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The Hausdorff Dimension of the Limit Sets of Infinitely Generated Kleinian Math. Proc. Camb. Phil. Soc. (2000), 128, 123 123 Printed in the United Kingdom c 2000 Cambridge Philosophical Society The Hausdorff dimension of the limit sets of infinitely generated Kleinian groups By KATSUHIKO MATSUZAKI Department of Mathematics, University of Michigan Ann Arbor, MI 48109-1109, U.S.A. and Department of Mathematics, Ochanomizu University Otsuka 2-1-1, Bunkyo-ku, Tokyo 112, Japan e-mail: [email protected] (Received 15 September 1997; revised 26 November 1998) Abstract In this paper we investigate the Hausdorff dimension of the limit set of an infinitely generated discrete subgroup of hyperbolic isometries and obtain conditions for the limit set to have full dimension. 1. Introduction Let Γ be a Kleinian group acting on the upper half space model HD+1 of (D + 1)- dimensional hyperbolic space and let Λ(Γ) and Λc(Γ) denote the limit set and the conical limit set of Γ respectively. A limit point x lies in Λc(Γ) if the orbit under Γ of any point in HD+1 approaches x inside a (Euclidean) non-tangential cone with vertex at x. The Hausdorff dimension of Λc(Γ) has geometric interpretations via the exponent of convergence of Γ; for example as the bottom of the spectrum of the hyperbolic Laplacian and the hyperbolic isoperimetric constant on the associated hy- D+1 perbolic manifold NΓ ÷ H =Γ (cf. [1]). In the case of geometrically finite Kleinian groups the limit set consists of Λc(Γ) and at most a countable set (cf. [8]). Thus the dimensions of Λ(Γ) and Λc(Γ) coincide. However, this is not the case in general, and the manner in which the dimension of Λ(Γ) reflects the geometric structure of NΓ (and vice versa) is not well understood. In this paper we begin by obtaining a necessary geometric condition under which the limit set of a general Kleinian group has full Hausdorff dimension. The `geometric condition' depends on points of the convex core CΓ of NΓ lying within a certain distance of the boundary @CΓ. The distance between any two points on NΓ (and therefore CΓ) is measured in terms of the metric on NΓ induced by the hyperbolic metric in HD+1. More precisely, by generalizing an argument of Tukia [13], we prove the following result. Proposition. Let Γ be a Kleinian group acting on HD+1. If there exists a constant 124 Katsuhiko Matsuzaki L< such that all points of the convex core C lie within a distance L of @C , then 1 Γ Γ dim Λ(Γ) 6 α<D; where α is a constant depending only on L and D. In fact, the hypothesis of the proposition can be weakened. By specifying a re- movable set A in NΓ, which is a union of certain regions having simple structure, the conclusion of the proposition remains valid if all points of C A lie within a Γ − bounded distance of @CΓ. We state this result for Fuchsian groups (i.e. for Kleinian groups with D = 1) as Theorem 1 in Section 2 and give a proof in Section 3. There are no difficulties in generalizing the proof of Theorem 1 to higher dimensions and we state the Kleinian group result as Theorem 4 in Section 5. Applications of these theorems to the limit sets of Schottky groups generated by infinite circle packings are given in Sections 5 and 6. As an application of recent work by Bishop and Jones [1], in Section 4 we obtain a sufficient condition under which the limit set of a Fuchsian group Γ has full dimen- sion. Roughly speaking, if there is a sequence of points going away from @CΓ and if @CΓ (possibly with infinitely many components) is not too large then dim Λ(Γ) = 1. In Section 7 we give concrete examples of Kleinian groups which satisfy dim Λc(Γ) <Dbut dim Λ(Γ) = D while the D-dimensional measure is zero. In Section 8 we illustrate our theorems using planar Riemann surfaces. 2. Results for Fuchsian groups In this section we state our theorems for Fuchsian groups. Let Γ be a Fuchsian group acting on the upper half plane model H2 of hyperbolic space. The quotient 2 space H =Γ is denoted by NΓ. For the sake of simplicity we assume that Γ has no elliptic elements, so that NΓ is a hyperbolic surface. This assumption is not essential and can be dispensed with. The limit set Λ(Γ) of Γ is a subset of R^ = R x . Its Hausdorff dimension f1g will be denoted by dim Λ(Γ) and can be regarded as an index which measures the size of the ideal boundary of N relative to the border (R^ Λ(Γ))=Γ. However, in Γ − terms of hyperbolic geometry this border is at infinity and so one considers instead the relative boundary of the convex core CΓ of NΓ. Here CΓ is the smallest convex closed subregion of N such that the inclusion map C , N induces homotopy Γ Γ ! Γ equivalence. This can be represented as CΓ = C(Λ(Γ))=Γ, where the convex hull C(Λ(Γ)) is the smallest convex closed set in H2 whose Euclidean closure contains Λ(Γ). First we obtain a sufficient condition under which dim Λ(Γ) is strictly less than 1. It is known that a Fuchsian group of a compact bordered surface satisfies this relation, and even when CΓ is not compact, the Proposition (in the Introduction) implies that dim Λ(Γ) < 1 if all points of CΓ are within a bounded distance of @CΓ. However, the hypothesis of the Proposition is rather restrictive in that if the hyperbolic surface NΓ has cuspidal ends (arising from parabolic elements of Γ) then the hypothesis is always violated. We wish to allow for regions with simple structure as well as cusp neighbourhoods even if they are far from the boundary of the convex core CΓ.In order to allow for such regions the following notion of removable sets is required. Hausdorff dimension of infinitely generated Kleinian groups 125 2 Definition. For a Fuchsian group Γ, we say that A = n N An in H is a removable set if it satisfies the following conditions: 2 2 S (a) each An is an open set in H which is either a hyperbolic disk, a horodisk tangential to R^ , or a neighbourhood of a complete geodesic within a constant distance; (b) the set A is a disjoint union of A , and A is invariant under Γ. f ng The projection A=Γ is also called a removable set in NΓ. This definition now allows us to weaken the hypothesis of the Proposition without affecting its conclusion. Theorem 1. Let Γ be a Fuchsian group acting on H2. If there exist a constant L< 1 and a removable set A H2 such that all points of C(Λ(Γ)) A lie within a distance L ⊂ − of @C(Λ(Γ)), then dim Λ(Γ) 6 α<1; where α is a constant depending only on L. Next we consider a sufficient condition so that dim Λ(Γ) = 1. For analytically finite Kleinian groups, Bishop and Jones [1] have recently obtained a condition for Λ(Γ) to have full Hausdorff dimension. We modify their argument to analytically infinite groups and prove the following: Theorem 2. Let N be a hyperbolic surface of infinite topological type and c Γ f ngn=1;2;::: the components of @C ( N ). If the hyperbolic lengths l(c ) satisfy Γ ⊂ Γ n 1 l(c ) 2 < n 1 n X then dim Λ(Γ) = 1. 3. Proof of Theorem 1 This section is entirely devoted to a proof of Theorem 1. The proof is based on arguments used by Tukia [13] to prove that the dimension of the limit set of a geo- metrically finite Kleinian group of the second kind is strictly less than the dimension of the sphere at infinity. We first establish some notation. We denote by K the set of all closed intervals Q on the real axis R.ForQ K, let a and b be the end points of Q with 2 Q Q b >a , and d(Q) be the Euclidean length b a .ForQ K and q N,we Q Q Q − Q 2 2 define K(Q; q)( K) to be the set of the q subintervals of Q obtained by dividing ⊂ Q equally. The following are defined for an interval Q and for a given positive constant L; however we omit the index L, for it does not change throughout the proof. We take 2 two points αQ and βQ in H such that their real coordinates satisfy Re αQ =(3aQ + bQ)=4 and Re βQ =(aQ +3bQ)=4 and such that the hyperbolic distance from α (β ) to the geodesic Re z = a Q Q f Qg ( Re z = b )is2L. Then let Q denote the closed rectangle with four vertices at f Qg + z =(aQ; 0), (bQ; 0), (bQ; Im βQ) and (aQ; Im αQ). Note that for any Q and Q0 in K, Q+ and Q+0 are equivalent under a Euclidean similarity. We define a positive number 126 Katsuhiko Matsuzaki (aQ, kd(Q))aQQ bQ (bQ, kd(Q)) Q * Q+ Q [s, t] (aQ, 0)Q (bQ, 0) Fig. 1. Rectangle Q+. k independent of Q as the ratio of the lengths of the vertical side of Q+ to the horizontal side. A smaller closed rectangle in Q+, with four vertices at z = αQ, βQ, 1 1 (Re βQ; 2 kd(Q)) and (Re αQ; 2 kd(Q)), is denoted QM, where kd(Q)=ImαQ =ImβQ (Fig.
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