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 ∞ Γ Γ dim Λ(Γ) 6 α 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) 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 Rˆ = R . Its Hausdorff dimension {∞} 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 (Rˆ 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 S (a) each An is an open set in H which is either a hyperbolic disk, a horodisk tangential to Rˆ , 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. { n} 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< ∞ 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 { n}n=1,2,... the components of ∂C ( N ). If the hyperbolic lengths l(c ) satisfy Γ ⊂ Γ n 1 l(c ) 2 < n ∞ 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 ∈ Q Q b >a , and d(Q) be the Euclidean length b a .ForQ K and q N,we Q Q Q − Q ∈ ∈ 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 { Q} ( Re z = b )is2L. Then let Q denote the closed rectangle with four vertices at { Q} + 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. 1). We call QM the face of Q. For 0 6 s 6 t 6 1, let Q[s,t] be the part of Q+ between the horizontal lines Im z = skd(Q) and Im z = tkd(Q) including the lines. Set Q = Q[1,1]. Let l = l(L) be the hyperbolic diameter of Q. Here we show a simple but crucial lemma in our argument. Let A = n N An be the removable set. For technical reasons we thin out each A by L and we∈ consider n S a region B = z A ρ(z,∂A ) >L , n { ∈ n | n } where the hyperbolic distance between two sets X and Y is denoted by ρ(X, Y ). This region may be empty. Set B = n N Bn. Then we obtain the following: ∈ Lemma 1. Under the same assumptionsS as Theorem 1, there is an integer q = q(L) such that for any interval Q K with QM = B, there exists Q0 K(Q, q) with Q0 Λ(Γ) = J. ∈ ∈ Proof. We may assume that Q C(Λ(Γ)). Let q be the smallest integer for M ⊂ Q which there exists Q0 K(Q, qQ) with Q0 Λ(Γ) = J. Put ∈ tQ = inf t>0 ∂C(Λ(Γ)) Q+ Q[0,t] . { | ⊂ } It is clear that if q tends to then t tends to 0. Suppose that t is so small that Q ∞ Q Q ρ(z,∂C(Λ(Γ)) Q+) > 2L for every z Q . On the other hand, we see that ∈ M ρ(z,∂C(Λ(Γ)) Q ) > 2L − + since the shortest geodesic from z to ∂C(Λ(Γ)) Q must cut across one of the − + geodesics Re z = a and Re z = b . Therefore ρ(z,∂C(Λ(Γ))) > 2L for any { Q} { Q} z Q (Fig. 2). However this is a contradiction, because ρ(z,∂C(Λ(Γ))) 2L for ∈ M 6 z Q B according to the prior assumption. Thus q is uniformly bounded. ∈ M − Q Hereafter we fix the integer q = q(L) arising from Lemma 1 and continue with our Hausdorff dimension of infinitely generated Kleinian groups 127 {Re z = aQ} 2L Q * cC (K(C)) 2L Q [0, t] Q′ Fig. 2. Distance from QM. definitions. For Q K we define a subset of the indices for the removable regions as ∈ IQ = n N Bn Q[ 1 , 1 ] J , { ∈ | q q } and set B(Q)= n I Bn. Since An are mutually disjoint, and since Bn disappear ∈ Q { } for thin An, there exists an integer N = N(L) such that #IQ 6 N for any Q K. S ∈ Here and throughout the paper # will denote the cardinality of a given set. We define the following set of subintervals of Q: ∞ i L(Q)= Q0 K(Q, 2 q) Q0 satisfies (d) and (m) , ( ∈ ) i=0 [ where (d) Q0 Λ(Γ) J and QM0 = B(Q); i (m) there is no Q00 ∞ K(Q, 2 q) such that Q00 contains Q0 properly and satisfies ∈ i=0 condition (d). S Moreover, we set L0(Q) ÷ L(Q) K(Q, q). − For each B we define its base point(s) v Rˆ as follows. For a horodisk, v is the n n ∈ n tangential point; for a hyperbolic disk, vn is the vertical projection of its centre to the + real axis; for a neighbourhood of a complete geodesic there are two base points vn and vn− which are the end points of the geodesic. We denote by V the set of all base points for B and by VQ the set of all base points for B(Q). Note that #VQ 6 2#IQ 6 2N. Lemma 2. The intervals L(Q)= Q0 cover Q (Λ(Γ) VQ). { } − Proof. Take an arbitrary point x in Q Λ(Γ). If no intervals in L(Q)coverx, we have an infinite chain of the faces Q0,Q1,... ,Qi ,... which are all contained { M M M } in B(Q) and converge to x, where Qi K(Q, 2iq) and Qi Qi+1 (Fig. 3). However, ∈ ⊃ since the chain is connected, this is possible only if the chain is contained in one of the elements of B(Q), and hence x must be its base point. This proves the lemma. 128 Katsuhiko Matsuzaki Q 1 1 [ q, q] 0 Q * 1 Q * 2 Q * x Fig. 3. Chain of faces. Lemma 3. For any Q0 L(Q) we have Q0 = B. ∈ M Proof. If Q0 K(Q, q) then Q0 has non-empty intersection with Q 1 1 . Thus the M [ q , q ] ∈ j condition QM0 B implies that QM0 B(Q). If Q0 L0(Q) then Q0 K(Q, 2 q)for ⊂ ⊂ j ∈1 ∈ some j 1 and thus there exists Q00 K(Q, 2 − q) such that Q00 Q0. Since Q0 > ∈ ⊃ satisfies condition (m), Q00 does not satisfy condition (d). Since Q00 Λ(Γ) J we see that Q00 B(Q): there is a Bn for some n IQ such that Q00 Bn. Since Q00 Q0 J M ⊂ ∈ M ⊂ M M the condition Q0 B actually implies Q0 B B(Q). M ⊂ M ⊂ n ⊂ We construct the hierarchy of coverings of Λ(Γ) V inductively. Without loss of − generality we may assume ^ Λ(Γ). First, let L0 be a family consisting of one ∞ interval that contains the whole Λ(Γ). For i > 0 we define Li+1 = L(Q). Q L [∈ i By Lemma 2 we know that L covers Λ(Γ) V for every i 1. i − > At this stage we have the following result, as a consequence of Lemmas 1–3. Corollary 1. Let Γ be a Fuchsian group satisfying the assumption of Theorem 1. Then the 1-dimensional measure of Λ(Γ) is zero. Proof. By Lemma 1 and the definition of L(Q), if Q K satisfies the condition ∈ QM = B then 1 d(Q0) 1 d(Q). 6 − q Q0 L(Q) X∈ By Lemma 3, for any Q Li (i 1) we have that QM = B. Thus ∈ > 1 d(Q0) 1 d(Q). 6 − q Q0 Li+1 Q Li X∈ X∈ Since L covers Λ(Γ) V for every i 1 by Lemma 2, and since V is countable, we i − > see that the 1-dimensional measure of Λ(Γ) is zero. Remark. We have not yet utilized condition (a) of a removable set on the shape (disk, horodisk or geodesic neighbourhood). Hence, only to see that the 1-dimensional measure of Λ(Γ) is zero as in Corollary 1, we may choose a removable set A so Hausdorff dimension of infinitely generated Kleinian groups 129 Bn Q * 2 y = m(x – vn) ′′ Q[ 1, 1] Q q q * Q ′ vn Fig. 4. Size of Bn. 2 that each component An is a domain in H with a Euclidean closure that has an intersection with R of zero measure. We shall discuss this fact more precisely in Section 6. We need more preliminary results in order to prove Theorem 1. Lemma 4. There are positive constants α and c strictly less than 1, depending only on L, such that α α d(Q) 6 cd(Q) Q L(Q) 0X∈ for any Q K with QM = B. ∈ The proof of the lemma makes use of the following observation. Lemma 5. If Q0 L0(Q) then there is an index n = n(Q0) I such that Q is within ∈ ∈ Q 0 distance l of Bn. j j 1 Proof. If Q0 K(Q, 2 q) for some j 1 then there exists Q00 K(Q, 2 − q) such ∈ > ∈ that Q Q00 J and Q00 B(Q), as is seen in the proof of Lemma 3. Since the 0 M M ⊂ diameter of Q0 is l, we obtain the result. Proof of Lemma 4. By changing the scaling we may assume that d(Q)=1. Lemma 5 asserts that we can define a map n: L0(Q) I by the correspondence → Q Q0 n(Q0), where Q is in l-neighbourhood of B . We will find a region in which 7→ 0 n(Q0) this neighbourhood lies. First we assume that the component Bn = Bn(Q0) is a disk or horodisk. Since Bn > QM, while Bn Q 1 1 J, we see that the Euclidean radius of Bn cannot be [ q , q ] too large. Hence there exists a constant m = m(L) > 0 independent of n such that the l-neighbourhood of B is above a parabola y = m(x v )2 touching R at the n − n base point of B . Thus any point (x, y) Q satisfies y m(x v )2 in this case n ∈ 0 > − n(Q0) (Fig. 4). Next we assume that Bn = Bn(Q0) is an r-neighbourhood of a geodesic for some r>0. We want to show that Q is either in the region y m(x v+ )2 or in the 0 > n(Q0) 2 − region y > m(x vn−(Q )) , which we do by choosing a smaller positive constant m to − 0 130 Katsuhiko Matsuzaki replace the present m if necessary. If r is uniformly bounded from above, this task is easy. Thus we need only consider the case where r is greater than a sufficiently large constant. Then, as before where Bn is a horodisk, Bn cannot be too large in the Euclidean sense. Hence we can also find a desired constant m in this case. Summing up, we have shown that for any point x Q0 with Q0 L0(Q) ∈ ∈ 2 kd(Q0) m(x v ) , (1) > − Q0 + where v is either v or v− according to the parabola over which Q lies if Q0 n(Q0) n(Q0) 0 Bn(Q0) is a neighbourhood of a geodesic, and otherwise vQ0 is vn(Q0). Next, for any (0, 1/q) and any v V , we define a subset of L0(Q)as ∈ ∈ Q L (Q; )= Q0 L0(Q) v = v, d(Q0) v { ∈ | Q0 6 } and the finite union over v as L(Q; )= Lv(Q; ). v [ For x Q0 L (Q; ), the inequality (1) yields ∈ ∈ v 2 k kd(Q0) m(x v) , (2) > > − or in other words, 1 Q0 x x v (k/m) 2 . ⊂{ || − | 6 } α Now we take the sum of d(Q0) for 0 <α<1. By (2), α α 1 d(Q0) = d(Q0) − dx Q L (Q;) Q Q0 0∈Xv X0 Z α 1 m 2 − 6 (x v) dx x v (k/m)1/2 k − Z| − |6 1 n o 2 2 k α 1 = − 2 . 2α 1 m − Since #VQ 6 2N, we obtain that 1 2 α 2 k α 1 d(Q0) 2N − 2 . (3) 6 2α 1 m Q0 L(Q;) ∈X − We now describe how to choose α and . First choose (0, 1 ) such that ∈ q 1 2 2 k α 1 1 2N − 2 (4) 2α 1 m 6 3q − holds for any α> 2 . After this, choose α ( 2 , 1) so that 3 ∈ 3 1 α 1+ . (5) 6 3q We fix these numbers. Finally we consider the sum of d(Q0) taken over all the Q0 in L(Q). If Q0 ∈ Hausdorff dimension of infinitely generated Kleinian groups 131 L(Q) L(Q; ), then d(Q0) >. Hence, by (5), − α 1 1 1 d(Q0) 1+ d(Q0) 1+ 1 . (6) 6 3q 6 3q − q Q0 L(Q) L(Q;) Q0 ∈ X− X Setting c =1 (1/3q) < 1, we obtain from (3), (4) and (6) the result that − α d(Q0) = + Q L(Q) L(Q;) L(Q) L(Q;) 0X∈ X X− 1 1 1 + 1+ 1 6 3q 3q − q 1 1 = c = cd(Q). 6 − 3q This completes the proof. From this lemma and the fact that each L covers Λ(Γ) V , we see immediately, i − in a manner similar to Corollary 1, that the Hausdorff dimension of Λ(Γ) is not greater than α. Thus we obtain Theorem 1. 4. Proof of Theorem 2 Theorem 2 is proved in the following form. The argument is similar to the proof of theorem 5 2in[1]. · Theorem 3. Let NΓ be a hyperbolic surface such that Γ is infinitely generated and dim Λ (Γ) < 1. Let c be the components of ∂C N . If the hyperbolic lengths c { n}n=1,2,... Γ ⊂ Γ l(cn) satisfy 1 l(c ) 2 < n ∞ n X then the 1-dimensional measure of Λ(Γ) is positive and in particular dim Λ(Γ) = 1. D+1 First we review briefly a geometric interpretation of dim Λc(Γ). Fix x, y H ∈ and let δ(Γ) denote the critical exponent of convergence for a (D + 1)-dimensional Kleinian group Γ: δ(Γ) = inf s [0,D] exp( sρ(x, γ(y))) < . { ∈ | − ∞} γ Γ X∈ This is independent of the choice of points x and y. A crucial fact is that dim Λc(Γ) = δ(Γ) for any non-elementary Kleinian group. (See [8] for Fuchsian groups and [1]for the general case.) Further, let λ0(Γ) denote the bottom of the spectrum of the hyperbolic Laplacian ∆ on NΓ, namely the infimum of the eigenvalues of ∆ with 2 L -eigenfunctions on NΓ. The Elstrodt–Patterson–Sullivan theorem ([12]) gives the relation between δ(Γ) and λ0(Γ) as D2/4 δ(Γ) 6 D/2 λ0(Γ) = δ(Γ)(D δ(Γ)) δ(Γ) D/2 . ( − > 132 Katsuhiko Matsuzaki Therefore, for a non-elementary Fuchsian group Γ, the condition dim Λc(Γ) < 1is equivalent to the condition λ0(Γ) > 0. We also use the following characterization of hyperbolic surfaces of infinite topo- logical type in the proof of Theorem 3. Lemma 6. If Γ is infinitely generated, then there exists a discrete set of points in the convex core CΓ where the injectivity radii are uniformly bounded away from zero. Proof. There is a universal constant r0 > 0 (the Margulis constant) such that each component of the thin part of NΓ, where the injectivity radius is less than r0, is either a cusp neighborhood or an annulus. Since CΓ minus the thin part is still non-compact, we can choose the required set of points. Proof of Theorem 3. Let K(x, y, t) be the heat kernel of ∆ on NΓ and G(x, y) Green’s ∞ function, namely G(x, y)= 0 K(x, y, t)dt.Ifλ0(Γ) > 0 and if ρ(x, y) is sufficiently large, Green’s function satisfies an inequality R 1 1 cρ(x,y) G(x, y) 6 C area (B(x, 1))− 2 area (B(y, 1))− 2 e− , where c>0 and C< are constants depending only on λ and B(x, r) is a portion ∞ 0 of NΓ within r of x ([1]). By Lemma 6, we can take a discrete set of points x in C where the injectivity { m} Γ radii are uniformly bounded away from zero. We may also assume that each xm has a sufficiently large distance from ∂CΓ. Let U be an edge of ∂CΓ with width 1 in the exterior of CΓ. We define Em as the expected time that a Brownian motion beginning at xm spends in U, namely, ∞ Em = K(xm,y,t)dm(y)dt = G(xm,y)dm(y) , Z0 ZU ZU where dm is the hyperbolic area element. By the above estimate of Green’s function and the fact that area(B(xm, 1)) is bounded away from zero, we obtain 1 1 cρ(xm,y) Em 6 C0 area (B(y, 1))− 2 e− dm(y) 6 C0 area (B(y, 1))− 2 dm(y) ZU ZU by taking another constant C0 < . It is easy to see that this integral over U is ∞ bounded by some constant multiple of the line integral over ∂CΓ = cn. In addition, there exists a constant η>0, not depending on n, such that area(B(y, 1)) > ηl(cn) 1 S for any y c . Then by the assumption l(c ) 2 < , the intergal above converges. ∈ n n ∞ Therefore by the dominated convergence theorem, P 1 cρ(x ,y) lim area (B(y, 1))− 2 e− m dm(y) m →∞ ZU 1 cρ(x ,y) = area (B(y, 1))− 2 lim e− m dm(y) U m Z →∞ and, since limm ρ(xm,y) for each y, it follows that Em 0asm . →∞ →∞ → →∞ Assume that the 1-dimensional measure of Λ(Γ) is zero. This is equivalent to the condition that the harmonic measure of the ideal boundary of CΓ is zero. Let β be the union of the core curves in the annuli of U. Then this assumption also implies that the probability of a Brownian motion hitting β is equal to 1 independent of Hausdorff dimension of infinitely generated Kleinian groups 133 the starting point xm. However the expected time that a Brownian motion starting at y β spends in U is bounded away from zero independent of where y is and ∈ thus the Markov property implies that Em is also bounded away from zero. This contradiction proves that the 1-dimensional measure of Λ(Γ) is positive. 5. Higher dimensional case Tukia [13] proved that for an arbitrary D > 1, the Hausdorff dimension of the limit set of a (D + 1)-dimensional geometrically finite Kleinian group of the second kind is strictly less than D. Our proof of Theorem 1 for Fuchsian groups have been obtained by generalizing Tukia’s argument and the dimension is of no matter in the proof. In other words, the higher dimensional generalization of Theorem 1 can be also obtained by the same argument. We now state it without proof. D+1 Definition. For a Kleinian group Γ acting on H , we say that A = n N An in ∈ HD+1 is a removable set if it satisfies the following conditions: D+1 S (a) each An is an open set in H which is either a hyperbolic ball, a horoball tangential to ∂HD+1 = RD , or a tubular neighbourhood of a complete {∞} geodesic at constant radius; (b) the set A is a disjoint union of A and A is invariant under Γ. { n} The projection A/Γ is also called a removable set in NΓ. Theorem 4. Let Γ be a Kleinian group acting on HD+1. If there exist a constant L< and a removable set A HD+1 such that all points of C(Λ(Γ)) A lie within a ∞ ⊂ − distance L of ∂C(Λ(Γ)), then dim Λ(Γ) 6 α Corollary 2. Let B ∞ be a family of mutually disjoint D-dimensional open { n}n=1 disks in the Euclidean space RD (D > 2), whose radii are greater than k>0 and less than K< . Let Γ be a discrete group generated by the reflections with respect to the ∞ circles Sn = ∂Bn. Then dim Λ(Γ) 6 β D + 1 ( A1 (P – A1)EC (¤(C)) cC (¤(C)) Sˆ n Fig. 5. Hemispheres and ∂C(Λ(Γ)). Hereafter we prove the assertion only for the case D = 2 for the sake of simplicity. Similar arguments are applicable to the case D>2. Letting ωn be an annular edge of width k/100 inside each circle Sn, we first consider an easier situation that Γ has no limit points in ωn. In this case, for any x in the common exterior of Sn , there is a t1 > k/100 such that z1 =(x, t1) ∂C(Λ(Γ)); { } S ∈ hence if a point z =(x, t) (P A1) C(Λ(Γ)) has such an x-coordinate then the ∈ − hyperbolic distance from z to ∂C(Λ(Γ)) is not greater than t0 100K ρ(z,z1) 6 log 6 log , t1 k and all points in (P A1) C(Λ(Γ)) lie within a bounded distance of corresponding − points z, which itself depends only on k and K. In the case that there is a limit point of Γ in ωn,sayinω1, there must be another circle, say S , that has non-empty intersection with the reflected image of ω with 2 S 1 respect to S1. Hence the Euclidean distance between S1 and S2 is less than about k/100 and between them there is a narrow interstice where the other circles are forbidden. The extreme case is that S1 and S2 are tangential at a point q. In this case q is a 2 limit point of Γ and there are two open disks R1 and R2 in R , mutually tangential at q, such that R1 and R2 contain no limit points of Γ. Moreover we can choose R1 and R2 uniformly large because the radii of S1 and S2 are bounded below by k. Note that C(Λ(Γ)) is contained in the common exterior of Rˆ 1 and Rˆ 2, where Rˆ i is the 3 hemisphere in H spanning ∂Ri (i =1, 2). Hence (P A1) C(Λ(Γ)) is contained in − a region bounded by four hemispheres Sˆ1, Sˆ2, Rˆ 1 and Rˆ 2. Since this region becomes thinner near q in the hyperbolic sense and since part of ∂C(Λ(Γ)) runs towards q within this region, ρ(z,∂C(Λ(Γ))) 0 as a point z (P A1) C(Λ(Γ)) tends to → ∈ − q. Let δ be a fixed positive constant; then we can take a uniformly large Euclidean neighbourhood U of q such that ρ(z,∂C(Λ(Γ))) δ for z U (P A1) C(Λ(Γ)). 6 ∈ − If z is outside U we apply the same argument as in the paragraph above to conclude that ρ(z,∂C(Λ(Γ))) is bounded. Even if S1 and S2 are not tangential, a similar argument works. We replace the tan- gential point q with a segment l joining the fixed points of that hyperbolic transfor- mation which is the composition of the reflections with respect to S1 and S2. Instead of R1 and R2 we can take truncated disks R10 and R20 which are amalgamated along l 2 (Fig. 6) so that C(Λ(Γ)) is contained in the convex hull of R (R0 R0 ). For a fixed − 1 2 δ>0 we take a Euclidean r-neighbourhood U of l such that ρ(z,∂C(Λ(Γ))) 6 δ for z U (P A1)C(Λ(Γ)). We can keep r bounded away from zero uniformly in the ∈ − limiting process towards the tangential case. We also apply the previous argument outside U and so find that ρ(z,∂C(Λ(Γ))) is bounded in any case. Hausdorff dimension of infinitely generated Kleinian groups 135 S1 S2 ′ R1 l ′ R2 Fig. 6. Truncated disks. 6. The D-dimensional measure As we mentioned in the Remark after Corollary 1, the condition on the shape of a removable set is not necessary to prove Corollary 1 alone. In this section we state a weaker assumption from which nullity of the D-dimensional measure of the limit set follows. As in the previous section, our statement does not necessarily hold in the Fuchsian case. We now define a generalized removable set for which the proofs of Lemmas 1–3 hold. D+1 Definition. For a Kleinian group Γ acting on H , we say that A = n N An in ∈ HD+1 is a weakly removable set if it satisfies the following conditions: S D+1 (a0) each An is a domain in H such that the D-dimensional measure of the D intersection of the Euclidean closure An with R is zero; (b0) the set A is a disjoint union of A and A is invariant under Γ. { n} Thus, as in the proof of Corollary 1, we have the following result: Theorem 5. Let Γ be a Kleinian group acting on HD+1. If there exist a constant L< and a weakly removable set A HD+1 such that all points of C(Λ(Γ)) A lie ∞ ⊂ − within a distance L of ∂C(Λ(Γ)), then the D-dimensional measure of Λ(Γ) is zero. As a corollary to this theorem, we obtain that the D-dimensional measure of Λ(Γ) is zero if Γ is generated by reflections with respect to circles, which are boundaries of D-dimensional disks, that are packed in RD in a certain way. This claim was proved (for D =2)by[5] in the course of an argument on the order of the circle packing constant. We state a more general result without restriction to circle packing; one circle need not be tangential to another. D Corollary 3. Let S ∞ be a family of circles in R (D 2) with mutually dis- { n}n=1 > joint interiors and without a point of accumulation, that satisfies the following condition: (u) there is a constant κ>1 such that, for any circle Sn, the ratio of the distance between the centres of Sn and any other circle to the radius of Sn is greater than κ. Let Γ be a discrete group generated by the reflections with respect to S . Then the { n} D-dimensional measure of Λ(Γ) is zero. 136 Katsuhiko Matsuzaki Proof. We use the same notation as in the proof of Corollary 2; Sˆn is the hemisphere D+1 ˆ in H spanning Sn and P is the common exterior of all the hemispheres Sn , which { } is a fundamental region of Γ in HD+1. However, in the present proof we set A1 = Int(P C(Λ(Γ))). Then we may regard A =Γ(A1) as a weakly removable set because the intersection D of the Euclidean closure of A1 with R consists of tangent points of the circles S , which is a countable set. Thus, by Theorem 5, if we show that all points of { n} ∂P C(Λ(Γ)) lie within a bounded distance of ∂C(Λ(Γ)), we obtain the desired result. Picking up an arbitrary component Sˆ of ∂P, we have only to prove the claim above for Sˆ C(Λ(Γ)). We may change the scale so that the radius of S is 1. We choose an exterior annular edge ω0 of S with width less than (κ 1)/2. Then condition (u) − implies that if another circle has non-empty intersection with ω0, its radius must be greater than (κ 1)/2. Thus the radii of circles close to S are uniformly bounded − away from zero and the arguments in the proof of Corollary 2 are applicable. We conclude that there exists a constant L< depending only on κ and D such that ∞ all points of Sˆ C(Λ(Γ)) lie within L of ∂C(Λ(Γ)). If a circle packing of R2 has bounded valency, meaning that the number of tan- gential circles to any circle is uniformly bounded by a constant, the ring lemma [10] implies that the packing circles satisfy condition (u). Thus the result of [5] follows from Corollary 3. 7. Examples for dim Λc(Γ) dim Λ(Γ) We believe that the assumptions in Theorem 2 are far from the sharp condition for dim Λ(Γ) = 1. In fact, from an example exhibited in this section, we can see that there is a Fuchsian group Γ which satisfies dim Λ(Γ) = 1 but not the assumption of Theorem 2. We utilize the Cheeger isoperimetric constant h(Γ) for a hyperbolic manifold NΓ, which is defined as area (∂W) h(Γ) = inf , vol (W ) where the infimum is taken over all the relatively compact domains W in NΓ with smooth boundary ∂W. When h(Γ) is positive we say that NΓ satisfies the hyperbolic isoperimetric inequality. It is known that h(Γ) > 0 if and only if λ0(Γ) > 0 (cf. [2, 4]). Recall that these conditions are equivalent to dim Λc(Γ) < 1. Theorem 6. There exists a Fuchsian group Γ such that dim Λ(Γ) = 1, dim Λc(Γ) < 1 and the 1-dimensional measure of Λ(Γ) is zero. Proof. Our example Γ is a subgroup of a Fuchsian group of a closed Riemann surface R0. 2 For the first step we construct a Riemann surface R1 = H /Γ1 such that the 1- dimensional measure of Λ(Γ1) is zero and dim Λ(Γ1) = 1 as follows (Fig. 7): first we take a Z-cover of R0 by cutting along the meridian α0 of a handle of R0 and connecting its copies infinitely. Next we divide this Z-cover into two parts by a copy α of α0 and replace one of them with its annular cover. We denote the resulting Riemann surface R1. Since the Z-cover does not admit a Green’s function, we can Hausdorff dimension of infinitely generated Kleinian groups 137 R0 Z-cover R1 R α0 αα Fig. 7. Covering of R0. see that the harmonic measure of the ideal boundary of R1 is zero, which implies that the 1-dimensional measure of Λ(Γ1) is zero. We can also see that R1 does not satisfy the hyperbolic isoperimetric inequality. Thus λ0(Γ1) is zero and in particular dim Λ(Γ1)=1. For the second step we take a planar normal cover R of R1 that is associated with the normal closure of the homotopy class of α in the fundamental group of R1. The Fuchsian group of this R is our desired Γ. Since Γ is normal in Γ1 we have Λ(Γ) = Λ(Γ1). Thus the 1-dimensional measure of Λ(Γ) is zero and the Hausdorff dimension is one. Since R is a planar cover of a closed Riemann surface, the injectivity radii of R are uniformly bounded away from zero, implying that h(Γ) > 0 (cf. [4]). Thus λ0(Γ) > 0 and equivalently dim Λc(Γ) < 1. We can also construct an example as we did in Theorem 6 for higher dimensional cases. Theorem 7. For any D > 2, there exists a (D +1)-dimensional Kleinian group Γ such that dim Λ(Γ) = D, dim Λc(Γ) Proof. We first choose a Kleinian group Γ1 of the second kind such that dim Λ(Γ1)= D and the D-dimensional measure of Λ(Γ1) is zero. The existence of Γ1 for any dimension is seen from the following argument. As in the proof of Theorem 6, if we have a closed hyperbolic manifold R0 with an embedded totally geodesic subman- ifold α of codimension 1 such that R α is connected, we can take its covering 0 0 − 0 manifold R1 in the same way. Let Γ1 be the Kleinian group of R1. Then, by the same reasoning as in the previous proof, Γ1 has the required property. The existence of R0 is guaranteed by [7]. If we are restricted to D = 2 we may also take Γ1 to be a totally degenerate group, since the 2-dimensional measure of Λ(Γ1) is zero (a well-known result as a partial answer to the Ahlfors measure conjecture) and dim Λ(Γ1)=2([1]). D In a fundamental domain of Γ1 in R , we take two circles with disjoint interiors. Let g beaMobius¨ transformation that maps the interior of one circle to the exte- rior of the other. Then, by the classical Klein combination theorem, the group G generated by Γ1 and g is also Kleinian and it is clear that dim Λ(G)=D. Moreover, the D-dimensional measure of Λ(G)iszero([6]). Our example Γ is a subgroup of G 1 generated by γgγ− γ Γ . { } ∈ 1 We prove that Γ fulfills the three requirements. The inclusion Λ(Γ ) Λ(Γ) is 1 ⊂ easily seen from the definition of limit set; thus dim Λ(Γ) = D. Since Γ is a subgroup of G, the D-dimensional measure of Λ(Γ) is zero. The remaining task is to show that 138 Katsuhiko Matsuzaki dim Λc(Γ) 8. Planar Riemann surfaces We now apply our theorems to particular planar domains. Throughout this section Riemann surfaces are assumed to be of the form R = C E, where E is a closed − subset consisting of a countable number of connected components that are isolated from the others. Corollary 4. Let R = C E be a Riemann surface as above and Γ the Fuchsian − group of R. For each component en (n =1, 2,...) of E we consider all the doubly connected regions in R having en as a boundary component and define m(en) as the supremum of the moduli of these. If 1 m(e )− 2 < n ∞ n X then dim Λc(Γ)=1; equivalently, R does not satisfy the hyperbolic isoperimetric inequal- ity. Proof. If all the en are singletons then R does not admit a Green’s function and it follows that dim Λc(Γ) = 1. Therefore we may assume that E has a continuum, so that the 1-dimensional measure of Λ(Γ) is zero. On the other hand, the assumption concerning convergence of the sum of moduli implies the condition given in The- orem 3 on lengths of boundary curves of the convex core. The result now follows immediately from Theorem 3. Remark. If the injectivity radii of a planar Riemann surface R are uniformly bounded away from zero, then R satisfies the hyperbolic isoperimetric inequality. Although the converse of this statement is not true in general, a Riemann surface R of the sort in Corollary 4, in which the infimum of injectivity radii is zero, does not satisfy the hyperbolic isoperimetric inequality. We further assume a Riemann surface R to be a Denjoy domain C E, where E − is a disjoint union of closed slits e ∞ in the real axis R. In this case we can state { n}n=1 both a sufficient condition and a necessary one in terms of the Euclidean lengths of slits e and intervals i R between two consecutive slits. Modification of Corollary n m ⊂ 4 easily yields a sufficient condition for Λ(Γ) to have full dimension. As a necessary condition, we obtain the following: Corollary 5. For a Denjoy domain R as above, if there exist constants k>0 and K< such that the length of each slit e is greater than k and the length of each interval ∞ n im is less than K, then the Fuchsian group Γ of R satisfies dim Λ(Γ) < 1. Hausdorff dimension of infinitely generated Kleinian groups 139 This can be proved by applying Theorem 1; take two horodisks tangent at in ∞ the upper half plane and in the lower half plane respectively as removable sets. 9. Conclusion We compare the main result in this paper with a result of [1]. They proved in particular the following: D+1 Proposition A. Let NΓ = H /Γ be a (D +1)-dimensional hyperbolic manifold such that the boundary of the convex core CΓ is compact. 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