Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 81–86

ABELIAN COVERINGS, POINCARÉ EXPONENT OF CONVERGENCE AND HOLOMORPHIC DEFORMATIONS

K. Astala and M. Zinsmeister

University of Helsinki, Department of Mathematics P.O. Box 4 (Hallituskatu 15), FIN-00014 Helsinki, Finland University of Orléans, Department of Mathematics B.P. 6759, F-45067 Orleans Cedex 2, France

Abstract. It is shown that the bottom spectrum λ_{0} of a hyperbolic 3-manifold M and the exponent of convergence δ of the corresponding Kleinian group Γ need not vary real analytically under a holomorphic deformation of the manifold or the group.

1. Introduction

This work is the continuation of [AZ2], [AZ3] where we applied the main analytic result of [AZ1] to study Fuchsian groups Γ and their holomorphic deformations, or families of Kleinian groups Γ_{t} acting on the Riemann sphere with isomorphisms φ_{t}, t ∈ Δ = {|z| < 1}, from Γ onto Γ_{t} with the property that φ_{0} = Id (hence Γ_{0} = Γ) and that the function t ⟼ φ_{t}(γ) is holomorphic for every γ ∈ Γ.

The starting point of the story is the fundamental work of Bowen [B]: If Γ is a cocompact Fuchsian group and {Γ_{t}}_{t∈Δ} a holomorphic deformation of Γ, then for a fixed t, either Γ_{t} is Fuchsian or

(1)  d_{t} > 1

where d_{t} denotes the Hausdorff dimension of the limit set L(Γ_{t}) of Γ_{t}. The author proves this result by reducing the problem to the thermodynamic formalism; he actually ends up with an explicit formula involving the dimension from which he concludes (1). Later Ruelle [R] has shown that the thermodynamic formalism also leads to the important result that

(2)  t ⟼ d_{t} is real analytic in Δ.

In the sequel we study general, not necessarily cocompact Fuchsian groups and we shall refer to (1) and (2) as the Bowen and Ruelle property for d_{t}, respectively. Indeed, in the case of general groups the behaviour of the limit set is more problematic: As a consequence of [AZ1], we proved in [AZ3] that both properties are false if Δ/Γ is conformally equivalent to a Carleson–Denjoy domain Ω, i.e. a domain of the form Ω = C\K where K is a compact subset of R satisfying

1991 Mathematics Subject Classification: Primary 30C62; Secondary 30F40, 58G25.

∀x ∈ K, ∀t ∈]0, diam(K)[, |K ∩ [x − t, x + t]| ≥ εt

for some constant ε > 0, where | · | stands for the Lebesgue measure.

However, there exists also another, and often more natural, possible generalization of the Bowen and Ruelle properties. Recall that each of the groups Γ_{t} admits the canonical Poincaré extension to a group of isometries of the hyperbolic 3-space and then, every group G of isometries of H^{3} has a natural real number attached to it, namely the Poincaré exponent of convergence δ(G). This can be defined, for instance, as the infimum of the real numbers δ such that

(3)  ∑_{g∈G} exp(−δh(g(x), x)) < ∞,  x ∈ H^{3},

where h(x, y) denotes the hyperbolic metric in H^{3} . In the ball model B^{3} , (3) becomes ∑_{g∈G}(1−∥g(0)∥)^{δ} < ∞. In geometric terms, δ(G) measures the volume growth in the manifold H^{3}/G, i.e. if V (x, R) denotes the hyperbolic volume of the ball with center x and radius R, then

δ(G) = 2 − \overline{lim}_{R→∞} \frac{log V (x, R)}{2R}.

If (Γ_{t}) is a holomorphic deformation of the Fuchsian group Γ, let us denote by δ_{t} the exponent of convergence of Γ_{t}; it is always true that δ_{t} ≤ d_{t} ([N]) with equality if Γ is cocompact. However, in general it is not even true that δ_{0} = d_{0} ; the first number can be shown to be the dimension of the conical limit set of the Fuchsian group Γ and it can be much smaller than the whole limit set. This is the case, for example, for groups uniformizing the Carleson–Denjoy domains (see [F]).

The importance of the exponent of convergence lies in its connections to the potential theory on the hyperbolic manifold H^{3}/Γ_{t} and, in particular, with the spectral properties of the Laplace operator Δ. Recall that [S1] for a Riemannian manifold M, the bottom spectrum or the lowest eigenvalue λ_{0}(M) of −Δ on L^{2}(M) is obtained as the infimum of

\begin{matrix}∫ & M & |∇φ|^{2} \\ ∫ & M & |φ|^{2} \end{matrix}

over the class of C^{∞} functions with compact support on M. Note that this number is also the supremum of the real λ for which there exists a positive smooth function f on M satisfying

Δf = λf.

Any holomorphic deformation Γ_{t} is quasiconformally conjugate to the initial Fuchsian group Γ_{0} and hence in particular, Γ_{t} can be expressed as a countable union of geometrically finite groups. For such Kleinian groups acting in H^{3} we have then, see [N], the following result known as the Elstrodt–Patterson–Sullivan theorem:

Theorem 0. If δ is the Poincaré exponent of Γ and M = H^{3}/Γ, then

λ_{0}(M) = \{ \begin{matrix}1, & if δ ≤ 1, \\ δ(2 − δ), & if δ > 1. \end{matrix}

The aim of this paper is to consider the analyticity properties of the exponent δ_{t} or the bottom spectrum λ_{0}(M_{t}), M_{t} = H^{3}/Γ_{t}, for deformations of non-cocompact Fuchsian groups. As the main result we show that the Bowen and Ruelle properties (1), (2) both fail also for δ_{t}, even in case of surfaces that are in a sense very close to compact surfaces.

To be more precise, let S_{0} be a compact surface of genus 3 and G_{0} its uniformizing group; let L_{0} be a normal subgroup of G_{0} such that G_{0}/L_{0} is isomorphic to Z^{3} and let C_{0} = Δ/L_{0} . Then C_{0} is the so called infinite “jungle gym”, that is, C_{0} can be quasi-isometrically embedded into R^{3} as a surface C which is invariant under translations t_{j}, 1 ≤ j ≤ 3, in three orthogonal directions. Moreover, S_{0} ≃ C/⟨t_{1}, t_{2}, t_{3}⟩.

We shall then prove the following

Theorem 1. There exists a holomorphic deformation (Γ_{t}) of L_{0} such that δ_{t} > 1 for t close to 1 and δ_{t} ≡ 1 for t close to 0. As a consequence, neither the Bowen nor the Ruelle property holds for δ_{t}.

Remarks. 1. Theorems 0 and 1 combined prove the existence of a “holomorphic” family of hyperbolic manifolds M_{t}, such that t ⟼ λ_{0}(M_{t}) is not real analytic. The manifolds are uniformly hyperbolic in the sense that the lengths of nontrivial hyperbolic geodesics have a uniform lower bound.

2. An interesting question is to decide if the same property holds for abelian coverings by Z^{d} when d = 1 or 2; the main difference is of course that then the uniformizing group is of divergence type and the convergence property in the case d = 3 will be crucial for our argument.

2. Proof of Theorem 1

Let S_{0} ∼ Δ/G_{0} be as above, let S_{1} ∼ Δ/G_{1} be a second compact surface of genus 3 and choose a quasiconformal homeomorphism f from S_{0} onto S_{1} . This lifts to a homeomorphism \overline{f}: Δ ⟶ Δ conjugating G_{0} to G_{1} . We may assume that f is nontrivial; in other words, that the pair (S_{1}, f) is a non-trivial element of the Teichmüller space T (S_{0}). We obtain therefore a new group L_{1} = \overline{f} ∘ L_{0} ∘ \overline{f}^{−1} and a new gym C_{1} = Δ/L_{1} .

Next, let g be the quasiconformal homeomorphism of the sphere \overline{C} which is conformal outside Δ and whose complex dilatation is equal to the dilatation of \overline{f} inside Δ. Then g maps the unit disk to a Jordan domain Ω and conjugates G_{0} to a quasifuchsian group G which leaves Ω invariant. It follows that Ω/G and ^{c}\overline{Ω}/G are conformally equivalent to S_{1} and S_{0}^{∗}, respectively, where S_{0}^{∗} ∼ ^{c}\overline{Δ}/G_{0} is the mirror image of S_{0} . In fact, we have just described Bers’ simultaneous uniformisation of the two surfaces. By construction, g conjugates L_{0} to a Kleinian group L with ∂Ω as its limit set and we have as well that Ω/L, ^{c}Ω/L are conformally equivalent to C_{1} and C_{0}^{∗}, respectively.

By cocompactness, the Poincaré exponent δ(G) of G is equal to the Hausdorff dimension of ∂Ω and hence by Bowen’s theorem it is strictly greater than 1. Moreover, according to a theorem of M. Rees ([Re]) this quantity is also equal to the Poincaré exponent δ(L) of L, since the group G is an abelian extension of L.

The second step in the proof consists of replacing the gym C_{1} by a surface C_{M} which will be more suitable for our purposes. In order to do so, we first note that there is a natural partition C_{j}^{m}, m ∈ Z^{3} , of C_{j}, j = 0, 1. Namely, as above we can represent C_{j} as a quasi-isometric copy of a surface C ⊂ R^{3} which is invariant under the orthogonal translations t_{1} , t_{2} , t_{3} . Considering the translation group ⟨t_{1}, t_{2}, t_{3}⟩ and its (relatively compact) fundamental domain F in C, we see that the translates of F tile the surface C. Taking the images of the tiles under the corresponding quasi-isometries C ⟶ C_{j} we obtain the desired partition. If then M is a large real number and π: Δ ⟶ C_{0} = Δ/L_{0} is the covering, we build a quasiconformal map f_{M}: Δ ⟶ Δ by requiring that the complex dilatation of f_{M} equals the dilatation of \overline{f} in π^{−1}(⋃_{m∈[−M,M]^{3}} C_{0}^{m}) and vanishes elsewhere in Δ. By construction f_{M} conjugates L_{0} to a Fuchsian group with quotient C_{M}, a surface quasiconformally equivalent to C_{0} and C_{1} .

We are here interested in the simultaneous uniformization of C_{0} and C_{M} which is constructed as in the previous step: Consider the qc homeomorphism g_{M} of the plane which is conformal outside Δ and has the same complex dilatation as f_{M} inside Δ. Using g_{M} we can put

L_{M} = g_{M} ∘ L_{0} ∘ g_{M}^{−1}.

Finally, since the homeomorphisms g_{M} are uniformly quasiconformal and since the mappings f_{M} are constructed so that the complex dilatations of g_{M} converge pointwise to the dilatation of g, after appropriate normalizations the elements g_{M} ∘ γ ∘ g_{M}^{−1} of the group L_{M} converge uniformly on C to g ∘ γ ∘ g^{−1} , i.e. to the corresponding element of L. We can then apply a theorem of Sullivan [S2], or more precisely the proof of Theorem 7 there, to get the estimate

\underline{lim}_{M⟶∞} δ(L_{M}) ≥ δ(L)

for the corresponding Poincaré exponents.

From now on we then fix M large enough so that δ(L_{M}) > 1 and to clarify the notations, we drop the subscript M.

In conclusion, we have shown the existence of a quasifuchsian conjugate L of the original gym group L_{0} such that firstly, δ(L) > 1 and secondly, the complex dilatation of the conjugating quasiconformal mapping g is compactly supported (mod L_{0} ).

For the third and main step, we begin by noticing that the group L_{0} is of convergence type, i.e. its Poincaré series converges at the exponent δ = 1 (see [LS]). Let μ be the dilatation of g; by construction, for a large (hyperbolic) radius R, this function is supported in ⋃_{γ∈L_{0}} B(γ(0), R). Hence the same reasoning as in [AZ2] implies that for some constant C > 0,

∀r > 0, ∀z ∈ ∂Δ, ∫_{D(z,r)} \frac{|μ(ζ)|^{2}}{1 − |ζ|} dζ d\overline{ζ} ≤ Cr,

or in other words, |μ|^{2}/(1 − |ζ|) dζ d\overline{ζ} is a Carleson measure in Δ.

We shall then define a holomorphic deformation of L_{0} by

(4)  𝒢_{t} = F_{t} ∘ L_{0} ∘ F_{t}^{−1},

where F_{t} is the (normalized) qc mapping with dilatation tμ. Hence 𝒢_{0} = L_{0} , 𝒢_{1} = L and so for this deformation δ_{1} > 1, δ_{0} = 1. On the other hand, the Carleson measure condition implies (see [Se]) that for t near 0 the limit set L(𝒢_{t}) = F_{t}(∂Δ) is rectifiable and therefore has dimension d_{t} = 1. This bounds also the Poincaré exponent δ_{t} = δ(𝒢_{t}) which remains always smaller than the dimension of the limit set. To conclude the proof, we must hence show that in fact δ_{t} ≥ 1 for |t| small.

We first argue that the conical limit set 𝒞_{t} of 𝒢_{t} is equal to F_{t}(𝒞_{0}), the image under F_{t} of the conical limit set 𝒞_{0} of L_{0} . This follows, for instance, from a result of Tukia (as presented in [DE], Theorem 5): For |t| small, F_{t} can be extended to a hyperbolic quasi-isometry of H^{3} in such a way that we still have the conjugation (4) for the corresponding groups acting on H^{3} . An alternative way to prove F_{t}(𝒞_{0}) = 𝒞_{t}, pointed out by the referee, is to apply the characterization of 𝒞_{t} as the set of points of approximation of 𝒢_{t}. That is, by [BM] z ∈ 𝒞_{t} if and only if there is a sequence {g_{m}}_{1}^{∞} ⊂ 𝒢_{t} such that |g_{m}(z) − g_{m}(x)| ≥ ε > 0, uniformly on compact subsets of \overline{C}\z. This notion is clearly invariant even under a topological conjugacy.

Next, recall that in all dimensions, for a Kleinian group the Hausdorff dimension of the conical limit set is smaller than or equal to the Poincaré exponent and that the equality holds in case of Fuchsian groups ([N, pp. 154 and 159]). Therefore we can deduce from M. Rees’ theorem that dim_{H}(𝒞_{0}) = δ(L_{0}) = δ(G_{0}) = 1. But now, as a mapping of \overline{C}, F_{t} is conformal outside Δ and so according to a theorem of Makarov [M], dim_{H}(F_{t}E) ≥ 1 for any set E ⊂ ∂Δ of Hausdorff dimension 1. In particular, we obtain that δ_{t} ≥ dim_{H}(𝒞_{t}) ≥ 1 for all |t| sufficiently small.

86 K. Astala and M. Zinsmeister

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Received 2 June 1993