Circle packings, kissing reflection groups and critically fixed anti-rational maps
CCIRCLE PACKINGS, KISSING REFLECTION GROUPSAND CRITICALLY FIXED ANTI-RATIONAL MAPS
RUSSELL LODGE, YUSHENG LUO, AND SABYASACHI MUKHERJEE
Abstract.
In this paper, we establish an explicit correspondence be-tween kissing reflection groups and critically fixed anti-rational maps.The correspondence, which is expressed using simple planar graphs, hasseveral dynamical consequences. As an application of this correspon-dence, we give complete answers to geometric mating problems for criti-cally fixed anti-rational maps. We also prove the analogue of Thurston’scompactness theorem for acylindrical hyperbolic 3-manifolds in the anti-rational map setting.
Contents
1. Introduction 12. Circle Packings 83. Kissing Reflection Groups 94. Critically Fixed Anti-rational Maps 295. Blaschke and Anti-Blaschke Products 446. Pared Deformation Space 54References 601.
Introduction
Ever since Sullivan’s translation of Ahlfors’ finiteness theorem into a solu-tion of a long standing open problem on wandering domains in the 1980s [47],many more connections between the theory of Kleinian groups and the studyof dynamics of rational functions on (cid:98) C have been discovered. These analo-gies between the two branches of conformal dynamics, which are commonlyknown as the Sullivan’s dictionary , not only provide a conceptual frameworkfor understanding the connections, but motivates research in each field aswell.In this paper, we extend this dictionary by establishing a strikingly ex-plicit correspondence between • Kissing reflection groups : groups generated by reflections along thecircles of finite circle packings P (see § • Critically fixed anti-rational maps : proper anti-holomorphic self-maps of (cid:98) C with all critical points fixed (see § a r X i v : . [ m a t h . D S ] J u l R. LODGE, Y. LUO, AND S. MUKHERJEE
This correspondence can be expressed by a combinatorial model: a simpleplanar graph Γ. A graph Γ is said to be k -connected if Γ contains more than k vertices and remains connected if any k − Theorem 1.1.
The following three sets are in natural bijective correspon-dence: • { -connected, simple, planar graphs Γ with d + 1 vertices up to pla-nar isomorphism } , • { Kissing reflection groups G of rank d + 1 with connected limit setup to QC conjugacy } , • { Critically fixed anti-rational maps R of degree d up to M¨obius conjugacy } .Moreover, if G Γ and R Γ correspond to the same graph Γ , then the limit set Λ( G Γ ) is homeomorphic to the Julia set J ( R Γ ) via a dynamically naturalmap. The correspondence between graphs and kissing reflection groups comesfrom the well-known circle packing theorem (see Theorem 2.1): given akissing reflection group G with corresponding circle packing P , the graphΓ associated to G in Theorem 1.1 is the contact graph of the circle packing P . The 2-connectedness condition for Γ is equivalent to the connectednesscondition for the limit set of G (see Figure 1.1).On the other hand, given a critically fixed anti-rational map R , we con-sider the union T of all fixed internal rays in the invariant Fatou compo-nents (each of which necessarily contains a fixed critical point of R ), knownas the Tischler graph (cf. [49]). We show that the planar dual of T is a 2-connected, simple, planar graph. The graph Γ we associate to R in Theorem1.1 is the planar dual T ∨ of T (see Figure 1.1).We remark that the correspondence between G Γ and R Γ through thegraph Γ is dynamically natural. Indeed, we associate a map N Γ to thegroup G Γ with the properties that N Γ and G Γ have the same grand orbits(cf. [37, 2]), and the homeomorphism between Λ( G Γ ) and J ( R Γ ) conjugates N Γ to R Γ . See § d + 1) with connected limit set and hyperbolic componentshaving critically fixed anti-rational maps (of degree d ) as centers. The geometric mating problems.
In complex dynamics, polynomialmating is an operation first introduced by Douady in [7] that takes twosuitable polynomials P and P , and constructs a richer dynamical systemby carefully pasting together the boundaries of their filled Julia sets so as IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 3 (a)
A 2-connected simple planargraph Γ with five vertices. (b)
The limit set of a kissing reflection group G Γ . The four outermost circles together withthe central circle form a finite circle packing P whose contact graph is Γ. The kissing re-flection group G Γ is generated by reflectionsalong the five circles in P . (c) The Julia set of the degree 4 critically fixed anti-rational map R Γ . TheTischler graph T , which is drawn schematically in red with two finite verticesand one vertex at infinity, is the planar dual of Γ. Figure 1.1.
An example of the correspondence.
R. LODGE, Y. LUO, AND S. MUKHERJEE to obtain a copy of the Riemann sphere, together with a rational map R from this sphere to itself (see Definition 4.16 for the precise formulation). Itis natural and important to understand which pairs of polynomials can bemated, and which rational maps are matings of two polynomials. The anal-ogous question in the Kleinian group setting can be formulated in terms ofCannon-Thurston maps for degenerations in the Quasi-Fuchsian space (see[15] and § n vertices is outerplanar if ithas a face with all n vertices on its boundary. It is said to be Hamiltonian if there exists a Hamiltonian cycle, i.e., a closed path visiting every vertexexactly once.
Theorem 1.2.
Let Γ be a -connected, simple, planar graph. Let G Γ and R Γ be a kissing reflection group and a critically fixed anti-rational map as-sociated with Γ . Then the following hold true. • Γ is outerplanar ⇔ R Γ is a critically fixed anti-polynomial ⇔ G Γ isa function group. • Γ is Hamiltonian ⇔ R Γ is a mating of two polynomials ⇔ G Γ is amating of two function groups ⇔ G Γ is in the closure of the quasi-conformal deformation space of the regular ideal polygon reflectiongroup. It is known that a rational map may arise as the geometric mating of morethan one pair of polynomials (in other words, the decomposition/unmating of a rational map into a pair of polynomials is not necessarily unique). Thisphenomenon was first observed in [53], and is referred to as shared matings (see [43]). In our setting, we actually prove that each Hamiltonian cycle ofΓ gives an unmating of R Γ into two anti-polynomials. Thus, we get manyexamples of shared matings coming from different Hamiltonian cycles in theassociated graphs.We now address the converse question of mateability in terms of lamina-tions. Let us first note that the question of mateability for kissing reflectiongroups can be answered using Thurston’s double limit theorem and thehyperbolization theorem. In the reflection group setting, possible degener-ations in the Quasi-Fuchsian space (of the regular ideal polygon reflectiongroup) are listed by a pair of geodesic laminations on the two conformalboundaries which are invariant under some orientation reversing involution σ (see § σ -invariant laminations turn out to be multicurveson the associated conformal boundaries. A pair of simple closed curves issaid to be parallel if they are isotopic under the natural orientation reversingidentification of the two conformal boundary components. A pair of lami-nations is said to be non-parallel if no two components are parallel. If welift a multicurve to the universal cover, we get two invariant laminations onthe circle. Then they are are non-parallel if and only if the two laminations IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 5 share no common leaf under the natural identification of the two copies ofthe circle. Thurston’s hyperbolization theorem asserts that in our setting,the degeneration along a pair of laminations exists if and only if this pair isnon-parallel.For a marked anti-polynomial, we can also associate a lamination via theB¨otthcher coordinate at infinity. Similar as before, we say a pair of (anti-polynomial) laminations are non-parallel if they share no common leaf underthe natural identification of the two copies of the circle. When we glue twofilled Julia sets using the corresponding laminations, the resulting topologi-cal space may not be a 2-sphere. We call this a
Moore obstruction . We provethe following more general mateability theorem for post-critically finite anti-polynomials, which in particular answers the question of mateability of twocritically fixed anti-polynomials.
Theorem 1.3.
Let P and P be two marked anti-polynomials of equal degree d ≥ , where P is critically fixed and P is postcritically finite, hyperbolic.Then there is an anti-rational map R that is the geometric mating of P and P if and only if there is no Moore obstruction.Consequently, if both P and P are critically fixed, then they are geomet-rically mateable if and only if the corresponding laminations are non-parallel. Acylindrical manifolds and gasket sets.
A circle packing is a connectedcollection of (oriented) circles in (cid:98) C with disjoint interiors. We say that aclosed set Λ is a round gasket if • Λ is the closure of some infinite circle packing; and • the complement of Λ is a union of round disks which is dense in (cid:98) C .We will call a homeomorphic copy of a round gasket a gasket .Many examples of kissing reflection groups and critically fixed anti-rationalmaps have gasket limit sets and Julia sets (see Figure 1.2). The correspon-dence allows us to classify all these examples (cf. [17, Theorem 28] and[16]). Theorem 1.4.
Let Γ be a -connected, simple, planar graph. Then, G Γ has gasket limit set ⇐⇒ R Γ has gasket Julia set ⇐⇒ Γ is -connected. The 3-connectedness for the graph Γ also has a characterization purely interms of the inherent structure of the hyperbolic 3-manifold associated with G Γ . Given a kissing reflection group G Γ , the index 2 subgroup (cid:101) G Γ consistingof orientation preserving elements is a Kleinian group. We say that G Γ is acylindrical if the hyperbolic 3-manifold for (cid:101) G Γ is acylindrical (see § Theorem 1.5.
Let Γ be a -connected, simple, planar graph. Then, G Γ is acylindrical ⇐⇒ Γ is -connected. Motivated by this theorem, we say that a critically fixed anti-rational map R Γ is acylindrical if Γ is 3-connected. R. LODGE, Y. LUO, AND S. MUKHERJEE
Thurston’s compactness theorem.
The acylindrical manifolds play avery important role in three dimension geometry and topology. Relevant toour discussion, Thurston proved that the deformation space of an acylin-drical 3-manifold is bounded [51], which is a key step in Thurston’s hyper-bolization theorem for Haken manifolds.The analogue of deformation spaces in holomorphic dynamics is hyperboliccomponents . A(n) (anti-)rational map with degree d is said to be hyperbolicif all critical points converge to attracting or super-attracting cycles underiteration. Hyperbolic (anti)-rational maps form an open set in the modulispace M ± d of (anti-)rational maps of degree d , and a connected componentof this open set is called a hyperbolic component. In the convex cocompactcase, it is well known that a hyperbolic 3-manifold with non-empty bound-ary is acylindrical if and only if its limit is a Sierpinski carpet (see [30] or[41, Theorem 1.8]). Based on this analogy, McMullen conjectured that thehyperbolic component of a rational map with Sierpinski carpet Julia set isbounded (see [30, Question 5.3]). This conjecture, which stimulated a lotof work towards understanding the topology of hyperbolic components, stillremains wide open.It is quite natural to ask the same question in our setting. However,the most naive translation is not true: the hyperbolic component H ( R ) ofa critically fixed anti-rational map R is never bounded (see the discussionin § R correspond to cusps of G . For Kleinian groups with parabolics, Thurston’s compactness theoremholds only for deformation spaces of pared manifolds , i.e., when the cuspsremain cusps under deformation. On the other hand, we have more freedomto deform anti-rational maps in H ( R ): the multipliers (which are analoguesof lengths of closed geodesics in holomorphic dynamics) of the correspondingrepelling fixed points can grow to infinity.This motivates the definition of the pared deformation space for R . Wefirst note that the dynamics on any critically fixed Fatou component can beconjugated to the dynamics of an anti-Blaschke product on the unit disk D ,which we call the uniformizing model . Let K >
0, we define X ( K ) ⊆ H ( R )to be the set of anti-rational maps R ∈ H ( R ) where the multiplier of anyrepelling fixed point in any uniformizing model remains bounded by K (see § Theorem 1.6.
Let R be a critically fixed anti-rational map. The pareddeformation space X ( K ) is bounded if and only if R is acylindrical.More precisely, if R is acylindrical, then for any K , the space X ( K ) isbounded in M − d . Conversely, if R is cylindrical, then there exists some K such that X ( K ) is unbounded in M − d . The uniform bounds on the multipliers of the fixed points allow us toconstruct a quasi-invariant tree capturing all the interesting dynamics. The
IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 7 proof of the fact that acylindricity of the anti-rational map implies bound-edness of the pared deformation space uses this tree as a crucial ingredient.The other direction is proved by constructing a particular pinching degen-eration in the corresponding pared deformation space.We now summarize these correspondences in the following table.
Simple planar graph Kissing reflection group Critically fixed anti-rational map
Connected Kissing reflection group -2-connected Connected limit set Critically fixed anti-rational map3-connected/Polyhedral Gasket limit set/Acylindrical/ Bounded QC space Gasket Julia set/ Acylindrical/Bounded pared deformation spaceOuterplanar Function kissing reflection group Critically fixed anti-polynomialHamiltonian Mating of two function groups/Closure of Quasi-Fuchsian space Mating of two anti-polynomialsA marked Hamiltonian cycle A pair of non-parallel multicurves A pair of non-parallel anti-polynomial laminations
Notes and references.
Aspects of the Sullivan’s dictionary were alreadyanticipated by Fatou [10, p. 22]. Part of the correspondence in Theorem1.1 has been observed in [23] where the Tischler graphs were assumed to betriangulations. A classification of critically fixed anti-rational maps has alsobeen obtained in [11]. There is also a connection between Kleinian reflectiongroups, anti-rational maps, and Schwarz reflections on quadrature domainsexplained in [21, 23, 19, 20]. In the holomorphic setting, critically fixedrational maps have been studied in [4, 13].Many critically fixed anti-rational maps have large symmetry groups. Theexamples corresponding to the five platonic solids are listed in Figure 1.2.The counterparts in the holomorphic setting were constructed in [8] (see also[3, 14]).Results on unboundedness of hyperbolic components were obtained in[25, 48]. On the other hand, various hyperbolic components of quadratic ra-tional maps were shown to be bounded in [9]. More recently, Nie and Pilgrimextended some boundedness results to higher degrees [35, 36]. The pareddeformation space we consider can be generalized to the case of periodiccycles. Many techniques used in this paper can be adapted to prove bound-edness/unboundedness of such more general pared deformation spaces.The connections between number theoretic problems for circle packingsand equidistribution results for Kleinian groups can be found in [18, 38, 17].
Structure of the paper.
We collect various known circle packing theo-rems in §
2. Based on this, we prove the connection between kissing reflectiongroups and simple planar graphs in §
3. In particular, the group part of The-orem 1.1, Theorem 1.2, Theorem 1.4, and Theorem 1.5 are proved in Propo-sition 3.4, Propositions 3.18, 3.20, 3.21, Proposition 3.10, and Proposition3.6 respectively.Critically fixed anti-rational maps are studied in §
4. The anti-rationalmap part of Theorem 1.1 is proved in Proposition 4.10. Once this is estab-lished, the anti-rational map part of Theorem 1.2 and Theorem 1.4 follow
R. LODGE, Y. LUO, AND S. MUKHERJEE (a)
Tetrahedron: z z +1 (b) Octahedron: z +1 z +5 z (c) Cube: z +7 z z +1 (d) Icosahedron:11 z +66 z − z +66 z − z (e) Dodecahedron: z − z +247 z +57 z − z +247 z +171 z +1 Figure 1.2.
Critically fixed anti-rational maps associated toPlatonic solids. The Fatou components are colored accordingto their grand orbit. The Tischler graph T , which is theplanar dual of Γ, is visible in the Figures by connecting thecenters of critical fixed Fatou components.from their group counterparts as explained in Corollary 4.17. Theorem 1.3is proved in Proposition 4.21 and Corollary 4.22.We study the uniformizing model of (anti-)Blaschke products in §
5, wherevarious geometric bounds are proved. With these bounds, we prove Theorem1.6 in § Acknowldgements.
The authors would like to thank Curt McMullen foruseful suggestions. The third author was supported by an endowment fromInfosys Foundation. 2.
Circle Packings
Recall that a circle packing P is a connected collection of (oriented) circlesin (cid:98) C with disjoint interiors. Unless stated otherwise, the circle packings inthis paper are assumed to contain at least three, but finitely many circles.The combinatorics of configuration of a circle packing can be described byits contact graph Γ: we associate a vertex to each circle, and two vertices areconnected by an edge if and only if the two associated circles intersect. Theembedding of the circles in (cid:98) C gives an embedding of its contact graph, thus IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 9 a planar graph . We remark that in this paper, a planar graph will mean agraph together with an embedding in (cid:98) C . Thus, we identify two planar graphsif there is an orientation preserving homeomorphism of (cid:98) C which induces thegraph isomorphism. It can also be checked easily that the contact graph ofa circle packing is simple. This turns out to be the only constraint for thegraph (See [50, Chapter 13]). Theorem 2.1 (Circle Packing Theorem) . Every connected, simple, planargraph is the contact graph of some circle packing. k -connected graphs. A graph Γ is said to be k -connected if Γ containsmore than k vertices and remains connected if any k − polyhedral if Γ is the 1-skeleton of a convexpolyhedron. According to Steinitz’s theorem, a graph is polyhedral if andonly if it is 3-connected and planar.Given a polyhedral graph, we have a stronger version of the circle packingtheorem [44] (cf. midsphere for canonical polyhedron). Theorem 2.2 (Circle Packing Theorem for polyhedral graphs) . Suppose Γ is a polyhedral graph, then there is a pair of circle packings whose contactgraphs are isomorphic to Γ and its planar dual. Moreover, the two circlepackings intersect orthogonally at their points of tangency.This pair of circle packings is unique up to M¨obius transformations. Marked contact graphs.
In many situations, it is better to work with amarking on the graph as well as the circle packing. A marking of a graph Γis a choice of the isomorphism φ : G −→ Γ , where G is the underlying abstract graph of Γ. We will refer to the pair(Γ , φ ) as a marked graph .Given two planar marked graphs (Γ , φ ) and (Γ , φ ) with the same un-derlying abstract graph G , we say that they are equivalent if φ ◦ φ − :Γ −→ Γ is a planar isomorphism.Similarly, a circle packing P is said to be marked if the associated contactgraph is marked. 3. Kissing Reflection Groups
Let Γ be a marked connected simple planar graph. By the circle packingtheorem, Γ is the contact graph of some marked circle packing P = { C , ..., C n } . We define the kissing reflection group associated to this circle packing P as G P := (cid:104) g , ..., g n (cid:105) , where g i is the reflection along the circle C i . Note that since a kissing reflection group is a discrete subgroup of thegroup Aut ± ( (cid:98) C ) of all M¨obius and anti-M¨obius automorphisms of (cid:98) C , defi-nitions of limit set and domain of discontinuity can be easily extended tokissing reflection groups. We shall use (cid:101) G P to denote the index two subgroupof G P consisting of orientation preserving elements. Note that (cid:101) G P lies onthe boundary of Schottky groups.We remark that if P (cid:48) is another circle packing realizing Γ, since the graphΓ is marked, there is a canonical identification of the circle packing P (cid:48) with P . This gives a canonical isomorphism between the kissing reflection groups G P (cid:48) and G P , which is induced by a quasiconformal map. We refer to Γ asthe contact graph associated to G P .3.1. Limit set and domain of discontinuity of kissing reflectiongroups.
Let P = { C , ..., C n } be a marked circle packing, and D i be the as-sociated open disks for C i . Let P be the set consisting of points of tangencyfor the circle packing P . LetΠ = (cid:98) C \ (cid:32) n (cid:91) i =1 D i ∪ P (cid:33) . Then Π is a fundamental domain of the action of G P on the domain ofdiscontinuity Ω( G P ). Denote Π = (cid:83) ki =1 Π i where Π i is a component of Π.Note that each component Π i is a closed ideal polygon in the correspondingcomponent of the domain of discontinuity bounded by arcs of finitely manycircles in the circle packing.We say that an element g ∈ G P is of generation l , if the word length | g | = l with respect to the standard generating set S = { g , ..., g n } . The followinglemma follows directly by induction. Lemma 3.1.
Let g = g i ...g i l be an element of generation l , then g · Π ⊆ D i . The domain of discontinuity can also be built up from generations. Weset Π := (cid:83) ni =1 g i · Π and Π j +1 = (cid:83) ni =1 g i · (Π j \ D i ). For consistency, we alsoset Π := Π. We call Π l the tiling of generation l . We have the followinglemma justifying this terminology. Lemma 3.2. Π l = (cid:91) | g | = l g · Π . Proof.
We will prove by induction. The base case is satisfied by the definitionof Π . Assume that Π j = (cid:83) | g | = j g · Π. Let g = g i g i ...g i j +1 be of generation j + 1. Note i (cid:54) = i . By Lemma 3.1, g i g i ...g i j +1 · Π does not intersect D i ,thus g · Π ⊆ g i · (Π j \ D i ). So (cid:83) | g | = j +1 g · Π ⊆ Π j +1 . The reverse inclusioncan be proved similarly. (cid:3) IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 11
By Lemma 3.2, we haveΩ( G P ) = (cid:91) g ∈ G P g · Π = ∞ (cid:91) i =0 Π i . Similarly, we set D i = (cid:98) C \ (cid:83) ij =0 Π j . We remark that D i is neither opennor closed: it is a finite union of open disks together with the orbit of P under the group elements of generation up to i on the boundaries of thesedisks.Let D i be its closure. Note that each D i is a union of closed disks forsome (possibly disconnected) finite circle packing. Indeed, D = (cid:98) C \ Π = n (cid:91) i =1 D i is the union of the closed disks corresponding to the original circle packing P . By induction, we have that at generation i + 1, D i +1 = n (cid:91) j =1 g j · D i \ D j (3.1)is the union of the images of the generation i disks outside of D j under g j .We also note that the sequence D i is nested, and thus the limit setΛ( G P ) = ∞ (cid:92) i =0 D i = ∞ (cid:92) i =0 D i . Therefore, we have the following expansive property of the group action onΛ( G P ). Lemma 3.3.
Let r n be the maximum spherical diameter of the disks in D n .Then r n → .Proof. Otherwise, we can construct a sequence of nested disks of radiusbounded from below implying that the limit set contains a disk, which is acontradiction. (cid:3)
We now prove the group part of Theorem 1.1.
Proposition 3.4.
The kissing reflection group G P has connected limit setif and only if the contact graph Γ of P is -connected.Proof. If Γ is not 2-connected, then there exists a circle (say C ) such thatthe circle packing becomes disconnected once we remove it. Then we see D ⊆ (cid:98) C is disconnected by Equation 3.1. This forces the limit set to bedisconnected as well (see Figure 3.1).On the other hand, if Γ is 2-connected, then D is connected by Equation3.1. Now by induction and Equation 3.1 again, D i is connected for all i .Thus, Λ( G P ) = (cid:84) ∞ i =0 D i is also connected. (cid:3) Proposition 3.4 and the definition of kissing reflection groups show thatthe association of a 2-connected simple planar graph with a kissing reflectiongroup with connected limit set is well defined and surjective. To verify thatthis is indeed injective, we remark that if P and P (cid:48) are two circle packingsassociated to two planar non-isomorphic contact graphs, then the closuresof the fundamental domains Π and Π (cid:48) are not homeomorphic. Note thatthe touching patterns of different components of Π or Π (cid:48) completely deter-mine the structures of the pairing cylinders of the associated 3-manifoldsat the cusps (See [26, § H / (cid:101) G P and H / (cid:101) G P (cid:48) with the pairing cylinder structures are not the same.Thus, the two kissing reflection groups G P and G P (cid:48) are not quasiconformallyisomorphic. Figure 3.1.
A disconnected limit set for a kissing reflectiongroup G with non 2-connected contact graph. G is generatedby reflections along the 5 visible large circles in the figure.3.2. Acylindrical kissing reflection groups.
Recall that (cid:101) G P is the index2 subgroup of G P consisting of orientation preserving elements. We set M ( G P ) := H ∪ Ω( G P ) / (cid:101) G P to be the associated 3-manifold with boundary. Note that the boundary ∂ M ( G P ) = Ω( G P ) / (cid:101) G P is a finite union of punctured spheres. Each punctured sphere correspondsto the double of a component of Π. IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 13
A compact 3-manifold M with boundary is called acylindrical if M contains no essential cylinders and is boundary incompressible. Here anessential cylinder C in M is a closed cylinder C such that C ∩ ∂M = ∂C ,the boundary components of C are not homotopic to points in ∂M and C is not homotopic into ∂M . M is said to be boundary incompressible ifthe inclusion π ( R ) (cid:44) −→ π ( M ) is injective for every component R of ∂M .(We refer the readers to [27, § § M ( G P ) is not a compact manifold as there are parabolicelements (cusps) in (cid:101) G P . Thurston [51] introduced the notion of pared man-ifolds to work with Kleinian groups with parabolic elements. In our setting,we can also use an equivalent definition without introducing pared mani-folds. To start the definition, we note that for a geometrically finite group,associated with the conjugacy class of a rank one cusp, there is a pair ofpunctures p , p on ∂M . If c , c are small circles in ∂M retractable to p , p , then there is a pairing cylinder C in M , which is a cylinder boundedby c and c (see [26, § Definition 3.5.
A kissing reflection group G P is said to be acylindrical if M ( G P ) is boundary incompressible and every essential cylinder is homotopicto a pairing cylinder.Note that M ( G P ) is boundary incompressible if and only if each com-ponent of Ω( G P ) is simply connected if and only if the limit set Λ( G P ) isconnected. We also note that the acylindrical condition is a quasiconformalinvariant, and hence does not depend on the choice of the circle packing P realizing a simple, connected, planar graph Γ. In the remainder of thissection, we shall prove the following characterization of acylindrical kissingKleinian reflection groups. Proposition 3.6.
The kissing reflection group G P is acylindrical if andonly if the contact graph Γ of P is -connected. This proposition will be proved after the following lemmas. Let Γ be a2-connected simple planar graph, and P = { C , ..., C n } be a realization ofΓ. Let G P be the kissing reflection group, with generators g , ..., g n givenby reflections along C , ..., C n . Note that a face F of Γ corresponds to acomponent Π F of Π, which also corresponds to a component R F of ∂ M ( G P ).Any two non-adjacent vertices v, w of the face F give rise to an essentialsimple closed curve (cid:101) γ Fvw on R F . More precisely, let g v , g w be the reflectionsassociated to the two vertices, then g v g w is a loxodromic element under theuniformization of R F which gives the simple closed curve (cid:101) γ Fvw on R F . Notethat g v g w itself may not be loxodromic as the vertices v, w may be adjacentin some other faces. If this is the case, then we have an accidental parabolicelement (see [27, p. 198, 3-17]).We first prove the following graph theoretic lemma. Lemma 3.7.
Let Γ be a -connected simple planar graph. If Γ is not -connected, then there exist two vertices v, w so that v, w lie on the inter-section of the boundaries of two faces F and F . Moreover, they are non-adjacent for at least one of the two faces. vv , = v , v , v , v , v , v , = v , v , v , = v , F F F Figure 3.2.
A schematic picture of the potentially non-simple cycle around v . Proof.
As Γ is not 3-connected, there exist two vertices v, w so that Γ \{ v, w } is disconnected. Let F , ..., F k be the list of faces that contain v as a vertex.Since Γ is planar, we may assume that F i s are ordered around v counterclockwise. Since Γ is planar and 2-connected, each face F i is a Jordandomain. Let v i, = v, v i, , ..., v i,j i be the vertices of F i ordered counterclockwise. Since F i s are ordered counterclockwise, we have that v i,j k = v i +1 , . We remark that there might be additional identifications. Then v , → v , → ... → v ,j (= v , ) → v , → ... → v k,j k = v , form a (potentially non-simple) cycle C (see Figure 3.2). Since Γ is 2-connected, Γ \ { w } is connected. Thus, in particular, any vertex p is con-nected to C \ { w } in Γ \ { w } . Therefore, if w appears only once in the cycle C , then C \ { w } is connected. This would imply that Γ \ { v, w } is connected,which is a contradiction. Since each face is a Jordan domain, w must appearon the boundaries of at least 2 faces F i and F i .Since Γ is simple, w is adjacent to v in at most 2 faces, in which case w = v i,j i = v i +1 , , i.e., it contributes to only one point in C . Therefore, thereexists a face on which w is not adjacent to v . This proves the lemma. (cid:3) We can now prove one direction of Proposition 3.6.
Lemma 3.8. If G P is acylindrical, then Γ is -connected. IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 15
Proof.
Note that Γ must be 2-connected as Λ( G P ) is connected (by theboundary incompressibility condition). We will prove the contrapositive,and assume that Γ is not 3-connected. Let v, w be the two vertices given byLemma 3.7. There are two cases.If v, w are non-adjacent vertices in two faces F , F , then g v g w gives apair of essential simple closed curves on R F and R F in ∂ M ( G P ). Thispair bounds an essential cylinder (see the Cylinder Theorem in [27, § g C g C (cid:48)(cid:48) inFigure 3.4).If v, w are non-adjacent vertices in F but adjacent vertices in F , then g v g w corresponds to an essential simple closed curve in R F , and a simpleclosed curve homotopic to a puncture in R F . Then g v g w is an acciden-tal parabolic, and the two curves bound an essential cylinder which is nothomotopic to a pairing cylinder (see g C g C (cid:48) in Figure 3.4).Therefore, in either case, G P is cylindrical. (cid:3) Gasket limit set.
Recall that a closed set Λ is a round gasket if • Λ is the closure of some infinite circle packing; and • the complement of Λ is a union of round disks which is dense in (cid:98) C .We call a homeomorphic copy of a round gasket a gasket .If Γ is 3-connected, then Γ is a polyhedral graph. Let Γ ∨ be the planardual of Γ. Then, Theorem 2.2 gives a (unique) pair of circle packings P and P ∨ whose contact graphs are isomorphic to Γ and Γ ∨ (respectively)such that P and P ∨ intersect orthogonally at their points of tangency (seeFigure 3.3). Let G P be the kissing reflection group associated with P . Sincethe circle packing P ∨ is dual to P , we have that (cid:91) g ∈ G P (cid:91) C ∈P ∨ g · C is an infinite circle packing, and the limit is the closureΛ( G P ) = (cid:91) g ∈ G P (cid:91) C ∈P ∨ g · C. Since Λ( G P ) is nowhere dense, and the complement is a union of rounddisks, we conclude that Λ( G P ) is a round gasket.Note that each component of Ω( G P ) is of the form g · D where g ∈ G P and D is a disk in the dual circle packing P ∨ . By induction, we have thefollowing. Lemma 3.9. If Γ is -connected, then the closure of any two different com-ponents of Ω( G P ) only intersect at cusps. We have the following characterization of gasket limit set for kissing re-flection groups.
Proposition 3.10.
Let Γ be a simple planar graph, then Λ( G P ) is a gasketif and only if Γ is -connected. ; Figure 3.3.
The limit set of a kissing reflection group G with a 3-connected contact graph. Proof.
Indeed, from the above discussion, if Γ is 3 connected, then Λ( G P )is a gasket.Conversely, if Γ is not 2-connected, then Λ( G P ) is disconnected by Propo-sition 3.4, so it is not a gasket. On the other hand, if Γ is 2-connected butnot 3-connected, by Lemma 3.7, we have two vertices v, w so that v, w lie onthe intersection of the boundaries of two faces F and F . If v, w are non-adjacent vertices in both F and F , then the corresponding componentsΩ F and Ω F touch at two points corresponding to the two fixed points ofthe loxodromic element g v g w (see g C g C (cid:48)(cid:48) in Figure 3.4). Thus, Λ( G P ) isnot a gasket. If v, w are non-adjacent vertices in F but adjacent verticesin F , then g v g w gives an accidental parabolic element. The correspondingcomponent Ω F is not a Jordan domain as the the unique fixed point of theparabolic element g v g w corresponds to two points on the ideal boundary ofΩ F (see g C g C (cid:48) in Figure 3.4). Therefore Λ( G P ) is not a gasket. (cid:3) In the course of the proof, we have actually derived the following charac-terization which is worth mentioning.
Proposition 3.11.
Let P be a circle packing whose contact graph Γ is not -connected, and G P be the associated kissing reflection group, then either • there exists a component of Ω( G P ) which is not a Jordan domain;or • there exist two components of Ω( G P ) whose closures touch at leastat two points. IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 17 CC (cid:48) C (cid:48)(cid:48) F F Figure 3.4.
The limit set of a kissing reflection group G with Hamiltonian but non 3-connected contact graph. Theunique Hamiltonian cycle of the associated contact graph Γdivides the fundamental domain Π( G ) into two parts Π ± ,which are shaded in grey and blue. With appropriate mark-ings, G is the mating of two copies of the group H shown inFigure 3.6.We are now able to prove the other direction of Proposition 3.6: Lemma 3.12. If Γ is -connected, then G P is acylindrical.Proof. Since Γ is 3-connected, it follows that the closures of any two differentcomponents of Ω( G P ) intersect only at cusps. This means that there are noessential cylinder other than the pairing cylinders of the rank one cusps. So G P is acylindrical. (cid:3) Proof of Proposition 3.6.
This follows from Lemma 3.8 and 3.12. (cid:3)
We remark that the unique configuration of pairs of circle packings givenin Theorem 2.2 gives a kissing reflection group with totally geodesic bound-ary . This unique point in the deformation space of acylindrical manifolds isguaranteed by a Theorem of McMullen [29].3.3.
Algebraic deformation space of kissing reflection groups.Definition of AH ( G ) . Let G be a finitely generated discrete subgroup ofAut ± ( (cid:98) C ). A representation ξ : G −→ Aut ± ( (cid:98) C ) is said to be weakly typepreserving if (1) ξ ( g ) ∈ Aut + ( (cid:98) C ) if and only if g ∈ Aut + ( (cid:98) C ), and(2) if g ∈ Aut + ( (cid:98) C ), then ξ ( g ) is parabolic whenever g is parabolic.Note that a weakly type preserving representation may send a loxodromicelement to a parabolic one. Definition 3.13.
Let G be a finitely generated discrete subgroup of Aut ± ( (cid:98) C ).AH( G ) := { ξ : ξ : G −→ G is a weakly type preserving isomorphism toa discrete subgroup G of Aut ± ( (cid:98) C ) } / ∼ , where ξ ∼ ξ if there exists a M¨obius transformation M such that ξ ( g ) = M ◦ ξ ( g ) ◦ M − , for all g ∈ G . AH( G ) inherits the quotient topology of algebraic convergence. Indeed,we say that a sequence of weakly type preserving representations { ξ n } con-verges to ξ algebraically if { ξ n ( g i ) } converges to ξ ( g i ) as elements of Aut ± ( (cid:98) C )for (any) finite generating set { g i } . Quasiconformal deformation space of G.
Recall that a Kleinian groupis said to be geometrically finite if it has a finite sided fundamental polyhe-dron. We say a finitely generated discrete subgroup of Aut ± ( (cid:98) C ) is geomet-rically finite if the index 2 subgroup is geometrically finite.Let G be a finitely generated, geometrically finite, discrete subgroup ofAut ± ( (cid:98) C ). A group G is called a quasiconformal deformation of G if there is aquasiconformal map F : S −→ S that induces an isomorphism ξ : G −→ G such that F ◦ g ( z ) = ξ ( g ) ◦ F ( z ) for all g ∈ G and z ∈ S . Such a group G is necessarily geometrically finite and discrete. The map F is uniquelydetermined (up to normalization) by its Beltrami differential on the domainof discontinuity Ω( G ) (See [27, § quasiconformal deformation space of G as T ( G ) = { ξ ∈ AH( G ) : ξ is induced by a quasiconformal deformation of G } . By definition, T ( G ) ⊆ AH( G ). Kissing reflection groups.
We now specialize to the case of a kissingreflection group. Recall that different realizations of a fixed marked, con-nected, simple, planar graph Γ as circle packings P give canonically isomor-phic kissing reflection groups G P . Hence, we can and will use the notationsAH(Γ) and T (Γ) to denote AH( G P ) and T ( G P ), respectively, for a fixed(but arbitrary) realization P of Γ.Let Γ be a simple planar graph. The goal of this section is to describethe algebraic deformation space AH(Γ ) and the closure T (Γ ) (of the qua-siconformal deformation space) in AH(Γ ).To fix the notations, we choose a circle packing { C , ..., C n } realizing Γ ,and denote the reflection along C i as ρ i . We define the kissing reflectiongroup G as the group generated by the reflections ρ i . Throughout this IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 19 subsection, the deformation spaces AH(Γ ) and T (Γ ) will stand for AH( G )and T ( G ).Let G be a discrete subgroup of Aut ± ( (cid:98) C ), and let ξ : G −→ G be aweakly type preserving isomorphism. Since the union of the circles C , ..., C n is connected, for each ρ i , there exists ρ j so that ρ i ◦ ρ j is parabolic. Thisimplies that no ξ ( ρ i ) is the antipodal map. Hence, ξ ( ρ i ) is also a circularreflection. Assume that ξ ( ρ i ) is the reflection along some circle C i . If ρ i ◦ ρ j is parabolic, then ξ ( ρ i ) ◦ ξ ( ρ j ) is also parabolic. Thus, C i is tangent to C j as well. This motivates the following definition. Definition 3.14.
Let Γ be a simple planar graph with the same number ofvertices as Γ , we say that Γ abstractly dominates Γ , denoted by Γ ≥ a Γ ,if there exists an embedding ψ : Γ −→ Γ as abstract graphs.We say that Γ dominates Γ , denoted by Γ ≥ Γ , if there exists an em-bedding ψ : Γ −→ Γ as planar graphs.We also define T a := { (Γ , ψ ) : Γ ≥ a Γ and ψ : Γ −→ Γ is an embedding } , and T := { (Γ , ψ ) : Γ ≥ Γ and ψ : Γ −→ Γ is a planar embedding } . ≥ a (cid:3) Figure 3.5.
The graph on the left abstractly dominates thegraph on the right, but no embedding of the right graph intothe left graph respects the planar structure.In other words, a simple planar graph Γ abstractly dominates Γ if Γ (asan abstract graph) can be constructed from Γ by introducing new edges;and it dominates Γ if one can do this while respecting the planar structure.We emphasize that the graph Γ is always assumed to be planar, but theembedding ψ may not respect the planar structure in T a (See Figure 3.5).We also remark that an element (Γ , ψ ) is identified with (Γ (cid:48) , ψ (cid:48) ) in T a or T if ψ (cid:48) ◦ ψ − extends to a planar isomorphism between Γ and Γ (cid:48) .We have the following lemma. Lemma 3.15.
Let G be a discrete subgroup of Aut ± ( (cid:98) C ) and let ξ : G −→ G be a weakly type preserving isomorphism. Then the simple planar graph Γ associated with G abstractly dominates Γ .Conversely, if Γ is a simple planar graph abstractly dominating Γ and G is a kissing reflection group associated with Γ , then there exists a weaklytype preserving isomorphism ξ : G −→ G .Proof. Let G = ξ ( G ) be a discrete faithful weakly type preserving repre-sentation, then the reflections ξ ( ρ i ) along C i generate G . If there were anon-tangential intersection between some C i and C j , it would introduce anew relation between ξ ( ρ i ) and ξ ( ρ j ) by discreteness of G (cf. [52, Part II,Chapter 5, § ξ is an iso-morphism. Similarly, if there were circles C i , C j , C k touching at a point,then discreteness would give a new relation among ξ ( ρ i ), ξ ( ρ j ) and ξ ( ρ k ),which would again lead to a contradiction. The above observations com-bined with the discussion preceding Definition 3.14 imply that { C , ..., C n } is a circle packing with associated contact graph Γ abstractly dominatingΓ .Conversely, assume that G is a kissing reflection group associated witha simple planar graph Γ abstractly dominating Γ . Let C i be the circlecorresponding to C i under a particular embedding of Γ into Γ, and g i bethe reflection along C i . Defining ξ : G −→ G by ξ ( ρ i ) = g i , it is easy tocheck that ξ is a weakly type preserving isomorphism. (cid:3) Note that for kissing reflection groups, the double of the polyhedronbounded by the half-planes associated to the circles C i is a fundamentalpolyhedron for the action of the index two Kleinian group on H . Thus, wehave the following corollary which is worth mentioning. Corollary 3.16.
Every group in AH (Γ ) is geometrically finite. Let (Γ , ψ ) ∈ T (respectively, in T a ). Let us fix a circle packing C , Γ , ..., C n, Γ realizing Γ, where C i, Γ corresponds to C i under the embedding ψ of Γ .Let g i be the associated reflection along C i, Γ , and G Γ = (cid:104) g , ..., g n (cid:105) . ThenLemma 3.15 shows that ξ (Γ ,ψ ) : G −→ G Γ ρ i (cid:55)→ g i is a weakly type preserving isomorphism. Thus, T (Γ) ∼ = T ( G Γ ) can beembedded in AH(Γ ) ∼ = AH( G ). Indeed, if ξ : G Γ −→ G represents anelement in T (Γ), then ξ ◦ ξ (Γ ,ψ ) : G −→ G is a weakly type preserving isomorphism. It can be checked that the map ξ (cid:55)→ ξ ◦ ξ (Γ ,ψ ) gives an embedding of T (Γ) into AH(Γ ). We shall identifythe space T (Γ) with its image in AH(Γ ) under this embedding. IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 21
Proposition 3.17. AH (Γ ) = (cid:91) (Γ ,ψ ) ∈T a T (Γ) , and T (Γ ) = (cid:91) (Γ ,ψ ) ∈T T (Γ) . The proof of this proposition will be furnished after a discussion of pinch-ing deformations. Once the connection with pinching deformation is estab-lished, the result can be derived from [39].
The perspective of pinching deformation.
Let Γ be a 2-connected sim-ple planar graph, and G P be an associated kissing reflection group (where P is a circle packing with contact graph isomorphic to Γ). Let F be a faceof Γ, and R F be the associated component of ∂ M ( G P ). Let C ∈ P be somecircle on the boundary of Π F . Then the reflection along C descends to ananti-conformal involution on R F , which we shall denote as σ F : R F −→ R F . Note that different choices of the circle descend to the same involution.It is known that for boundary incompressible geometrically finite Kleiniangroups, the quasiconformal deformation space is the product of the Te-ichm¨uller spaces of the components of the conformal boundary (see [27,Theorem 5.1.3]): T ( (cid:101) G P ) = (cid:89) F face of Γ Teich( R F ) . We denote by Teich σ F ( R F ) ⊆ Teich( R F ) those elements corresponding to σ F -invariant quasiconformal deformations of R F . Then T ( G P ) = (cid:89) F face of Γ Teich σ F ( R F ) . Indeed, any element in (cid:81) F face of Γ Teich σ F ( R F ) is uniquely determined bythe associated Beltrami differential on Π F , which can be pulled back by G P to produce a G P -invariant Beltrami differential on (cid:98) C . Such a Beltrami dif-ferential can be uniformized by the Measurable Riemann mapping Theorem.Recall that Π F is an ideal polygon. Note that the component of Ω( G P )containing Π F is simply connected and induces the hyperbolic metric on Π F .Then R F is simply the double of Π F . We claim that the only σ F -invariantgeodesics are those simple closed curves (cid:101) γ Fvw , where v, w are two non-adjacentvertices on the boundary of F . Indeed, any σ F -invariant geodesic wouldintersect the boundary of the ideal polygon perpendicularly, and the (cid:101) γ Fvw are the only geodesics satisfying this property. We denote the associatedgeodesic arc in Π F by γ Fvw .We define a multicurve on a surface as a disjoint union of simple closedcurves, such that no two components are homotopic. It is said to be weighted if a positive number is assigned to each component. We shall identify twomulticurves if they are homotopic to each other.Let T be a triangulation of F obtained by adding new edges connectingthe vertices of F . Since each additional edge in this triangulation connectstwo non-adjacent vertices of F , we can associate a multicurve (cid:101) α F T on R F consisting of all (cid:101) γ Fvw , where vw is a new edge in T . A marking of the graphΓ also gives a marking of the multicurve. We use α F T to denote the multi-arcin the hyperbolic ideal polygon Π F . Since T is a triangulation, a standardargument in hyperbolic geometry shows that the hyperbolic ideal polygonΠ F is uniquely determined by the lengths of the multi-arc α F T . If F has m sides, then any triangulation of F has m − σ F ( R F ) = R m − by assigning to each element of Teich σ F ( R F ) the lengths of the markedmulti-arc α F T (or equivalently the lengths of the marked multicurve (cid:101) α F T ).Note that different triangulations yield different coordinates.In order to degenerate in Teich σ F ( R F ), the length of some arc γ Fvw mustshrink to 0. The above discussion implies that the closure of Teich σ F ( R F )in the Thurston’s compactification consists only of weighted σ F -invariantmulticurves (cid:101) α F . If S k ∈ Teich σ F ( R F ) converges to a weighted σ F -invariantmulticurve (cid:101) α F , then we say that S k is a pinching deformation on (cid:101) α F . Moregenerally, we say that G k ∈ T ( G P ) is a pinching deformation on (cid:101) α = (cid:83) (cid:101) α F if for each face F , the conformal boundary associated to F is a pinchingdeformation on (cid:101) α F .Given two curves (cid:101) γ Fvw ⊆ R F and (cid:101) γ F (cid:48) ut ⊆ R F (cid:48) , we say that they are parallel if they are homotopic in the 3-manifold M ( G P ). Since the curve (cid:101) γ Fvw cor-responds to the element g v g w , we get that (cid:101) γ Fvw and (cid:101) γ F (cid:48) ut are parallel if andonly if (possibly after switching the order) v = u and w = t .Let (cid:101) α = (cid:83) (cid:101) α F be a union of σ F -invariant multicurves. We say that (cid:101) α isa non-parallel multicurve if no two components are parallel. We note thatnon-parallel multicurves (cid:101) α for G P are in one-to-one correspondence with thesimple planar graphs that dominate Γ. Indeed, any multicurve (cid:101) α = (cid:83) (cid:101) α F corresponds to a planar graph that dominates Γ. The non-parallel conditionis equivalent to the condition that the graph is simple.We are now ready to prove Proposition 3.17. Proof of Proposition 3.17.
The first equality follows from Lemma 3.15.We first show (cid:83) (Γ ,ψ ) ∈T T (Γ) ⊆ T (Γ ). Let Γ be a simple planar graphthat dominates Γ with planar embedding ψ : Γ −→ Γ. It follows from thediscussion on pinching deformations that there exists a non-parallel multi-curve (cid:101) α ⊆ ∂ M ( G P ) associated to Γ. We complete Γ to a triangulation T by adding edges. As before, T gives a σ -invariant multicurve (cid:101) β ⊆ ∂ M ( G P )which contains (cid:101) α . We set (cid:101) β = (cid:101) α (cid:116) (cid:101) α (cid:48) . Then the lengths of the multicurve (cid:101) α (cid:48) gives a parameterization of T (Γ), and the lengths of the multicurve (cid:101) β gives a IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 23 parameterization of T (Γ ). Now given any element (cid:126)l = ( l γ : γ ∈ (cid:101) α (cid:48) ) ∈ T (Γ),we show that it can be realized as the algebraic limit of a sequence in T (Γ ).Such a construction is standard, and follows directly from [39, Theo-rem 5.1] (see also [28]). For completeness, we sketch the proof here. Let G ∈ T (Γ ) be so that the length of the multicurve (cid:101) α (cid:48) is (cid:126)l . We assume that (cid:101) β is realized by hyperbolic geodesics in ∂ M ( G ). Let A ⊆ ∂ M ( G ) be an (cid:15) -neighborhood of (cid:101) α . We choose (cid:15) small enough so that A is disjoint from (cid:101) α (cid:48) , and each component contains only one component of (cid:101) α . We construct asequence of quasiconformal deformations supported on A so that the mod-ulus of each annulus in A tends to infinity while (cid:101) α remains as core curves.Since the multicurve (cid:101) α is non-parallel, Thurston’s Hyperbolization Theoremguarantees a convergent subsequence (see [39] for more details). This alge-braic limit is a kissing reflection group. By construction, the contact graphof such a limiting kissing reflection group is Γ, and the lengths of (cid:101) α (cid:48) is (cid:126)l .Conversely, if a sequence of quasiconformal deformations ξ n : G −→ G n of G converges to ξ ∞ : G −→ G algebraically, then G is a kissing reflectiongroup. Since { ξ n ( g i ) } converges, where g i are the standard generators, thecontact graph for G dominates Γ . The embedding ψ : Γ −→ Γ (respectingthe planar structure) comes from the identification of the generators. (cid:3)
Quasi-Fuchsian space and mating locus.
The above discussionapplies to the special case of quasiFuchsian space. This space is relatedto the mating locus for critically fixed anti-rational maps.Let Γ d be the marked d + 1-sided polygonal graph, i.e., Γ d contains d + 1vertices v , ..., v d +1 with edges v i v i +1 where indices are understood modulo d + 1. We choose the most symmetric circle packing P d realizing Γ d . Moreprecisely, consider the ideal ( d +1)-gon in D ∼ = H with vertices at the ( d +1)-st roots of unity. The edges of this ideal ( d + 1)-gon are arcs of d + 1 circles,and we label these circles as P d := { C , ..., C d +1 } , where we index themcounter-clockwise such that C passes through 1 and e πi/ ( d +1) . Let G d bethe kissing reflection group associated to P d . Note that (cid:101) G d is a Fuchsiangroup. We remark that since any embedding of the polygonal graph Γ d intoa graph Γ abstractly dominating Γ d respects the planar structure, we havethat AH(Γ d ) = T (Γ d ).Recall that a graph Γ is said to be Hamiltonian if there exists a cyclewhich passes through each vertex exactly once. Since a simple planar graphwith d +1 vertices is Hamiltonian if and only if it dominates Γ d , the followingproposition follows immediately from Proposition 3.17. Proposition 3.18.
Let Γ be a simple planar graph with d + 1 vertices, thenany kissing reflection group G with contact graph Γ is in the closure T (Γ d ) if and only if Γ is Hamiltonian. Let Γ be a marked Hamiltonian simple planar graph with a Hamiltoniancycle C = ( v , ..., v d +1 ). Let G := (cid:104) g , ..., g d +1 (cid:105) be a kissing reflection group with contact graph Γ. The Hamiltonian cycle divides the fundamental do-main Π ⊆ (cid:98) C into two parts, and we denote them as Π + and Π − , where weassume that the Hamiltonian cycle is positively oriented on the boundaryof Π + and negatively oriented on the boundary of Π − (see Figure 3.4). Wedenote Ω ± := (cid:91) g ∈ G g · Π ± . Since each Ω ± is G -invariant, we have Λ( G ) = ∂ Ω + = ∂ Ω − .As in § , ± = d +1 (cid:91) i =1 g i · Π ± , and Π j +1 , ± = d +1 (cid:91) i =1 g i · (cid:0) Π j, ± \ D i (cid:1) . For consistency, we also set Π , ± = Π ± . Then the arguments of the proof ofLemma 3.2 show that Π l, ± = (cid:91) | g | = l g · Π ± . Lemma 3.19.
The closures Ω + and Ω − are connected.Proof. We shall prove by induction that the closures (cid:83) ni =0 Π i, + and (cid:83) ni =0 Π i, + \ D j are connected for all n and j .Indeed, the base case is true as Π + and Π + \ D j are connected for all j .Assume that (cid:83) ni =0 Π i, + and (cid:83) ni =0 Π i, + \ D j connected for all j . Note that n +1 (cid:91) i =0 Π i, + = Π + ∪ d +1 (cid:91) j =1 g j · (cid:32) n (cid:91) i =0 Π i, + \ D j (cid:33) . By induction hypothesis, g j · ( (cid:83) ni =0 Π i, + \ D j ) is connected. Since each g j · (cid:32) n (cid:91) i =0 Π i, + \ D j (cid:33) intersects Π + along an arc of C j = ∂D j , we have that (cid:83) n +1 i =0 Π i, + is alsoconnected.Similarly, n +1 (cid:91) i =0 Π i, + \ D j = Π + ∪ (cid:91) k (cid:54) = j g k · (cid:32) n (cid:91) i =0 Π i, + \ D k (cid:33) is connected for any j .Connectedness of Ω − is proved in the same way. (cid:3) IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 25
Matings of function kissing reflection groups.
We say that a kissingreflection group G with connected limit set is a function kissing reflectiongroup if there is a component Ω of Ω( G ) invariant under G . This terminol-ogy was traditionally used in complex analysis as one can construct Poincar´eseries, differentials and functions on it.We say that a simple planar graph Γ with n vertices is outerplanar ifit has a face with all n vertices on its boundary. We also call this face the outer face . Note that a 2-connected outerplanar graph is Hamiltonian with aunique Hamiltonian cycle. The following proposition characterizes functionkissing reflection groups in terms of their contact graphs. Proposition 3.20.
A kissing reflection group G is a function group if andonly if its contact graph Γ is 2-connected and outerplanar.Proof. If Γ is outerplanar, then let Ω F be the component of Ω( G ) associatedto the outer face F . It is easy to see that the standard generating set fixesΩ F , so Ω F is invariant under G .Conversely, assume that Γ is not outerplanar. Since Π is a fundamentaldomain of the action of G on Ω( G ), any G -invariant component of Ω( G )must correspond to some face of Γ. Let F be a face of Γ. Since Γ is notouterplanar, there exists a vertex v which is not on the boundary of F .Thus, g v · Π F is not in Ω F , and hence Ω F is not invariant under G . (cid:3) Figure 3.6.
The limit set of a function kissing reflectiongroup H with an outerplanar contact graph. The grey re-gion represents the part Π b of the fundamental domain Π( H )corresponding to the non-outer faces of Γ( H ). The kissingreflection group G shown in Figure 3.4 can be constructedby pinching a simple closed curve for H .For a function kissing reflection group G , we set Ω b := Ω( G ) \ Ω , whereΩ is G -invariant. Similarly, we shall use the notations Π b := Π ∩ Ω b (see Figure 3.6), and Π ib := Π i ∩ Ω b , for i ≥ b = (cid:98) C − Ω the filled limit set for the function kissing reflectiongroup G , and denote it by K ( G ).Let G ± be two function kissing reflection groups with the same numbervertices in their contact graphs. We say that a kissing reflection group G isa geometric mating of G + and G − if we have • a decomposition Ω( G ) = Ω + ( G ) (cid:116) Ω − ( G ) with Λ( G ) = ∂ Ω + ( G ) = ∂ Ω − ( G ); • weakly type preserving isomorphisms φ ± : G ± −→ G ; • continuous surjections ψ ± : K ( G ± ) −→ Ω ± ( G ) which are conformalbetween the interior ˚ K ( G ± ) and Ω ± ( G ) such that for any g ∈ G ± , ψ ± ◦ g | K ( G ± ) = φ ± ( g ) ◦ ψ ± .We shall now complete the proof of the group part of Theorem 1.2. Proposition 3.21.
A kissing reflection group G is a geometric mating oftwo function kissing reflection groups if and only if the contact graph Γ( G ) of G is Hamiltonian.Proof. If G is a geometric mating of G ± , then we have a decompositionΩ( G ) = Ω + ( G ) (cid:116) Ω − ( G ) with Λ( G ) = ∂ Ω + ( G ) = ∂ Ω − ( G ). This gives adecomposition of Π( G ) = Π + (cid:116) Π − , and thus a decomposition of the contactgraph Γ( G ) = Γ + ∪ Γ − . More precisely, the vertex sets of Γ ± coincide withthat of Γ( G ), and there is an edge in Γ ± connecting two vertices if and onlyif the point of intersection of the corresponding two circles lies in ∂ Π ± . Sincethe action of G ± on K ( G ± ) is semi-conjugate to G on Ω ± ( G ), it follows thatΓ ± is planar isomorphic to Γ( G ± ). Thus, the intersection of Γ + and Γ − (which is the common boundary of the outer faces of Γ + and Γ − ) gives aHamiltonian cycle for Γ( G ).Conversely, if Γ( G ) is Hamiltonian, then a Hamiltonian cycle yields adecomposition Γ( G ) = Γ + ∪ Γ − , where Γ + and Γ − are outerplanar graphswith vertex sets equal to that of Γ( G ). We construct function kissing re-flection groups G ± with contact graphs Γ( G ± ) = Γ ± . Note that we havea natural embedding of Γ( G ± ) into Γ( G ), which gives an identification ofvertices and non-outer faces. Thus, we have weakly type preserving iso-morphisms φ ± : G ± −→ G coming from the identification of vertices andProposition 3.17. By quasiconformal deformation, we may assume that foreach non-outer face F ∈ Γ( G ± ), the associated conformal boundary com-ponent R F ( G ± ) ⊆ ∂ M ( G ± ) is conformally equivalent to the correspondingconformal boundary component R F ( G ) ⊆ ∂ M ( G ).The existence of the desired continuous map ψ + can be derived from amore general statement on the existence of Cannon-Thurston maps (see [34,Theorem 4.2]). For completeness and making the proof self-contained, wegive an explicit construction of ψ + : K ( G + ) −→ Ω + ( G ). This constructionis done in generations: we start with a homeomorphism ψ +0 : Π b ( G + ) −→ Π , + . IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 27
This can be chosen to be conformal on the interior as the associated confor-mal boundaries are assumed to be conformally equivalent. Assuming that ψ + i : i (cid:91) j =0 Π jb ( G + ) −→ i (cid:91) j =0 Π j, + is constructed, we extend ψ + i to ψ + i +1 by setting ψ + i +1 ( z ) := φ + ( g k ) ◦ ψ + i ◦ g k ( z )if z ∈ Π i +1 b ( G + ) ∩ D k . It is easy to check by induction that ψ + i +1 is a home-omorphism which is conformal on the interior (see Figure 3.4 and Figure3.6).Let P ( G + ) := Π b ( G + ) \ Π b ( G + ), then P ( G + ) consists of cusps wherevarious components of Π b ( G + ) touch. Let P ∞ ( G + ) := (cid:83) g ∈ G + g · P ( G + ).Then, P ∞ ( G + ) is dense in Λ( G + ). We define P ∞ , + similarly (where, P , + :=Π , + \ Π , + ), and note that P ∞ , + is dense in ∂ Ω + ( G ) = Λ( G ).Note that ψ + i = ψ + j on P i ( G + ) for all j ≥ i . Thus, we have a well definedlimit ψ + : P ∞ ( G + ) −→ P ∞ , + . We claim that ψ + is uniformly continuouson P ∞ ( G + ). For an arbitrary (cid:15) >
0, there exists N such that all disks in D N ( G ) have spherical diameter < (cid:15) by Lemma 3.3. Choose δ so that anytwo non-adjacent disks in D N ( G + ) are separated in spherical metric by δ .Then if x, y ∈ P ∞ ( G + ) with d ( x, y ) < δ , they must lie in two adjacent disksof D N ( G + ). Thus, ψ + i ( x ) and ψ + i ( y ) (whenever defined) lie in two adjacentdisks of D N ( G ) for all i ≥ N . Hence, d ( ψ + i ( x ) , ψ + i ( y )) < (cid:15) for all i ≥ N .This shows that ψ + is uniformly continuous on P ∞ ( G + ).Thus, we have a continuous extension ψ + : K ( G + ) −→ Ω + ( G ). It is easyto check that ψ + is surjective, conformal on the interior, and equivariantwith respect to the actions of G + and G .The same proof gives the construction for ψ − . This shows that G is ageometric mating of G + and G − . (cid:3) Note that from the proof of Proposition 3.21, we see that each (marked)Hamiltonian cycle H gives a decomposition of the graph Γ = Γ + ∪ Γ − and hence an unmating of a kissing reflection group. From the pinchingperspective as in the proof of Proposition 3.17, this (marked) Hamiltoniancycle also gives a pair of non-parallel multicurves. This pair of multicurvescan be constructed explicitly from the decomposition of Γ. Indeed, each edgein Γ ± − H gives a pair of non-adjacent vertices in H . The corresponding σ -invariant curve comes from this non-adjacent vertices in H . If a marking isnot specified on H , this pair of multicurves is defined only up to simultaneouschange coordinates on the two conformal boundaries.3.5. Nielsen maps for kissing reflection groups.
Let Γ be a simpleplanar graph, and G Γ be a kissing reflection group with contact graph Γ.Recall that D = (cid:83) nj =1 D j is defined as the union of the closure of the disks for the associated circle packing. We define the Nielsen map N Γ : D −→ (cid:98) C by N Γ ( z ) = g j ( z ) if z ∈ D j . Let us now assume that Γ is 2-connected, then the limit set Λ( G Γ ) isconnected. Thus, each component of Ω( G Γ ) is a topological disk. Let F bea face with d + 1 sides of Γ, and let Ω F be the component of Ω( G ) containingΠ F . By a quasiconformal deformation, we can assume that the restriction of G Γ on Ω F is conformally conjugate to the regular ideal d + 1-gon reflectiongroup on D ∼ = H .Fo ther regular ideal d + 1-gon reflection group G d (whose associatedcontact graph is Γ d ), the d + 1 disks yield a Markov partition for the actionof the Nielsen map N d ≡ N Γ d on the limit set Λ( G d ) = S . The diametersof the preimages of these disks under N d shrink to 0 uniformly by Lemma3.3, and hence the Nielsen map N d is topologically conjugate to z (cid:55)→ z d on S (cf. [23, § N Γ is topologically conjugate to z d on theideal boundary of Ω F . This allows us to replace the dynamics of N Γ on Ω F by z d for every face of Γ, and obtain a globally defined orientation reversingbranched covering G Γ : (cid:98) C −→ (cid:98) C .More precisely, let φ : D −→ Ω F be a conformal conjugacy between N d and N Γ , which extends to a topological conjugacy from the ideal boundary S = I ( D ) = ∂ D onto I (Ω F ). Let ψ : D −→ D be an arbitrary homeomorphicextension of the topological conjugacy between z d | S and N d | S fixing 1. Wedefine G Γ := (cid:26) N Γ on D \ (cid:83) F Ω F , ( φ ◦ ψ ) ◦ m − d ◦ ( φ ◦ ψ ) − on Ω F , where m − d ( z ) = z d .The map z d has d + 1 invariant rays in D connecting 0 with e πi · jd +1 byradial line segments. We shall refer to these rays as internal rays . For eachΩ F , we let T F be the image of the union of these d + 1 internal rays (under φ ◦ ψ ), and we call the image of 0 the center of Ω F . This graph T F is adeformation retract of Π F fixing the ideal points. Thus, each ray of Ω F lands exactly at a cusp where two circles of the circle packing touch. Wedefine T ( G Γ ) := (cid:91) F T F , and endow it with a simplicial structure such that the centers of the facesare vertices. Then T ( G Γ ) is the planar dual to the graph Γ. We will nowsee that G Γ acts as a ‘reflection’ on each face of T ( G Γ ). Lemma 3.22.
Let P be an open face of T ( G Γ ) , then G Γ is a homeomor-phism sending P to the interior of its complement.Proof. Since Γ is 2-connected, the complement of P is connected, so P is aJordan domain. Since P contains no critical point, and G Γ ( ∂P ) = ∂P , it IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 29 follows that G Γ sends P homeomorphically to the interior of its complement. (cid:3) We shall see in the next section that this branched covering G Γ is topo-logically conjugate to a critically fixed anti-rational map R , and this dualgraph T ( G Γ ) is the Tischler graph associated with R .4. Critically Fixed Anti-rational Maps
In this section, we shall show how critically fixed anti-rational maps arerelated to circle packings and kissing reflection groups.We define an anti-polynomial P of degree d as P ( z ) = a d z d + a d − z d − + ... + a where a i ∈ C and a d (cid:54) = 0. An anti-rational map R of degree d is the ratio oftwo anti-polynomials R ( z ) = P ( z ) Q ( z )where P and Q have no common zeroes and the maximum degree of P and Q is d . An anti-rational map of degree d is an orientation reversing branchedcovering of (cid:98) C . It is said to be critically fixed if all of its critical points arefixed. The Julia set and Fatou set of anti-rational maps can be defined asin the rational setting.4.1. Tischler graph of critically fixed anti-rational maps.
Let R be acritically fixed anti-rational map of degree d . Let c , · · · , c k be the distinctcritical points of R , and the local degree of R at c i be m i ( i = 1 , · · · , k ).Since R has (2 d −
2) critical points counting multiplicity, we have that k (cid:88) i =1 ( m i −
1) = 2 d − ⇒ k (cid:88) i =1 m i = 2 d + k − . Suppose that U i is the invariant Fatou component containing c i ( i =1 , · · · , k ). Then, U i is a simply connected domain such that R | U i is con-formally conjugate to z m i | D [32, Theorem 9.3]. This defines internal raysin U i , and R maps the internal ray at angle θ ∈ R / Z to the one at angle − m i θ ∈ R / Z . It follows that there are ( m i + 1) fixed internal rays in U i .A straightforward adaptation of [32, Theorem 18.10] now implies that allthese fixed internal rays land at repelling fixed points on ∂U i .We define the Tischler graph T of R as the union of the closures of thefixed internal rays of R . Lemma 4.1. R has exactly ( d + 2 k − distinct fixed points in (cid:98) C of which ( d + k − lie on the Julia set of R .Proof. Note that R has no neutral fixed point and exactly k attracting fixedpoints. The count of the total number of fixed points of R now follows fromthe Lefschetz fixed point theorem (see [22, Lemma 6.1]). (cid:3) Lemma 4.2.
The fixed internal rays of R land pairwise.Proof. As R is a local orientation reversing diffeomorphism in a neighbor-hood of the landing point of each internal ray, it follows that at most twodistinct fixed internal rays may land at a common point.Note that the total number of fixed internal rays of R is k (cid:88) i =1 ( m i + 1) = 2( d + k − , while there are only ( d + k −
1) landing points available for these rays byLemma 4.1. Since no more than two fixed internal rays can land at a commonfixed point, the result follows. (cid:3)
In our setting, it is more natural to put a simplicial structure on T sothat the vertices correspond to the critical points of R . We will refer to therepelling fixed points on T as the midpoints of the edges. To distinguish anedge of T from an arc connecting a vertex and a midpoint, we will call thelatter an internal ray of T . Corollary 4.3.
The valence of the critical point c i (as a vertex of T ) is ( m i + 1) ( i = 1 , · · · , k ). The repelling fixed points of R are in bijectivecorrespondence with the edges of T . The proof of Lemma 4.2 implies the following (see [13, Corollary 6] forthe same statement in the holomorphic setting).
Corollary 4.4.
Each fixed point of R lies on (the closure of ) a fixed internalray. Lemma 4.5.
The faces of T are Jordan domains.Proof. Let F be a face of the Tischler graph T . Then a component of theideal boundary I ( F ) consists of a sequence of edges e , ..., e m of T , orientedcounter clockwise viewed from F . Note that a priori, e i may be equal to e j in T for different i and j .Since each vertex has valence at least 3, we have that e i (cid:54) = e i +1 . Thus,we can choose a component U of (cid:98) C \ F so that at most one pair of adjacentedges of ∂U are not adjacent on the ideal boundary I ( F ); i.e. there are atmost two adjacent edges of ∂U whose point of intersection has more thanone accesses from F (see Figure 4.1). Let S = ∂U .Since R is orientation reversing and each edge of T is invariant underthe map, R reflects the two sides near each open edge of S . Thus, there isa connected component V of R − ( U ) with ∂V ∩ ∂U (cid:54) = ∅ and V ∩ F (cid:54) = ∅ .It is easy to see from the local dynamics of z m i at the origin that near acritical point, R sends the region bounded by two adjacent edges of T to itscomplement. Therefore, by our choice of U , we have ∂U ⊆ ∂V . Since ∂F is R -invariant, V ∩ ∂F = ∅ . Thus, V ⊆ F and V contains no critical pointsof R . As U is a disk, the Riemann-Hurwitz formula now implies that V is a IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 31 F US = ∂U Figure 4.1.
An a priori possible schematic picture of a com-ponent of the boundary of F .disk, and R is a homeomorphism from ∂V to ∂U . Therefore, ∂U = ∂V and V = F ; so F is a Jordan domain. (cid:3) In particular, we have the following.
Corollary 4.6.
The Tischler graph T is connected. In fact, the Tischler graph gives a topological model for the map R . Corollary 4.7.
Let F be a face of T , and F c = (cid:98) C \ F be the closure ofits complement. Then R : F −→ F c is an orientation reversing homeomor-phism. Lemma 4.8.
Let F , F be two faces of T , then the boundaries share atmost one edge.Proof. Suppose that F and F share two or more edges. For i = 1 ,
2, let γ i be the hyperbolic geodesic arc in ˚ F i connecting the two midpoints of theedges, and let γ = γ ∪ γ . Since each vertex has valence at least 3, wesee that γ is essential in (cid:98) C \ V ( T ) = (cid:98) C \ P ( R ), where P ( R ) stands for thepost-critical set of R . By Corollary 4.7, there exists γ (cid:48) ∼ γ in (cid:98) C \ P ( R ) suchthat R : γ (cid:48) −→ γ is a homeomorphism. This gives a Thurston’s obstruction(more precisely, a Levy cycle) for the anti-rational map R (see the discussionin § F and F share two or more edges is false. (cid:3) The following result, where we translate the above properties of the Tiis-chler graph to a simple graph theoretic property of its planar dual, plays acrucial role in the combinatorial classification of critically fixed anti-rationalmaps.
Lemma 4.9.
Let Γ be the planar dual of the Tischler graph T of a criticallyfixed anti-rational map R . Then Γ is simple and -connected.Proof. By Lemma 4.8, no two faces of T share two edges on their boundary.Hence, the dual graph contains no multi-edge. Again, as each face of T is a Jordan domain by Lemma 4.5, the dual graph contains no self-loop.Therefore Γ is simple.The fact that each face of T is a Jordan domain also implies that thecomplement of the closure of each face is connected. So the dual graphΓ remains connected upon deletion of any vertex. In other words, Γ is2-connected. (cid:3) Constructing critically fixed anti-rational maps from graphs.
Let Γ be a 2-connected simple planar graph, and G Γ be an associated kissingreflection group. In §
3, we constructed a topological branched covering G Γ from the Nielsen map of G Γ . In the remainder of this section, we will usethis branched covering G Γ to promote Lemma 4.9 to a characterization ofTischler graphs of critically fixed anti-rational maps. Proposition 4.10.
A graph T is the Tischler graph of a critically fixedanti-rational map R if and only if the dual (planar) graph Γ is simple and -connected. Moreover, R is topologically conjugate to G Γ . We will first introduce some terminologies. A post-critically finite branchedcovering (possibly orientation reversing) of S is called a Thurston map . Wedenote the post-critical set of a Thurston map f by P ( f ). Two Thurstonmaps f and g are equivalent if there exist two orientation-preserving home-omorphisms h , h : ( S , P ( f )) → ( S , P ( g )) so that h ◦ f = g ◦ h where h and h are isotopic relative to P ( f ).A set of pairwise non-isotopic, essential, simple, closed curves Σ on S \ P ( f ) is called a curve system . A curve system Σ is called f -stable if forevery curve σ ∈ Σ, all the essential components of f − ( σ ) are homotopicrel P ( f ) to curves in Σ. We associate to a f -stable curve system Σ the Thurston linear transformation f Σ : R Σ −→ R Σ defined as f Σ ( σ ) = (cid:88) σ (cid:48) ⊆ f − ( σ ) f : σ (cid:48) → σ ) [ σ (cid:48) ] Σ , where σ ∈ Σ, and [ σ (cid:48) ] Σ denotes the element of Σ isotopic to σ (cid:48) , if it ex-ists. The curve system is called irrreducible if f Σ is irreducible as a lineartransformation. It is said to be a Thurston obstruction if the spectral radius λ ( f Σ ) ≥ λ in S \ P ( f ) is an embedding of (0 ,
1) in S \ P ( f ) withend-points in P ( f ). It is said to be essential if it is not contractible in S fixing the two end-points. A set of pairwise non-isotopic essential arcs Λ iscalled an arc system . The Thurston linear transformation f Λ is defined in a IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 33 similar way, and we say that it is irreducible if f Λ is irreducible as a lineartransformation.Although Thurston’s topological characterization theorem of rational maps(see [6, Theorem 1]) has not been extended to anti-rational maps, the fol-lowing proposition makes up for the lack of an analogue. It is proved byconsidering the second iterate of the Thurston’s pullback map on the Te-ichm¨uller space of S \ P ( f ). Proposition 4.11. [23, Proposition 6.1]
Let f be an orientation reversingThurston map so that f ◦ f has hyperbolic orbifold. Then f is equivalent toan anti-rational map if and only if f ◦ f is equivalent to a rational map ifand only if f ◦ f has no Thurston’s obstruction. Moreover, if f is equivalentto an anti-rational map, the map is unique up to M¨obius conjugacy. For a curve system Σ (respectively, an arc system Λ), we set (cid:101)
Σ (respec-tively, (cid:101)
Λ) as the union of those components of f − (Σ) (respectively, f − (Λ))which are isotopic to elements of Σ (respectively, Λ). We will use Σ · Λ todenote the minimal intersection number between them. We will be usingthe following theorem excerpted and paraphrased from [42, Theorem 3.2].
Theorem 4.12. [42, Theorem 3.2]
Let f be an orientation preserving Thurstonmap, Σ an irreducible Thurston obstruction in ( S , P ( f )) , and Λ an irre-ducible arc system in ( S , P ( f )) . Assume that Σ intersect Λ minimally, theneither • Σ · Λ = 0 ; or • Σ · Λ (cid:54) = 0 and for each λ ∈ Λ , there is exactly one connected compo-nent λ (cid:48) of f − ( λ ) such that λ (cid:48) ∩ (cid:101) Σ (cid:54) = ∅ . Moreover, the arc λ (cid:48) is theunique component of f − ( λ ) which is isotopic to an element of Λ . With these preparations, we are now ready to show that G Γ is equivalentto an anti-rational map. The proof is similar to [23, Proposition 6.2]. Lemma 4.13.
Let Γ be a -connected, simple, planar graph. Then G Γ isequivalent to a critically fixed anti-rational map R Γ .Proof. It is easy to verify that G Γ ◦ G Γ has hyperbolic orbifold, except whenΓ = Γ d , in which case G Γ has only two fixed critical points each of which isfully branched, and G Γ is equivalent to z d . Thus, by Proposition 4.11, wewill prove the lemma by showing that there is no Thurston’s obstruction for G Γ ◦ G Γ .We will assume that there is a Thurston obstruction Σ, and arrive at acontradiction. After passing to a subset, we may assume that Σ is irre-ducible. Isotoping the curve system Σ, we may assume that Σ intersects thegraph T ( G Γ ) minimally. Let λ be an edge in T ( G Γ ), then Λ = { λ } is anirreducible arc system. Let (cid:101) Σ be the union of those components of G − (Σ)which are isotopic to elements of Σ.We claim that (cid:101) Σ does not intersect G − ( λ ) \ λ . Indeed, applying Theo-rem 4.12, we are led to the following two cases. In the first case, we have Σ · λ = 0; hence G − (Σ) cannot intersect G − ( λ ), and the claim follows.In the second case, since G Γ ( λ ) = λ , we conclude that λ is the uniquecomponent of G − ( λ ) isotopic to λ . Thus, the only component of G − ( λ )intersecting (cid:101) Σ is λ , so the claim follows.Applying this argument on all the edges of T ( G Γ ), we conclude that (cid:101) Σdoes not intersect G − ( T ( G Γ )) \ T ( G Γ ).Since Γ is 2-connected, and T ( G Γ ) is dual to Γ, the graph obtained byremoving the boundary edges of any face of T ( G Γ ) is still connected. ByLemma 3.22, we deduce that ( G − ( T ( G Γ )) \ T ( G Γ )) ∪ V ( T ( G Γ )) is con-nected. Thus, ( G − ( T ( G Γ )) \ T ( G Γ )) ∪ V ( T ( G Γ )) is a connected graphcontaining the post-critical set P ( G Γ ◦ G Γ ) = V ( T ( G Γ )). This forces Σ tobe empty, which is a contradiction.Therefore, G Γ is equivalent to an anti-rational map R Γ . Since G Γ is criti-cally fixed, so is R Γ . (cid:3) Let h , h : ( S , P ( G Γ )) → ( S , P ( R Γ )) be two orientation preservinghomeomorphisms so that h ◦ G Γ = R Γ ◦ h where h and h are isotopicrelative to P ( G Γ ) = V ( T ( G Γ )).We now use a standard pullback argument to prove the following. Lemma 4.14. h ( T ( G Γ )) is isotopic to the Tischler graph T ( R Γ ) .Proof. Let α be an edge of T ( G Γ ), then G Γ ( α ) = α . Consider the sequenceof homeomorphisms h i with h i − ◦ G Γ = R Γ ◦ h i . Each h i is normalized so that it carries P ( G Γ ) to P ( R Γ ). Note by induction, β i = h i ( α ) is isotopic to β relative to the end-points. Applying isotopy, wemay assume that there is a decomposition β = β − + γ + β +0 , where β ± aretwo internal rays and γ does not intersect any invariant Fatou component.Then, every β i = β − i + γ i + β + i has such a decomposition too.Note that R Γ ( γ i +1 ) = γ i and R Γ : γ i +1 → γ i is injective for each i . Since R Γ is hyperbolic, and the curves γ i are uniformly bounded away from thepost-critical point set of R Γ , we conclude that the sequence of curves { γ i } shrinks to a point. Thus, if β is a limit of β i in the Hausdorff topology, then β is a union of two internal rays and it is isotopic to β . Therefore, h ( α ) isisotopic to an edge of T ( R Γ ).Since this is true for every edge of T ( G Γ ) and h is an orientation pre-serving homeomorphism, by counting the valence at every critical point, weconclude that h ( T ( G Γ )) is isotopic to T ( R Γ ). (cid:3) Therefore, after performing isotopy, we may assume that h = h : T ( G Γ ) −→ T ( R Γ )gives a conjugacy between G Γ and R Γ . Hence, both h and h send a faceof T ( G Γ ) homomorphically to the corresponding face of T ( R Γ ). IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 35
As an application of the above, we will show that the dynamics on thelimit set and Julia set are topologically conjugate (cf. [23, Theorem 6.11]).
Lemma 4.15.
There is a homeomorphism h : Λ( G Γ ) −→ J ( R Γ ) satisfying h ◦ G Γ = R Γ ◦ h .Proof. Let P consist of the points of tangency of the circle packing, P i +1 := G − ( P i ), and P ∞ := (cid:83) ∞ i =0 P i . Then P ∞ corresponds to the G Γ orbit of thecusps, and hence is dense in Λ( G Γ ). Let Q := h ( P ), which is also the setof repelling fixed points of R Γ , and Q i +1 := R − ( Q i ). Then Q ∞ := (cid:83) ∞ i =0 Q i is dense in J ( R Γ ).Recall that for each face F of Γ, the dynamics of G Γ (respectively, of R Γ )on the corresponding component of Ω( G Γ ) (respectively, on the correspond-ing critically fixed Fatou component of R Γ ) can be uniformized to z d on D .Let Π be the closed ideal d +1-gon in D with ideal vertices at the fixed pointsof z d on S . Let L G ,F and L R ,F be the image of Π under the uniformizingmap in the dynamical plane of G Γ and R Γ respectively, and L G and L R bethe union over all faces.Let E := (cid:98) C \ L G and H := (cid:98) C \ L R . Inductively, let E i +1 := G − ( E i )and H i +1 := R − ( H i ). Note that these sets are the analogues of D i forkissing reflection groups. Since G Γ (respectively, R Γ ) send a face of T ( G Γ )(respectively, a face of T ( R Γ )) to its complement univalently, our construc-tion guarantees that E ⊆ E and H ⊆ H . Inductively, we see that {E i } and {H i } are nested sequences of closed sets. Thus, we haveΛ( G Γ ) = ∞ (cid:92) i =0 E i and J ( R Γ ) = ∞ (cid:92) i =0 H i . Since R Γ is hyperbolic, and H i contains no critical value, the diametersof the components of H i ∩ J ( R Γ ) shrink to 0 uniformly. On the other hand,since G Γ = N Γ on Λ( G Γ ), the diameters of the components of E i ∩ Λ( G Γ )shrink to 0 uniformly by Lemma 3.3.After isotoping h , we may assume that h ( E ) = H . We consider thepullback sequence { h i } with h i − ◦ G Γ = R Γ ◦ h i , where each h i carries P ( G Γ ) to P ( R Γ ). Since we have assumed h = h on P , inductively, we have h j = h i on P i and h j ( P i ) = Q i for all j ≥ i . Wedefine h ( x ) := lim h i ( x ) for x ∈ P ∞ .Then, a similar argument as in the proof of Proposition 3.21 implies that h is uniformly continuous on P ∞ . Indeed, given (cid:15) >
0, we can choose N so that the diameter of any component of H N ∩ J ( R Γ ) is less than (cid:15) . Wechoose δ so that any two non-adjacent component of E N ∩ Λ( G Γ ) are at least δ distance away. If x, y ∈ P ∞ are at most δ distance apart, they lie in twoadjacent components of E N , so h j ( x ) , h j ( y ) lie in two adjacent componentsof H N for all j ≥ N . Hence, d ( h j ( x ) , h j ( y )) < (cid:15) .Since P ∞ and Q ∞ are dense on Λ( G Γ ) and J ( R Γ ) (respectively), we geta unique continuous extension h : Λ( G Γ ) −→ J ( R Γ ) . Applying the same argument on the sequence { h − j } , we get the contin-uous inverse of h . Thus, h is a homeomorphism. Since h conjugates G Γ to R Γ on P ∞ , it also conjugates G Γ to R Γ on Λ( G Γ ). (cid:3) We are now ready to prove Proposition 4.10 (cf. [23, Proposition 6.13]).
Proof of Proposition 4.10.
By Lemma 4.9, the dual graph of a Tischler graph(of a critically fixed anti-rational map) is 2-connected, simple and planar.Conversely, given any 2-connected, simple, planar graph Γ, by Lemma4.13 and Lemma 4.14, we can construct a critically fixed anti-rational map R Γ whose Tischler graph is the planar dual of Γ.It remains to show that G Γ and R Γ are topologically conjugate. Let h be the topological conjugacy on Λ( G Γ ) produced in Lemma 4.15. Let U be a critically fixed component G Γ . Then by construction, there existsΦ : D −→ U conjugating m − d ( z ) = z d to G Γ . The map extends to asemiconjugacy between S and ∂U .Similarly, let V be the corresponding critically fixed Fatou component of R Γ . Then we have Ψ : D −→ V conjugating m − d ( z ) = z d to R Γ , whichextends to a semiconjugacy. Note that there are d + 1 different choices ofsuch a conjugacy, we may choose one so that h ◦ Φ = Ψ on S . Thus, wecan extend the topological conjugacy h to all of U by setting h := Ψ ◦ Φ − .In this way, we obtain a homeomorphic extension of h to all criticallyfixed components of G Γ . Lifting these extensions to all the preimages of thecritically fixed components of G Γ , we get a map defined on (cid:98) C . Since the di-ameters of the preimages of the critically fixed components (of G Γ ) as well asthe corresponding Fatou components (of R Γ ) go to 0 uniformly, we concludethat all these extensions paste together to yield a global homeomorphism,which is our desired topological conjugacy. (cid:3) In light of Proposition 4.10, we see that the association of the planar iso-morphism class of a 2-connected, simple, planar graph to the M¨obius conju-gacy class of a critically fixed anti-rational map is well defined and surjective.To verify that this is indeed injective, we remark that if two graphs Γ andΓ (cid:48) are planar isomorphic, then the associated Tischler graphs are planar iso-morphic as well. This means that the corresponding pair of critically fixedanti-rational maps are Thurston equivalent. Thus by Thurston’s rigidityresult, they are M¨obius conjugate.As in the kissing reflection group setting, we define a geometric matingof two (anti-)polynomials as follows.
IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 37
Definition 4.16.
We say that a rational map (or an anti-rational map) R is a geometric mating of two polynomials (or two anti-polynomials) P ± withconnected Julia sets if we have • a decomposition of the Fatou set F ( R ) = F + (cid:116) F − with J ( R ) = ∂ F + = ∂ F − ; and • two continuous surjections on the filled Julia sets ψ ± : K ( P ± ) → F ± that are conformal between Int K ( P ± ) and F ± so that ψ ± ◦ P ± = R ◦ ψ ± . As a corollary, we have
Corollary 4.17. (1) A critically fixed anti-rational map R is an anti-polynomial if andonly if the dual graph Γ of T ( R ) is outerplanar.(2) A critically fixed anti-rational map R is a geometric mating of twoanti-polynomials if and only if the dual graph Γ of T ( R ) is Hamil-tonian.(3) A critically fixed anti-rational map R has a gasket Julia set if andonly if the dual graph Γ of T ( R ) is -connected.Proof. The first statement follows from Proposition 4.10 as the graph Γ isouterplanar if and only if there is a vertex in the dual graph Γ ∨ = T ( R )with maximal valence if and only if R has a fully branched fixed criticalpoint.For the second statement, let Γ be a 2-connected, simple, planar graph.Let G Γ be an associated kissing reflection group, G Γ and R Γ be the associatedbranched covering and anti-rational map. By Proposition 4.10, G Γ and R Γ are topologically conjugated by h .If R Γ is a geometric mating of two anti-polynomials P ± which are neces-sarily critically fixed, then the conjugacy gives a decomposition of Ω( G Γ ) =Ω + (cid:116) Ω − . It is not hard to check that the Γ-actions on Ω ± are conjugateto the actions of the function kissing reflection groups G ± on their filledlimit sets, where G ± correspond to P ± . Thus, G Γ is a geometric matingof the two function kissing reflection groups G ± , so Γ is Hamiltonian byProposition 3.21.Conversely, if Γ is Hamiltonian, then G Γ is a geometric mating of twofunction kissing reflection groups G ± , and we get a decomposition of Ω( G Γ )by Proposition 3.21. The conjugacy h transports this decomposition to theanti-rational map setting. One can now check directly that R Γ is a geometricmating of the anti-polynomials P ± that are associated to the function kissingreflection groups G ± .The third statement follows immediately from Propositions 4.10 and 3.10. (cid:3) Dynamical correspondence.
Let G Γ and R Γ be a kissing reflectiongroup and the critically fixed anti-rational map associated to a 2-connectedsimple planar graph Γ. Here we summarize the correspondence betweenvarious dynamical objects associated with the group and the anti-rationalmap. • Markov partitions for limit and Julia sets:
The associated cir-cle packing P gives a Markov partition for the group dynamics onthe limit set (more precisely, for the action of the Nielsen map N Γ on Λ( G Γ )). On the other side, the faces of the Tischler graph deter-mine a Markov partition for the action of R Γ on the Julia set. Thetopological conjugacy respects this pair of Markov partitions andthe itineraries of points with respect to the corresponding Markovpartitions. • Cusps and repelling fixed points:
The conjugacy classes of cuspsof G Γ bijectively correspond to pairs of adjacent vertices, or equiv-alently, to the edges of Γ. On the other side, the repelling fixedpoints of R Γ are in bijective correspondence with the edges of theTischler graph T = Γ ∨ , and thus also with edges of the Γ. They arenaturally identified by the topological conjugacy. • σ -invariant curves and -cycles: Each σ -invariant simple closedcurve on the conformal boundary ∂ M ( G Γ ) corresponds to a pairof non-adjacent vertices on a face of Γ. On the other side, theMarkov partition for R Γ | J ( R (Γ)) shows that each 2-cycle on the idealboundary of an invariant Fatou component corresponds to a pair ofnon-adjacent vertices on the associated face of Γ. These σ -invariantcurves and 2-cycles are naturally identified by the topological con-jugacy. We remark that it is possible that a σ -invariant curve yieldsan accidental parabolic element, in which case the corresponding2-cycle on the ideal boundary of the invariant Fatou component co-alesce, and gives rise to a repelling fixed point of R Γ . This happensif and only if the two corresponding vertices are adjacent in Γ. • Question mark conjugacy:
If we choose our group G Γ so thateach conformal boundary is the double of a suitable regular idealpolygon, then the topological conjugacy between an invariant Fatoucomponent and the associated component Ω( G Γ ) gives a homeomor-phism between the ideal boundaries φ : S −→ S that conjugates z e to the Nielsen map N e of the regular ideal ( e + 1)-gon reflec-tion group. This conjugacy φ is a generalization of the Minkowskiquestion mark function (see [21, § § • Function groups and anti-polynomials: If G Γ is a function kiss-ing reflection group (of rank d + 1), then it can be constructed bypinching a σ -invariant multicurve α on one component of ∂ M ( G d ) IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 39 (recall that G d is the regular ideal polygon group of rank d + 1). Un-der the natural orientation reversing identification of the two com-ponents of ∂ M ( G d ), the multicurve α gives a σ -invariant multicurve α (cid:48) on the component of ∂ M ( G Γ ) associated to the G Γ -invariant do-main Ω ⊆ Ω( G Γ ). The multicurve α (cid:48) consists precisely of the simpleclosed curves corresponding to the accidental parabolics of G Γ . Onthe other side, the corresponding 2-cycles on the ideal boundary ofthe unbounded Fatou component (i.e., the basin of infinity) generatethe lamination for the Julia set of the anti-polynomial R Γ . • QuasiFuchsian closure and mating: If G Γ lies in the closureof the quasiFuchsian deformation space of G d , then G Γ is obtainedby pinching two non-parallel σ -invariant multicurves α + and α − onthe two components of ∂ M ( G d ); equivalently, G Γ is a mating of twofunction kissing reflection groups G + and G − . On the other side, thecritically fixed anti-rational map R Γ is a mating of the critically fixedanti-polynomials P + and P − , which correspond to the groups G + and G − (respectively). The topological conjugacy between N Γ | Λ( G Γ ) and R Γ | J ( R Γ ) is induced by the circle homeomorphism that conju-gates N d to z d .4.4. Mating of two anti-polynomials.
In this subsection, we shall dis-cuss the converse question of mateablity in terms of laminations. Let P ( z ) = z d + a d − z d − + a d − z d − + ... + a be a monic centered anti-polynomial with connected Julia set. We denotethe filled Julia set by K ( P ). There is a unique B¨ottcher coordinate ψ : C \ D −→ C \ K ( P )with derivative 1 at infinity that conjugates z d to P . We shall call a moniccentered anti-polynomial equipped with this preferred B¨ottcher coordinatea marked anti-polynomial. Note that when the Julia set is connected, thismarking is equivalent to an affine conjugacy class of anti-polynomials to-gether with a choice of B¨ottcher coordinate at infinity. If we further assumethat the Julia set is locally connected, the map ψ extends to a continuoussemi-conjugacy ψ : S −→ J ( P ) between z d and P . This semi-conjugacygives rise to a z d - invariant lamination on the circle S for the marked anti-polynomial. The coordinate on S ∼ = R / Z will be called external angle . Theimage of { re πiθ ∈ C : r ≥ } under ψ will be called an external ray atangle θ , and will be denoted by R ( θ ). If x ∈ J ( P ) is the intersection oftwo external rays, then x is a cut-point of J ( P ) (equivalently, a cut-pointof K ( P )).If we denote C = C ∪ { ( ∞ , w ) : w ∈ S } as the complex plane togetherwith the circle at infinity, then a marked anti-polynomial extends continu-ously to C by the formula P ( ∞ , w ) = ( ∞ , w d ). Let P and Q be two marked degree d anti-polynomials. To distinguishthe two domains C , we denote them by C P and C Q . The quotient S P,Q := ( C P ∪ C Q ) / { ( ∞ P , w ) ∼ ( ∞ Q , w ) } defines a topological sphere. Denote the equivalence class of ( ∞ P , w ) by( ∞ , w ), and define the equator of S P,Q to be the set { ( ∞ , w ) : w ∈ S } . The formal mating of P and Q is the degree d branched cover P ⊥⊥ Q : S f,g → S f,g defined by the formula( P ⊥⊥ Q )( z ) = P ( z ) z ∈ C P , ( ∞ , z d ) for ( ∞ , z ) ,Q ( z ) z ∈ C Q . Suppose that P and Q have connected and locally connected Julia sets.The closure in S P,Q of the external ray at angle θ in C P is denoted R P ( θ ),and likewise the closure of the external ray at angle θ in C Q is denoted R Q ( θ ). Then, R P ( θ ) and R Q ( − θ ) are rays in S P,Q that share a commonendpoint ( ∞ , e πiθ ). The extended external ray at angle θ in S P,Q is definedas R ( θ ) := R P ( θ ) ∪ R Q ( − θ ). Each extended external ray intersects each of K ( P ), K ( Q ), and the equator at exactly one point.We define the ray equivalence ∼ ray as the smallest equivalence relationon S P,Q so that two points x, y ∈ S P,Q are equivalent if there exists θ so that x, y ∈ R ( θ ). A ray equivalence class γ can be considered as an embeddedgraph: the vertices are the points in γ ∩ ( K ( P ) ∪ K ( Q )), and the edges arethe external extended rays in γ . We also denote by [ x ] the ray equivalenceclass of x .An equivalence relation ∼ on S is said to be Moore-unobstructed if thequotient S / ∼ is homeomorphic to S . The following theorem is due to A.Epstein (see [40, Proposition 4.12]). Proposition 4.18.
Let P and Q be two anti-polynomials of equal degree d ≥ with connected and locally connected Julia sets. If the equivalence classesof ∼ ray are ray-trees having uniformly bounded diameter, then S P,Q / ∼ ray isMoore-unobstructed. We remark that the proof of this theorem uses Moore’s theorem. It isoriginally stated only for polynomials, but the proof extends identically tothe case of anti-polynomials. There is also a partial converse of the abovestatement which we will not be using here.For a critically fixed anti-polynomial P , we have the following descriptionof the cut-points of K ( P ). The result directly follows from the dynamicalcorrespondence between function kissing reflection groups and critically fixedanti-polynomials discussed in § Lemma 4.19.
Let P be a critically fixed anti-polynomial. Then, any cut-point of K ( P ) is eventually mapped to a repelling fixed point that is thelanding point of a -cycle of external rays. IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 41
An extended external ray that contains a repelling fixed point of a crit-ically fixed anti-polynomial is called a principal ray . The ray equivalenceclass of a principal ray is called a principal ray class . In our setting, theMoore obstructions can be detected simply by looking at these principal rayclasses.
Lemma 4.20.
Let P and Q be two marked anti-polynomials of equal degree d ≥ , where P is critically fixed and Q is postcritically finite and hyperbolic.(1) If a principal ray equivalence class of ∼ ray contains more than onecut-point of K ( P ) , it must contain a 2-cycle or a 4-cycle.(2) The equivalence ∼ ray is Moore-obstructed if and only if some prin-cipal ray equivalence class contains a 2-cycle or a 4-cycle.Proof. Suppose we have two distinct repelling fixed points w, z of P whichare cut-points and w ∼ ray z . Without loss of generality, we can assumethat w and z have distance two in their equivalence class considered as agraph. In particular there will exist two extended external rays α z and α w that land at z and w and share a common endpoint v ∈ K ( Q ). Let γ bethe concatenation of these two rays. Each of α z , α w must have dynamicalperiod 2 since they land at repelling fixed points of P . If v is a fixed point,then the graph α w ∪ ( P ⊥⊥ Q )( α w ) is a cycle of length 2 for ∼ ray . If v hasperiod 2, then γ ∪ ( P ⊥⊥ Q )( γ ) is a cycle of length 4 for ∼ ray . This concludesthe proof of statement 1.If a principal ray equivalence contains a cycle, it is immediate that ∼ ray is Moore-obstructed. Now assume that no principal ray equivalence classcontains a 2-cycle or a 4-cycle. This means that the diameter of a principalray equivalence class is bounded by 4 as there can be at most one cut-pointin this class. Any principal ray equivalence class must evidently be a tree.Now let z ∈ ∂ K ( P ). If there is no cut-point in the equivalence class[ z ], then [ z ] is a tree and has diameter at most 2. Otherwise, by Lemma4.19, it is eventually mapped to a principal ray equivalence class. Sinceboth P and Q are hyperbolic, the map P ⊥⊥ Q is a local homeomorphism on J ( P ) and J ( Q ). Since all the principal ray equivalence classes are trees, inparticular, simply connected, any component of its preimage under P ⊥⊥ Q is homeomorphic to it. Thus by induction, we conclude that [ z ] is a treeand has diameter at most 4. Now by Proposition 4.18, the mating is Moore-unobstructed. (cid:3) We now prove that for a large class of pairs of anti-polynomials, theabsence of Moore obstruction is equivalent to the existence of geometricmating.
Proposition 4.21.
Let P and Q be marked anti-polynomials of equal degree d ≥ , where P is critically fixed and Q is postcritically finite and hyperbolic.There is an anti-rational map that is the geometric mating of P and Q ifand only if ∼ ray is not Moore-obstructed. Proof.
The only if part of the statement is immediate. To prove the if part,we will first show that the absence of Moore obstruction can be promoted tothe absence of Thurston obstruction. Note that the only case where ( P ⊥⊥ Q ) has non-hyperbolic orbifold is when P and Q are both power maps, in whichcase the conclusion of the theorem is immediate. Thus we assume for theremainder of the proof that ( P ⊥⊥ Q ) has hyperbolic orbifold.Suppose ( P ⊥⊥ Q ) has a Thurston obstruction Γ. We may assume thatΓ is irreducible. Since Q has no Thurston obstruction, there is some edge λ in the Hubbard tree H p so that Γ · λ (cid:54) = 0. Note that λ is an irreduciblearc system consisting of one arc, so it follows from [46, Theorem 3.9] that Γmust also be a Levy cycle consisting of one essential curve γ .Since P ⊥⊥ Q is hyperbolic, the Levy cycle is not degenerate. Thus thereis a periodic ray class that contains a cycle [46, Theorem 1.4(2)]. By thehypothesis that there is no Moore obstruction, no such periodic ray classexists. This is a contradiction, and so no Thurston obstruction exists for( P ⊥⊥ Q ) . By Proposition 4.11, it follows that P ⊥⊥ Q is also unobstructedand Thurston equivalent to a rational map.Having shown that P ⊥⊥ Q has no Thurston obstruction, the Rees-Shishikuratheorem (which extends directly to orientation reversing maps) implies thatthe geometric mating of P and Q exists [45, Theorem 1.7]. (cid:3) Recall that a pair of laminations on S for marked anti-polynomials is saidto be non-parallel if they share no common leaf under the natural orientationreversing identification of the two copies of S . This is equivalent to sayingthat the ray equivalence classes contain no 2-cycle. We immediately havethe following corollary from Lemma 4.20 and Proposition 4.21. Corollary 4.22.
A marked critically fixed anti-polynomial P and a markedpost-critically finite, hyperbolic anti-polynomial Q are geometrically mateableif and only if the principal ray equivalence classes contain no -cycles or -cycles.Two marked critically fixed anti-polynomials P and Q are geometricallymateable if and only if their laminations are non-parallel.Proof. The first statement is immediate.To see the second one, we note that according to Lemma 4.19, the lamina-tion of a marked critically fixed anti-polynomial is generated by the 2-cyclesof rays landing at the repelling fixed points (which are cut-points). In lightof this, it is easy to see that for two marked critically fixed anti-polynomials,no principal ray equivalence class contain a 4-cycle. The second statementnow follows from this observation and the first statement. (cid:3)
IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 4378 3801 18 1434 58 1278 3801 18 1434 58 12
Figure 4.2.
Two cubic anti-polynomials together with allrays that have period two or smaller. The first map is acritically fixed antipolynomial so that the 1/8 and 5/8 raysco-land. To second map is produced by tuning the first mapwith basilicas so that the pairs (0 / , /
4) and (1 / , /
4) co-land. The unique principal ray equivalence class has no cycle.Their mating is depicted below using the software [1]. Blaschke and Anti-Blaschke Products
For d ≥
2, let B + d := { f ( z ) = z d − (cid:89) i =1 z − a i − a i z : | a i | < } and B − d := { f ( z ) = z d − (cid:89) i =1 z − a i − a i z : | a i | < } be the space of (normalized) marked (anti-)Blaschke products. Each mapin B ± d has an attracting fixed point at the origin.Let m d ( z ) := z d and m − d ( z ) := z d . We shall identify S := R / Z . Underthis identification, we have that m d ( t ) = d · t and m − d ( t ) = − d · t . Notethat for each f ∈ B ± d , there exists a unique quasisymmetric homeomorphism η f : S −→ S with(1) η f ◦ m ± d = f ◦ η f , and(2) f (cid:55)→ η f is continuous on B ± d with φ m ± d = id.The conjugacy η f provides with a marking of the repelling periodic pointsfor f ∈ B ± d . Let F be the set of repelling fixed points for m ± d . Then themultiplier of the repelling fixed point of f corresponding to x ∈ F is L f ( x ) := log | f (cid:48) ( η f ( x )) | . Note that f (cid:48) is understood as ∂∂z f if f is anti-holomorphic.We define the following pared deformation space Definition 5.1.
Let
K >
0, we define B ± d ( K ) := { f ∈ B ± d : L f ( x ) ≤ K for all x ∈ F } . Let M ± d denote the M¨obius conjugacy classes of (anti-)rational maps ofdegree d , we also let [ B ± d ( K )] := { [ f ] ∈ M ± d : f ∈ B ± d ( K ) } .Let D be the unit disk equipped with the hyperbolic metric d . For f ∈ B ± d ,we denote the set of all critical points of f in D by C ( f ). We will prove thefollowing. Proposition 5.2.
There exists M = M ( d, K ) such that for all f ∈ B ± d ( K ) and all c ∈ C ( f ) , we have d ( c, f ( c )) < M .Conversely, let f ∈ B ± d be such that d ( c, f ( c )) < M for some M > andall c ∈ C ( f ) , then f ∈ B ± d ( K ) for some K = K ( d, M ) . The proof will be given after some qualitative and quantitative boundsfor (anti-)Blaschke products are established.
IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 45
Algebraic limit.
Let Rat ± d denote the space of rational maps and anti-rational maps of degree d . By fixing a coordinate system, a rational map (oran anti-rational map) can be expressed as a ratio of two homogeneous poly-nomials (or homogeneous anti-polynomials) f ( z : w ) = ( P ( z, w ) : Q ( z, w )),where P and Q have degree d with no common divisors. Thus, using thecoefficients of P and Q as parameters, we haveRat ± d = P d +1 \ V (Res) , where Res is the resultant of the two polynomials P and Q , and V (Res) isthe hypersurface for which the resultant vanishes. This embedding gives anatural compactification Rat ± d = P d +1 , which will be called the algebraiccompactification . Maps f ∈ Rat ± d can be written as f = ( P : Q ) = ( Hp : Hq ) , where H = gcd( P, Q ). We set ϕ f := ( p : q ) , which is a rational map (or an anti-rational map) of degree at most d . Thezeroes of H in P are called the holes of f , and the set of holes of f is denotedby H ( f ).If { f n } ⊆ Rat ± d converges to f ∈ Rat ± d , we say that f is the algebraic limit of the sequence { f n } . It is said to have degree k if ϕ f has degree k . Abusingnotations, sometimes we shall refer to ϕ f as the algebraic limit of { f n } . Astraightforward modification of [5, Lemma 4.2] to the anti-holomorphic caseyields the following result. Lemma 5.3. If f n converges to f algebraically, then f n converges to ϕ f uniformly on compact subsets of (cid:98) C \ H ( f ) . Qualitative bounds for (anti-)Blaschke products.
The normalization f (0) = 0 imposed on our (anti-)Blaschke products is useful essentially be-cause of the following lemma (cf. [31, Proposition 10.2]). Lemma 5.4.
Any sequence { f n } ∈ B ± d has a subsequence converging alge-braically to f ∈ Rat ± d where ϕ f ∈ B ± k with ≤ k ≤ d .Proof. This follows directly from computation and the fact that f ∈ B ± d hasthe form f ( z ) = z (cid:81) d − i =1 z − a i − a i z or f ( z ) = z (cid:81) d − i =1 z − a i − a i z . (cid:3) The fact that algebraic limits of sequences in B ± d always have degree atleast one gives the following corollary. Corollary 5.5.
Let f n ∈ B ± d be a sequence converging algebraically to f .Then for any hole p ∈ H ( f ) , there exists a sequence of critical points c n of f n converging to p . Proof.
Let 0 / ∈ U (cid:48) be a small disk centered at ϕ f ( p ), and let C (cid:48) = ∂U (cid:48) .Since the hole p ∈ S , so ϕ f ( p ) is not a critical value for ϕ f . Thus, we mayassume that U (cid:48) contains no critical values of ϕ f . Let U be the component of ϕ − f ( U (cid:48) ) that contains p , and C = ∂U . Suppose for contradiction that thereexists no sequence of critical points c n of f n converging to p . Shrinking U ifnecessary, we may assume U contains no critical points of f n or any otherholes.By Lemma 5.3, f n converges uniformly to ϕ f on a neighborhood of C . Let C n be the component of f − n ( C (cid:48) ) that converges to C in Hausdorff topology,and U n be the component bounded by C n that contains p . For sufficientlylarge n , U n contains no critical point of f n , so f n : U n −→ U (cid:48) is univalent.This is a contradiction as there exists a sequence of zeroes of f n convergingto the hole p , but 0 / ∈ U (cid:48) . (cid:3) Let { f n } ⊆ B ± d . We assume that we have a marking of the critical pointsin D , i.e., for each f n , we have an ordered list c ,n , ..., c d − ,n ∈ D of thecritical points of f n . It is understood that a critical point with multiplicity k appears in the list k times.After possibly passing to a subsequence, we may assume that for any pair( i, j ), lim n →∞ d ( c i,n , c j,n )exists (which can possibly be ∞ ). We can thus construct a partition { , ..., d − } = I (cid:116) I (cid:116) ... (cid:116) I m so that • lim n →∞ d ( c i,n , c j,n ) < ∞ if i and j are in the same partition set; and • lim n →∞ d ( c i,n , c j,n ) = ∞ if i and j are in different partition sets.We call C k,n := { ( c i,n ) : i ∈ I k } a critical cluster for f n . We say that thecritical cluster C i,n has degree d i,n = (cid:80) m j,n + 1 where m j,n is the order ofthe critical point and the sum is taken over all critical points in C i,n . Afterpassing to a subsequence, we can assume that d i,n and m j,n are all constant.Therefore, we will simply denote the degree as d i .After passing to a subsequence, we can assumelim n →∞ d ( c i,n , f n ( c i,n ))exists (which can possibly be ∞ ). We say that the critical point c i,n is active if lim n →∞ d ( c i,n , f n ( c i,n )) = ∞ , and inactive otherwise. By Schwarz-Pick lemma, if c i,n is active (respectively, inactive), then all of the criticalpoints in the critical cluster containing c i,n are active (respectively, inactive).Thus, we say that a critical cluster C k,n is active if any of the critical pointcontained in C k,n is active, and inactive otherwise.We are ready to prove Proposition 5.2. Proof of Proposition 5.2.
Let us fix d ≥
2, and
K >
0. By way of contradic-tion, suppose that there does not exist any
M >
IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 47 i.e., there exists a sequence { f n } ⊆ B ± d ( K ) and c n ∈ C ( f n ) with d ( c n , f n ( c n )) → ∞ . We mark all the critical points of f n . After passing to a subsequence, let C ,n , ..., C m,n be the critical clusters of f n .As we will be using different coordinates later, we will use to denotethe common fixed point of each f n in D . Let d ( , C k,n ) be the sequence ofdistances between and the critical points in C k,n . Our assumption impliesthat the sequence { f n } has at least one active critical cluster. After passingto a subsequence and reindexing, we assume that C ,n is a furthest activecritical cluster: d ( , C ,n ) ≥ d ( , C k,n ) for all active k and all n . Assume c ,n ∈ C ,n ; note that by Schwarz-Pick lemma, d ( , c ,n ) → ∞ as is fixed.Let L n , M n ∈ Isom( D ) ⊆ PSL ( C ) with L n (0) = c ,n and M n (0) = f n ( c ,n ). Note that M − n ◦ f n ◦ L n (0) = 0, and thus M − n ◦ f n ◦ L n ∈ B ± d .After passing to a subsequence, we may assume that { M − n ◦ f n ◦ L n } con-verges algebraically to f ∈ Rat ± d . By counting the critical points, we knowthe degree of ϕ f is d , in particular deg( ϕ f ) ≥ M n and L n change the non-dynamical coordinates intodynamical (original) ones. Thus, the inverses M − n and L − n are appliedto dynamically meaningful points. As d ( c ,n , f n ( c ,n )) → ∞ , we can as-sume after passing to a subsequence that lim n M − n ( c ,n ) = p ∈ S . Sim-ilarly, as d ( , c ,n ) → ∞ , we can assume after passing to a subsequencethat L − n ( ) and L − n ( f n ( c ,n )) both converge in S . Since ϕ f ∈ B ± k with k ≥
2, there are at least two preimages q, q (cid:48) ∈ ϕ − f ( p ). Possibly after re-naming q and q (cid:48) , we can assume that lim n L − n ( ) (cid:54) = q . We claim thatlim n L − n ( f n ( c ,n )) = lim n L − n ( ), thus, lim n L − n ( f n ( c ,n )) (cid:54) = q as well. In-deed, otherwise lim n L − n ( f n ( c ,n )) (cid:54) = lim n L − n ( ). By standard hyperbolicgeometry, this means that d ( , f n ( c ,n )) − d ( , c ,n ) = d ( L − n ( ) , L − n ( f n ( c ,n ))) − d ( L − n ( ) , L − n ( c ,n ))= d ( L − n ( ) , L − n ( f n ( c ,n ))) − d ( L − n ( ) , → + ∞ . In particular d ( , c ,n ) < d ( , f n ( c ,n )) for sufficiently large n , which is acontradiction to the Schwarz lemma.There are now two cases to consider. If q is not a hole, then M − n ◦ f n ◦ L n converges uniformly to ϕ f on a disk neighborhood U of q . We may alsoassume that ϕ f : U −→ ϕ f ( U ) is univalent. Since lim n M − n ( c ,n ) = p = ϕ f ( q ) and c ,n = L n (0) by definition, in the dynamical coordinate, we have L n ( U ) ⊆ M n ( ϕ f ( U ))for sufficiently large n . Since lim n d ( c ,n , f n ( c ,n )) = ∞ , the modulus of theannulus M n ( ϕ f ( U )) \ L n ( U ) → ∞ . This means that there exists a repellingfixed point of f n in L n ( U ), and since M n ( ϕ f ( U )) \ L n ( U ) is an ‘approximatefundamental annulus’ of this repelling fixed point, the multipliers of these Figure 5.1.
Asymptotically, c ,n is in the direction p ∈ S when viewed from f n ( c ,n ). The point q ∈ S which is notpointing towards is mapped to p by the rescaling limit ϕ f .The bottom two figures give a schematic picture in M n and L n coordinates. ϕ f ( U ) converges to the complement of a inthe L n coordinates, thus L n ( U ) ⊆ M n ( ϕ f ( U )).fixed points tend to ∞ . But this is a contradiction to the fact that { f n } ⊆ B ± d ( K ).If q is a hole, then by Corollary 5.5, there exists a sequence, say { L − n ( c ,n ) } converging to q (see Figure 5.1). Since lim n L − n ( ) ∈ S \ { q } , we have d ( , c ,n ) − d ( , c ,n ) → ∞ . Again, as lim n L − n ( f n ( c ,n )) ∈ S \ { q } , we have d ( c ,n , f n ( c ,n )) − d ( c ,n , c ,n ) → ∞ . IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 49
Then by Schwarz-Pick lemma, we have d ( c ,n , f n ( c ,n )) − d ( f n ( c ,n ) , f n ( c ,n )) → ∞ , so d ( c ,n , f n ( c ,n )) → ∞ . Thus, c ,n is an active critical point of f n , whichcontradicts our assumption that C ,n is a furthest active critical cluster.We now proceed to prove the converse part. By way of contradiction,suppose that there exists M >
0, a sequence { f n } ⊆ B ± d with d ( c, f n ( c )) Let f ∈ B ± d ( K ) and K := Cvx Hull C ( f ) ∪ F ( f ) where C ( f ) is the set of critical points of f in the unit disk D , and F ( f ) is the set ofrepelling fixed points of f on S . There exists a constant M = M ( d, K ) suchthat d ( x, f ( x )) < M for all x ∈ K .Proof. Let x ∈ K , then x lies in some (ideal) triangle with vertices in C ( f ) ∪ F ( f ). Since hyperbolic triangles are thin, x is within a uniformly boundeddistance from the geodesic γ := [ p , p ] where p , p ∈ C ( f ) ∪ F ( f ). If p i ∈ F ( f ), since f has modulus of the multipliers bounded by K , for all points q i ∈ γ sufficiently close to p i in Euclidean metric, we have d ( q i , f ( q i )) < M for some constant M depending only on K . If p i ∈ C ( f ), we let q i = p i . Let γ (cid:48) = ( q , q ) be any truncated geodesic segment with q i as above.We now conclude by Proposition 5.2 that q i is moved at most distance M = M ( d, K ) by f . Thus if y ∈ ( q , q ), by Schwarz-Pick lemma, wehave d ( f ( y ) , q i ) ≤ d ( y, q i ) + max { M , M } . We normalize so that y = 0, and( q , q ) ⊆ R . Let H ± be the horocycles based at ± − , 1) at ∓ a where a > d (0 , a ) = max { M , M } , and let D ± be the associated horo-disks. Then by standard hyperbolic geometry, we concludethat f ( y ) ∈ D + ∩ D − . Therefore, d ( y, f ( y )) ≤ M for some constant M depending only on d and K . Since d ( x, γ ) is uniformly bounded, once againby Schwarz-Pick lemma, we have that d ( x, f ( x )) < M for all x ∈ K , where M = M ( d, K ). (cid:3) Quasi-invariant tree. Corollary 5.6 and general properties of hyperbolicpolygons can be used to construct an embedded tree T approximating K which is quasi-invariant by f (see [31] and [24]). Such an embedding is farfrom being canonical. To fix the construction, we now give a particularembedding that will be used later. Although similar techniques are usedfor both Blaschke products and anti-Blaschke products, the quasi-invarianttrees that can arise are quite different. Thus, to be more specific, from nowon, we will work exclusively for degenerating sequences of anti-Blaschkeproducts.Let { f n } be a sequence in B − d ( K ), with markings on the critical points in D , and repelling fixed points p ,n , ...., p d +1 ,n . After passing to a subsequence,we assume we have critical clusters C ,n , ..., C m,n with degrees d , ..., d m . Let b i,n be the hyperbolic barycenter of the critical points in C i,n . Let B n = { b i,n : i = 1 , ..., m } , F n = { p i,n : i = 1 , ..., d + 1 } and V n = B n ∪ F n . Since we have a marking on the critical clusters and the repelling fixedpoints, the first indices of b i,n and p i,n are always labeled according to thismarking. We will now connect vertices V n using geodesics in an appropriateway to get a sequence of trees T n ⊆ K n .Let b ,n ∈ B n . We normalize by L n ∈ Isom( D ) so that L n (0) = b ,n .Then after possibly passing to a subsequence, { L − n ◦ f n ◦ L n } convergesalgebraically to a degree d Blaschke product ϕ f , where d is the degreeof the critical cluster C ,n associated to b ,n . Let Q = { q , ..., q d +1 } bethe repelling fixed points of ϕ f , then L − n ( b i,n ) ( i (cid:54) = 1) and L − n ( p i,n ) allconverge to Q as the convex hull K n is quasi-invariant by Corollary 5.6. If q i is not a hole, then there exists a unique repelling fixed point, say p j,n so that L − n ( p j,n ) → q i , and we connect b ,n to p j,n by the geodesic ray[ b ,n , p j,n ]. If q i is a hole, then by a modification of Corollary 5.5, thereexist barycenters b j ,n , ..., b j l ,n with L − n ( b j k ,n ) → q i , for k = 1 , · · · , l . Afterpassing to a subsequence, we can assume that d ( b j ,n , b ,n ) = min { d ( b j k ,n , b ,n ) : k = 1 , · · · l } for all n . We claim that for each k ∈ { , · · · , l } , the angles ∠ b ,n b j ,n b j k ,n areuniformly bounded away from 0, in particular d ( b j k ,n , b ,n ) − d ( b j ,n , b ,n ) n −→∞ for each k ∈ { , · · · , l } . Proof of the claim. Suppose by way of contradiction that we can find, say b j ,n so that ∠ b ,n b j ,n b j ,n → 0. Consider the triangle T n := b ,n b j ,n b j ,n where each side has length going to infinity; note that ∠ b j ,n b ,n b j ,n → IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 51 { L − n ( b j k ,n ) } converges to q i for all k ∈ { , · · · , l } . Moreover, since d ( b j ,n , b ,n ) ≤ d ( b j ,n , b ,n ) for all n , we have that ∠ b ,n b j ,n b j ,n → 0. Sinceall three angles of T n go to 0, it follows that the triangles T n degenerate into a‘tripod’. More precisely, there exists a n ∈ T n so that the angles ∠ b ,n a n b j ,n , ∠ b j ,n a n b j ,n and ∠ b j ,n a n b ,n are all uniformly bounded from below. Thepoints a n can also be chosen so that the distance to any critical point goesto ∞ . Choose M n ∈ Isom( D ) so that M n (0) = a n ; since a n is in the convexhull K n , d ( a n , f n ( a n )) is uniformly bounded by Corollary 5.6. So, possiblyafter passing to a subsequence, { M − n ◦ f n ◦ M n } converges to a degree 1 anti-Blaschke product ϕ g (the degree is 1 as all the critical points ‘escape’ for therescaling limit). We also note that M − n ( b ,n ), M − n ( b j ,n ) and M − n ( b j ,n )all converge to 3 distinct points on S as there is a uniform lower bound onthe angles. These three limit points must all be fixed points of ϕ g by quasi-invariance. But this is not possible as any degree 1 anti-Blaschke producthas exactly 2 fixed points on the circle S . (cid:3) Back to the construction, we connect b ,n to b j ,n by the geodesic segment[ b ,n , b j ,n ]. Inductively, we perform such construction for all barycenters,and we get a graph with vertices in V and geodesic edges. Using the claim,it is easy to verify that if b i,n is connected to b j,n when we do the above localconstruction at b i,n , then b j,n is connected to b i,n for the local constructionat b j,n . Since at each vertex, the angles between two edges are uniformlybounded below, and the lengths of the edges go to infinity, we conclude thatthe graph we get is a tree T n . Since T n ⊆ K n , the tree is quasi-invariant byCorollary 5.6.We summarize some of the properties of T n from our construction. Proposition 5.7. Let { f n } ⊆ B − d ( K ) with markings on the critical points.After passing to a subsequence, we can associate quasi-invariant critical trees T n with vertex set V n = B n ∪ F n so that • F is the set of ends for T n ; • The vertex b i,n has valence d i + 1 ; • Each edge is a geodesic segment with length → ∞ as n → ∞ , or ageodesic ray; • There exists M = M ( d, K ) so that for any point x ∈ T n , we have d ( x, f n ( x )) < M . Degenerations in B − d ( K ) via pinching -cycles. We now introduce aparticular type of degeneration in B − d ( K ). The degeneration will be called pinching at a -cycle , and will be used in § B − d .The d +1 repelling fixed points divide the circle S into d +1 intervals. Theseintervals will be called dynamical intervals . By the dynamics of f ∈ B − d , itis easy to see that the 2-cycles are in one-to-one correspondence with the(unordered) pairs of non-adjacent intervals. Therefore, allowing differentmarking for the repelling fixed points on S , 2-cycles can be identified with unordered pairs ( d , d ) with d + d = d + 1, d i ≥ 2, such that the twocomplementary components of a 2-cycle in S contain d and d fixed pointsrespectively.By a classical result of Heins (see [12] and [54]), given any set of d − D , there exists a Blaschke product, unique up to post-compositionwith elements of Isom( D ), with critical points (in D ) precisely at the given d − z , we have the sameresult for anti-Blaschke products as well.Thus, let r n = 1 − n and we consider a sequence of anti-Blaschke products f n with a critical point at − r n of order d − r n oforder d − 1. We assume that( d − 1) + ( d − 1) = d − , so f n has degree d . Post-composing f n with suitable members of Isom( D ),we get two sequences { [ f ± n ] } ⊆ (cid:2) B − d (cid:3) normalized so that f + n ( r n ) = r n and f + n ( − r n ) < f − n ( − r n ) = − r n and f − n ( r n ) > d ( − r n , r n ) − d ( f ± n ( − r n ) , f ± n ( r n )) ≤ M for some universal constant M = M ( d ). Therefore, d ( f + n ( − r n ) , − r n ) ≤ M and d ( f − n ( r n ) , r n ) ≤ M. Thus, we get two degenerating sequences { [ f ± n ] } ⊆ [ B − d ( K )] for some K by Proposition 5.2. The two sequences { f ± n } are rather asymmetric: f + n prefers the critical point at r n while f − n prefers the critical point at − r n .Ultimately, the pinching degeneration will be an interpolation between thesetwo maps. To get to the construction, we first study some properties of thetwo sequences { f ± n } .Let L ,n , L ,n ∈ Isom( D ) be two sequences fixing ± L ,n (0) = − r n and L ,n (0) = r n . More precisely, we have L ,n ( z ) = z − r n − r n z , and L ,n ( z ) = z + r n r n z . Then one can easily verify from our construction that L − ,n ◦ f + n ◦ L ,n → ϕ +1 ( z ) = z d + a a z d with a hole at the fixed point 1 where a > 0, and L − ,n ◦ f − n ◦ L ,n → ϕ − ( z ) = z d with a hole at 1.Similarly, L − ,n ◦ f + n ◦ L ,n → ϕ +2 ( z ) = ( − d − z d with a hole at the fixed point − L − ,n ◦ f − n ◦ L ,n → ϕ − ( z ) = ( − d − z d − a − a z d IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 53 with a hole at − a > T ± n can be described quite explicitly forthese two sequences. T ± n has exactly two interior vertices − r n and r n . Thevertex − r n is connected to d repelling fixed points which converge to thefixed points of ϕ ± other than 1, and r n is connected to d repelling fixedpoints which converge to the fixed points of ϕ ± other than − P that separates the d fixed pointsconnected to − r n from the d fixed points connected to r n . Let { q ± ,n , q ± ,n } denote this 2-cycle where q ± ,n has positive imaginary part. We show that Lemma 5.8. Let u ± i,n be the orthogonal projection of q ± i,n onto [ − r n , r n ] .Then, after passing to a subsequence, lim n →∞ d ( u − i,n , r n ) < ∞ , and lim n →∞ d ( u + i,n , − r n ) < ∞ . Proof. Indeed, the situations are symmetric, and we shall only prove theclaim for u − ,n . To simply the notation, we now denote u n := u − ,n . Let M n ∈ Isom( D ) fixing the real line and M n (0) = u n . Then by Proposition5.7, d ( u n , f − n ( u n )) ≤ M .We first claim that lim n →∞ d ( u n , − r n ) = ∞ . Otherwise, after passing toa subsequence, we have L − ,n ( q − ,n ) converges to q (cid:54) = 1, thus q is not a hole.Since all the fixed points connecting to − r n lie on the same side of q ,n and q ,n , we conclude that q and ϕ − ( q ) lie in two adjacent dynamical intervals of ϕ − ( z ) = z d touching at 1, which is a contradiction, as there are no 2-cycleof ϕ − there.Now suppose for contradiction that lim n →∞ d ( u n , r n ) = ∞ , then { M − n ◦ f − n ◦ M n } converges algebraically to a degree 1 map ϕ with holes at ± ϕ fixes ± 1, and ϕ fixes i = M − n ( q − ,n ). This forces ϕ ( z ) = z , thus d ( u n , f − n ( u n )) = d ( M n (0) , f − n ( M n (0)) → d (0 , ϕ (0)) = 0. But ϕ − ( t ) = t d onthe real axis, so f − n moves some points on ( − r n , u n ) at least log d towards − r n . But this is impossible by Schwarz-Pick Lemma as the hyperbolic dis-tance between such a point on ( − r n , u n ) and u n cannot increase under f − n .Thus, the lemma follows. (cid:3) An anti-Blaschke product is called hyperbolic if there is an attracting fixedpoint on S . Although we will not be using it in this paper, the followinginteresting phenomenon is worth mentioning. Corollary 5.9. The algebraic limits ϕ +1 and ϕ − are hyperbolic anti-Blaschkeproducts.Proof. By Lemma 5.8, lim n d ( u − i,n , r n ) < ∞ . So L − ,n ( q − ,n ) converges to q (cid:54) = − 1, thus q is not a hole. Therefore q and ϕ − ( p ) form a two cycle for ϕ − 24 R. LODGE, Y. LUO, AND S. MUKHERJEE in two adjacent dynamical intervals touching at − 1. By counting the totalnumber of fixed points and 2-cycles, we see that ϕ − neither has a fixed pointin D nor has a parabolic fixed point. Thus ϕ − must be a hyperbolic anti-Blaschke product. Similarly, ϕ +1 is a hyperbolic anti-Blaschke product. (cid:3) We proceed with the construction of the pinching degeneration alongthe 2-cycle P . Note that since both f + n and f − n were defined by post-composing f n with suitable M¨obius maps, there exists A n ∈ Isom( D ) satis-fying f + n = A n ◦ f − n . The map A n is unique once we require A n ( ± 1) = ± A n ( − r n ) = f + n ( − r n ). Since { d ( − r n , f + n ( − r n )) } is a bounded sequence, { A n } lies in a compact family. For t ∈ [0 , s t,n linearly parame-terizes the interval [ − r n , f + n ( − r n )]. Let A t,n be a family of isometries of D fixing the real axis and satisfying A t,n ( − r n ) = s t,n . Note that A ,n = id and A ,n = A n . Consider the family f t,n := A t,n ◦ f − n . Note that there exists K so that [ f t,n ] ∈ [ B − d ( K )] for all t and n as the critical points are moveduniformly bounded distances away by f t,n . For sufficiently large n , the pro-jection u t ,n of the associated 2-periodic point q t ,n onto [ − r n , r n ] moves con-tinuously with respect to t . Thus, by Lemma 5.8 and the intermediate valuetheorem, we can find t ( n ) ∈ (0 , 1) such that for g n := f t ( n ) ,n = A t ( n ) ,n ◦ f − n ,the associate projection u t ( n )1 ,n is the origin.The sequence of maps { g n } , which can be regarded as suitable averagesof the maps f + n and f − n , is referred to as the pinching degeneration along the -cycle P . To summarize, we have the following property for g n that will beused later. Lemma 5.10. Let { [ g n ] } ⊆ [ B − d ( K )] be a pinching degeneration along the -cycle P , and γ n be the geodesic connecting the -cycle P . Then sup x ∈ γ n d ( x, g n ( x )) → . Proof. Abusing notations, we denote the 2-cycle P of g n by { q ,n , g n ( q ,n ) } .By construction, q ,n = i . Note that { g n } converges algebraically to a degree1 map ϕ which fixes ± 1. Since ϕ ( i ) = i , we have ϕ ( z ) = z . This meansthat q ,n → − i .Note that the geodesics γ n converge to the interval [ i, − i ]. Since ± i arenot holes of ϕ , it follows that the multiplier g (cid:48) n ( q ,n ) converges to ϕ (cid:48) ( i ) = 1.Hence, if x ∈ B (cid:15) ( q ,n ) ∩ γ n , then d ( x, g n ( x )) → g n converges locally uniformly to z on (cid:98) C \ {± } , yield the result. (cid:3) Pared Deformation Space If R is a hyperbolic rational map with connected Julia set, then it is well-known that the marked hyperbolic components can be modeled by the spaceof Blaschke products for the critical and post-critical Fatou components (see § IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 55 In our setting, let R be a critically fixed anti-rational map of degree d ,and H be the hyperbolic component containing R in M − d = Rat − d / PSL ( C ).We will say that a subset of A ⊆ H is bounded if the closure A in M − d iscompact.We first note that the hyperbolic component H for a critically fixed anti-rational map R associated to the graph Γ is never bounded. Indeed, let γ be a simple loop in the Tischler graph. If γ intersects a Fatou component U ,then γ ∩ U is the union of two internal rays connecting the super attractingfixed point c ∈ U to two repelling fixed point q , q ∈ ∂U (see Figure 6.1).We can perturb the map slightly in all such Fatou components so that c becomes attracting, and we still have invariant rays connecting c and q , q (see [25, Proposition 3.1] for the holomorphic setting). Then a standardpinching degeneration along the union of all these rays gives a divergingsequence in H (see [48] for the holomorphic setting). Figure 6.1. The contact graph Γ is drawn in black. TheTischler graph contains a loop with bold blue edges. Therestriction of this loop to each Fatou component is eitherempty or a union of two internal rays.Therefore, the naive translation of Thurston’s compactness theorem isfalse here. As discussed in the introduction, this is not very surprisingas these repelling fixed points correspond to cusps in the group setting,and we do not change the local geometry of the cusps when we deformthe corresponding kissing reflection groups. This motivates the followingdefinition of pared deformation spaces.To start, we list the critical Fatou components of R ∈ H as U , ..., U m where R | U i has degree d i , i = 1 , · · · , m . We say that q is a boundary marking for R , if it assigns to each critical Fatou component U i a fixed point q ( U i ) onthe ideal boundary S = I ( U ). In other words, q is an m -tuple ( x , ..., x m ) where x i is a fixed point on I ( U i ). We will consider the space of pairs (cid:101) H = { ( R, q ) : R ∈ H, and q is a boundary marking } . Let (cid:101) X be a component of (cid:101) H . Then a similar argument as in the proof of[33, Theorem 5.7] gives that (cid:101) X is homeomorphic to B − d × B − d × .... × B − d m .Let K > 0, we define the marked pared deformation space as (cid:101) X ( K ) = B − d ( K ) × B − d ( K ) × .... × B − d m ( K ) , and the pared deformation space X ( K ) ⊆ M − d as the projection of (cid:101) X ( K ).In the remainder of this section, we shall prove Theorem 1.6. Enriched Tischler graph. Let { R n } ⊂ (cid:101) X ( K ) = B − d ( K ) × .... × B − d m ( K ).Then we get m sequences of Blaschke products { f ,n } , · · · , { f m,n } . We in-troduce a marking on the critical points, and after passing to a subsequence,we assume that T ,n , ..., T m,n are the quasi-invariant critical trees for thesesequences of Blaschke products. By the uniformization map, we may regard T i,n as a subset of the corresponding critical Fatou component U i,n of R n .We let T Enn = (cid:83) mi =1 T i,n . Since for each fixed i and all n , the trees T i,n areplanar isomorphic, we denote by T En the planar isomorphism class of T Enn ,and call it the enriched Tischler graph corresponding to the sequence { R n } .The enriched Tischler graph T Enn can be constructed from the Tischlergraph T of the critically fixed anti-rational map R by replacing each vertexby the corresponding quasi-invariant critical tree where the identificationcomes from the boundary marking. Similar to the convention we adoptedfor Tischler graphs, we define the vertices of T En , denoted by V ( T En ), asthose corresponding to the interior vertices of T i,n .The enriched Tischler graph T En is closely related to T . We first notethe faces of T En are in one-to-one correspondence with the faces of T . Theedges of T En can be divided into two categories: we say that an edge of T En is a crossing edge if it connects (the barycenters of) a pair of critical clustersin two different Fatou components; and a non-crossing edge otherwise. Notethat the crossing edges of T En are in one-to-one correspondence with theedges of T . The non-crossing edges of T En correspond to vertices of T .Two faces A, B of T En share a common crossing edge on their boundariesif and only if the corresponding faces in T share the corresponding edgeon their boundaries. On the other hand, if two faces A, B of T En sharea common non-crossing edge on their boundaries, then the correspondingfaces in T share the corresponding vertex on their boundaries.We shall now fix an embedding of T En in (cid:98) C . A simple closed curve γ in (cid:98) C \ V ( T En ) is said to be essential if it separates vertices of T En . Anessential closed curve γ in (cid:98) C \ V ( T En ) is said to cut an edge if the two end-points of the edge lie in two different components of (cid:98) C \ γ . The followinglemma is a key point in the proof of Theorem 1.6, and the only place wherethe acylindrical condition is used. IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 57 Lemma 6.1. Let R be an acylindrical critically fixed anti-rational map,and T En be the enriched Tischler graph associated to a sequence in thecorresponding marked pared deformation space. Then, any essential simpleclosed curve γ ⊂ (cid:98) C \ V ( T En ) cuts at least edges.Proof. Let γ be an essential simple closed curve. Since it is essential, it cutsat least two edge. By way of contradiction, we suppose that γ cuts exactlytwo edges. This means that there exist two faces A, B of T En with exactlytwo common edges on their boundaries. These two edges cannot be adjacentas each vertex has valence at least 3 and each face is a Jordan domain. Thesetwo edges cannot be non-crossing in the same Fatou component. Indeed,since each vertex of the tree T i,n has valence at least 3, no two componentsof D \ T i,n share two common edges on their boundaries. Therefore, if A (cid:48) , B (cid:48) are the corresponding faces of T , then (cid:98) C \ A (cid:48) ∪ B (cid:48) is disconnected. Butthis means that the dual graph of T is not 3-connected, so R is cylindrical,which is a contradiction. (cid:3) We shall now prove one direction of Theorem 1.6. We will be frequentlyusing the following estimate of the hyperbolic metric ρ U on a topologicaldisk U (i.e., a proper, simply connected subset of C ). The proof followsimmediately by the Schwarz lemma and Koebe distortion theorem.12 d ( z, ∂U ) ≤ ρ U ( z ) ≤ d ( z, ∂U ) (6.1) Proposition 6.2. If R is acylindrical, then for any K > , the pared de-formation space X ( K ) is bounded in M − d .Proof. By way of contradiction, suppose that { R n } ⊂ X ( K ) is a degenerat-ing sequence, i.e., { R n } escapes every compact subset of M − d . Introducingboundary markings, we may assume that { R n } ⊂ (cid:101) X ( K ). Let { f i,n } be theassociated sequence in B − d i ( K ), and U i,n be the critical Fatou componentsof R n . Note that R n is conjugate to f i,n on U i,n . After passing to a subse-quence, we get the quasi-invariant critical trees T i,n and the enriched Tischlergraphs T Enn . Let v n be a vertex of T Enn . We may assume that v n is the hy-perbolic barycenter of some critical cluster of degree e in U ,n . Conjugatingby a sequence in PSL ( C ), we may also assume that v n = 0, D ⊆ U ,n and1 / ∈ U ,n . After passing to a subsequence, the pointed disks ( U ,n , 0) con-verge to some pointed disk ( U, 0) in the Carath´eodory topology, and R n | U ,n converges to some anti-holomorphic map R ∞ : U → U as d U ,n (0 , R n (0)) isuniformly bounded by Proposition 5.2. Since there are exactly e − U ,n ,we conclude that the degree of R ∞ : U → U is e . Therefore, after pass-ing to a subsequence, the anti-rational maps R n converge algebraically tosome f ∈ Rat − d such that ϕ f = R ∞ on U . So in particular, we have thatdeg( ϕ f ) ≥ e .We claim that ϕ f has degree d . Proof of the claim. Suppose that this is not true. Then, f has a hole atsome points a ; i.e., a ∈ H ( f ). Let B ( a, (cid:15) ) be the (cid:15) neighborhood of a , and C (cid:15) := ∂B ( a, (cid:15) ). By shrinking (cid:15) , we may assume that B ( a, (cid:15) ) \ { a } containsneither holes nor fixed points of ϕ f . Since a is a hole and ϕ f has degree ≥ e , the arguments of Corollary 5.5 show that there is a sequence of criticalpoints of R n converging to a . Hence, the circle C (cid:15) is an essential closed curvefor T Enn for large n . Thus, by Lemma 6.1, C (cid:15) cuts at least 3 edges of T Enn .After passing to a subsequence, we may assume that in these coordinates, T Enn converges in Hausdorff topology to T En ∞ .Let x n ∈ C (cid:15) ∩ T Enn be a point of intersection of the curve C (cid:15) with acutting edge. After passing to a subsequence, we may assume that x n → x and x n ∈ U k,n for some 1 ≤ k ≤ m . By Lemma 5.3, we have that R n ( x n )converges to ϕ f ( x ). Since B ( a, (cid:15) ) \ { a } contains no fixed point of ϕ f , we con-clude that the spherical distances d S ( x n , R n ( x n )) stay uniformly boundedaway from 0. On the other hand, we know that d U k,n ( x n , R n ( x n )) ≤ M byProposition 5.7. By Inequality 6.1, we conclude that the spherical distance d S ( x n , ∂U k,n ) is uniformly bounded away from 0. Thus, after possibly pass-ing to a subsequence, ( U k,n , x n ) converges in Carath´eodory topology to somepointed disk ( V, x ).Consider T V := T En ∞ ∩ V . Then if y ∈ T V , there exists a sequence y n → y with y n ∈ T Enn . Since d U k,n ( y n , R n ( y n )) ≤ M , we conclude that d V ( y, ϕ f ( y )) ≤ M . This means that any limit point x of T V on ∂V is fixedby ϕ f . Since C (cid:15) cuts the edge containing x n , and there are no fixed pointsof ϕ f in B ( a, (cid:15) ) \ { a } , we conclude that a is a limit point of T V on ∂V , and ϕ f ( a ) = a .If a is a critical point of ϕ f , then near a , the map ϕ f behaves like z k forsome k ≥ 2. Since a ∈ ∂V , using Inequality 6.1 and the fact (cid:90) tt k y dy = log 1 t k − → ∞ as t → 0, we conclude that there exists x ∈ T V with d V ( x, ϕ f ( x )) > M ,which is a contradiction.Thus, a is not a critical point of ϕ f . Since C (cid:15) cuts at least 3 edges of T Enn , we have three distinct edges e , e , and e of T V , T V , and T V with a ∈ e ∩ e ∩ e . Note that the union of any two edges divides a neighborhood U a of a into two components. Since ϕ f behaves like reflection near a , therecan be at most two invariant directions to a , and hence, there exists an edge,say e , which is mapped to the component of U a \ e ∪ e not containing e .This means that there exists x ∈ e with d V ( x, ϕ ( x )) > M , which is acontradiction. The claim now follows. (cid:3) The fact that deg ϕ f = d contradicts the supposition that { R n } escapesevery compact subset of M − d . Thus, the proposition follows. (cid:3) IRCLE PACKINGS, REFLECTION GROUPS AND ANTI-RATIONAL MAPS 59 Pinching parallel -cycles. We shall now prove the other direction ofTheorem 1.6. The proof is morally similar to unboundedness of deformationspaces of cylindrical 3-manifolds (associated with kissing reflection groups).In both worlds, the proof boils down to availability of parallel simple closedcurves or 2-cycles (cf. Subsection 4.3), which can be pinched to obtainescaping sequences in the parameter spaces. Proposition 6.3. If R is cylindrical, then there exists K > such that X ( K ) is unbounded in M − d Proof. Since R is cylindrical, the dual graph Γ of the Tischler graph T isnot 3-connected. By Proposition 3.11 and Proposition 4.10, we have twocases. Case 1: There exist two critical Fatou components U, V touching at acommon 2-cycle P of R . The 2-cycle P of R gives rise to a pair of 2-cycles P U and P V on S for the anti-Blaschke products associated with R| U and R| V . Let γ U and γ V be the geodesics in D connecting the two points in P U and P V , respectively. Abusing notations, we will denote the imagesof these geodesics (under the uniformizations) in U, V by γ U , γ V as well.Then, γ U ∪ γ V is a closed curve in the dynamical plane of R . Introducingan appropriate boundary marking, we consider a sequence { R n } in X ( K )(for some K > 0) where we pinch along P U and P V while keeping thedynamics in the other critical Fatou components as z d k . Suppose that afterconjugating and passing to a subsequence, { R n } converges to R ∞ ∈ Rat − d .After passing to a subsequence, we may assume that γ U n and γ V n convergein the Hausdorff topology to Y U and Y V , where Y U and Y V are continua. ByLemma 5.10, sup x ∈ γ Un d U n ( x, R n ( x )) → . Using equation 6.1 and the fact { R n } converges to R ∞ uniformly on (cid:98) C , weconclude that d S ( x, R ∞ ( x )) = 0 for all x ∈ Y U (cf. [48, Theorem A] ). Thus,the spherical diameter of Y U is 0 . Similarly, the spherical diameter of Y V is 0. But this is a contradiction as γ U n ∪ γ V n cuts (cid:98) C into two dynamicallynon-trivial regions. Case 2: There exist a critical Fatou component U , and a 2-cycle P U ⊆ I ( U ) for the anti-Blaschke product associated with R| U such that P U isidentified in the dynamical plane of R . As in the previous case, consider thegeodesic in D connecting the two points in P U , and denote its image in U by γ U . 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Department of Mathematics and Computer Science, Indiana State Univer-sity, Terre Haute, IN 47809, USA E-mail address : [email protected] Institute for Mathematical Sciences, Stony Brook University, 100 NicollsRd, Stony Brook, NY 11794-3660, USA E-mail address : [email protected] School of Mathematics, Tata Institute of Fundamental Research, 1 HomiBhabha Road, Mumbai 400005, India E-mail address ::