On the local connectivity of limit sets of Kleinian groups

Transcription

1 On the local connectivity of limit sets of Kleinian groups James W. Anderson and Bernard Maskit Department of Mathematics, Rice University, Houston, TX Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY Abstract AMS no. 30F40 The main result here is the fact that the limit set of a geometrically finite Kleinian group with connected limit set is locally connected. For an analytically finite Kleinian group with connected limit set, we give necessary and sufficient conditions for the limit set to be locally connected. 1 Introduction The purpose of this note is to fill a gap in the current literature by providing a proof of the fact that the limit set of a geometrically finite Kleinian group with connected limit set is locally connected (Corollary 3.3). We actually prove a bit more; we give a necessary and sufficient condition for the limit set of an analytically finite Kleinian group with connected limit set to be locally connected. The condition is that the limit set of every degenerate structure subgroup must be locally connected. For an analytically finite Kleinian group a structure subgroup is a maximal subgroup of the stabilizer of a component of the set of discontinuity, where this subgroup, as a Kleinian group, has a simply connected invariant component on which it acts without accidental parabolic elements. A finitely generated Kleinian group is degenerate if its set of discontinuity is connected, simply-connected and hyperbolic. We do not consider the difficult, and mostly unresolved, conjecture that the limit set of a degenerate group is locally connected; we do show that this conjecture is the only obstacle to locally connectivity of the limit set of a general analytically finite Kleinian group with connected limit set. A survey of the consequences of a positive resolution of this conjecture is given in Abikoff [1]. In particular, there are strong connections with Thurston s ending lamination conjecture. It has been shown by Cannon and Thurston [5] and by Minsky [10] that certain classes of degenerate groups have locally connected limit set. Roughly speaking, Cannon and Thurston consider those degenerate groups which arise from pseudo-anosov homeomorphisms of a surface, while Minsky Research supported in part by NSF Grant

2 considers those for which there is a lower bound on the injectivity radius of the corresponding hyperbolic 3-manifold. The authors wish to thank Curt McMullen for bringing this gap in the literature to our attention and for several important suggestions. McMullen has also sketched a similar proof (see [9]), based in part on the work of Floyd [6]. We also thank him for pointing out (oral communication) that there is a closely related result in the theory of rational maps proved by Tan and Yin [12]. Basic Definitions A Kleinian group is a discrete subgroup of PSL 2 (C), which can be viewed as acting either on hyperbolic 3-space H 3 via isometries or on the Riemann sphere C via Möbius transformations. The action of Γ partitions C into two sets. The domain of discontinuity Ω(Γ) is the largest open subset of C on which Γ acts properly discontinuously. Its complement, the limit set Λ(Γ), is the smallest non-empty closed subset of C which is invariant under the action of Γ. If Λ(Γ) contains at least three points, we say Γ is non-elementary. In this case, the universal cover of any component of Ω(Γ) is the upper half plane H 2, and the stabilizer st Γ ( ) = {γ Γ γ( ) = } of in Γ lifts to a group Φ of isometries of H 2. Hence, the Poincaré metric on H 2 descends to a complete metric of curvature 1 on and on /st Γ ( ). Say that Γ is analytically finite if Ω(Γ)/Γ has finite area in this metric; in particular, Ω(Γ)/Γ is the union of finitely many Riemann surfaces, each of which is a compact surface with a finite number of punctures. Ahlfors finiteness theorem [3] states that every finitely generated Kleinian group is analytically finite. There are several classes of Kleinian groups which are of special interest. A quasifuchsian group is a finitely generated Kleinian group Γ whose limit set Λ(Γ) is a Jordan curve, such that no element of Γ interchanges the components of C Λ(Γ). A degenerate group is a non-elementary finitely generated Kleinian group Γ whose domain of discontinuity Ω(Γ) and limit set Λ(Γ) are both nonempty and connected. A function group is a Kleinian group Γ whose domain of discontinuity Ω(Γ) contains a component that is invariant under the action of Γ. A B-group is a function group for which the invariant component is simply-connected; equivalently, a B-group is a function group with connected limit set. Quasifuchsian groups and degenerate groups are examples of B-groups. A Kleinian group is geometrically finite if there exists a finite sided fundamental polyhedron for its action on H 3. For a detailed discussion of equivalent formulations of geometric finiteness, see [4]. By way of example, quasifuchsian groups are geometrically finite, while degenerate groups are not. 2 B-groups In this section, we show that a finitely generated B-group has locally connected limit set if and only if each of its degenerate structure subgroups has locally connected limit set. The main tools we use 2

3 are a decomposition of function groups due to Maskit and a characterization of local connectivity that can be found in Whyburn [13]. We start with a description of the decomposition for a B- group; a more detailed treatment, for the more general case of a function group, can be found in chapters IX and X of [8]. Let Γ be an analytically finite B-group, and let be an invariant component of Ω(Γ). Since Γ is analytically finite, the stabilizer of in Γ is finitely generated; also, since Λ(Γ) is connected, is necessarily simply connected. Note that is unique unless Γ is quasifuchsian, in which case there are two choices, either of which does equally well. A Riemann map ϕ : H 2 conjugates the action of Γ on to the action of a Fuchsian group Φ = ϕ Γ ϕ 1 on H 2. A parabolic element ξ Γ is accidental if the corresponding element ϕ ξ ϕ 1 Φ is hyperbolic. In this case, let l be the (hyperbolic) line in H 2 invariant under ϕ ξ ϕ 1 (that is, l is the axis of ϕ ξ ϕ 1 ), and define the axis of ξ to be a = ϕ 1 (l). Let ξ be an accidental parabolic element of Γ, acting on, with axis a, and consider the full axis A = a fix(ξ). Then A is a Jordan curve that is precisely invariant under st Γ (A) in Γ; that is, A is invariant under the action of the subgroup st Γ (A) and γ(a) is disjoint from A for every element γ Γ st Γ (A). The only possibilities for the subgroup st Γ (A) are that it is parabolic cyclic, or that it is a Z 2 -extension of a parabolic cyclic group; in the latter case, there are fixed points of elliptic elements of order 2 on A. The full axes of accidental parabolic elements of Γ are called structure loops. Since Γ is analytically finite, there are only finitely many inequivalent structure loops, and any two distinct structure loops are disjoint. Further, for any one structure loop A, the spherical diameters of any sequence of distinct translates of A go to zero, and so it follows that the spherical diameters of any sequence of distinct structure loops go to zero. In general, a function group contains multiple different accidental parabolics. Let ξ 1,..., ξ p be a maximal collection of primitive accidental parabolic elements of Γ that generate non-conjugate cyclic subgroups, and let A j be the full axis of ξ j. Normalize so that and so that does not lie on any structure loop. Then each structure loop A has a well-defined inside and outside, where the inside is the bounded component of C A and the outside contains. The components of the complement of the union of the structure loops are the panels of Γ; the stabilizers of the panels are the structure subgroups of Γ. The crucial properties of this decomposition are as follows. There are only finitely many conjugacy classes of structure subgroups in Γ, and each structure subgroup either is quasifuchsian or is degenerate without accidental parabolics; in particular, the structure subgroups are finitely generated. The closure of each panel contains the limit set of its stabilizer (the corresponding structure subgroup) in fact, for each panel R, Λ(Γ) R = Λ(st Γ (R)). If x Λ(Γ) is not a limit point of any structure subgroup, then x is the limit of a nested sequence of translates of some A j. Finally, for any two distinct panels, P 1 and P 2, either their closures are disjoint and separated by a structure loop, or their closures intersect in a structure loop which lies on the boundary of each. Proposition 2.1 Let Γ be an analytically finite B-group. Then, Γ has locally connected limit set 3

4 if and only if the limit set of every degenerate structure subgroup is locally connected. set. Proof We suppose first that every degenerate structure subgroup has locally connected limit We have already noted that every structure subgroup is either quasifuchsian or degenerate. Since the limit set of a quasifuchsian group is a Jordan curve, it is connected and locally connected. Hence our hypothesis implies that every structure subgroup has locally connected limit set. We recall the following. A set X has property S if, for each ε > 0, X is the union of a finite number of connected sets, each having diameter less than ε. It is well known ([13], p. 23) that a continuum X is locally connected if and only if X has property S. Let ε > 0 be given. Normalize Γ so that and so that does not lie on any structure loop. Since the diameters of any sequence of distinct structure loops go to zero, there are only finitely many structure loops whose diameter is greater than any given number. Hence, there are a finite number of structure loops, call them B 1,..., B s, whose diameter is greater than 1 4 ε. Since there are only finitely many B i, there are only finitely many panels which contain at least one of the B i in their boundary. Call these panels R 1,..., R t. Each R j has diameter at least 1 4 ε and, except for the finitely many B i on its boundary, each structure loop on the boundary of each R j has diameter less than 1 4 ε. Among the remaining structure loops, let C 1,..., C n,... be the outside structure loops of diameter less than 1 4 ε; that is, no C i separates any C j from, and every structure loop of diameter less than 1 4 ε either is one of the C i or is separated from by some C i. Note that, by our choice of R 1,..., R t above, each C i lies on the boundary of some R j. For each i, let C i be the intersection of Λ(Γ) with the closed bounded disc in C determined by C i. Since each structure loop intersects Λ(Γ) in exactly one point, one easily sees that each C i is connected. We now have that ( t ) Λ(Γ) = Λ(Γ) R j. j=1 C i i=1 Since the limit set of each structure subgroup is connected and locally connected, for each j, the limit set Λ(st Γ (R j )) = Λ(Γ) R j of the stabilizer, st Γ (R j ), of R j is connected and locally connected. Since there are finitely many R j, we can find a finite number of connected subsets D 1,..., D m of Λ(Γ) so that each D k has diameter less than 1 4 ε and m t D k Λ(Γ) R j. k=1 j=1 Since every limit point of Γ either lies in the closure of some R j, or lies inside one of the C i, we can enlarge each of the D k by adjoining all the C i such that the parabolic fixed point on C i lies in D k. Call these new sets D k, k = 1,..., m. Since the C i are all disjoint and of diameter at most 1 4 ε, each D k has diameter less than ε. 4

5 We need to show that each D k is connected. Assume not. Then, there are two disjoint open sets in C, U and V, whose union contains D k. Since D k is connected, we can assume that D k U. Then, for every i, the parabolic fixed point on the boundary of C i is contained in U. Then, since C i is connected, C i U. Hence D k U. We have shown that Λ(Γ) can be written as a finite union of sets D k, each connected and of diameter less than ε. Hence, Λ(Γ) has property S, and so is locally connected. Now, suppose that Γ has locally connected limit set; we will show that Λ(Φ) is locally connected, where Φ is any structure subgroup of Γ. We start with the observation that, if x Λ(Γ) and x Λ(Φ), then there is a structure loop separating x from Λ(Φ). Since each structure loop intersects Λ(Γ) in a single point, it follows that X Λ(Φ) is connected whenever X is a connected subset of Λ(Γ). Since Λ(Γ) has property S, given ε > 0, we can write Λ(Γ) as the union Y 1 Y m, where each Y t is connected and has diameter at most ε; by taking closures, we may assume that each Y t is closed. Intersecting each Y t with Λ(Φ) gives a decomposition of Λ(Φ) as the union of finitely many connected sets of diameter at most ε. Hence, Λ(Φ) has property S, and hence is locally connected. 3 Analytically finite groups with connected limit set We now consider the general case of an analytically finite Kleinian group with connected limit set. An E-continuum is a plane continuum X such that, for every ε > 0, there are at most a finite number of complementary domains of X of diameter greater than ε. It is well known ([13], p. 113) that a plane continuum X is locally connected if and only if X satisfies the following two properties: (i) X is an E-continuum, and, (ii) for every complementary domain of X, is locally connected. If Γ is an analytically finite Kleinian group with connected limit set, then every component of Ω(Γ) is simply connected; it is well known and easy to see that, since Γ is analytically finite, the stabilizer of any component of Ω(Γ) is a finitely generated B-group. Proposition 3.1 Let Γ be an analytically finite Kleinian group with connected limit set. Then Λ(Γ) is locally connected if and only if every component stabilizer of Γ has locally connected limit set. Proof It was shown by Maskit [7] that Λ(Γ) is an E-continuum for every analytically finite Kleinian group Γ with connected limit set. For an analytically finite group, the boundaries of the components are exactly the limit sets of the component stabilizers. For function groups, a structure subgroup is a maximal subgroup Φ of Γ which has a simply connected invariant component, contains no accidental parabolic elements, and contains every 5

6 parabolic element whose fixed point lies in its limit set Λ(Φ) (see [8], p. 268). A structure subgroup of a more general analytically finite Kleinian groups is a structure subgroup of a stabilizer of a component of Ω(Γ). For an analytically finite Kleinian group Γ, every component stabilizer is a finitely generated function group. Combining 3.1 with 2.1, we obtain the following. Theorem 3.2 Let Γ be an analytically finite Kleinian group with connected limit set. Then Λ(Γ) is locally connected if and only if every degenerate structure subgroup of Γ has locally connected limit set. Corollary 3.3 Let Γ be a geometrically finite Kleinian group with connected limit set. Then, Λ(Γ) is locally connected. Proof It was shown by Thurston (see [11]) that every component stabilizer of a geometrically finite Kleinian group is again geometrically finite. It is well known (see for example Maskit [8] p. 289) that every structure subgroup of a geometrically finite function group is either quasifuchsian or elementary; in particular, every structure subgroup has locally connected limit set. 4 Analytically finite groups We now consider an analytically finite Kleinian group Γ whose limit set is not necessarily connected. It follows from work of Abikoff and Maskit [2] that, for such a group Γ, there exists a finite collection Φ 1,..., Φ p of subgroups of Γ that are analytically finite and have connected limit set so that, if x Λ(Γ), then either x is a translate of a point of some Λ(Φ j ) or x is the limit of a nested sequence of translates of a Jordan curve c lying entirely in Ω(Γ). It follows that every connected component of Λ(Γ) is either the limit set of an analytically finite Kleinian group with connected limit set, or is a single point. These subgroups of Γ are unique up to conjugation in Γ; a structure subgroup of Γ is a structure subgroup of a conjugate of one of these subgroups, Φ i. Hence, we have the following. Corollary 4.1 Let Γ be an analytically finite Kleinian group. Every connected component of Λ(Γ) is locally connected if and only if every structure subgroup of Γ has locally connected limit set. Corollary 4.2 Every component of the limit set of a geometrically finite Kleinian group is locally connected. We close by noting that the construction given in Section VIII.A of [8] can easily be used to construct an infinitely generated Kleinian group whose limit set is connected but not locally connected. 6

7 References [1] W. Abikoff, Kleinian groups geometrically finite and geometrically perverse, in Geometry of Group Representations, ed. by Wm. M. Goldman and Andy R. Magid, Contemporary Mathematics, vol. 74, American Mathematical Society, Providence, RI, 1988, [2] W. Abikoff and B. Maskit, Geometric decomposition of Kleinian groups, Amer. J. Math. 99 (1977), [3] L. Ahlfors, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), [4] B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Functional Analysis 113 (1993), [5] J. Cannon and W. P. Thurston, Group invariant Peano curves, preprint, [6] W. J. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), [7] B. Maskit, Intersections of Component Subgroups of Kleinian Groups, in Discontinuous Groups and Riemann Surfaces, ed. by L. Greenberg, Annals of Mathematics Studies, no. 79, Princeton University Press, Princeton, 1974, [8] B. Maskit, Kleinian groups, Springer Verlag, New York, [9] C. McMullen, The classification of conformal dynamical systems, in Current Developments in Mathematics, 1995, International Press, to appear. [10] Y. Minsky, On rigidity, limit sets, and end invariants of hyperbolic 3-manifolds, J. Amer. Math. Soc. 7 (1994), [11] J.W. Morgan, On Thurston s uniformization theorem for three-dimensional manifolds, in The Smith Conjecture, ed. by J. Morgan and H. Bass, Academic Press, 1984, [12] L. Tan and Y. Yin, Local connectivity of the Julia set for geometrically finite rational maps, ENS Lyon preprint UMPA , [13] G. T. Whyburn, Analytic Topology, American Mathematical Society, Providence, RI,