: On Pommerenke’s inequality for the eigenvalue of a fixed point. - Colloq. Math. (Wroclaw) (to appear).

[L-H] Lehto, O., and K.I. Virtanen: Quasiconformal mappings in the plane, Second edition.

- Springer-Verlag, 1973.

[M1] Maskit, B.: Panelled web groups. - In: Kleinian groups and related topics. Proceedings of the Conference held in Oaxtapec, 1981. Lecture Notes in Mathematics 971, Springer-Verlag, Berlin, 1983, 79–108.

[M2] Maskit, B.: Kleinian groups. - Springer-Verlag, 1988.

[McM] McMullen, C.: Iteration on Teichmüller space. - Invent Math. 99, 1990, 425–454.

[Pe] Petersen, C.L.: On the Pommerenke–Levin–Yoccoz inequality. - Preprint, IHES, 1991.

[Po] Pommerenke, Ch.: On conformal mapping and iteration of rational functions. - Complex Variables Theory Appl. 5, 2-4, 1986.

[Y] Yoccoz, J.C.: Sur la taille des membres de l’ensemble de Mandelbrot. - Manuscript, Orsay, 1987.

Received 6 September 1991