LLinear slices close to a Maskit slice
Kentaro ItoOctober 18, 2018
Abstract
We consider linear slices of the space of Kleinian once-puncturedtorus groups; a linear slice is obtained by fixing the value of the traceof one of the generators. The linear slice for trace 2 is called the Maskitslice. We will show that if traces converge ‘horocyclically’ to 2 thenassociated linear slices converge to the Maskit slice, whereas if the tracesconverge ‘tangentially’ to 2 the linear slices converge to a proper subsetof the Maskit slice. This result will be also rephrased in terms of complexFenchel-Nielsen coordinates. In addition, we will show that there is alinear slice which is not locally connected.
One of the central issues in the theory of Kleinian groups is to understand thestructures of deformation spaces of Kleinian groups. In this paper we considerKleinian punctured torus groups, one of the simplest classes of Kleinian groupswith a non-trivial deformation theory.Let S be a once-punctured torus and let R ( S ) be the space of conjugacyclasses of representations ρ : π ( S ) → PSL(2 , C ) which takes a loop surround-ing the cusp to a parabolic element. The space AH ( S ) of Kleinian puncturedtorus groups is the subset of R ( S ) of faithful representations with discreteimages. Although the interior of AH ( S ) is parameterized by a product ofTeichm¨uller spaces of S , its boundary is quite complicated. For example, Mc-Mullen [Mc2] showed that AH ( S ) self-bumps, and Bromberg [Br] showed that AH ( S ) is not even locally connected. We refer the reader to [Ca] for moreinformation on the topology of deformation spaces of general Kleinian groups.In this paper we investigate the shape of AH ( S ) form the point of viewof the trace coordinates. Let us fix a pair a, b of generators of π ( S ). Then every1 a r X i v : . [ m a t h . G T ] M a r epresentation ρ in R ( S ) is essentially determined by the data (tr ρ ( a ) , tr ρ ( b )) =( α, β ) ∈ C . Thus we identify R ( S ) with C in this introduction (see Section2 for more accurate treatment). We want to understand when ( α, β ) ∈ C corresponds to a point of AH ( S ). More precisely, we consider in this paperthe shape the linear slice L ( β ) := { α ∈ C : ( α, β ) ∈ AH ( S ) } of AH ( S ) when β close to 2. Note that L (2) is known as the Maskit slice, cor-responding to the set of representations ρ ∈ AH ( S ) such that ρ ( b ) is parabolic.It is natural to ask the following question: “When β tends to 2, does L ( β ) con-verge to L (2)?” Parker and Parkkonen [PP] studied this question in the casethat a real number β > β ∈ C \ [ − ,
2] tends to 2, and obtain the completeanswer to this question. In fact, the answer depends on the manner how β tends to 2.To describe our results, we need to introduce the notion of complex length.Let ρ ∈ R ( S ) and assume that β = tr ρ ( b ) is close to 2. Then the complexlength λ of ρ ( b ) is determined by the relation β = 2 cosh( λ/
2) and the normal-ization Re λ > , Im λ ∈ ( − π, π ]. We denote this λ by λ ( β ). Note that β → λ ( β ) →
0. We say that a sequence β n ∈ C \ [ − ,
2] converges horocyclically to 2 if for any disk in the right-half plane C + touching at zero, λ ( β n ) are eventually contained in this disk. On the other hand, we say thatthe sequence β n converges tangentially to 2 if there is a disk in C + touchingat zero which does not contain any λ ( β n ). Now we can state our main result.(See Theorems 6.6 and 6.8 for more precise statements. See also Figure 3.) Theorem 1.1.
Suppose that a sequence β n ∈ C \ [ − , converges to . If β n → horocyclically, then L ( β n ) converge to L (2) in the sense of Hausdorff. On theother hand, if β n → tangentially, then L ( β n ) converge (up to subsequence)to a proper subset of L (2) in the sense of Hausdorff. We now sketch the essential idea which is underlying this phenomenon.Especially we explain the reason why the limit of linear slices is a propersubset of L (2) in the case where β n → β n → α n ∈ L ( β n )converges to α ∈ C . We will explain that α should lie in a proper subsetof L (2). Let us take a sequence ρ n ∈ AH ( S ) such that (tr ρ n ( a ) , tr ρ n ( b )) =2 α n , β n ). Since ( α n , β n ) → ( α,
2) as n → ∞ , and since AH ( S ) is closed, wehave ( α, ∈ AH ( S ), and hence α ∈ L (2). By taking conjugations, we mayassume that ρ n ( a ) → A α and ρ n ( b ) → B in PSL(2 , C ), where A α = (cid:18) α − i − i (cid:19) and B = (cid:18) (cid:19) . In addition, by pass to a subsequence if necessary, we may also assume thatthe sequence ρ n ( π ( S )) converges geometrically to a Kleinian group Γ, whichcontains the algebraic limit (cid:104) A α , B (cid:105) . From the assumption that β n → (cid:104) ρ n ( b ) (cid:105) converge geometrically torank-2 abelian group (cid:104) B, C (cid:105) , where C is of the form C = (cid:18) ζ (cid:19) for some ζ ∈ C \ R , see Theorem 6.5. Therefore the geometric limit Γ containsthe group (cid:104) A α , B, C (cid:105) . For any given integer k , one see from C k A α = A α − kiζ that the group (cid:104) A α − kiζ , B (cid:105) is a subgroup of the Kleinian group Γ. Hence thegroup (cid:104) A α − kiζ , B (cid:105) is discrete and thus α − kiζ ∈ L (2). Therefore α should becontained in the intersection (cid:92) k ∈ Z ( kiζ + L (2)) , which is a proper subset of L (2).In the proof of Theorem 1.1, we will make an essential use of Bromberg’stheory in [Br]. In fact, Bromberg obtained in [Br] a coordinate system forrepresentations in AH ( S ) close to the Maskit slice. The poof of Theorem 1.1 isthen obtained by comparing Bromberg’s coordinates and the trace coordinates.Some other topics and computer graphics of linear slices can be fond in[Mc2], [MSW] and [KY], as well as [PP].This paper is organized as follows; In section 2, we recall some basic factabout spaces of representations and their subspaces. In section 3, we in-troduce the trace coordinates for the space R ( S ) of representations of theonce-punctured torus group. In section 4, we recall Bromberg’s theory in [Br]which gives us a local model of the space AH ( S ) of Kleinian once-puncturedtorus groups near the Maskit slice. In section 5, we consider relation betweenBromberg’s coordinates and the trace coordinates, and obtain an estimatewhich will be used in the proofs of the main results. We will show our main3esults, Theorems 6.6 and 6.8, in section 6. We also show that there is a linearslice which is not locally connected. In section 7, we translate our main resultsin terms of the complex Fenchel-Nielsen coordinates.The following is the mainstream of this paper, where the top (resp. bottom)line is corresponding to the tangential (resp. horocyclic) convergence:Theorem 5.1 (cid:43) (cid:51) (cid:36) (cid:44) Proposition 5.2 (cid:43) (cid:51)
Lemma 6.7 (cid:43) (cid:51)
Theorem 6.6Proposition 5.3 (cid:43) (cid:51)
Lemma 6.9 (cid:43) (cid:51)
Theorem 6.8
Acknowledgements.
The author would like to thank Hideki Miyachi for hismany helpful discussions. He is also grateful to Keita Sakugawa for developinga computer program drawing linear slices, which was very helpful to proceedthis research. All computer-generated figures of linear slices of this paper aremade by this program.
In this section, we recall the definitions of spaces we will work with.Let (
M, P ) be a paired manifold; that is, M is a compact, hyperbolizable3-manifold with boundary and P is a disjoint union of tori and annuli in ∂M .Especially, every torus component of ∂M is contained in P . Let R ( M, P ) := Hom irr P ( π ( M ) , PSL(2 , C ))denote the set of all type-preserving, irreducible representations of π ( M ) intoPSL(2 , C ). Here a representation ρ : π ( M ) → PSL(2 , C ) is said to be type-preserving if ρ ( γ ) is parabolic or identity for every γ ∈ π ( P ). The space ofrepresentations R ( M, P ) := R ( M, P ) / PSL(2 , C )is the set of all PSL(2 , C )-conjugacy classes [ ρ ] of representations ρ in R ( M, P ).We endow this space R ( M, P ) with the algebraic topology; that is, a sequence[ ρ n ] converges to [ ρ ] if there are representatives ρ n in [ ρ n ] and ρ in [ ρ ] suchthat for every g ∈ π ( M ) the sequence ρ n ( g ) converges to ρ ( g ) in PSL(2 , C ).The conjugacy class [ ρ ] of a representation ρ is also denoted by ρ if there is noconfusion. We are interested in the topological nature of the space AH ( M, P ) := { ρ ∈ R ( M, P ) : ρ is faithful, discrete } . N, P ) , ( N, P (cid:48) ) and ( ˆ
N , ˆ P ) (from left to right).It is known by Jørgensen [Jø] that AH ( M, P ) is closed in R ( M, P ). Let
M P ( M, P ) denote the subset of AH ( N, P ) consists of representations ρ whichare minimally parabolic (i.e., ρ ( g ) is parabolic if and only if g ∈ π ( P )) andgeometrically finite. It is known by Marden [Mar] and Sullivan [Su] that M P ( N, P ) is equal to the interior of AH ( N, P ) as a subset of R ( N, P ). Re-cently, it was shown by Brock, Canary and Minsky [BCM] that the closure of
M P ( M, P ) is equal to AH ( M, P ).In this paper, we only consider the following three paired manifolds(
N, P ) , ( N, P (cid:48) ) , ( ˆ N , ˆ P )which are constructed as follows (see Figure 1): Let S be a torus with oneopen disk removed. Throughout of this paper, we fix a pair a, b of generatorsof π ( S ) such that the geometric intersection number equals one. Then thecommutator [ a, b ] = aba − b − is homotopic to ∂S . Now we set N := S × [0 , P := ∂S × [0 , . We next set P (cid:48) := P ∪ A , where A ⊂ S × { } is an annulus whose core curveis freely homotopic to b ∈ π ( S ). Finally, we let( ˆ N , ˆ P ) := ( N \ W, P ∪ T ) , where W is a regular tubular neighborhood of b × { / } in N = S × [0 ,
1] and T := ∂W .Note that AH ( N, P (cid:48) ) lies in the boundary of AH ( N, P ); in fact ρ ∈ AH ( N, P ) lies in AH ( N, P (cid:48) ) if and only if ρ ( b ) is parabolic. This space5 H ( N, P (cid:48) ) is called the
Maskit slice of AH ( N, P ). It is known by Minsky[Mi] that AH ( N, P (cid:48) ) has exactly two connected components. Bromberg’s the-ory in [Br] gives us an information about the topology of AH ( N, P ) near AH ( N, P (cid:48) ). The aim of this paper is to understand the topology of AH ( N, P )near AH ( N, P (cid:48) ) from the view point of the trace coordinates, which is ex-plained in the next section. AH ( N, P ) In this section, we introduce a trace coordinate system on a subset of R ( N, P )containing R ( N, P (cid:48) ).Recall that (
N, P ) = ( S × [0 , , ∂S × [0 , S is a torus with oneopen disk removed. In this case, the space R ( N, P ) consists of all PSL(2 , C )-conjugacy classes of representations ρ : π ( S ) = (cid:104) a, b (cid:105) → PSL(2 , C )which satisfy the condition tr( ρ ([ a, b ])) = −
2. Note that the trace of thecommutator [ a, b ] is well defined, although the traces of ρ ( a ) and ρ ( b ) aredetermined up to sign.As we will see below, for any given ( α, β ) ∈ C , there is a representation ρ ∈ R ( N, P ) which satisfies tr ρ ( a ) = α , tr ρ ( b ) = β , and this ρ is determineduniquely up to pre-composition of automorphism ( a, b ) (cid:55)→ ( a, b − ) of π ( N ).Therefore the subset D tr := { ( α, β ) ∈ C : ∃ ρ ∈ AH ( N, P ) s.t. tr ρ ( a ) = α , tr ρ ( b ) = β } of C is well-defined. Note that the set D tr is symmetric under the action( α, β ) (cid:55)→ ( β, α ). For a given β ∈ C , the slice L ( β ) := { α ∈ C : ( α, β ) ∈ D tr } . of D tr is called the linear slice for β . Note that L ( β ) is symmetric under theaction of z (cid:55)→ − z . The aim of this paper is to understand the shape of L ( β )when β is close to 2.To study the shape of linear slices, it would be convenient if we couldidentify R ( N, P ) with C simply by ρ (cid:55)→ (tr ρ ( a ) , tr ρ ( b )). But the thing is notso simple. One reason is that traces of ρ ( a ) , ρ ( b ) are determined up to sign,and the other reason is that, for a given ( α, β ) ∈ C , there exist two candidate6f representations ρ which satisfy (tr ρ ( a ) , tr ρ ( b )) = ( α , β ). Therefore, inthis section, we will choose an appropriate open domain Ω ⊂ R ( N, P ) so thatthere exists an embedding Tr : Ω → C such that Tr( ρ ) = ( α, β ) satisfies(tr ρ ( a ) , tr ρ ( b )) = ( α , β ) for every ρ ∈ Ω.We begin by identifying R ( N, P (cid:48) ) with C . For a given α ∈ C , let ρ α be therepresentation in R ( N, P (cid:48) ) defined by ρ α ( a ) := (cid:18) α − i − i (cid:19) , ρ α ( b ) := (cid:18) (cid:19) . Then we have the following lemma. (See Lemma 4.3 in [Br]. Note that we areassuming that every element of R ( N, P (cid:48) ) is irreducible.)
Lemma 3.1.
The map ψ : C → R ( N, P (cid:48) ) defined by α (cid:55)→ ρ α is a homeomor-phism. Note that the map ψ in Lemma 3.1 induces a homeomorphism from L (2)onto the Maskit slice AH ( N, P (cid:48) ).In the next lemma, we will show that the homeomorphism ψ − : R ( N, P (cid:48) ) → C naturally extends to an embedding from an open domain Ω ⊂ R ( N, P ) con-taining R ( N, P (cid:48) ) into C . Lemma 3.2.
There exist an open, connected, simply connected domain Ω ⊂ R ( N, P ) and a homeomorphism Tr : Ω → C which satisfy the following:1. Ω contains R ( N, P (cid:48) ) , and Tr takes R ( N, P (cid:48) ) onto C × { } . In addition,we have Tr( ρ α ) = ( α, for every α ∈ C .2. For every ρ ∈ Ω , Tr( ρ ) = ( α, β ) satisfies tr ρ ( a ) = α and tr ρ ( b ) = β . Throughout of this paper, we fix such a domain Ω. We call Tr the tracecoordinate map and ( α, β ) = Tr( ρ ) the trace coordinates of ρ ∈ Ω. The rest ofthis section is devoted to the proof of this lemma. The commutative diagram(3.2) should be helpful for understanding the arguments. The reader may skipthis proof by admitting Lemma 3.2.To show Lemma 3.2, it is convenient to consider the space (cid:101) R ( N, P ) ofrepresentations of π ( N ) into SL(2 , C ), instead of PSL(2 , C ). More precisely,7he set (cid:101) R ( N, P ) consists of SL(2 , C )-conjugacy classes of representations ˜ ρ of π ( S ) into SL(2 , C ) which satisfy the condition tr( ˜ ρ ([ a, b ])) = −
2. TheSL(2 , C )-conjugacy class of ˜ ρ is also denoted by ˜ ρ if there is no confusion. It iswell known that an element ˜ ρ of (cid:101) R ( N, P ) is uniquely determined by the triple(tr ˜ ρ ( a ) , tr ˜ ρ ( b ) , tr ˜ ρ ( ab )) of complex number (see for example [Bo] or [Go]): Lemma 3.3.
The map (cid:102)
Tr : (cid:101) R ( N, P ) → Ξ := { ( α, β, γ ) ∈ C : α + β + γ = αβγ } \ { (0 , , } defined by ˜ ρ (cid:55)→ (tr ˜ ρ ( a ) , tr ˜ ρ ( b ) , tr ˜ ρ ( ab )) is a homeomorphism. By using this lemma, we often identify (cid:101) R ( N, P ) with the subset Ξ of C .For ( α, β ) ∈ C , the numbers γ satisfying α + β + γ = αβγ are given by γ = 12 (cid:16) αβ ± (cid:112) α β − α + β ) (cid:17) . Therefore the projection Π : Ξ → C \ { (0 , } defined by ( α, β, γ ) (cid:55)→ ( α, β ) is a two-to-one branched covering map. If wedenote by γ , γ the solutions of the equation α + β + γ = αβγ on γ , wehave γ + γ = αβ . On the other hand, we havetr( AB ) + tr( AB − ) = tr A tr B for every A, B ∈ SL(2 , C ). Therefore one see that if two representations ˜ ρ , ˜ ρ in (cid:101) R ( N, P ) have the same image under the map Π ◦ (cid:102) Tr, they are only differingby pre-composition of the automorphism ( a, b ) (cid:55)→ ( a, b − ) of π ( S ).Now let π : (cid:101) R ( N, P ) → R ( N, P )be the natural projection, which is a four-to-one covering map. The groupof covering transformation for π is isomorphic to Z × Z which is generatedby ( α, β, γ ) (cid:55)→ ( − α, β, − γ ) and ( α, β, γ ) (cid:55)→ ( α, − β, − γ ), where (cid:101) R ( N, P ) isidentified with Ξ ⊂ C as in Lemma 3.3.Now let us take an open, connected and simply connected domain ∆ ⊂ C \ { (0 , } which satisfy the following:1. ∆ contains the set C × { } , and 8. ∆ lies in the set { ( α, β ) ∈ C : Re β > , α β (cid:54) = 4( α + β ) } .Here, the condition α β (cid:54) = 4( α + β ) is equivalent to the condition that thepair ( α, β ) is not a critical value of the projection Π : Ξ → C \ { (0 , } .Throughout of this paper, we fix such a domain ∆.Since α β (cid:54) = 4( α + β ) for every ( α, β ) ∈ ∆, and since ∆ is connectedand simply connected, one can take a univalent branch of the square root of α β − α + β ) on ∆. We take the branch such that the value for ( α, ∈ ∆is equal to − i . Then we obtain the univalent branch of γ = γ ( α, β ) = 12 (cid:16) αβ + (cid:112) α β − α + β ) (cid:17) (3.1)on ∆, and hence the univalent branch θ : ∆ → Ξ of Π − on ∆. Lemma 3.4.
The map π ◦ (cid:102) Tr − ◦ θ : ∆ → R ( N, P ) is a homeomorphism ontoits image.Proof. We only need to show that the orbit of θ (∆) under the action of Z × Z on Ξ are mutually disjoint. Take two points ( α, β ) , ( α (cid:48) , β (cid:48) ) ∈ ∆. Supposefor contradiction that ( α, β, γ ( α, β )) , ( α (cid:48) , β (cid:48) , γ ( α (cid:48) , β (cid:48) )) ∈ Ξ are equivalent un-der the action of non-trivial element of the covering transformation group Z × Z . Since Re β > β (cid:48) >
0, one see that ( α (cid:48) , β (cid:48) , γ ( α (cid:48) , β (cid:48) )) =( − α, β, − γ ( α, β )). Then from (3.1) we have γ ( α (cid:48) , β (cid:48) ) = 12 (cid:16) α (cid:48) β (cid:48) + (cid:112) α (cid:48) β (cid:48) − α (cid:48) + β (cid:48) ) (cid:17) = 12 (cid:16) − αβ + (cid:112) α β − α + β ) (cid:17) . But this with γ ( α (cid:48) , β (cid:48) ) = − γ ( α, β ) implies (cid:112) α β − α + β ) = 0, whichcontradicts to ( α, β ) ∈ ∆.Now let Ω := π ◦ (cid:102) Tr − ◦ θ (∆)and Tr := (cid:16) π ◦ (cid:102) Tr − ◦ θ (cid:17) − : Ω → ∆ . Then we obtain the following commutative diagram: (cid:101) R ( N, P ) π (cid:15) (cid:15) (cid:102) Tr (cid:47) (cid:47) Ξ R ( N, P ) ⊃ Ω Tr (cid:47) (cid:47) ∆ . θ (cid:79) (cid:79) (3.2)9o show that this Ω and Tr satisfy the desired property in Lemma 3.2, we onlyneed to show that Tr( ρ α ) = ( α,
2) for every α ∈ C . This can be seen from thefollowing two facts: (i) If we regard ρ α = ψ ( α ) as an element of (cid:101) R ( N, P ), wehave (cid:102)
Tr( ρ α ) = ( α, , α − i ). (ii) From our choice of the branch θ , we have θ ( α,
2) = ( α, , α − i ). Thus we complete the proof of Lemma 3.2. AH ( N, P ) This section is devoted to explain the theory of Bromberg in [Br], which tells usthe topology of AH ( N, P ) near the Maskit slice AH ( N, P (cid:48) ). In fact, Brombergconstruct a subset of C × ˆ C such that AH ( N, P ) is locally homeomorphic tothis set at every point in
M P ( N, P (cid:48) ). Given µ ∈ C , we define a representation σ µ ∈ R ( N, P (cid:48) ) by σ µ ( a ) := (cid:18) − iµ − i − i (cid:19) , σ µ ( b ) := (cid:18) (cid:19) . This representation σ µ is nothing but the representation ρ α with α = − iµ ,which is defined in the previous section. The subset M := { µ ∈ C : σ µ ∈ AH ( N, P (cid:48) ) } of C is also called the Maskit slice . Since that the map C → R ( N, P (cid:48) ) definedby µ (cid:55)→ σ µ is a homeomorphism from Lemma 3.1, M is homeomorphic to AH ( N, P (cid:48) ), and the interior int( M ) of M is homeomorphic to M P ( N, P (cid:48) ).Since µ ∈ M if and only if − iµ ∈ L (2), we have L (2) = i M = { iµ : µ ∈ M} . Note that M is invariant under the translation µ (cid:55)→ µ + 2. We refer the readerto [KS] for basic properties of M . It is known by Minsky (Theorem B in [Mi])that M has two connected components M + , M − , where M + contained in theupper half-plane and M − is the complex conjugation of M + .2 Coordinates for AH ( ˆ N , ˆ P ) We now introduce a coordinate system on the space AH ( ˆ N , ˆ P ). Recall that ˆ N is N minus a regular tubular neighborhood W of b × { / } , and ˆ P is a unionof P and T = ∂W . Bromberg’s idea in [Br] is that the space AH ( ˆ N , ˆ P ) canbe used as a local model of AH ( N, P ) near a point of AH ( N, P (cid:48) ).The fundamental group of ˆ N is expressed as π ( ˆ N ) = (cid:104) a, b, c : [ b, c ] = id (cid:105) , where a, b is the pair of generators of the fundamental group of S × { } ⊂ ˆ N ,and c is freely homotopic to an essential simple closed curve on T that boundsa disk in W . We regard π ( T ) = (cid:104) b, c (cid:105) . The space R ( ˆ N , ˆ P ) of representationsfor ( ˆ N , ˆ P ) is expressed as R ( ˆ N , ˆ P ) = { ρ : π ( ˆ N ) → PSL(2 , C ) : tr ρ ([ a, b ]) = − , tr ρ ( c ) = 4 } / PSL(2 , C ) . For a given ( µ, ζ ) ∈ C , we define a representation ˆ σ µ,ζ ∈ R ( ˆ N , ˆ P ) byˆ σ µ,ζ ( a ) := σ µ ( a ) , ˆ σ µ,ζ ( b ) := σ µ ( b ) , ˆ σ µ,ζ ( c ) := (cid:18) ζ (cid:19) . Then we have the following:
Lemma 4.1 (Lemma 4.5 in [Br]) . The map C → R ( ˆ N , ˆ P ) defined by ( µ, ζ ) (cid:55)→ ˆ σ µ,ζ is a homeomorphism.Remark. Following the rule of notation in [Br], the representation ˆ σ µ,ζ shouldbe written as σ µ,ζ . But we reserve the notation σ µ,ζ for another representation,which will be defined in the next subsection.We define a subset B of C by B := { ( µ, ζ ) ∈ C : ˆ σ µ,ζ ∈ AH ( ˆ N , ˆ P ) } . Then, by the above lemma, the map
B → AH ( ˆ N , ˆ P )defined by ( µ, ζ ) (cid:55)→ ˆ σ µ,ζ is a homeomorphism. Note that ( µ, ζ ) ∈ B implies µ ∈ M since the restriction of ˆ σ µ,ζ to the subgroup (cid:104) a, b (cid:105) of π ( ˆ N ) is equal to σ µ . Note also that if Im ζ = 0 then ( µ, ζ ) (cid:54)∈ B ; in fact, if Im ζ = 0, it violatesdiscreteness or faithfulness of the representation ˆ σ µ,ζ .11or any ( µ, ζ ) ∈ B , the quotient manifold ˆ M = H / ˆ σ µ,ζ ( π ( ˆ N )) is homeo-morphic to the interior of ˆ N , and has a rank-2 cusp whose monodromy groupis the rank-2 parabolic subgroup of PSL(2 , C ) generated by ˆ σ µ,ζ ( b ) and ˆ σ µ,ζ ( c ).Since ˆ σ µ,ζ ( c − k a ) = (cid:18) − i ( µ − kζ ) − i − i (cid:19) and ˆ σ µ,ζ ( b ) = (cid:18) (cid:19) , one can see that if ( µ, ζ ) ∈ B then µ − kζ ∈ M for every k ∈ Z . Brombergshowed that the converse is also true if Im ζ (cid:54) = 0 (see Proposition 4.7 in [Br]): Theorem 4.2 (Bromberg) . Let ( µ, ζ ) ∈ C with Im ζ (cid:54) = 0 . Then ( µ, ζ ) ∈ B ifand only if µ − kζ ∈ M for every integer k . AH ( N, P ) Following [Br], we now introduce a coordinate system on AH ( N, P ) by usingthe coordinate system on AH ( ˆ N , ˆ P ) introduced in the previous subsection.Now let B + := { ( µ, ζ ) ∈ B : Im ζ > } and define a set A ⊂ C × ˆ C by A := B + ∪ ( M × {∞} ) . The following theorem due to Bromberg claim that the set A can be used fora local model of AH ( N, P ) at every point of
M P ( N, P (cid:48) ) ⊂ AH ( N, P ). Theorem 4.3 (Bromberg (Theorem 4.13 in [Br])) . For any ν ∈ int( M ) , thereexist a neighborhood U of ( ν, ∞ ) in A , a neighborhood V of σ ν in AH ( N, P ) ,and a homeomorphism Φ : U → V. Remark.
Although Bromberg restricted to the case that ν ∈ int( M + ) in [Br],it is obvious that the same argument works well for ν ∈ int( M − ).In this situation, we say that ( µ, ζ ) ∈ U is Bromberg’s coordinates of therepresentation Φ( µ, ζ ) ∈ V . In what follows, we also write σ µ,ζ := Φ( µ, ζ ) . We now briefly explain the definition of the map Φ : U → V, ( µ, ζ ) (cid:55)→ σ µ,ζ to what extent we need in the following argument. See [Br] for the full details.12iven ν ∈ int( M ), a neighborhood U of ( ν, ∞ ) in A is chosen sufficiently smallso that the following argument works well. Let ( µ, ζ ) ∈ U . If ζ = ∞ then σ µ, ∞ is defined to be σ µ . If ζ (cid:54) = ∞ , the quotient manifoldˆ M µ,ζ = H / ˆ σ µ,ζ ( π ( ˆ N ))has a rank-2 cusp whose monodromy group is generated by ˆ σ µ,ζ ( b ) and ˆ σ µ,ζ ( c ).Since we are choosing U sufficiently small, it follows from the filling theoremdue to Hodgson, Kerckhoff and Bromberg (see Theorem 2.5 in [Br]) that thereexists a c -filling M µ,ζ of ˆ M µ,ζ for every ( µ, ζ ) ∈ U with ζ (cid:54) = ∞ . More precisely,there is a complete hyperbolic manifold M µ,ζ homeomorphic to the interior of N and an embedding φ µ,ζ : ˆ M µ,ζ → M µ,ζ which satisfy the following properties:1. the image of φ µ,ζ is equals to M µ,ζ minus the geodesic representative of( φ µ,ζ ) ∗ (ˆ σ µ,ζ ( b )),2. ( φ µ,ζ ) ∗ (ˆ σ µ,ζ ( c )) is trivial in π ( M µ,ζ ), and3. φ µ,ζ extends to a conformal map between the conformal boundaries ofˆ M µ,ζ and M µ,ζ .The map φ µ,ζ is called the c -filling map . We will define σ µ,ζ to be an elementin AH ( N, P ) associated to M µ,ζ . To this end, we need to determine a marking N → M µ,ζ . Since the restriction of the representation ˆ σ µ,ζ to the subgroup (cid:104) a, b (cid:105) ⊂ π ( ˆ N ) is equal to σ µ , the manifold M µ = H /σ µ ( π ( N )) covers ˆ M µ,ζ .The covering map is denoted byΠ µ,ζ : M µ → ˆ M µ,ζ . Let f µ : N → M µ be a homotopy equivalence which induces σ µ . Then σ µ,ζ is defined to be a representation of π ( N ) into PSL(2 , C ) induced form φ µ,ζ ◦ Π µ,ζ ◦ f µ ; N f µ −−−−→ marking M µ Π µ,ζ −−−−→ covering ˆ M µ,ζ φ µ,ζ −−−→ filling M µ,ζ . This σ µ,ζ is faithful, and hence, is contained in AH ( N, P ) (see Lemma 3.6 in[Br]). Note from the construction of σ µ,ζ that the geodesic in M µ,ζ associatedto σ µ,ζ ( a ) is homotopic to the image of the geodesic ˆ M µ,ζ associated to ˆ σ µ,ζ ( a )by φ µ,ζ . 13 Relation between the trace coordinates andBromberg’s coordinates
Let us consider the situation in Theorem 4.3. We may assume that V iscontained in the domain Ω of the trace coordinate map. In this section, wewill study the relation between Bromberg’s coordinates ( µ, ζ ) ∈ U of σ µ,ζ ∈ V and its trace coordinates ( α, β ) = Tr( σ µ,ζ ). More precisely, we will observein Theorem 5.1 that ( µ, ζ ) is approximated by ( iα, πi/λ ( β )), where λ ( β ) =2 cosh − ( β/
2) is the complex length of σ µ,ζ ( b ). For any element g ∈ PSL(2 , C ), its complex length l ( g ) ∈ C is a value whichsatisfies tr g = 4 cosh (cid:18) l ( g )2 (cid:19) . If g is not parabolic, this is equivalent to say that g is conjugate to the M¨obiustransformation z (cid:55)→ e l ( g ) z . For a loxodromic element g ∈ PSL(2 , C ), its com-plex length l ( g ) determined uniquely if we take it in the setΛ := { z ∈ C : Re z > , − π < Im z ≤ π } . In what follows, we always assume that l ( g ) ∈ Λ for loxodromic transformation g . We now want to fix one-to-one correspondence between the complex length l ( g ) of loxodromic element g ∈ SL(2 , C ) and its trace tr g . Note that the map z (cid:55)→ z/
2) takes the interior of Λ into the right-half plane C + := { z ∈ C : Re z > } . We define a map λ : C + \ (0 , → Λas its inverse. Then we have λ ( z ) = 2 cosh − (cid:16) z (cid:17) = 2( z − / + o ( z −
2) ( z → , where the real part of a square root is chosen positive. We have λ (tr g ) = l ( g )for every loxodromic element g ∈ SL(2 , C ) with tr g ∈ C + \ (0 , .2 Main estimates The following theorem tells us a relation between Bromberg’s coordinates andthe trace coordinates for representations close to the Maskit slice.
Theorem 5.1.
Let ν ∈ int( M ) . For any (cid:15) > , we can choose a neighborhoods U of ( ν, ∞ ) and V of σ ν in Theorem 4.3 so that they also satisfy the following: V is contained in the domain Ω of the trace coordinate map, and for any ( µ, ζ ) ∈ U with ζ (cid:54) = ∞ , we have1. | µ − iα | ≤ (cid:15) , and2. | ζ − πi/λ ( β ) | ≤ (cid:15) Im ζ ,where ( α, β ) = Tr( σ µ,ζ ) is the trace coordinates of σ µ,ζ Remark.
These estimates 1 and 2 follow from the fact that we can choose the c -filling map φ µ,ζ : ˆ M µ,ζ → M µ,ζ close to the isometry outside a neighborhoodof the rank-2 cusp. Then the estimates 1 and 2 are obtained from estimatesdue to McMullen (Lemma 3.20 in [Mc1]) and Magid (Theorem 1.2 in [Mag]),respectively. Proof of Theorem 5.1.
Let us take a neighborhood U of ( ν, ∞ ) in A , a neigh-borhood V of σ ν in AH ( N, P ) and a homeomorphism Φ : U → V as in thestatement of Theorem 4.3. We may assume that V ⊂ Ω. We will show belowthat estimates 1 and 2 are obtained if we modify U sufficiently small.For ( µ, ζ ) ∈ U with ζ (cid:54) = ∞ , let φ µ,ζ : ˆ M µ,ζ → M µ,ζ be the c -filling map. To control the distortion of the map φ µ,ζ , we need torecall the notion of normalized length.Suppose that δ > T δ ( T ) denote the component of δ -thin part of ˆ M µ,ζ associ-ated to the rank-2 cusp. We endow the boundary ∂ T δ ( T ) of T δ ( T ) with thenatural Euclidean metric. The marking map ˆ N → ˆ M µ,ζ induces a markingmap T → ∂ T δ ( T ). Via this marking, the pair of generators b, c of π ( T ) arealso regarded as the pair of generators of π ( ∂ T δ ( T )). In this setting, the normalized length L ( c ) of the free homotopy class of c ⊂ ∂ T δ ( T ) is defined by L ( c ) := length( c (cid:48) ) (cid:112) Area( ∂ T δ ( T )) , c (cid:48) ) is the Euclidean length of the geodesic representative c (cid:48) of c in ∂ T δ ( T ). This L ( c ) does not depend on the choice of δ . Sinceˆ σ µ,ζ ( b ) = (cid:18) (cid:19) and ˆ σ µ,ζ ( c ) = (cid:18) ζ (cid:19) , the normalized length L ( c ) can be calculated concretely as L ( c ) = | ζ |√ ζ . For any given
K >
0, we can choose the neighborhood U sufficiently smallso that for all ( µ, ζ ) ∈ U with ζ (cid:54) = ∞ , the normalized length L ( c ) of c at therank-2 cusp of ˆ M µ,ζ is greater than K . We will show below that if we takesuch K sufficiently large, the estimates 1 and 2 hold.We may assume that there is a uniform upper bound of | µ | for ( µ, ζ ) ∈ U .Then, since tr ˆ σ µ,ζ ( a ) = − µ , there is an upper bound R > a ∗ of ˆ σ µ,ζ ( a ) in ˆ M µ,ζ for all ( µ, ζ ) ∈ U with ζ (cid:54) = ∞ . Therefore we can take δ > N ( a ∗ ,
1) of a ∗ in ˆ M µ,ζ does not intersect the δ -thin part T δ ( T ) ⊂ ˆ M µ,ζ for all ( µ, ζ ) ∈ U with ζ (cid:54) = ∞ .It follows from the filling theorem due to Hodgson, Kerckhoff and Bromberg(see Theorem 2.5 in [Br]) that for δ > (cid:15) > K > c in ˆ M µ,ζ is greaterthan K then the c -filling map can be chosen so that it restricts to a (1 + (cid:15) )-bi-Lipschitz diffeomorphism φ µ,ζ : ˆ M µ,ζ \ T δ ( T ) → M µ,ζ \ T δ ( b ∗ ) , where T δ ( b ∗ ) ⊂ M µ,ζ is the δ -Margulis tube of the geodesic representative b ∗ of σ µ,ζ ( b ); i.e., the core curve of the filled torus in M µ,ζ . We can now apply atheorem of McMullen (Lemma 3.20 in [Mc1]) to obtain | tr (ˆ σ µ,ζ ( a )) − tr ( σ µ,ζ ( a )) | < C ( R ) (cid:15) , where C ( R ) > R . (Recall that R isthe upper bounds of the hyperbolic length of a ∗ .) Since tr ˆ σ µ,ζ ( a ) = − µ ,tr σ µ,ζ ( a ) = α , and since α is close to − iµ , we obtain | µ − iα | < (cid:15) (cid:15) > K large enough. Thus we obtain the first estimate.We next show the second estimate. One can expect to obtain this kindof estimate since the Teichm¨uller parameter of the torus ∂ T δ ( T ) ⊂ ˆ M µ,ζ withrespect to the generators b, c is equal to ζ/ ∂ T δ ( b ∗ ) ⊂ M µ,ζ is equalto 2 πi/λ ( β ), and since there is a bi-Lipschitz map of small distortion betweenthese tori. Magid accomplish this estimate in [Mag]. In fact, by simplifyinghis estimates (ii) and (iv) of Theorem 1.2 in [Mag], we see that there is someconstant C > L ( c ) = | ζ | / √ ζ of c issufficiently large, we have (cid:12)(cid:12)(cid:12)(cid:12) λ ( β ) − πiζ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (Im ζ ) | ζ | = 4 CL ( c ) . (5.1)One can also see from L ( c ) = | ζ | / √ ζ that Re(4 πi/ζ ) = 2 π/L ( c ) . Com-bining this with (5.1), we have | λ ( β ) | > (cid:12)(cid:12)(cid:12)(cid:12) πiζ (cid:12)(cid:12)(cid:12)(cid:12) = 2 π | ζ | (5.2)for L ( c ) large enough. Finally, multiplying | ζ/λ ( β ) | on both sides of (5.1) andusing the estimate (5.2), we obtain (cid:12)(cid:12)(cid:12)(cid:12) ζ − πiλ ( β ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:48) (Im ζ ) | ζ | = C (cid:48) L ( c ) Im ζ < (cid:15) Im ζ for a given (cid:15) > L ( c ) is large enough. Thus we obtain the second estimate. A and D Since the shape of A is well understood from Theorem 4.2, we can expectto understand the shape of D tr from that of A . To apply Theorem 5.1, it isconvenient to consider the image of the set D tr by the transformation ( α, β ) (cid:55)→ ( iα, πi/λ ( β )). More precisely, we define a map F : C × ( C + \ (0 , → C × ˆ C by F ( z, w ) := (cid:18) iz, πiλ ( w ) (cid:19) and set D +tr := { ( α, β ) ∈ D tr : β ∈ C + } D := F ( D +tr ) . Note that, since we are interested in the shape of D tr where the second entryis close to 2, we may restrict our attention to D +tr . Note also that F ( z,
2) =( iz, ∞ ) for every z .Let us now consider the situation of Theorem 5.1. We define a homeomor-phism ϕ from U onto its domain by ϕ = F ◦ Tr ◦ Φ; ϕ : U Φ −−−→ V Tr −−−→ D tr F −−−→ D . Then by definition we have ϕ ( µ, ∞ ) = ( µ, ∞ ) for any ( µ, ∞ ) ∈ U . It followsfrom Theorem 5.1 the point ϕ ( µ, ζ ) is close to ( µ, ζ ) ∈ U even if ζ (cid:54) = ∞ . There-fore, we expect that the shape of A is similar to that of D in a neighborhoodof ( ν, ∞ ) for every ν ∈ int( M ).We will justify this expectation in Propositions 5.2 and 5.3 below. In whatfollows, we denote by B (cid:15) ( z ) the (cid:15) -neighborhood of z in C , and by B (cid:15) ( z, w ) the (cid:15) -neighborhood of ( z, w ) in C . Proposition 5.2.
For any ν ∈ int( M ) , there exists (cid:15) > which satisfy thefollowing: For any < (cid:15) < (cid:15) and I > , there exists K > such that for all z ∈ C with | z | > K and < Im z < I , B (cid:15) ( ν, z ) ⊂ A implies B (cid:15)/ ( ν, z ) ⊂ D . Proof.
For any fixed ν ∈ int( M ), let us take neighborhoods U ⊂ A , W ⊂ D of ( ν, ∞ ) such that ϕ = F ◦ Tr ◦ Φ is a homeomorphism from U onto W . Wemay assume that U is of the form U = A ∩ { ( µ, ζ ) ∈ C : | µ − ν | < (cid:15) , | ζ | > K/ } for some (cid:15) > K >
0. Let us take 0 < (cid:15) < (cid:15) and I > K large enough, wemay also assume that d C ( ϕ ( µ, ζ ) , ( µ, ζ )) < (cid:15) µ, ζ ) ∈ U with 0 < Im ζ < I . (Note that if | ζ | → ∞ then | ζ | / √ ζ → ∞ .)Now let us take z ∈ C with | z | > K and 0 < Im z < I , and suppose that B (cid:15) ( ν, z ) ⊂ A . Then B (cid:15) ( ν, z ) ⊂ U and 0 < Im ζ < I for every ( µ, ζ ) ∈ B (cid:15) ( ν, z )(since K and I are larger than (cid:15) ). Thus the inequality (5.3) holds for every( µ, ζ ) ∈ B (cid:15) ( ν, z ). Using this fact, we will show that B (cid:15)/ ( ν, z ) ⊂ ϕ ( B (cid:15) ( ν, z )) , B (cid:15)/ ( ν, z ) ⊂ D .Suppose for contradiction that there exists some p ∈ B (cid:15)/ ( ν, z ) \ ϕ ( B (cid:15) ( ν, z )).Let consider a line segment γ ( t ) := (1 − t ) ϕ ( p ) + tp, t ∈ [0 , C which joins ϕ ( p ) to p . Since d C ( ϕ ( p ) , p ) < (cid:15)/ p ∈ B (cid:15)/ ( ν, z ), wehave γ ([0 , ⊂ B (cid:15)/ ( ν, z ). Now let t ∞ := inf { t : γ ( t ) (cid:54)∈ ϕ ( B (cid:15) ( ν, z )) } . Since ϕ ( p ) lies in ϕ ( B (cid:15) ( ν, z )) but p does not, and since ϕ ( B (cid:15) ( ν, z )) is open,one see that 0 < t ∞ ≤
1. Let take an increasing sequence t n → t ∞ ( n → ∞ )and let q n := ϕ − ( γ ( t n )) ∈ B (cid:15) ( ν, z ). Since ϕ ( q n )(= γ ( t n )) lie in B (cid:15)/ ( ν, z )and d C ( ϕ ( q n ) , q n ) < (cid:15)/
8, we have q n ∈ B (cid:15)/ ( ν, z ) for all n . Therefore anaccumulation point q ∞ of { q n } lies in B (cid:15) ( ν, z ). It follows form the continuity of ϕ that ϕ ( q ∞ ) = γ ( t ∞ ). Since ϕ is local homeomorphism at q ∞ , this contradictsthe definition of t ∞ . Thus we obtain B (cid:15)/ ( ν, z ) ⊂ ϕ ( B (cid:15) ( ν, z )) ⊂ D .We set B (cid:15),I ( ν ) := B (cid:15) ( ν ) × { z ∈ C : Im z > I } . Proposition 5.3.
For any ν ∈ int( M ) , there exists (cid:15) > which satisfy thefollowing: For any < (cid:15) < (cid:15) there is I > such that B (cid:15),I ( ν ) ⊂ A implies B (cid:15)/ , I ( ν ) ⊂ D . Proof.
The proof is almost parallel to that of Proposition 5.2. For any fixed ν ∈ int( M ), let us consider the homeomorphism ϕ : U → W as in the proofof Proposition 5.2. One can see from Theorem 4.2 that there exist (cid:15) > I > B (cid:15) ,I ( ν ) ⊂ U . Now let us take 0 < (cid:15) < (cid:15) arbitrarily.Theorem 5.1 implies that if we choose I large enough, we have the following:for any ( µ, ζ ) ∈ B (cid:15),I ( ν ), ( µ (cid:48) , ζ (cid:48) ) := ϕ ( µ, ζ ) satisfies | µ (cid:48) − µ | < (cid:15)/ | ζ (cid:48) − ζ | < ( (cid:15)/
8) Im ζ . Using this fact, we can show that B (cid:15)/ , I ( ν ) ⊂ ϕ ( B (cid:15),I ( ν )) , which implies that B (cid:15)/ , I ( ν ) ⊂ D . The remaining argument is almost thesame to that of Proposition 5.2, so we leave it for the reader.19 Main Results
In this section, we will show our main results, Theorems 6.6 and 6.8. Moreprecisely, for a given sequence β n ∈ C \ [ − ,
2] converging to 2, we consider theHausdorff limit of the linear slices L ( β n ) and the Carath´eodory limit of theinteriors int( L ( β n )) of the linear slices. A We first consider horizontal slices of A , which will appear as limits of linearslices. Let M ( ζ ) denote the slice of A by fixing the second entry ζ ∈ C ∪ {∞} in the product structure; that is, M ( ζ ) := { µ ∈ C : ( µ, ζ ) ∈ A} . By definition of A , one see that (i) M ( ζ ) lies in M for every ζ , (ii) M ( ζ ) isempty if Im ζ ≤
0, and that (iii) M ( ∞ ) = M . It follows from Theorem 4.2that if Im ζ > M ( ζ ) can be written as M ( ζ ) = (cid:92) k ∈ Z ( kζ + M ) , (6.1)where kζ + M = { kζ + µ : µ ∈ M} . (Note that (6.1) does not hold if Im ζ ≤ M ( ζ ) is invariant under the action of (cid:104) z + 2 , z + ζ (cid:105) . It is known byWright [Wr] that the stripe { z ∈ C : − ≤ Im z ≤ } does not intersect M .Therefore one see that M ( ζ ) = ∅ if 0 < Im ζ ≤ A and linearslices, or horizontal slices of D tr . By definition, we have α ∈ L ( β ) ⇐⇒ ( α, β ) ∈ D tr ⇐⇒ (cid:18) iα, πiλ ( β ) (cid:19) ∈ D and (cid:18) iα, πiλ ( β ) (cid:19) ∈ A ⇐⇒ α ∈ i M (cid:18) πiλ ( β ) (cid:19) . Recall from Theorem 5.1 that ( µ, ζ ) ∈ A is almost equivalent to ( µ, ζ ) ∈ D if µ lies in int( M ) and | ζ | is large enough. Therefore we may expect that L ( β ) issimilar to i M (4 πi/λ ( β )) when β is close to 2. We will justify this observationbelow. To this end, we first recall the definitions of Hausdorff convergence andCarath´eodory convergence. 20 efinition 6.1 (Hausdorff convergence) . Let F n ( n ∈ N ) , F ∞ be closed sub-sets in C . We say that the sequence F n converges F ∞ in the sense of Hausdorffif the following two conditions are satisfied:1. For any x ∞ ∈ F ∞ , there is a sequence x n ∈ F n such that x n → x ∞ .2. If there is a sequence x n j ∈ F n j such that x n j → x ∞ , then x ∞ ∈ F ∞ . Definition 6.2 (Carath´eodory convergence) . Let Ω n ( n ∈ N ) , Ω ∞ be opensubsets in C . We say that the sequence Ω n converges to Ω ∞ in the sense ofCarath´eodory if the following two conditions are satisfied:1. For any compact subset X of Ω ∞ , X ⊂ Ω n for all large n .2. If there is an open subset O of C and an infinite sequence { n j } ∞ j =1 suchthat O ⊂ Ω n j , then O ⊂ Ω ∞ .Note that closed subsets F n ⊂ C converge to F ∞ ⊂ C in the sense ofHausdorff if and only of their complements C \ F n converge to C \ F ∞ in thesense of Carath´eodory.The next lemma implies that M ( ζ n ) converge to M if and only if Im ζ n →∞ , which is a direct consequence of (6.1): Lemma 6.3.
Suppose that a sequence { ζ n } ∞ n =1 in C with Im ζ n > convergesto ∞ in ˆ C . Then the followings are equivalent:1. Im ζ n → ∞ as n → ∞ .2. M ( ζ n ) converge to M in the sense of Hausdorff as n → ∞ .3. int( M ( ζ n )) converge to int( M ) in the sense of Carath´eodory as n → ∞ . To describe our main theorems, we also need the following definition (seeFigure 1):
Definition 6.4.
Suppose that a sequence { λ n } ∞ n =1 in the right-half plane C + = { z ∈ C | Re z > } converges to 0. We say that λ n → horocyclically if forany (cid:15) > , | λ n − (cid:15) | < (cid:15) for all large n , and that λ n → tangentially if there isa constant (cid:15) > such that | λ n − (cid:15) | > (cid:15) for all n . λ n → | Im(2 πi/λ n ) | → ∞ , and thattangentially if and only if | Im(2 πi/λ n ) | are uniformly bounded above.When a sequence β n ∈ C \ [ − ,
2] converges to 2, the limit of the sequence L ( β n ) depends on whether λ ( β n ) → G n of PSL(2 , C ) converges geometrically to a subgroup G of PSL(2 , C ) if G n converge to G in the senseof Hausdorff as closed subsets of PSL(2 , C ). Theorem 6.5.
Suppose that a sequence B n of loxodromic elements convergesto B = (cid:18) (cid:19) in PSL(2 , C ) . Let λ n denote the complex length of B n . Then we have thefollowing:1. If λ n → horocyclically, then the sequence (cid:104) B n (cid:105) converges geometricallyto (cid:104) B (cid:105) .2. Suppose that λ n → tangentially. We further assume that there existsa complex number ξ with Im ξ ≥ and a sequence m n of integers with | m n | → ∞ such that lim n →∞ (cid:18) πiλ n − m n (cid:19) = ξ. In this situation, we have lim n →∞ B − m n n = C := (cid:18) ξ (cid:19) . n addition, if Im ξ (cid:54) = 0 , the sequence (cid:104) B n (cid:105) converges geometrically tothe rank- parabolic group (cid:104) B, C (cid:105) .Remark.
When λ n → M > < Im(2 πi/λ n ) < M for every n . Therefore we may assume that, by pass toa subsequence if necessary, the sequence 2 πi/λ ( β n ) converges to some ξ ∈ C with Im ξ ≥ z (cid:55)→ z + 1. We can now state our main theorem for linear slices L ( β n ) such that λ ( β n )converge tangentially to 0. See Figure 3, left column. Theorem 6.6.
Let { β n } ∞ n =1 be a sequence in C \ [ − , which converges to as n → ∞ . Suppose that λ ( β n ) converge tangentially to . We further assumethat there exists a complex number ξ with Im ξ ≥ and a sequence m n ofintegers with | m n | → ∞ such that lim n →∞ (cid:18) πiλ ( β n ) − m n (cid:19) = ξ. Then we have the following:1. L ( β n ) converge to i M (2 ξ ) in the sense of Hausdorff as n → ∞ .2. int( L ( β n )) converge to int( i M (2 ξ )) in the sense of Carath´eodory as n →∞ . The following lemma is an essential part of the proof of Theorem 6.6.
Lemma 6.7.
Under the same assumption as in Theorem 6.6, we have thefollowing: For any α ∈ int( i M (2 ξ )) there exists (cid:15) > such that B (cid:15) ( α ) ⊂ int( L ( β n )) for all large n .Proof. Suppose that α ∈ int( i M (2 ξ )). Then ( iα, ξ ) ∈ int( A ). Let (cid:15) > ν = iα . Since ( iα, ξ ) ∈ int( A ), one canfind 0 < (cid:15) < (cid:15) such that B (cid:15) ( iα, ξ ) ⊂ A . Since the set A is invariant underthe action ( z, w ) (cid:55)→ ( z, w + 2), we have B (cid:15) ( iα, ξ + 2 m n ) ⊂ A . Let K > (cid:15) > I = Im(2 ξ ) + 1.Since | m n | → ∞ as n → ∞ , we have | ξ + 2 m n | > K for all large n . Then byProposition 5.2, we have B (cid:15)/ ( iα, ξ + 2 m n ) ⊂ D L ( β ) (gray parts) for β close to 2, restricted to the square of width 24 centered at 0. Left columncorresponds to tangential convergence λ ( β ) →
0, where λ ( β ) are points onthe circle | z − | = 1 whose imaginary part equal 0 . . . λ ( β ) → λ ( β ) equal 0 . . i (top), 0 . . i (middle) and 0 . . i (bottom).24or all large n . On the other hand, since the sequence { πi/λ ( β n ) − m n } converges to ξ as n → ∞ , we have (cid:12)(cid:12)(cid:12)(cid:12) πiλ ( β n ) − (2 ξ + 2 m n ) (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15)/ n . Therefore B (cid:15)/ (cid:18) iα, πiλ ( β n ) (cid:19) ⊂ D hold for all large n . Thus we obtain B (cid:15)/ ( α ) ⊂ int( L ( β n )) for all large n . Proof of Theorem 6.6.
We need to prove the following four conditions (H1),(H2), (C1) and (C2), where (H1) and (H2) are corresponding to the Haus-dorff convergence and (C1) and (C2) are corresponding to the Carath´eodoryconvergence: (H1)
For any α ∈ i M (2 ξ ) there exists a sequence α n ∈ L ( β n ) such that α n → α . (H2) If α n j ∈ L ( β n j ) and α n j → α then α ∈ i M (2 ξ ). (C1) For any compact subset X ⊂ int( i M (2 ξ )), X ⊂ int( L ( β n )) for all large n . (C2) If there exist an open subset O ⊂ C and a infinite sequence { n j } ∞ j =1 suchthat O ⊂ int( L ( β n j )), then O ⊂ int( i M (2 ξ )).Proof of (H1): For any α ∈ i M (2 ξ ), there exists a sequence { α ( j ) } ∞ j =1 inthe interior of i M (2 ξ ) such that α ( j ) → α as j → ∞ . It follows from Lemma6.7 that for every j , there exists positive constant N ( j ) such that α ( j ) ∈ L ( β n )for all n ≥ N ( j ). Thus we obtain the result. To be more precise, let us choose { N ( j ) } ∞ j =1 so that N ( j + 1) > N ( j ) and N ( j ) → ∞ as j → ∞ , and set α n := α ( j ) for every N ( j ) ≤ n < N ( j ). Since j → ∞ as n → ∞ , we obtain α n ∈ L ( β n ) and α n → α as n → ∞ .Proof of (C1): Let X be a compact subset of int( i M (2 ξ )). For every α ∈ X ,it follows from Lemma 6.7 that there exist (cid:15) ( α ) > N ( α ) > B (cid:15) ( α ) ( α ) ⊂ int( L ( β n )) for all n ≥ N ( α ). Since (cid:91) α ∈ X B (cid:15) ( α ) ( α )25s an open covering of the compact set X , we can choose finite set of points { α j } lj =1 ⊂ X such that (cid:91) ≤ j ≤ l B (cid:15) ( α j ) ( α j )is also an open covering of X . Set N := max ≤ j ≤ l N ( α j ). Then B (cid:15) ( α j ) ( α j ) ⊂ int( L ( β n )) for all n ≥ N and all 1 ≤ j ≤ l . Thus we obtain X ⊂ int( L ( β n ))for all n ≥ N .Proof of (H2): For simplicity, we denote { n j } by { n } and assume that α n ∈ L ( β n ) converge to α as n → ∞ . Take ρ n ∈ AH ( N, P ) ∩ Ω such thatTr( ρ n ) = ( α n , β n ). Since α n → α and β n →
2, the sequence { ρ n } ∞ n =1 convergesalgebraically to the conjugacy class of σ iα in AH ( N, P ). We may assume thatthe representatives of the conjugacy classes ρ n , which are also denoted by ρ n ,converge algebraically to σ iα .We now consider representations χ n of π ( ˆ N ) = (cid:104) a, b, c : [ b, c ] = id (cid:105) intoPSL(2 , C ) defined by χ n ( a ) := ρ n ( a ) , χ n ( b ) := ρ n ( b ) , χ n ( c ) := ( ρ n ( b )) − m n . One can see form Theorem 6.5 that the sequence χ n converges algebraicallyto χ ∞ := ˆ σ iα, ξ , which is defined in 4.2. We now claim that χ ∞ = ˆ σ iα, ξ isfaithful and discrete. If this is true, we obtain ( iα, ξ ) ∈ B . Especially wehave Im ξ (cid:54) = 0. It then follows from Im ξ ≥ iα, ξ ) ∈ A , and thus α ∈ i M (2 ξ ). Therefore we only have to show the claim above.Since π ( ˆ N ) is finitely generated and since the image χ n ( π ( ˆ N )) of χ n isequal to the discrete group ρ n ( π ( N )), it follows from the theorem due toJørgensen and Klein in [JK] that χ ∞ ( π ( ˆ N )) is discrete and that there existgroup homomorphisms ψ n : χ ∞ ( π ( ˆ N )) → χ n ( π ( ˆ N ))satisfying χ n = ψ n ◦ χ ∞ . Now suppose for contradiction that there is a non-trivial element g in ker χ ∞ . Then it must lie in ker χ n for all n . Since ker χ n is normally generated by a word b m n c , and since the word length of g ∈ π ( ˆ N )with respect to the generators a, b, c is bounded, we obtain a contradiction.Thus we obtain the claim.Proof of (C2): By the same argument as in the proof for (H2), we have α ∈ i M (2 ξ ) for every α ∈ O . Therefore O ⊂ i M (2 ξ ). Since O is open, wehave O ⊂ int( i M (2 ξ )). 26 .4 Main theorem for horocyclic convergence We now state our main theorem for linear slices L ( β n ) such that λ ( β n ) convergehorocyclically to 0. See Figure 3, right column. Theorem 6.8.
Let { β n } ∞ n =1 be a sequence in C \ [ − , which converges to as n → ∞ . Suppose that λ ( β n ) converge horocyclically to . Then we have thefollowing:1. L ( β n ) converge to i M in the sense of Hausdorff as n → ∞ .2. int( L ( β n )) converge to int( i M ) in the sense of Carath´eodory as n → ∞ . The following lemma is an essential part of the proof of Theorem 6.8.
Lemma 6.9.
Under the same assumption as in Theorem 6.8, we have thefollowing: For any α ∈ int( i M ) there exists (cid:15) > such that B (cid:15) ( α ) ⊂ int( L ( β n )) for all large n .Proof. Let (cid:15) > µ = iα ∈ int( M ).Let us take 0 < (cid:15) < (cid:15) such that B (cid:15) ( iα ) ⊂ int( M ). By Theorem 4.2, onesee that there exists I > B (cid:15),I ( iα ) ⊂ A . Then by Proposition5.3, if we choose I > B (cid:15)/ , I ( iα ) ⊂ D . Since λ ( β n ) → πi/λ ( β n )) > I for all large n . Thus, for every α (cid:48) ∈ B (cid:15)/ ( α ), we have ( iα (cid:48) , πi/λ ( β n )) ∈ D , or ( α (cid:48) , β n ) ∈ D tr . Therefore weobtain B (cid:15)/ ( α ) ⊂ int( L ( β n )) for all large n . Proof of Theorem 6.8.
The proof is almost parallel to that of Theorem 6.6.We need to show the following four conditions: (H1)
For any α ∈ i M there exists α n ∈ L ( β n ) such that α n → α . (H2) If α n j ∈ L ( β n j ) and α n j → α then α ∈ i M . (C1) For any compact subset X in int( i M ), X ⊂ int( L ( β n )) for all large n . (C2) If there exist an open subset O ⊂ C and a infinite sequence { n j } ∞ j =1 suchthat O ⊂ int( L ( β n j )) then O ⊂ int( i M ).Proof of (H1): For any α ∈ i M , there exists a sequence α ( j ) ∈ int( i M )such that α ( j ) → α ( j → ∞ ). It follows from Lemma 6.9 that for each j wehave α ( j ) ∈ L ( β n ) for all large n . Thus we obtain the claim.27igure 4: The linear slice L ( α ) (gray part) for α = 5 . i ; restricted to aneighborhood of 2 whose width is about 0 .
2. (The horizontal line thorough 2is the locus where the computer can not detect non-discreteness.)Proof of (C1): Let X ⊂ int( i M ) be a compact subset. For each α ∈ X ,it follows from Lemma 6.9 that there exist (cid:15) ( α ) > N ( α ) > B (cid:15) ( α ) ( α ) ⊂ int( L ( β n )) for all n ≥ N ( α ). Since (cid:83) α ∈ X B (cid:15) ( α ) ( α ) is an opencovering of X , we may choose a finite set of points { α j } ⊂ X such that (cid:83) j B (cid:15) ( α j ) ( α j ) is also an open covering. Since B (cid:15) ( α j ) ( α j ) ⊂ int( L ( β n )) for all n ≥ N := max j N ( α j ), we have X ⊂ int( L ( β n )) for all n ≥ N .Proof of (H2): For simplicity we denote { n j } by { n } , and assume that α n ∈L ( β n ) converge to α . Take ρ n ∈ AH ( N, P ) ∩ Ω such that Tr( ρ n ) = ( α n , β n ).Since α n → α , β n →
2, the sequence { ρ n } ∞ n =1 converges to σ iα ∈ R ( N, P ),and since AH ( N, P ) is closed, we have σ iα ∈ AH ( N, P ). Therefore we obtain iα ∈ M and hence α ∈ i M .Proof of (C2): By the same argument as in (H2), we have α ∈ i M forevery α ∈ O . Therefore O ⊂ i M . Since O is open, we have O ⊂ int( i M ). Here we will show that there exists a linear slice which is not locally connectedat their boundary (see Figure 4). This is a direct consequence of Bromberg’sargument in [Br] showing that AH ( N, P ) is not locally connected. This re-sult is concerned with vertical slices of A , whereas Theorems 6.6 and 6.8 areconcerned with horizontal slices of A . Theorem 6.10.
There exists α ∈ C such that L ( α ) is not locally connectedat ∈ ∂ L ( α ) ; that is, U ∩ L ( α ) is disconnected for any sufficiently small eighborhood U ⊂ C of .Proof. The homeomorphism F : D +tr → D defined in section 5.3 induces ahomeomorphism from L ( α ) ∩ C + = { β ∈ C : ( α, β ) ∈ D +tr } to a slice { ζ ∈ ˆ C : ( iα, ζ ) ∈ D} of D . Since F ( α,
2) = ( iα, ∞ ), to show that L ( α ) is not locally connected at 2for some α , it suffices to show that the set { ζ ∈ ˆ C : ( iα, ζ ) ∈ D} is not locallyconnected at ζ = ∞ . We will show this by using the fact observed in [Br] thatthe vertical slice { ζ ∈ ˆ C : ( iα, ζ ) ∈ A} of A is not locally connected at ζ = ∞ for some α .From the argument of Bromberg in the proof of Theorem 4.15 in [Br], thereexist µ ∈ int( M ), ζ ∈ C with Im ζ >
0, and (cid:15) > B (cid:15) ( µ, ζ + 2 n ) arecontained in different connected components of { ( ν, z ) ∈ A : | ν − µ | < (cid:15) } for every integer n . By Theorem 4.3, we can take neighborhoods U , W of( µ, ∞ ) in A , D , respectively, such that ϕ = F ◦ Tr ◦ Φ : U → W is a homeo-morphism. We may assume that U is of the form U = A ∩ { ( ν, z ) ∈ C : | ν − µ | < (cid:15), | z | > K } . Then for all large n , B (cid:15) ( µ, ζ + 2 n ) are contained in U , and thus contained indistinct connected components of U .By choosing (cid:15) > K > B (cid:15)/ ( µ, ζ + 2 n ) ⊂ D for all large n . Therefore, B (cid:15)/ ( µ, ζ + 2 n ) are contained in distinct connected components of W for alllarge n . Since W ⊂ D is a neighborhood of ( µ, ∞ ), we see that the set { ζ ∈ ˆ C : ( µ, ζ ) ∈ D} is not locally connected at ζ = ∞ . Letting α = − iµ , weobtain the result. In this section, we restate Theorems 6.6 and 6.8 in terms of the complexFenchel-Nielsen coordinates. We begin with recalling the definition of the realFenchel-Nielsen coordinates for Fuchsian representations.29igure 5: Fundamental domains of images of η λ (left) and η λ,τ (right).Given λ >
0, we define a representation η λ ∈ R ( N, P ) by η λ ( a ) := 1sinh( λ/ (cid:18) cosh( λ/ − − λ/ (cid:19) , η λ ( b ) := (cid:18) e λ/ e − λ/ (cid:19) . Then η λ ( π ( N )) acts properly discontinuously on the upper-half plane H ,and hence is a Fuchsian group (see Figure 5, left). Note that η λ ( a ) fixes − , η λ ( b ) fixes 0 , ∞ , and thus the axes of η λ ( a ) and η λ ( b ) are perpendicular to eachother. In addition, the complex length of η λ ( b ) is equal to λ ∈ R .Now we add a twisting parameter τ . Given ( λ, τ ) ∈ R + × R , we define afuchsian representation η λ,τ ∈ R ( N, P ) by η λ,τ ( a ) := (cid:18) e τ/ e − τ/ (cid:19) η λ ( a ) , η λ,τ ( b ) := η λ ( b ) . Note that the quotient surface H /η λ,τ ( π ( N )) is obtained by cutting the sur-face H /η λ ( π ( N )) along the geodesic representative of η λ ( b ), twisting by hy-perbolic length τ and re-glueing (see Figure 5, right).Now we obtain a map F N : R + × R → R ( N, P )defined by ( λ, τ ) (cid:55)→ η λ,τ . It is well-known that this map is a homeomorphismonto the space of Fuchsian representations. By allowing the parameters λ, τ to be complex numbers, we obtain a map F N : ( C \ πi Z ) × C → R ( N, P ) . We say that ( λ, τ ) is the complex Fenchel-Nielsen coordinates of the represen-tation η λ,τ . Note that if λ ∈ R and τ ∈ C , η λ,τ is the complex earthquake of η λ ,30ee [Mc2]. It is known by Kourouniotis [Ko] and Tan [Ta] that there is an opensubset of ( C \ πi Z ) × C containing R + × R such that the map F N induces ahomeomorphism from this set onto the quasifuchsian space
M P ( N, P ).Let D F N := { ( λ, τ ) ∈ ( C \ πi Z ) × C : η λ,τ ∈ AH ( N, P ) } . Since we have tr η λ,τ ( a ) = 4 coth (cid:18) λ (cid:19) cosh (cid:16) τ (cid:17) , tr η λ,τ ( b ) = 4 cosh (cid:18) λ (cid:19) , the map Θ : ( C \ πi Z ) × C → C defined byΘ( λ, τ ) := (cid:18) (cid:18) λ (cid:19) cosh (cid:16) τ (cid:17) , (cid:18) λ (cid:19)(cid:19) takes D F N onto D tr . For a given λ ∈ C \ πi Z , let (cid:101) L ( λ ) := { τ ∈ C : η λ,τ ∈ AH ( N, P ) } . We define a map f λ : C → C by f λ ( z ) := 2 coth (cid:18) λ (cid:19) cosh (cid:16) z (cid:17) so that we have Θ( λ, τ ) = ( f λ ( τ ) , λ/ f λ takes (cid:101) L ( λ )onto L ( β ) where β = 2 cosh( λ/ (cid:101) L ( λ ) is (cid:104) z + λ, z + 2 πi (cid:105) -invariant,where the translation z (cid:55)→ z + λ corresponds to the Dehn twist about b .We want to understand the shape of (cid:101) L ( λ ) by using the Maskit slice M when λ lies in C + and is close to zero. To this end, we normalize (cid:101) L ( λ ) so thatthe action of the Dehn twist about b corresponds to the translation z (cid:55)→ z + 2.(Recall that the Maskit slice M has this property.) Let us define a map g λ : C → C by g λ ( z ) := 2 λ ( z − πi )and set (cid:98) L ( λ ) := g λ ( (cid:101) L ( λ ))Then (cid:98) L ( λ ) is (cid:104) z + 2 , z + 4 πi/λ (cid:105) -invariant and the map h λ ( z ) := f λ ◦ g − λ ( z )31akes zero to zero and (cid:98) L ( λ ) onto L ( β ), where β = 2 cosh( λ/ h λ ( z ) = 2 coth (cid:18) λ (cid:19) cosh (cid:18) λz πi (cid:19) = 2 i coth (cid:18) λ (cid:19) sinh (cid:18) λz (cid:19) , one can see that if λ n → n → ∞ , then h λ n ( z ) → iz uniformly on anycompact subset of C . Thus we obtain the following corollary of Theorems 6.6and 6.8 (see Figure 6): Corollary 7.1.
Suppose that λ n ∈ C + , λ n → as n → ∞ .1. If λ n → horocyclically, then (cid:98) L ( λ n ) converge to M in the sense of Haus-dorff, and int( (cid:98) L ( λ n )) converge to int( M ) in the sense of Carath´eodory.2. Suppose that λ n → tangentially. In addition we assume that thereexist a sequence of integers { m n } ∞ n =1 such that the sequence πi/λ n − m n converges to some ξ ∈ C as n → ∞ . Then (cid:98) L ( λ n ) converge to M (2 ξ ) in the sense of Hausdorff, and int( (cid:98) L ( λ n )) converge to int( M (2 ξ )) in thesense of Carath´eodory.Proof. The statement for Hausdorff convergence can be easily seen. The state-ment for Carath´eodory convergence follows form Hausdorff convergence of thecomplements.
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