The classification of Kleinian groups of Hausdorff dimensions at most one and Burnside's conjecture
TThe classification of Kleinian groups ofHausdorff dimensions at most one andBurnside’s conjecture
Yong Hou ∗ Abstract
In this paper we provide the complete classification of convex co-compact Kleinian group of Hausdorff dimensions less than 1 . In partic-ular, we prove that every convex cocompact Kleinian group of Haus-dorff dimension ă ě
1. The upper bounds of Hausdorff dimensions of classicalSchottky groups has long been established by Phillips-Sarnak [15] andDoyle [7]. The prove of the theorem relies on the result of Hou [10].
We take Kleinian groups to be finitely generated, discrete subgroups ofPSL p , C q . The main theorem is:
Theorem 1.1 (Classification) . All convex cocompact Kleinian groups Γ withlimit set of Hausdorff dimension ă are classical Schottky groups. Thisbound is sharp. It was pointed out by Sarnak to me that, originally Burnside conjecturedthat all classical Schottky groups must have Hausdorff dimension at mostone. However Burnside’s conjecture was disapproved by Myrberg [14, 7].Theorem 1.1, implies the converse of Burnside’s conjecture is in fact true. ∗ Primary:57M50; 53C30. Secondary:22A05 . Supported by Ambrose Monell Fundation. a r X i v : . [ m a t h . G T ] F e b orollary 1.2 (Converse of Burnside’s conjecture [4]) . All non-classicalSchottky groups must have Hausdorff dimension ě . Finally we note that the proof of Theorem 1.1 relies on the result of [10]Theorem 1.3. The proof of 1.3 is completely different and of independentinterests from the current works.
Theorem 1.3 (Hou[10]) . There exists λ ą such that any Kleinian groupwith limit set of Hausdorff dimension ă λ is a classical Schottky group. Finally, we note that Theorem 1.1 completes the picture of lower spectrumof the Schottky groups dimensions which compliments the upper boundsestablished in [15, 7].
Denote J g to be the rank- g Schottky space. Denote J g,o the open subsetof classical Schottky groups of J g . For Γ P J g denote by D Γ the Hausdorffdimension of Λ Γ , the limit set of Γ . Let J g denote the open subset of J g consists of Γ Schottky groups with Hausdorff dimension D Γ ă . The mainclaim of our proof can then be stated as: J g,o X J g “ J g . Since J g,o X J g Ă J g is open subset of J g , our proof essentially consists of twomain parts as follows:(1) J g,o X J g is also closed in J g .(2) Every connected components of J g contains a point in J g,o X J g . Note that it follows from part p q that we have J g,o X J g consists of connectedcomponents of J g . The proof of part p q is done as follows. It is a result of Bowen [3] that,a Schottky group Γ has Hausdorff dimension ă , if and only if there exist a rectifiable Γ-invariant closed curve. Let R p S , W q be the space of boundedlength closed curves which intersects the compact set W Ă C and equippedwith Fr´echet metric. It is complete space, see Section 2. We show that ifHausdorff dimension ă n in R p S , W q . We also show that2f Γ is a Schottky group, then every quasi-circle of Γ has an open neighbor-hood in the relative topology of Ψ Γ (see section 3) such that, every elementof the open neighborhood is a quasi-circle of Γ . We also define linearity andtransversality invariant for quasi-circles, and show that quasi-circles of clas-sical Schottky groups preserve these invariants, and non-classical Schottkygroups do not have transverse linear quasi-circles.Given a quasi-circle of a Schottky group Γ, we show that there existsan open neighborhood (in the relative topology of space of rectifiable curveswith respect to Frechet metric) about the quasi-circle such that, every pointin the open neighborhood is a quasi-circle of Γ , see Lemma 3.9. Next assumethat we have a sequence of classical Schottky groups Γ n Ñ Γ to a Schottkygroup, and are all of Hasudorff dimensions less than one. We then studysingularity formations of classical fundamental domains of Γ n when Γ n Ñ Γ . These singularities are of three types: tangent, degenerate, and collapsing.We show that all these singularities will imply that there exists a quasi-circlesuch that, every open neighborhood about this quasi-circle will contain somepoints which is not a quasi-circle. Essentially, the existence of a singularitywill be obstruction to the existence of any open neighborhood that are ofquasi-circles, see Lemma 3.11. Hence it follows from these results that, ifΓ n Ñ Γ with Γ n classical and, all Hausdorff dimensions are of less than onethen Γ must be a classical Schottky group.The proof of part p q follows from our stronger result which is Lemma2.13. Where we show that one can connect any Schottky group Γ P J g bya continuous path in t Γ t u t Pr , s P J g to some classical Schottky group withnonincreasing Hausdorff dimension. The proof of this part relies on Theorem1.3 in [10] and Hausdorff dimension ă Schottky group Γ of rank g is defined as convex-cocompact discrete faithfulrepresentation of the free group F g in PSL p , C q . It follows that Γ is freelygenerated by purely loxodromic elements t γ i u g . This implies we can findcollection of open topological disks D i , D i ` g , ď i ď g of disjoint closure¯ D i X ¯ D i ` g “ H in the Riemann sphere B H “ C with boundary curves B ¯ D i “ c i , B ¯ D i ` g “ c i ` g . By definition c i , c i ` g are closed Jordan curves in Riemannsphere B H , such that γ i p c i q “ c i ` g and γ i p D oi q X D oi ` g “ H . Whenever thereexists a set t γ , ..., γ g u of generators with all t c i , c i ` g u g as circles, then it is3alled a classical Schottky group with t γ , ..., γ g u classical generators.Schottky space J g is defined as space of all rank g Schottky groups up toconjugacy by PSL p , C q . By normalization, we can chart J g by 3 g ´ J g is 3 g ´ p J g q group is Out p F g q , which is isomorphic to quotient ofthe handle-body group. Denote by J g,o the set of all elements of J g that areclassical Schottky groups. Note that J g,o is open in J g . On the other hand itis nontrivial result due to Marden that J g,o is non-dense subset of J g . How-ever, it follows from Theorem1.3, subset of J g with Hausdorff dimension lessthan some λ is in J g,o and is 3 g ´ Notations: • Given Γ a Kleinian group, we denote by Λ Γ and Ω Γ and D Γ its limit set,region of discontinuity, and Hausdorff dimension respectively through-out this paper. • J λg “ t Γ P J g | D Γ ă λ u and J λg,o “ t Γ P J g,o | D Γ ă λ u , for λ ą . • J λg,o defined as: the closure of set of all classical Schottky groups withHausdorff dimension ď λ in J g . • Given a fundamental domain F of Γ, we denote the orbit of F underactions of Γ by F Γ . We also say F Γ is a classical fundamental domainof classical Schottky group if B F are disjoint circles. Definition 2.1 (Quasi-circles) . Given a geometrically finite Kleinian groupΓ , a closed Γ-invariant Jordan curve that contains the limit set Λ Γ is called quasi-circle of Γ.Next we give a construction of quasi-circles of Γ which is a generalizationof the construction by Bowen [3].Let F be a fundamental domain of Γ, and t c i u g be the collection of2 g disjoint Jordan curves comprising B F . Let ζ denote collection of arcs ζ “ t ζ i u connecting points p i P c i , p i P c i ` g for 1 ď i ď g, and arcs on c i ` g that connects p i ` g to γ i p p i q and p i ` g P c i ` g to γ i p p i q . So ζ is a set of g disjointcurves connecting disjoint points on collection of Jordan curves of B F (Figure1 ). 4igure 1: Quasi-circle η Γ “ Λ Γ Y Y γ P Γ γ p ζ q defines a Γ-invariant closed curve containing Λ Γ . η Γ defines a quasi-circle of Γ. Obviously there are infinitely many quasi-circlesand different ζ gives a different quasi-circles. Note that, the simply connectedregions C z η Γ gives the Bers simultaneous uniformization of Riemann surfaceΩ Γ { Γ . Definition 2.2 (Generating curve) . Given a quasi-circle η Γ of Γ . We saya collection of disjoint curves ζ is a generating curve of η Γ , if η Γ can begenerated by ζ. Note that the quasi-circles constructed in [3], which requires that p i ` g isa imagine of p i under element of γ i , is a subset of the collection that we havedefined here. In fact, this generalization is also used for the construction ofquasi-circles of non-classical Schottky groups. Proposition 2.3.
Every quasi-circle η Γ of Γ is generated by some generatingcurves ζ. Proof.
Let η Γ be a quasi-circle of Γ . Let F be a fundamental domain of Γ . Set ξ “ F X η Γ and ¯ ξ “ B F X η Γ . Then ξ, ¯ ξ consists of collection of disjointcurves which only intersects along B F . Hence we have, γ p ξ q X γ p ξ q “ H for γ, γ P Γ with γ “ γ . Since Γ p ξ q´ Γ p ξ q “ Λ Γ we have have η Γ “ Λ Γ Y Γ p ξ Y ¯ ξ q , hence ξ Y ¯ ξ is a generating curve of η Γ . Definition 2.4 (Linear quasi-circle) . We call a quasi-circle η Γ linear if, η Γ z Λ Γ consists of points, circular arcs or lines.5ote that, if η Γ is linear then there exists F Γ such that, η Γ X F Γ and η Γ X B F Γ are piece-wise circular arcs or lines.We say an arc ζ Ă η Γ X F Γ is orthogonal if the tangents at intersections on B F Γ are orthogonal with B F Γ , and an arc ξ Ă η Γ X B F Γ is parallel if ξ Ă B F Γ . Definition 2.5 (Right-angled quasi-circle) . Given a linear quasi-circle η Γ ofΓ, if all linear arcs intersect at right-angle then we say η Γ is right-angledquasi-circle . Definition 2.6 (Transverse quasi-circle) . Given a quasi-circle η Γ of Γ, wesay η Γ is transverse quasi-circle if η Γ intersects B F Γ orthogonally for some F Γ and, η Γ have no parallel arc. Otherwise, we say η Γ is non-transverse. Definition 2.7 (Parallel quasi-circle) . Given a quasi-circle η Γ of Γ, we say η Γ is parallel quasi-circle if there exists some arc η of η Γ such that η Ă B F Γ Proposition 2.8.
Transverse quasi-circles always exists for a given Schottkygroup Γ . Proof.
Let F be bounded by 2 g distinct Jordan closed curves and take anycurve connecting p i P c i , p i ` g P c i ` g such that p i ` g “ γ i p p i q and p i ` g P c i ` g “ γ i p p i ` g q for 1 ď i ď g, that intersects c i , c i ` g orthogonally.It should be noted that a quasi-circle of Γ in general is not necessarily rectifiable . For instance, if we take ζ to be some non-rectifiable generatingcurves then, η Γ will be non-rectifiable. Recall a curve is said to be rectifiableif and only if the 1-dimensional Hausdorff measure of the curve is finite. Thisis not the only obstruction to rectifiability, in fact we have the following resultof Bowen: Theorem 2.9 ([3]) . For a given Schottky group Γ , the Hausdorff dimensionof limit set is D Γ ă , if and only if there exists a rectifiable quasi-circle for Γ . The proof of Theorem 2.9 relies on the fact that the Poincare series of Γconverges if and only if D Γ ă Proposition 2.10.
Let Γ be a Schottky group of D Γ ă . Suppose a givengenerating curve ζ is a rectifiable curve. Then η Γ is a rectifiable quasi-circleof Γ . roof. Let µ be the 1-dimensional Hausdorff measure. Since D Γ ă , wehave µ p Λ Γ q “ . Let Γ “ t γ k u k “ , let γ k denotes the derivative of γ k . Thenwe have, µ p η Γ q “ ÿ k “ µ p γ k p ζ qq— µ p ζ q ÿ k “ | γ k p z q| , z P ζ. Also since D Γ ă ř k “ | γ k | ă 8 . Thisimplies that η Γ is rectifiable if and only if ζ rectifiable.Let W Ă C be a compact set. Denote the space of closed curves with bounded length in C that intersect with W by: R p S , W q Ă t h : S Ñ C | h p S q X W “ H , h continuous rectifiable map u . For h , h P R p S , W q let (cid:96) p h q , (cid:96) p h q be it’s respective arclength. TheFr´echet distance is defined as, d F p h , h q “ inf t sup | h p σ q ´ h p σ q| ; σ , σ P Homeo p S qu ` | (cid:96) p h q ´ (cid:96) p h q| . For a given compact W Ă C , the space of closed curves with boundedlength R p S , W q is a metric space with respect to d F . Two curves in φ, ψ P R p S , W q are same if there exists parametrization σ such that ψ p σ q “ φ and (cid:96) p φ q “ (cid:96) p ψ q . The topology on R p S , W q is defined with respect to the metric d F , see [1] p388. Let ξ be a generating curve for η Γ . Fix a indexing of Γ, set ξ i “ Y i γ j p ξ q . Let σ i be a parametrization of ξ i such that σ “ Y i σ i define aparametrization of η Γ . Then d F p η , η q ď M p inf σ ,σ | ξ p σ k q ´ ζ p σ k q| ` | (cid:96) p ξ q ´ (cid:96) p ζ q|q for some M, k.
This implies continuity of η Γ with respect to generatingcurve ξ. Proposition 2.11.
For a given compact W Ă C , the space R p S , W q iscomplete metric space with respect to d F . Proof.
Let t φ i u Ă R p S , W q be a Cauchy sequence. φ i are rectifiable curvesof bounded length and so there exists t σ i u Lipschitz parameterizations withbounded Lipschitz constants, such that t φ i p σ i p t qqu are uniformly Lipschitz.Then completeness follows from the fact that all curves of R p S , W q arecontained within some large compact subset of C . J g was not known previously. In fact, very little isknown on the topological and geometrical structure of J g in general. In thisdirection we prove our next proposition. Proposition 2.12.
Every connected component of J g contains a point in J g,o X J g . The proof of Proposition 2.12 will follow from Lemma 2.13, a muchstronger statement. We shall construct and shows the existence of a path in J g such that it is non-increasing in Hausdorff dimensions. Lemma 2.13.
Let Γ a Schottky group with D Γ ă . There exists a path t Γ t u Ă J g , t P r , s in J g such that, Γ is classical Schottky group. The construction of t Γ t u relies on the condition of D Γ ă Proof.
Let B p Γ q be the space of unit ball Beltrami differentials of Γ. For µ P B p Γ q and we set µ (cid:15) “ (cid:15) µ for small (cid:15) ą
0, and let f (cid:15) be the correspondingquasi-conformal map. We will next construct new quasi-conformal ˜ f (cid:15) basedon f (cid:15) . Note that we take f to be the identity.Let F Γ Schottky domain bounded by disjoint closed Jordan curves t C i u g generated by t γ i u g . Denote by D i the closed topological disk bounded by C i “ B D i . Note that we also have some small disjoint open neighborhoods D i Ă U i . Let D i j be the topological disk bounded by image of C i underadmissible string of γ j , and similarly U i j denotes images of U i . Here wesay string γ i ...γ i l ...γ i j is admissible for D i if i j “ i , and | i l ´ i l ` | “ g . Set V i j “ U i j z D i j . Then the limit set Λ Γ is given by X k “ Y | j |“ k D i j for admissiblestring i j P t , ..., g u k . We define sequence of quasiconformal maps t f (cid:15) ,k u asfollows. Set f (cid:15) ,k “ f (cid:15) on C z Y | j |“ k U i j . Let B D τi j be continuous family ofJordan curves which are shrinks of B D i j to a point into the interior of D i j as τ Ñ 8 . Let φ τi j be conformal maps D i j Ñ D τi j . We define f (cid:15) ,k “ φ (cid:15) i j on D i j for all i j P t
I, ..., g u k . Let ψ (cid:15) i j be quasiconformal maps on V i j which connects f (cid:15) on C z Y | j |“ k U i j to φ (cid:15) i j on D i j . Note that the existence of ψ (cid:15) i j follows fromthe fact that f (cid:15) | B U ij and φ (cid:15) i j | B D ij are quasisymmetric maps, hence we havequasiconformal extension to annulis V i j . Finally we set f (cid:15) ,k “ ψ (cid:15) i j on V i j forall i j . The sequence of quaisconformal maps satisfies the consistence condition f (cid:15) ,k ´ “ f (cid:15) ,k on C z Y | j |“ k U i j where f (cid:15) , is defined to be f (cid:15) . By our con-struction we have for some M ą (cid:15) ,k are bounded K k ă M i.e. } µ (cid:15) k } ă ρ for some ρ ă k . Nowsince Λ Γ “ X k “ Y | j |“ k D i j which is of Hausdorff dimension ă
1, it followsfrom Theorem 35 . X k “ Y | j |“ k D i j is removable set for quaisconformalmaps. Hence it follows from uniform boundedness of dilations we have thelimit of ˜ f (cid:15) “ lim f (cid:15) ,k is quasiconformal map on C . In addition, by our construction we have lim l Ñ8 X lk Y | j |“ k f (cid:15) ,k p D i j q “ lim l Ñ8 X lk Y | j |“ k D (cid:15) i j Ă X k “ Y | j |“ k D i j which implies that, ˜ f (cid:15) p Λ Γ q Ă Λ Γ .This implies Hausdorff dimension D (cid:15) of ˜ f (cid:15) p Λ Γ q is D (cid:15) ď D Γ . We set Γ “ ˜ f (cid:15) Γ ˜ f ´ (cid:15) . So we have a path Γ s “ ˜ f s Γ ˜ f ´ s for s ď (cid:15) such that D Γ s ď D Γ for s ď (cid:15) . Now we repeat the above construction for Γ in placeof Γ to get Γ and so on. By this process we have constructed sequence of t Γ n u and a path of Γ s such that D Γ s ď D Γ for s ď n . Since D ni j convergeto points as n Ñ 8 , we have D Γ n Ñ
0, hence by continuity of Hausdorffdimension we have for sufficiently large s such that D Γ s ă λ where λ ą s Ă J g such that Γ t P J g,o for some large t . Let t Γ n u be a sequence of classical Schottky groups such that Γ n Ñ Γ . Denote QC p C q space of quasiconformal maps on C . It follows from quasiconformaldeformation theory of Schottky space, for Γ c classical Schottky group we canwrite , J g “ t f ˝ γ ˝ f ´ P PSL p , C q| f P QC p C q , γ P Γ c u{ PSL p , C q . Notations:
Set H to be the collection of all classical Schottky groups ofHausdorff dimension ď λ for some λ ă . Note that there exists a sequence of quasiconformal maps f n and f of C such that, we can write Γ n “ f n p Γ q and Γ “ f p Γ c q . Here we write f p Γ q : “t f ˝ g ˝ f ´ | g P Γ u for a given Kleinian group Γ and quasiconformal map f. Schottky space J g can also be considered as subspace of C g ´ . This pro-vides J g analytic structure as 3 g ´ Proposition 3.1.
Let t Γ n u be a sequence of Schottky groups with Γ n Ñ Γ to a Schottky group Γ . Let F be a fundamental domain of Γ . There exists asequence of fundamental domain t F n u of Γ n such that F n Ñ F . roof. Let Y gi “ C i “ B F be the Jordan curves which is the boundary of F . Set C n “ f ´ n pY gi “ C i q . Then C n is the boundary of a fundamental domain of f ´ n p Γ q . Hence we have a fundamental domain F n of Γ n defined by C n with F n Ñ F . Lemma 3.2.
Suppose f n Ñ f and Γ “ f p Γ c q . Every quasi-circle withbounded length of Γ n “ f n p Γ c q is in R p S , W q for some compact W. Proof.
Let η n be a quasi-circle of Γ n . Note Λ Γ n Ă η n . Since limit set Λ Γ n Ñ Λ Γ , and limit set Λ p Γ q is compact, and η n is rectifiable, we can find somecompact set W Ą Y n Λ Γ n Y Λ Γ . Given t Γ n u Ă H with Γ n Ñ Γ, let E Γ n denote the collection of all boundedlength quasi-circles of Γ n . We define Y E Γ n to be the closure of Y E Γ n . Notethat one can always get a non-simple closed curves given by non-simple gen-erating curves, hence we need to remove all these trivial singular curves in Y E Γ n , i.e. curves that is generated by some non-simple generating curves,which we call them trivial singular curves . Let E denote the collection ofall trivial singular curves in η P BY E Γ n , i.e. curves which have a singularity(non-simple) generating curve of η. We define Ψ Γ “ BY E Γ n ´ E . We define O p η q , open sets about η P Ψ Γ in relative topology given by O p η q X Ψ Γ for some open set O p η q Ă R p S , W q . Let Γ be a Schottky group. Let F be a fundamental domain of Γ . For O p η Γ q of η Γ P Ψ Γ , and suppose every element is quasi-circle, and let ζ ξ denote a generating curve of ξ P O p η Γ q with respect to F . Then we have, O p ζ q “ Y ξ P O p η Γ q ζ ξ the collection of all generating curves of the open set O p η Γ q gives a open set of generating curves of η Γ . On set of collection of allgenerating curves of elements of Ψ Γ , we define the topology as ξ η n Ñ ξ η , ifand only if η n Ñ η for η n , η P Ψ Γ . Remark . We will sometime denote by η P B Ψ Γ a curve which is the limitof rectifiable quasi-circles of t Γ n u . Proposition 3.4.
Let t Γ n u Ă H with Γ n Ñ Γ . Then every bounded lengthquasi-circle η Γ of Γ is in Ψ Γ . In addition, if η n are linear then η Γ is linearquasi-circles of Γ . Proof.
Let Γ n “ f ´ n p Γ q . Note that since D Γ n ď λ , we have D Γ ď λ ă . Define η n “ f ´ n p η Γ q , for all n. Then t η n u is a sequence of Jordan closedcurves. It follows from Proposition 2.3, we have a generating curve ζ of η Γ . So10 Γ “ Λ Γ YY γ P Γ γ p ζ q , and we have η n “ f ´ n p Λ Γ qY f ´ n pY γ P Γ γ p ζ qq . Since Λ Γ n “ f ´ n p Λ Γ q and f ´ n pY γ P Γ γ p ζ qq “ Y γ P Γ f ´ n γf n p f ´ n p ζ qq so its Y γ n P Γ n γ n p ζ n q , where ζ n “ f ´ n p ζ q . Hence ζ n is a generating curve of η n which are quasi-circles ofΓ n . Denote by ζ Γ a generating curve for η Γ . Since ζ is rectifiable curve,modify ζ n if necessary, we can assume t ζ n u are rectifiable curves.Let z P ζ n , since D Γ n ď λ , we have the 1-dimension Hausdorff measure µ p η n q : µ p η n q — µ p ζ n q ÿ γ P Γ n | γ p z q| ă 8 , where γ is the derivative of γ. Hence t η n u are rectifiable quasi-circles. Itfollows that there exists c ą µ p η n q ă cµ p η Γ q for large n , hence t η n u are bounded quasi-circles of R p S , W q , we have η n Ñ η Γ and η Γ P Ψ Γ . Finally, if η n are linear then ζ n are linear and since Mobius maps preserveslinearity, we have η Γ is linear. Definition 3.5 (Good-sequences) . For a given sequence t η n u of quasi circlesof t Γ n u Ă H with Γ n Ñ Γ, we say t η n u is a good-sequence of quasi-circlesif, it is convergent sequence and η is a quasi-circle of Γ . We also call t η n u (non)transverse good-sequence if all η n are also (non)transverse. Lemma 3.6 (Existence) . Let t Γ n u Ă H be a sequence of Schottky groupswith Γ n Ñ Γ . There exists a good-sequence t η n u of quasi-circles of t Γ n u . Inaddition, if t η n u is also (non)transverse then η is (non)transverse.Proof. Let f n be quasi-conformal maps such that Γ n “ f n p Γ q . Let η Γ be aquasi-circle of Γ . Then η n “ f n p η Γ q is a good-sequence of quasi-circles. The(non)transverse property obviously is preserved. Corollary 3.7 (Linear-invariant) . Let t η n u be a good-sequence of linearquasi-circles of t Γ n u Ă H . η is a linear quasi-circle of Γ . Proof.
Linearity is obviously preserved at η . Corollary 3.8 (Tranverse-invariant) . Let t η n u be a good-sequence of quasi-circles of t Γ n u Ă H . Then η is a (non)transverse linear quasi-circle of Γ ifand only if t η n u is a (non)transverse.Proof. emma 3.9 (Open) . Let t Γ n u Ă H . Let Γ n Ñ Γ be a Schottky group. Let η Γ be a quasi-circle of Γ . Then there exists a relative open neighboredood O p η Γ q of η Γ such that every element of η Γ in O p η Γ q is a quasi-circle of Γ . Proof.
Let F be a fundamental domain of Γ . Let ξ be the generating curveof η Γ with respect to F . Let F n be the fundamental domain of Γ n with F Ñ F n . By Proposition 3.4, we have f n : η Γ Ñ η n . Let U p ξ q be a smallopen neighborhood about ξ generating curve of η Γ and Let U p ξ n q to be theopen set about the generating curve ξ n of η n given by f n p U p ξ qq , and set O p ξ n q “ U p ξ n q X Ψ Γ . Then O p ξ n q is open sets of generating curves in Ψ Γ (Figure 2). Let O p ξ n q “ f n p O p ξ qq for O p ξ q a open set of generating curvesfor η Γ . Let ξ n P f n p O p ξ c qq and η n be generated by ξ n . Assuming O p ξ q issufficiently small neighborhood, we have length (cid:96) p η n q ă c(cid:96) p η Γ q for some small c ě n and η n generated by all ξ n P f n p O p ξ qq . Since η n Ñ η Γ , wehave η is quasi-circle of Γ with bounded length. This defines a open setwhich all elements are quasi-circles in Ψ Γ .Note that the above lemma essentially states that, for η n quasi-circles ofΓ n with η n Ñ η Γ and ξ n generating curves of η n , we have for some smalldeformations of ξ n which generates η n and converges in Ψ Γ is quasi-circles ofΓ. Figure 2: Open set of generating curves about ζ Corollary 3.10.
Let t Γ n u Ă H with Γ n Ñ Γ a Schottky group Γ . Let t η n u be agood-sequence of quasi-circles of t Γ n u . Then there exists an open neighborhood O p η q of η such that every element of O p η q is a quasi-circle of Γ . Proof.
Follows from Lemma 3.6 and Lemma 3.9.12ext we analyze the formations of singularities for a given sequence ofclassical Schottky groups converging to a Schottky group. These types ofsingularities has been studied in [9, 10].
Lemma 3.11 (Singularity) . Let t Γ n u Ă H with Γ n Ñ Γ a Schottky group.Assume that Γ is non-classical Schottky group. Then there exists η Γ suchthat, every O p η Γ q contains a deformation of η Γ which is non-quasi-circle. Here we say a closed curve is a non-quasi-circle , if it’s non-Jordan(containsa singularity point), or it’s not Γ-invariant.
Proof.
For each n , let F n be a classical fundamental domain of Γ n . Given asequence of classical fundamental domains t F n u the convergence is consideras follows: tB F n u is collection of 2 g circles t c i,n u i “ ,..., g in the Riemann sphere C , pass to a subsequence if necessary, then lim n c i,n , is either a point or acircle. We say t F n u convergents to G , if G is a region that have boundaryconsists of lim n c i,n for each i , which necessarily is either a point or a circle.Note that G is not necessarily a fundamental domain, nor it’s necessarilyconnected.Let lim F n “ G . By assumption that Γ is not a classical Schottky group,we have G is not a classical fundamental domain of Γ . We have B G consists ofcircles or points. However these circles may not be disjoint. More precisely,we have the following possible degeneration of circles of B F n which gives B G of at least one of following singularities types: • Tangency: Contains tangent circles. • Degeneration: Contains a circles degenerates into a point. • Collapsing: Contains two circles collapses into one circle. Here we havetwo concentric circles centered at origin and rest circles squeezed inbetween these two and these two collapse into a single circle in G . B G contains: Tangency.Let p be a tangency point. Let η Γ be a quasi-circle of Γ which passthrough point p. Assume the Lemma is false, then all sufficiently small openneighborhood O p η Γ q contains only quasi-circles of Γ . It follows from Propo-sition 3.2, we have a sequence with η n Ñ η Γ . Let ξ n be the generating curvesof η n with respect to F n . Define φ n as follows. Note that ξ n X B F n consistsof points or linear arcs. We define ζ n be a generating curve with ζ n X B F n
13o consists of linear arcs, with one of its end point to be the point that con-verges into a tangency in B G . In addition we also require the arcs on B F n thatwith a end points which converges to tangency point be be connect by a arcbetween the other end points. It is clear that every O p η n q X F n contains agenerating curve of this property. Let φ n be the quasi-circle generated by ζ n , and set φ “ lim φ n , then φ contains a loop singularity at a point of tangency(Figure 3). Hence φ is non-quasi-circle of Γ . Since every O p η Γ q contains sucha φ we must have the lemma to be true for this type of singularity.Figure 3: Tangency singularity B G contains: Degeneration.In this case, we must have any quasi-circle η n pass through a degenera-tion point. Note that we can’t have all circles degenerates into a single pointwhich would collapses quasi-circles to a point. To see this, let γ i,n Ñ γ i be thegenerating set of Γ n corresponding to circles t c i,n u and γ i it’s limit in Γ . Sinceby assumption Γ is Schottky group, so γ i must all be loxodromic, hence dis-tinct fixed points. But if all circles degenerates into a single point, and sinceall fixed points of γ i,n are contained within disjoint regions bounded by circles t c i,n u then, we can’t have all of t γ i u fixed points distinct, a contradiction.In general, even Γ P B J g we still can’t have all circles degenerates intoa single point. Since degenerating into a single point would imply Γ eitheris not discrete or not free group. But element of B J g is either cusps orgeometrically infinite group, and both types are discrete and free group.There are two possibilities to have degenerate points. Case (A): twocircles merge into a single degenerate point p . Case (B): A circle degeneratesinto a point on a circle.Consider (A). In this case, any quasi-circles η will have two possibleproperties, either there is a point q on some circle of B G such that everyquasi-circle must pass through q , or two separate arcs of η meet at p . The14econd possibility implies there exists a loop singularity at p , hence η cannot be a Jordan curve (Figure 4). Therefore we only have the first possibilityfor η . But it follows from Proposition 3.4, all rectifiable quasi-circles of Γis the limit of some sequence of quasi-circles of Γ n , and hence all must passthrough q . However q is not a limit point, hence we can have some quasi-circlenot passing through q, a contradiction.Figure 4: Degenerate singularitiesNow consider (B). Here we can assume that there exists at least two circlesthat do not degenerates into points. Otherwise, we will have the third type(collapsing) singularity, which we will consider next. Having some circlesdegenerates into a point on to a circle at p , we have a sequence of quasi-circles η n which passes through p. This is given by the generating curves thathave curves connecting the degenerating circle to point p n Ñ p convergingto linear arc intersecting orthogonally at boundary. But any neighboringquasi-circle of η n will converges to a η with a loop singularity at p, whichis not a Jordan curve (Figure 5).Figure 5: Degenerate singularitiesFinally, we note that if we have a degeneration point p which is not alimit point, then there exists a neighboring quasi-circle of η Γ that misses the15oint p. Hence any O p η Γ q will contains some curve which is the limit of asequence of quasi-circles that do not pass through the point. This gives anon-quasi-circle in O p η Γ q . B G contain: Collapsing.Let C denote the collapsed circle, i.e F n Ñ C . We will show then Λ Γ Ă C . Assume the contrary and suppose that there exists a fixed point not on C . Then there are infinitely many elements with fixed points not on C , sincelimit set is perfect set.Let γ n P Γ n with γ n Ñ γ. Since γ n p F n q Ñ γ p C q , we must have either γ p C q “ C or γ p C q X C “ H . By assumption we have a fixed point of a elementof Γ is R C , which implies infinitely many fixed points R C . So take threedistinct fixed points a, b, c which are respectively attractive fixed points of γ a , γ b , γ c P Γ such that a, b, c R C . Then γ kα p C q Ñ α for each α P t a, b, c u and k Ñ 8 . Let γ kn,α P Γ n with γ kn,α Ñ γ kα . Since F n Ñ C so we have γ kn,α p F n q Ñ γ kα p C q for each k. Now since γ kα p C q converges to distinct fixedpoints a, b, c so by choose k large enough we have for sufficiently large n , γ kn,α p F n q are contained within distinct regions bounded by circles of B β n p F n q for some β n P Γ n . To show this, we first choose a element β n P Γ n so that fixedpoint a is contained in a disk bounded by B β n p F n q and fixed points b, c arebounded in a region disjoint from the disk. Then choose β n P Γ n so that b, c fixed points lie within disjoint disks bounded by B β n p F n q , where β n “ β n β n . Hence for large k , we have γ kα p C q for α P t a, b, c u will lies in disjoint regionsbounded by circles of B β n p F n q , for sufficiently large n , see Figure 6.Figure 6: First figure have a γ kα p C q (in red) outside all circles which forbidcollapsing. Second have all three γ kα p C q in a big circle which forbid collapsing.The third don’t contain any and so collapses.Let β P Γ be the limit of β n . Since β n p F n q Ñ β p C q we have β n p F n q collapsing to a circle. However, γ kα p C q do not collapse to points for fixed k. Hence β n p F n q Ñ β p C q , a contradiction.Hence all fixed points of elements of Γ are Ă C , which implies Λ Γ Ă C . Since Γ is Schottky group, we must have Γ is Fuchsian schottky group. By165], Γ is classical Schottky group, a contradiction.
Lemma 3.12. η Γ in Lemma . is non-trivial singular Γ -invariant closedcurve.Proof. Consider tangency. In this case we will see that we have either: p a q tangencies are removable i.e. we can choose classical domain without tan-gency, or p b q one of the tangencies is a fixed point, or p c q there exists η Γ suchthat every singularities have four branches arcs (degree four singularity),such that η Γ curve is formed by linking collection of closed graphs of girth 3with every vertices of degree four in the graph. We will see that in (b) and(c) η Γ is not trivial singular curve. This can be seen as follows.First for simplicity suppose rank two ă α, β ą with corresponding circles t C α , C α , C β , C β u with tangency. If either C α , C α or C β , C β is tangent thenwe have a fixed point of the tangency.Now suppose we have just one of tangency from either C α , C β or C α , C β both not both, say C α , C β . Then we can just shrink C α the radius a bitand enlarge C α proportionally so that it remains disjoint from rest of circles.Better to way to see this is by conjugating C α , C α into circles centered onorigin, then one can easily see we can remove the tangency by choose anotherset of circles which gives classical domain, see Figure 7.Next suppose both C α , C β and C α , C β are tangent. Then the tangency iseither fixed point of α ´ β or it’s not a fixed point. Suppose the latter, thenthe singularity constructed in the proof of Lemma 3.11 gives (cid:53) -type singu-larity with bottom vertex the tangency of C α , C β . Similarly C α , C β producesame type of singularity. We choose the top vertices of (cid:53) to be images of tan-gency under α, β , since by assumption the tangency are not fixed points. Weconnect the two pair tangent circles by straight arcs correspondingly. Thenthese triangular curves tessellating into graph of (cid:53)(cid:52) attached through α, β maps, and converges to limit points of Γ in the curve, hence we have a closedgraph. In addition, it’s easy to see that every vertex of the graph is of degreefour with girth three in η Γ . Hence η Γ is formed by link (images of straightarcs) of collections of degree four closed graphs of girth 3, see Figure 8.Now we will see that η Γ is not trivial singular curve. Suppose the contrary,then we can choose some Schottky domain so that η Γ is given by singulargenerating curves ξ which are limit of generating curves of quasi-circles. Byedge and degree restrictions and girth of the graph, we have possible ξ givenby combinations of: Ý , ¯ (cid:52) see Figure 9. We label the triangular regions17igure 7: removable tangencyFigure 8: tangency singularitiesFigure 9: singular generating curvesaccording in Figure 9. In addition we label the tangency triangular regionin Figure 8 by a, b . There are three possible combinations of ξ : Ý Ý , ¯ (cid:52) ¯ (cid:52) , Ý ¯ (cid:52) . We show that none of them can form η as Γ invariant curve.Denote generators of the new Schottky domain by γ, ρ . Consider Ý Ý .We denote the triangular region of the other Ý by c , d respectively. Sincethey generate η , there exists ψ P Γ such that c “ ψ p a q , d “ ψβ p b q , ρ p c q “ ψβα ´ p a q , ρ p d q “ ψβα ´ β p b q , ργ ´ p c q “ ψβα ´ βα ´ p a q . This implies that ργ ´ p c q “ ψ p βα ´ q ψ ´ p c q . Hence ψ ´ ργ ´ ψ “ p βα ´ q . After simple com-putations this equality implies either ρ “ γ ´ , or rρ ´ r “ γ for some r P Γ . However since ρ, γ are free generators, this is impossible. Hence this relationimplies ρ, γ are not generators expressible in terms of β, α. (cid:52) ¯ (cid:52) we denote the triangular regions of second ¯ (cid:52) by f correspondingly. Then we have, f “ φ p a q , ρ p f q “ φβα ´ p a q , ργ ´ p f q “ φβα ´ βα ´ p a q for some φ P Γ . Hence we have ργ ´ “ φ p βα ´ q φ ´ . It followsfrom previous argument which implies ρ, γ are not generators of Γ.Finally consider Ý ¯ (cid:52) . There exists λ P Γ where we have, c “ λ p a q , d “ λβ p b q , ρ p e q “ λβα ´ p a q , ρ p f q “ λβα ´ β p b q , ρ p g q “ λβα ´ βα ´ p a q , ργ ´ p c q “ λβα ´ βα ´ β p b q , ργ ´ p d q “ λβα ´ βα ´ βα ´ p a q . Hence using these relationswe have, p βα ´ q ´ λ ´ ργ ´ λ “ λ ´ γρ ´ λ p βα ´ q . Set χ “ p βα ´ q ´ λ ´ ργ ´ λ. Then we have χ “ χ ´ αβ ´ . This implies χ “ αβ ´ , which is impossible.Hence it follows from above argument and the assumption of Lemma3.11, Γ is not classical Schottky group, we can only have the possibilityof having tangency as fixed points. However, one can not choose anotherSchottky domain of Jordan curves which would exclude fixed point outsidethese Jordan curves. Hence singularity at fixed point can not be trivializedby choose different domains i.e. they are non-removable .Similarly for k circles with tangency. Following previous argument, wehave either all pairs tangency or we can remove the tangency one by one asbefore by choose difference classical Schottky domain circles. For all pairstangency we have γ i , γ j gives tangency. It’s sufficient to assume i “ j andtangent point don’t maps into other tangent point, otherwise it gives fixedpoint. Similarly as before they generates η Γ with links of collection of closedgraphs of degree four edges. It follows from previous argument, we can nothave singular generating curves that gives η Γ .Consider degeneracy. In this case we have circles degenerating into points.If we have a singularity at degenerated point which is limit point then weknow it can not be trivialized by choose another Schottky domain. Now onthe other hand, suppose we have all the degenerate points are not as limitpoints and, these singularities are trivial. Then η Γ is given by singular gener-ating curves with respect to some Schottky domain of Γ bounded by disjointJordan curves C . This Schottky domain is the limit of sequence of Schot-tky domain bounded by Jordan curves C n of Γ n . Denote the correspondingsequence of quasi-circles by η n . Let C i,n be the Jordan curves that boundsthe part of the regions which encloses limit points of C i,n (the circles degen-erating to points). Then with respect to C n and since η n Ñ η Γ , we musthave η Γ contains infinite numbers of singularities within C i,n . However since C i,n degenerates into points, we can only have finite collection of singularitieswithin that region, which is contradiction. Hence we can not have η Γ givenby singular generating curves. 19 Proof of Theorem 1.1
Proof. (Theorem 1.1)First note that by Selberg Lemma we can just assume Kleinian group to betorsion-free.Now note that if a Kleinian group Γ of D Γ ă D Γ ă R “ Ω Γ { Γ in H { Γ . If R is incompressible then, subgroup π p R q Ă Γ have D π p R q “ R is compressible then, we can cut alongcompression disks. We either end with incompressible surface as before orafter finitely many steps of cutting we obtain topological ball, which implies H { Γ is handle-body, hence free.Let Γ P J g be a non-classical Schottky group with D Γ ă . Let t Γ n u Ă H ,and Γ n Ñ Γ . Since Γ is Schottky group, it follows from Lemma 3.9, thatthere exists an open set O p η Γ q such that every element is quasi-circle of Γ.Since Γ is a non-classical Schottky group, hence by Lemma 3.11, we musthave every open set about η Γ contains some non-quasi-circle, in-particularwe must have O p η Γ q contain a non-quasi-circle. Hence we must have Γ isa classical Schottky group. This implies J g,o X J g is both open and closedin J g i.e. connected component of J g . Now it follows from Proposition 2.12, J g,o X J g “ J g . Finally, sharpness comes from the fact that, there exists Kleinian groupswhich is not free of Hausdorff dimension equal to one. Hence we have ourresult.E-mail: [email protected]
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