The asymptotic directions of pleating rays in the Maskit embedding

THE ASYMPTOTIC DIRECTIONS OF PLEA TING RA YS IN THE MASKIT EMBEDDING. SARA MALONI Abstract. This article was b orn as a generalisation of the analy- sis made b y Series in [22] where she made the first attempt to plot a deformation space of Kleinian group of more than 1 complex dimension. W e use the T op T erms’ Relationship prov ed by the author and Series in [13] to determine the asymptotic directions of pleating ra ys in the Maskit embedding of a hyperb olic surface Σ as the b ending measure of the ‘top’ surface in the boundary of the con vex core tends to zero. The Maskit embedding M of a surface Σ is the space of geometrically finite groups on the b oundary of quasifuc hsian space for which the ‘top’ end is homeomorphic to Σ, while the ‘bottom’ end consists of triply punctured spheres, the remains of Σ when the pants curv es ha v e been pinched. Giv en a pro jectiv e measured lamination [ η ] on Σ, the ple ating r ay P = P [ η ] is the set of groups in M for whic h the b ending measure pl + ( G ) of the top comp onent ∂ C + of the b oundary of the con vex core of the asso ciated 3-manifold H 3 /G is in the class [ η ]. MSC classification: 30F40, 30F60, 57M50 1. Introduction Let Σ b e a surface of negative Euler c haracteristic together with a pan ts decomp osition P . Kra’s plumbing construction endo ws Σ with a pro jective structure as follows. Replace eac h pair of pan ts b y a triply punctured sphere and glue, or ‘plum b’, adjacent pants by gluing punc- tured disk neigh b ourho o ds of the punctures. The gluing across the i th pan ts curv e is defined by a complex parameter τ i ∈ C . The asso ciated holonom y representation ρ : π 1 (Σ) − → P S L (2 , C ) giv es a pro jectiv e structure on Σ which depends holomorphically on the τ i . In particular, the traces of all elements ρ ( γ ) , γ ∈ π 1 (Σ), are p olynomials in the τ i . In [13] the author and Series pro v ed a form ula, called T op T erms’ R elationship , which is Theorem 2.11 in Section 2.1.1, giving a simple linear relationship betw een the co efficien ts of the top terms of ρ ( γ ), as p olynomials in the τ i , and the Dehn–Th urston co ordinates of γ relative to P , see Section 2.1.1 for the definitions. This result generalises the Date : Octob er 23, 2018. 1 2 SARA MALONI previous results pro v ed b y Keen and Series in [11] in the case of the once punctured torus Σ 1 , 1 and b y Series in [22] for the twice punctured torus Σ 1 , 2 . These form ulas were used in the case Σ = Σ 1 , 1 , Σ 1 , 2 to determine the asymptotic directions of pleating ra ys in the Maskit em b edding of Σ as the b ending measure of the ‘top’ surface in the b oundary of the conv ex core tends to zero, see Section 2 for the definitions. In the presen t article we will use the general T op T erms’ Relationship to gen- eralise the description of asymptotic directions of pleating rays to the case of an arbitrary hyperb olic surface Σ, see Theorem 3.7 in Section 3. The Maskit embedding M of a surface Σ is the space of geomet- rically finite groups on the b oundary of quasifuchsian space for whic h the ‘top’ end is homeomorphic to Σ, while the ‘b ottom’ end consists of triply punctured spheres, the remains of Σ when the pan ts curv es ha v e b een pinched. As suc h representations v ary in the character v ari- et y , the conformal structure on the top side v aries o v er the T eic hm ¨ uller space T (Σ), see Section 2.4 for a detailed discussion. Let Σ = Σ g ,b , and supp ose we hav e a geometrically finite free and discrete representation ρ for which M ρ = Σ × R . Denote ξ = ξ (Σ) = 3 g − 3 + b the complexit y of the surface Σ. Fix disjoin t, non-trivial, non- p eripheral and non-homotopic simple closed curv es σ 1 , . . . , σ ξ whic h form a maximal pan ts decomposition of Σ. W e consider groups for whic h the conformal end ω − is a union of triply punctured spheres glued across punctures corresp onding to σ 1 , . . . , σ ξ , while ω + is a mark ed Rie- mann surface homeomorphic to Σ. Kra’s plumbing construction gives us an explicit parametrisation of a holomorphic family of representa- tion ρ τ : π 1 (Σ) − → G ( τ ) ∈ PSL(2 , C ) such that, for certain v alues τ = ( τ 1 , . . . , τ ξ ) ∈ C ξ of the parameters, ρ τ has the ab ov e geometry , see Section 2.2 for the definition of this construction. The Maskit em b edding is the map whic h sends a p oint X ∈ T (Σ) to the p oin t τ = ( τ 1 , . . . , τ ξ ) ∈ C ξ for which the group G ( τ ) has ω + = X . Denote the image of this map by M = M (Σ). Note that, with abuse of no- tation, we will also call Maskit emb e dding the image M of the map T (Σ) − → C ξ just describ ed. W e in v estigate M using the metho d of pleating ra ys. Giv en a pro- jectiv e measured lamination [ η ] on Σ, the ple ating r ay P = P [ η ] is the set of groups in M for which the b ending measure pl + ( G ) of the top comp onen t ∂ C + of the b oundary of the con v ex core of the asso ciated 3-manifold H 3 /G is in the class [ η ]. It is known that P is a real 1- submanifold of M . In fact w e can parametrise this ray b y θ ∈ (0 , c η ), where c η ∈ (0 , π ), so that we asso ciate to θ the group G θ ∈ P suc h that pl + ( G θ ) = θ η , see Theorem 6 in [21] for the case Σ = Σ 1 , 1 . Note THE ASYMPTOTIC DIRECTIONS 3 that this result relies on Thurston’s b ending conjecture which is solved for rational lamination by work of Otal and Bonahon and in the case of punctured tori b y work of Series. F or a general (irrational) lami- nation, an ywa y , w e can only conjecture that the real dimension of the asso ciated pleating ray is 1. Our main result is a formula for the as- ymptotic direction of P in M as the b ending measure tends to zero, in terms of natural parameters for the represen tation space R and the Dehn–Th urston co ordinates of the supp ort curv es to [ η ] relative to the pinc hed curves on the b ottom side. This leads to a metho d of lo cating M in R . W e restrict to pleating ra ys for which [ η ] is r ational , that is, sup- p orted on closed curves, and for simplicit y write P η in place of P [ η ] , although noting that P η dep ends only on [ η ]. F rom general results of Bonahon and Otal [3], for any pan ts decomp osition γ 1 , . . . , γ ξ suc h that σ 1 , . . . , σ ξ , γ 1 , . . . , γ ξ are mutually non-homotopic and fill up Σ (see section 2.5 for the definitions), and any pair of angles θ i ∈ (0 , π ), there is a unique group in M for whic h the b ending measure of ∂ C + is P ξ i =1 θ i δ γ i . (This extends to the case θ i = 0 for i ∈ I ⊂ { 1 , . . . , ξ } pro vided { σ 1 , . . . , σ ξ , γ j | j / ∈ I } fill up Σ and also to the case θ = π .) Th us given η = P ξ i =1 a i δ γ i , there is a unique group G = G η ( θ ) ∈ M with b ending measure pl + ( G ) = θ η for an y sufficien tly small θ > 0. Let S denote the set of homotop y classes of multiple lo ops on Σ, and let the pan ts curves defining P b e σ i , i = 1 , . . . , ξ . The Dehn– Thurston c o or dinates of γ ∈ S are i ( γ ) = ( q 1 , p 1 , . . . , q ξ , p ξ ), where q i = i ( γ , σ i ) ∈ N ∪ { 0 } is the geometric inte rsection num ber b et w een γ and σ i and p i ∈ Z is the t wist of γ ab out σ i . F or a detailed discus- sion ab out this parametrisation see Section 2.1.1 b elow or Section 3 of [13]. If η = P ξ i =1 a i δ γ i , the ab ov e condition of Bonahon and Otal on σ 1 , . . . , σ ξ , γ 1 , . . . , γ ξ is equiv alen t to ask q i ( η ) > 0 , ∀ i = 1 , . . . , ξ . W e call suc h laminations admissible . The main result of this pap er is the following. W e will state this result more precisely , as Theorem 3.7 in Section 3. Theorem A. Supp ose that η = P ξ i =1 a i δ γ i is admissible. Then, as the b ending me asur e pl + ( G ) ∈ [ η ] tends to zer o, the ple ating r ay P η appr o aches the line < τ i = p i ( η ) q i ( η ) , = τ 1 = τ j = q j ( η ) q 1 ( η ) . W e should note that, in contrast to Series’ statement, w e were able to disp ense with the h yp othesis ‘ η non exceptional’ (see [22] for the definition), b ecause we w ere able to improv e the original pro of. In 4 SARA MALONI addition, the definition of the line is different b ecause we ha v e corrected a misprin t in [22]. One migh t also ask for the limit of the h yp erb olic structure on ∂ C + ( G ) as the b ending measure tends to zero. The following result is an immediate consequence of the first part of the pro of of Theorem A. Theorem B. L et η = P ξ 1 a i δ γ i b e as ab ove. Then, as the b ending me asur e pl + ( G ) ∈ [ η ] tends to zer o, the induc e d hyp erb olic structur e of ∂ C + along P ξ c onver ges to the b aryc entr e of the laminations σ 1 , . . . , σ ξ in the Thurston b oundary of T (Σ) . This should b e compared with the result in [20], that the analogous limit through groups whose b ending laminations on the tw o sides of the con v ex hull b oundary are in the classes of a fixed pair of laminations [ ξ ± ], is a F uc hsian group on the line of minima of [ ξ ± ]. It can also b e compared with Theorem 1.1 and 1.2 in [6]. Finally , we wan ted to underline that the result achiev ed in The- orem 2.4 about the relationship b etw een the Thurston’s symplectic form and the Dehn–Th urston co ordinated for the curv es is very in- teresting in its o wn. It tells us that giv en tw o lo ops γ , γ 0 ∈ S whic h b elongs to the same chart of the standard train trac k, see Section 2.1.2 for the definition, then Ω Th ( γ , γ 0 ) = P ξ i =1 ( q i p 0 i − q 0 i p i ) , where the vec- tor i ( γ ) = ( q 1 , p 1 , . . . , q ξ , p ξ ) , i ( γ 0 ) = ( q 0 1 , p 0 1 , . . . , q 0 ξ , p 0 ξ ) are the Dehn– Th urston co ordinates of the curv es γ , γ 0 . The plan of the pap er is as follows. Section 2 provides an ov erview of all the background material needed for understanding and pro ving the main results whic h we will pro v e in Section 3. In particular, in Section 2 w e will discuss issues related to curv es on surfaces (for example we will recall the Dehn–Thurston co ordinates of the space of measured lamina- tions, Thurston’s symplectic structure, and the curve and the marking complexes). In the same section we will also review Kra’s plumbing construction which endows a surface with a pro jectiv e structure whose holonom y map gives us a group in the Maskit embedding, and w e will discuss the T op T erms’ Relationship. Then we will recall the definition of the Maskit em b edding and of the pleating ra ys. In Section 3, on the other hand, after fixing some notation, we will prov e the three main results stated ab o v e. W e will follows Series’ metho d [22]: we will state (without pro of ) the theorems whic h generalise straigh tforw ardly to our case, but w e will discuss the results whic h require further commen ts. In particular, man y pro ofs b ecome m uc h more complicated when we increase the complex dimension of the parameter space from 1 or 2 to the general ξ (Σ). It is worth noticing that, using a sligh tly differen t pro of in Theorem A, w e w ere able to extend Series’ result to the case THE ASYMPTOTIC DIRECTIONS 5 of ‘non-exceptional’ laminations, see Section 3 for the definition. W e also corrected some mistak es in the statement of Theorem A. 2. Back gr ound 2.1. Curv es on surfaces. 2.1.1. Dehn–Thurston c o or dinates. In this section w e review Dehn– Th urston co ordinates, which extend to global coordinates for the space of measure laminations M L (Σ). These co ordinates are effectively the same as the c anonic al c o or dinates in [22]. W e follow the description in [13]. First we need to fix some notation. Supp ose Σ is a surface of finite type, let S 0 = S 0 (Σ) denote the set of free homotopy classes of connected closed simple non-boundary parallel curv es on Σ, and let S = S (Σ) b e the set of m ulti-curv es on Σ, that is, the set of finite unions of pairwise disjoint curv es in S 0 . F or simplicity w e usually refer to elements of S as ‘curves’ rather than ‘multi-curv es’, in other words, a curv e is not required to b e connected. The geometric in tersection num ber i ( α, β ) b etw een α, β ∈ S is the least n umber of in tersections b et ween curves representing the tw o homotopy classes, that is i ( α, β ) = min a ∈ α, b ∈ β | a ∩ b | . Giv en a surface Σ = Σ b g of finite t yp e and negativ e Euler c harac- teristic, choose a maximal set P C = { σ 1 , . . . , σ ξ } of homotopically dis- tinct and non-b oundary parallel lo ops in Σ called p ants curves , where ξ = ξ (Σ) = 3 g − 3 + b is the complexit y of the surface. These con- nected curv es split the surface in to k = 2 g − 2 + b three-holed spheres P 1 , . . . , P k , called p airs of p ants . (Note that the b oundary of P i ma y include punctures of Σ.) W e refer to b oth the set P = { P 1 , . . . , P k } , and the set P C , as a p ants de c omp osition of Σ. No w supp ose we are given a surface Σ together with a pan ts decom- p osition P C as ab o v e. Giv en γ ∈ S , define q i = q i ( γ ) = i ( γ , σ i ) ∈ Z > 0 for all i = 1 , . . . , ξ . Notice that if σ i 1 , σ i 2 , σ i 3 are pan ts curves which together b ound a pair of pan ts whose interior is embedded in Σ, then the sum q i 1 + q i 2 + q i 3 of the corresp onding in tersection num bers is ev en. The q i are usually called the length p ar ameters of γ . T o define the twist p ar ameter tw i = t w i ( γ ) ∈ Z of γ about σ i , we first ha v e to fix a marking on Σ. (See D. Th urston’s preprint [24] for a detailed discussion ab out three different, but equiv alen t wa ys of fixing a marking on Σ.) A wa y of sp ecifying the marking is b y c ho osing a set of curv es D i , eac h one dual to a pants curve σ i , see next paragraph for the definition. Then, after isotoping γ in to a well-defined standard 6 SARA MALONI p osition relative to P and to the marking, the t wist tw i is the signed n um b er of times that γ in tersects a short arc transverse to σ i . W e mak e the conv en tion that if i ( γ , σ i ) = 0, then t w i ( γ ) > 0 is the num ber of comp onen ts in γ freely homotopic to σ i . Eac h pants curve σ is the common b oundary of one or t wo pairs of pan ts whose union we refer to as the mo dular surfac e asso ciated to σ , denoted M ( σ ). Nota that if σ is adjacent to exactly one pair of pants, M ( σ ) is a one holed torus, while if σ is adjacen t to t w o distinct pairs of pants, M ( σ ) is a four holed sphere. A curve D is dual to the pants curv e σ if it in tersect σ minimally and is completely contained in the mo dular surface M ( σ ). R emark 2.1 (Conv en tion on dual curv es) . W e shall need to consider dual curves to σ i ∈ P C . The intersection n um b er of suc h a connected curv e with σ i is 1 if M ( σ i ) a one-holed torus and 2 if it is a four- holed sphere. W e adopt a useful conv en tion introduced in [24] which simplifies the formulae, in suc h a wa y as to a v oid the need to distinguish b et ween these t w o cases. Namely , for those σ i for whic h M ( σ i ) is Σ 1 , 1 , we define the dual curve D i ∈ S to b e two parallel copies of the connected curve intersecting σ i once, while if M ( σ i ) is Σ 0 , 4 w e take a single copy . In this w a y we alwa ys hav e, by definition, i ( σ i , D i ) = 2. See Section 2 of [13] for a deep er discussion. There are v arious wa ys of defining the standar d p osition of γ , leading to differing definitions of the twist. In this pap er w e will alw a ys use the one defined by D. Th urston [24] (which w e will denote p i ( γ )), but we refer to our previous article [13] for a further discussion ab out the dif- feren t definitions of the twist parameter and for the precise relationship b et ween them (Theorem 3.5 [13]). With either definition, a classical theorem of Dehn [5], see also [19] (p 12), asserts that the length and t wist parameters uniquely determine γ ∈ S . This result was describ ed b y Dehn in a 1922 Breslau lecture [5]. Theorem 2.2. (Dehn ’s the or em, 1922) Given a marking ( P C ; D ) = ( σ 1 , . . . , σ ξ ; D 1 , . . . , D ξ ) on Σ , the map i = i ( P C ; D ) : S (Σ) − → Z ξ > 0 × Z ξ which sends γ ∈ S (Σ) to ( q 1 ( γ ) , . . . , q ξ ( γ ); tw 1 ( γ ) , . . . , tw ξ ( γ )) is an inje ction. The p oint ( q 1 , . . . , q ξ , t w 1 , . . . , t w ξ ) is in the image of i (and henc e c orr esp onds to a curve) if and only if: (i) if q i = 0 , then t w i > 0 , for e ach i = 1 , . . . , ξ . (ii) if σ i 1 , σ i 2 , σ i 3 ar e p ants curves which to gether b ound a p air of p ants whose interior is emb e dde d in Σ , then the sum q i 1 + q i 2 + q i 3 of the c orr esp onding interse ction numb ers is even. THE ASYMPTOTIC DIRECTIONS 7 One can think of this theorem in the following wa y . Supp ose given a curv e γ ∈ S , whose length parameters q i ( γ ) necessarily satisfy the parit y condition (ii), then the q i ( γ ) uniquely determine γ ∩ P j for each pair of pan ts P j , j = 1 , . . . , k , in accordance with the p ossible arrange- men ts of arcs in a pair of pants, see for example [19]. Now given tw o pan ts adjacen t along the curve σ i , w e ha v e q i ( γ ) p oints of intersection coming from eac h side and we hav e only to decide how to match them together to reco v er γ . The matching tak es place in the cyclic cov er of an annular neigh b ourho o d of σ i . The t wist parameter tw i ( γ ) sp ecifies whic h of the Z possible choices is used for the matching. In 1976 William Thurston redisco v ered Dehn’s result and extended it to a parametrisation of (Whitehead equiv alence classes of ) measured foliation of Σ, see F athi, Laudenbac h and P o ´ enaru [8] or Penner with Harer [19] for a detailed discussion. Penner’s approac h for parametris- ing M L (Σ) is through tr ain tr acks . Using them, Thurston also defined a symplectic form on M L (Σ), called Thurston ’s symple ctic form . Since it will b e useful later, w e will recall its definition and some prop erties in the next section. 2.1.2. Thurston ’s symple ctic form. W e will fo cus on Penner’s approach, follo wing Hamenstad’s notation [9]. W e will define train trac ks and some other related notions, so as to b e able to define the symplectic form. Then we will presen t an easy w a y to calculate it. A tr ain tr ack on the surface Σ is an em b edded 1–complex τ ⊂ Σ whose edges (called br anches ) are smo oth arcs with well–defined tan- gen t vectors at the endpoints. At an y vertex (called a switch ) the inciden t edges are m utually tangent. Through eac h switc h there is a path of class C 1 whic h is em b edded in τ and con tains the switc h in its in terior. In particular, the branc hes whic h are inciden t on a fixed switch are divided into “incoming” and “outgoing” branc hes ac- cording to their inw ard p ointing tangent vectors at the switc h. Eac h closed curv e component of τ has a unique biv alen t switc h, and all other switc hes are at least triv alen t. The complementary regions of the train trac k hav e negativ e Euler c haracteristic, which means that they are differen t from discs with 0, 1 or 2 cusps at the b oundary and different from annuli and once-punctured discs with no cusps at the b oundary . A train track is called generic if all switc hes are at most triv alen t. Note that in the case of a triv alen t vertex there is one incoming branch and t w o outgoing ones. Denote B = B ( τ ) the set of branc hes of τ . Then a function w : B − → R > 0 (resp. w : B − → R ) is a tr ansverse me asur e (resp. weighting ) for τ if it satisfies the switch c ondition , that is for all switches v , we wan t 8 SARA MALONI P i w ( e i ) = P j w ( E j ) where the e i are the incoming branches at v and E j are the outgoing ones. A train track is called r e curr ent if it admits a transverse measure whic h is p ositive on ev ery branc h. A train track τ is called tr ansversely r e curr ent if ev ery branc h b ∈ B ( τ ) is intersected b y an em b edded simple closed curve c = c ( b ) ⊂ Σ which intersects τ transv ersely and is such that Σ − τ − c do es not contain an embedded bigon, i.e. a disc with t w o corners on the b oundary . A recurrent and transversely recurrent train track is called bir e curr ent . A geo desic lamination or a train track λ is c arrie d by a train trac k τ if there is a map F : Σ − → Σ of class C 1 whic h is isotopic to the identit y and which maps λ to τ in suc h a w a y that the restriction of its differen tial dF to every tangent line of λ is non–singular. A generic transv ersely recurren t train trac k which carries a complete geo desic lamination is called c omplete , where w e define a geo desic lamination to b e complete if there is no geo desic lamination that strictly con tains it. Giv en a generic birecurren t train track τ ⊂ Σ, w e define V ( τ ) to b e the collection of all (not necessary nonzero) transverse measures supp orted on τ and let W ( τ ) b e the v ector space of all assignmen ts of (not necessary nonnegativ e) real n um b ers, one to eac h branch of τ , whic h satisfy the switch conditions. By splitting, we can arrange τ to b e generic. Since Σ is oriented, w e can distinguish the righ t and left hand outgoing branches, see Figure 1. If n , n 0 ∈ W ( τ ) are weigh tings on τ (representing p oin ts in M L (Σ)), then w e denote b y b v ( n ) , c v ( n ) the weigh ts of the left hand and righ t hand outgoing branches at v resp ectiv ely . The Thurston pr o duct is defined as Ω Th ( n , n 0 ) = 1 2 X v b v ( n ) c v ( n 0 ) − b v ( n 0 ) c v ( n ) . In Theorem 3.1.4 of Penner [19] it is prov ed that, if the train trac k τ ⊂ Σ is complete, then the interior ◦ V ( τ ) of V ( τ ) for a complete train trac k τ ⊂ Σ can b e though t of as a chart on the PIL manifold M L Q (Σ) of rational measured laminations, that is laminations supported on m ulti-curv es. (PIL is short for pie c ewise–inte gr al–line ar , see [19, Sec- tion 3.1] for the definition.) In addition, in this case, we can identify W ( τ ) with the tangen t space to M L Q (Σ) at a p oint in ◦ V ( τ ). The Th uston pro duct Ω Th defined ab ov e allo ws us to define a symplectic structure on the PIL manifold M L Q (Σ). It is in teresting to note that if τ is oriented, then there is a natural map h τ : W ( τ ) − → H 1 (Σ; R ), see Section 3.2 [19], which is related to the Thurston pro duct by the follo wing result. F or a generalisation of THE ASYMPTOTIC DIRECTIONS 9 incoming branc h c v ( n ) b v ( n ) v w b w ( n ) c w ( n ) Figure 1. W eigh ted branc hes at a switch. this result to the case of an arbitrary (not necessarily orientable) track τ ∈ Σ, see Section 3.2 [19]. Prop osition 2.3 (Lemma 3.2.1 and 3.2.2 [19]) . F or any tr ain tr ack τ , Ω Th ( · , · ) is a skew-symmetric biline ar p airing on W ( τ ) . In addition, if τ is c onne cte d, oriente d and r e curr ent, then for any n , n 0 ∈ W ( τ ) , Ω Th ( n , n 0 ) is the homolo gy interse ction numb er of the classes h τ ( n ) and h τ ( n 0 ) . In Prop osition 4.3 of [22], Series relates Th urston pro duct to the Dehn–Th urston co ordinates describ ed ab o v e, but her pro of works only for the case Σ = Σ 1 , 2 , since she uses a particular choice of train trac ks, called c anonic al tr ain tr acks . Our idea w as to use the standar d tr ain tr acks , as defined b y Penner [19] in Section 2.6. The Dehn-Thurston co ordinates, using Penner’s twist ˆ p i , giv e us a choice of a standard mo del and of sp ecific weigh ts on eac h edge of the trac k. Then one can calculate the Th urston’s pro duct, using the definition ab o v e, for a pair of curves γ , γ 0 ∈ S supp orted on a common standard train track. Finally , using the relationship b et w een P enner’s and D. Thurston’s t wist, as described by Theorem 3.5 by Maloni and Series [13], one can pro v e the following result, which will b e very imp ortant in the pro of of our main theorems. In particular, the standard train trac k are of tw o t yp es: the trac ks in the annuli around the pan ts curves and the trac ks in the pair of pants. The sum of the Thurston’s pro duct in the ann uli giv e us P ξ i =1 ( q i ˆ p 0 i − q 0 i ˆ p i ), using Penner’s t wists, while the sum of the pairs of pants give us some terms, so that the total sum giv e us the results that w e wan t, that is the pro duct P ξ i =1 ( q i p 0 i − q 0 i p i ), where we use D. Thurston’s twist. W e should notice that this result, although w e prov ed it b ecause we need it in our last section, is really interesting in its o wn and it is p ossible m uc h more can be said from it. Theorem 2.4. Supp ose that lo ops γ , γ 0 ∈ S b elongs to the same chart (and so ar e supp orte d on a c ommon standar d tr ain tr ack) and they ar e r epr esente d by c o or dinates i ( γ ) = ( q 1 , p 1 , . . . , q ξ , p ξ ) , i ( γ 0 ) = ( q 0 1 , p 0 1 , . . . , q 0 ξ , p 0 ξ ) . Then Ω Th ( γ , γ 0 ) = P ξ i =1 ( q i p 0 i − q 0 i p i ) . In addition, if γ , γ 0 ar e disjoint, then Ω Th ( γ , γ 0 ) = 0 . 10 SARA MALONI Notice that this symplectic form Ω Th ( · , · ) induces a map R 2 ξ − → R 2 ξ defined by x = ( x 1 , y 1 , . . . , x ξ , y ξ ) − → x ∗ = ( y 1 , − x 1 , . . . , − y ξ , x ξ ) such that Ω Th ( i ( γ ) , i ( δ )) = i ( γ ) · i ( δ ) ∗ where · is the usual inner pro duct on R 2 ξ . T o understand the meaning of the vector x ∗ b etter, w e should recall the last Proposition of Section 3.2 of [19] and some notation from Bonahon’s work (see his survey pap er [1] for a general in tro duction to the argument and for other further references). After rigorously defining the tangen t space T α M L (Σ) with α ∈ M L (Σ), Bonahon prov ed in [2] that w e can interpret any tangent v ector v ∈ T α M L (Σ) as a geo desic lamination with a transv erse H¨ older distribution. Note that the space W ( τ ) can b e seen as the space of H¨ older distributions on the track τ , since it is defined to b e the vector space of all assignmen ts of not necessary nonnegative real n um b ers, one to each branch of τ , which satisfy the switch conditions. He also c haracterised which geo desic laminations with transv erse distributions corresp ond to tangen t vectors to M L (Σ). Notice that if the lamination is carried b y the track τ , w e can lo cally identify T α M L (Σ) with W ( τ ) whic h is isomorphic to R 2 ξ . Theorem 2.5 (Theorem 3.2.4 [19]) . F or any surfac e Σ , the Thurston pr o duct is a skew-symmetric, nonde gener ate, biline ar p airing on the tangent sp ac e to the PIL manifold M L 0 (Σ) . 2.1.3. Complex of curves and marking c omplex. In this section, we re- view the definitions of the complex of curves and of the marking com- plex. W e will use this language in the last Section where we will pro v e our main Theorems. While it is not essential to use this language, we b eliev e most readers will already b e familiar with these definitions and will find easier to understand the ideas of our pro ofs. In addition, these to ols will shorten the pro ofs. W e summarise briefly the definition of simplicial complex and few related definitions which we will need later on, and w e refer to Hatcher [10] for a complete discussion. Definition 2.6. Giv en K (0) a set (of vertic es ), then K ⊂ P ( K (0) ), where P ( K (0) ) is the p o w er set of K (0) , is a simplicial c omplex if (1) ∅ / ∈ K ; (2) ∀ τ ⊂ σ ∈ K , τ 6 = ∅ ⇒ τ ∈ K. Giv en σ ∈ K , we define the link of σ to b e the set lk K ( σ ) = { τ ∈ K | τ ∩ σ = ∅ , τ ∪ σ ∈ K } . Definition 2.7. Giv en a surface Σ, let C (0) (Σ) b e the set of isotopy classes of essential, nonp eripheral, simple closed curves in Σ. Then THE ASYMPTOTIC DIRECTIONS 11 w e define the c omplex of curves C (Σ) as the simplicial complex with v ertex set C (0) (Σ) and where m ulticurv es gives simplices. In particular k –simplices of C (Σ) are ( k + 1)–tuples { γ 0 , . . . , γ k } of distinct nontrivial free homotopy classes of simple, nonp eripheral closed curv es, whic h can b e realised disjoin tly . Note that this complex is ob viously finite–dimensional b y an Euler c haracteristic argument, and is t ypically lo cally infinite. If Σ = Σ g ,b , then the dimension is dim ( C (Σ)) = ξ (Σ) − 1 = 3 g − 4 + b. Note that the cases of low er complexit y , which are called sp or adic b y Masur and Minsky [16], require a separate discussion. In particular if Σ = Σ 0 ,b with b 6 3, then C (Σ) is empty . If Σ = Σ 1 , 0 , Σ 1 , 1 , Σ 0 , 4 , using this definition, C (Σ) is disconnected (in fact, it is just an infinite set of vertices). So we slightly mo dify the definition, in suc h a w a y that edges are placed b etw een vertices corresp onding to curves of smallest p ossible intersection n um b er (1 for the tori, 2 for the sphere). Finally also in the case of an annulus, that is Σ = Σ 0 , 2 = A , C (Σ) needs to b e defined in a different wa y , which w e do not not discuss here as it is not needed. W e refer the interested to Masur and Minsky [17] for a detailed discussion. W e define no w the marking complex. Before defining it, w e need to giv e few additional definitions. Definition 2.8. Given a surface Σ, a c omplete cle an marking µ on Σ is a pan ts decomposition base( µ ) = { γ 1 , . . . , γ ξ } , called the b ase of the marking, together with the choice of dual curv es D i for eac h i = 1 , . . . , ξ suc h that D i ∩ γ j = ∅ for an y j 6 = i . There are tw o t yp es of elementary moves on a complete clean mark- ing: (1) Twist : Replace a dual curve D i b y another dual curv e D 0 i ob- tained from D i b y a Dehn–t wist or an half–twist around γ i . (2) Flip : Exc hange a pair ( γ i , D i ) with a new pair ( γ 0 i , D 0 i ) := ( D i , γ i ) and c hange the dual curves D j with j 6 = i so that they will satisfy the prop ert y describ ed in the Definition 2.8. This op eration is called cle aning the marking and it is not uniquely defined. W e will only need to use the base of the markings, so w e will not describ e these op erations more deeply . The interested reader can refer to [17] for a deep er analysis on this topic. Definition 2.9. Given a surface Σ, let MC (0) (Σ) b e the set of com plete clean markings in Σ. Then we define the marking c omplex MC (Σ) 12 SARA MALONI Figure 2. Deleting horocyclic neigh b ourho o ds of the punctures and preparing to glue. as the simplicial complex with v ertex set MC (0) (Σ) and where tw o v ertices are connected b y an edge if the tw o markings are connected b y an elemen tary mo v e. 2.2. Plum bing construction. In this section w e review the plumbing construction which gives us the complex parameters τ i for the Maskit em b edding. The idea of the plumbing construction is to manufacture Σ by gluing triply punctured spheres across punctures. There is one triply punctured sphere for each pair of pan ts P ∈ P , and the gluing across the pan ts curve σ j is implemented b y a sp ecific pro jective map dep ending on a parameter τ j ∈ C . The τ j will b e the parameters of the resulting holonomy representation ρ τ : π 1 (Σ) − → P S L (2 , C ) with τ = ( τ 1 , . . . , τ ξ ). More precisely , w e first fix an identification of the interior of each pair of pan ts P i to a standard triply punctured sphere P . W e endo w P with the pro jectiv e structure coming from the unique hyperb olic metric on a triply punctured sphere. Then the gluing is carried out b y deleting op en punctured disk neighbourho o ds of the t w o punctures in question and gluing horocyclic ann ular collars round the resulting tw o b oundary curv es, see Figure 2. 2.2.1. The gluing. First recall (see for example [18] p. 207) that an y triply punctured sphere is isometric to the standard triply punctured sphere P = H / Γ, where Γ = D  1 2 0 1  ,  1 0 2 1  E . Fix a standard fundamen tal domain for Γ, as shown in Figure 3, so that the three punctures of P are naturally lab elled 0 , 1 , ∞ . Let ∆ 0 b e the ideal triangle with vertices { 0 , 1 , ∞} , and ∆ 1 b e its reflection in the imaginary axis. W e sometimes refer to ∆ 0 as the white triangle and ∆ 1 as the blac k. THE ASYMPTOTIC DIRECTIONS 13 1 − 10 µ 0 µ 1 µ ∞ Figure 3. The standard fundamental domain for Γ. The white triangle ∆ 0 is unshaded. With our usual pan ts decomp osition P , fix homeomorphisms Φ i from the interior of eac h pair of pan ts P i to P . This identification induces a lab elling of the three b oundary comp onents of P i as 0 , 1 , ∞ in some or- der, fixed from now on. W e denote the b oundary lab elled  ∈ { 0 , 1 , ∞} b y ∂  P i . The identification also induces a colouring of the t w o righ t an- gled hexagons whose union is P i , one b eing white and one b eing blac k. Supp ose that the pan ts P , P 0 ∈ P are adjacen t along the pan ts curve σ meeting along b oundaries ∂  P and ∂  0 P 0 . (If P = P 0 then clearly  6 =  0 .) The gluing across σ will b e describ ed b y a complex parameter τ with = τ > 0, called the plumbing p ar ameter of the gluing. Let ∆ 0 ⊂ H b e the ideal ‘white’ triangle with v ertices 0 , 1 , ∞ . Notice that there is a unique orientation preserving symmetry Ω  of ∆ 0 whic h sends the v ertex  ∈ { 0 , 1 , ∞} to ∞ : Ω 0 =  1 − 1 1 0  , Ω 1 =  0 − 1 1 − 1  , Ω ∞ = Id =  1 0 0 1  . As describ ed in Figure 4, first we use the maps Ω  to reduce to the case  =  0 = ∞ . In that case, we first need to reverse the direction in the left triangle ∆ 0 , by the map J whic h is a rotation ab out the origin of an angle π , and then w e should translate it, b y the map T τ where J =  − i 0 0 i  , T τ =  1 τ 0 1 .  The gluing map b et w een the pan ts P , P 0 ∈ P is then describ ed by Ω − 1  J − 1 T − 1 τ Ω  0 . F or a general discussion, we refer to Section 4 and 5 of [13]. The recip e for gluing tw o pants apparen tly depends on the direction of 14 SARA MALONI T τ P ∂ � ( P ) P � ∂ � � ( P � ) Ω 0 J Ω 1 Figure 4. The gluing construction when  = 1 and  0 = 0. tra v el across their common b oundary . Lemma 4.2 in [13] shows that, in fact, the gluing in either direction is implemen ted b y the same recip e and uses the same parameter τ . R emark 2.10 (Relationship with Kra’s construction) . As explained in detail in Section 4.4 of [13], Kra’s plumbing construction (see Kra [12]) is essen tially iden tical to our construction. The difference is that we implemen t the gluing in the upp er half space H without first mapping to the punctured disk D ∗ . In particular the precise relationship betw een THE ASYMPTOTIC DIRECTIONS 15 our plum bing parameter τ and Kra’s one t K is giv en b y τ = − i π log t K . 2.3. T op T erms’ Relationship. W e can no w state the main result of our previous w ork whic h will b e fundamental for the pro of of our main theorems. The plumbing construction describ ed in Section 2.2 endo ws Σ with a pro jective structure whose asso ciated holonom y rep- resen tation ρ τ : π 1 (Σ) − → P S L (2 , C ) dep ends holomorphically on the plum bing parameters τ = ( τ 1 , . . . , τ ξ ). In particular, the traces of all elemen ts ρ ( γ ) , γ ∈ π 1 (Σ), are p olynomials in the τ i . Theorem A of [13] is a very simple relationship b etw een the co efficien ts of the top terms of ρ ( γ ), as p olynomials in the τ i , and the Dehn–Th urston co ordinates of γ relative to P . Theorem 2.11 (T op T erms’ Relationship) . L et γ b e a c onne cte d simple close d curve on Σ ,such that its Dehn–Thurston c o or dinates ar e i ( γ ) = ( q 1 , p 1 , . . . , q ξ , p ξ ) . If γ not p ar al lel to any of the p ants curves σ i , then T r ρ ( γ ) is a p olynomial in τ 1 , · · · , τ ξ whose top terms ar e given by: T r ρ ( γ ) = ± i q 2 h  τ 1 + ( p 1 − q 1 ) q 1  q 1 · · ·  τ ξ + ( p ξ − q ξ ) q ξ  q ξ + R, = ± i q 2 h τ q 1 1 · · · τ q ξ ξ + ξ X i =1 ( p i − q i ) τ q 1 1 · · · τ q i − 1 i · · · τ q ξ ξ ! + R wher e • q = P ξ i =1 q i > 0 ; • R r epr esents terms with total de gr e e in τ 1 · · · τ ξ at most q − 2 and of de gr e e at most q i in the variable τ i ; • h = h ( γ ) is the total numb er of scc -ar cs in the standar d r epr e- sentation of γ r elative to P , se e b elow. If q ( γ ) = 0 , then γ = σ i for some i , ρ ( γ ) is p ar ab olic, and T r ρ ( γ ) = ± 2 . The non-negative integer h = h ( γ ) is defined as follo ws. The curv e γ is first arranged to intersect eac h pan ts curve minimally . In this p osition, it in tersects a pair of pan ts P in a num ber of arcs joining the b oundary curves of P . W e call one of these an scc -ar c (short for same- (b oundary)-comp onent-connector) if it joins one b oundary comp onen t to itself, and denote by h the total n um b er of scc -arcs, tak en ov er all pan ts in P . Note that some authors call the scc-arcs waves . R emark 2.12 . As noted in Section 4.2 [13], with our conv en tion the base p oin t for the gluing construction is when < τ i = 1. It w ould b e more natural to hav e, as base p oint, < τ i = 0. That can b e achiev ed by 16 SARA MALONI c hanging the fundamental domain for the standard triply punctured sphere. In particular, one should ha v e as the white triangle ∆ 0 the set { z ∈ C |< z ∈ ( − 1 2 , 1 2 ) , | z | > 1 2 } . This new parameter, equal the old one minus 1, w ould also make the form ula ab o v e neater. In fact the form ula, with this new parameter, also called call τ i , b ecomes: T r ρ ( γ ) = ± i q 2 h  τ 1 + p 1 q 1  q 1 · · ·  τ ξ + p ξ q ξ  q ξ + R, = ± i q 2 h τ q 1 1 · · · τ q ξ ξ + ξ X i =1 p i τ q 1 1 · · · τ q i − 1 i · · · τ q ξ ξ ! + R F rom no w on w e will use this new parameter which is equal the τ i − − parameter in [13] minus 1. 2.4. Maskit em b edding. In this section w e recall the definition of the Maskit emb e dding of Σ, follo wing Series’ article [22], see also [15]. Let R (Σ) b e the set of represen tations ρ : π 1 (Σ) − → P S L (2 , C ) mo dulo conjugation in P S L (2 , C ). Let M ⊂ R b e the subset of represen tations for whic h: (i) the group G = ρ ( π 1 (Σ)) is discrete (Kleinian) and ρ is an iso- morphism, (ii) the images of σ i , i = 1 , . . . , ξ , are parab olic, (iii) all comp onen ts of the regular set Ω( G ) are simply connected and there is exactly one inv arian t comp onent Ω + ( G ), (iv) the quotient Ω( G ) /G has k + 1 comp onents (where k = 2 g − 2 + n if Σ = Σ ( g ,n ) ), Ω + ( G ) /G is homeomorphic to Σ and the other comp onen ts are triply punctured spheres. In this situation, see for example Section 3.8 of Marden [14], the corre- sp onding 3–manifold M ρ = H 3 /G is top ologically Σ × (0 , 1). Moreo v er G is a geometrically finite cusp group on the b oundary (in the alge- braic topology) of the set of quasifuchsian representations of π 1 (Σ). The ‘top’ comp onent Ω + /G of the conformal b oundary may b e iden- tified to Σ × { 1 } and is homeomorphic to Σ. On the ‘b ottom’ com- p onen t Ω − /G , iden tified to Σ × { 0 } , the pan ts curv es σ 1 , . . . , σ ξ ha v e b een pinched, making Ω − /G a union of k triply punctured spheres glued across punctures corresp onding to the curves σ i . The conformal structure on Ω + /G , together with the pinched curves σ 1 , . . . , σ ξ , are the end invariants of M ρ in the sense of Minsky’s ending lamination theorem. Since a triply punctured sphere is rigid, the conformal struc- ture on Ω − /G is fixed and indep enden t of ρ , while the structure on Ω + /G v aries. It follo ws from standard Ahlfors–Bers theory , using the Measurable Riemann Mapping Theorem (see again Section 3.8 of [14]), THE ASYMPTOTIC DIRECTIONS 17 that there is a unique group corresp onding to each p ossible conformal structure on Ω + /G . F ormally , the Maskit emb e dding of the T eic hm ¨ uller space of Σ is the map T (Σ) − → R which sends a p oin t X ∈ T (Σ) to the unique group G ∈ M for which Ω + /G has the mark ed conformal structure X . 2.4.1. R elationship b etwe en the plumbing c onstruction and the Maskit emb e dding. In the Section 2.2, giv en a pants decomp osition P C = { σ 1 , . . . , σ ξ } of Σ, w e constructed a family of pro jective structures on Σ, to each of whic h is asso ciated a natural holonomy representation ρ τ : π 1 (Σ) − → P S L (2 , C ). Proposition 4.4 of [13] pro v es that our plum bing construction describ ed ab o v e, for suitable v alues of the pa- rameters, giv es exactly the Maskit emb e dding of Σ. Prop osition 2.13 (Prop osition 4.4 [13]) . Supp ose that τ ∈ H ξ is such that the asso ciate d developing map D ev τ : ˜ Σ − → ˆ C is an emb e dding. Then the holonomy r epr esentation ρ τ is a gr oup isomorphism and G = ρ τ ( π 1 (Σ)) ∈ M . 2.5. Three manifolds and pleating ra ys. Let M b e a hyperb olic 3–manifold, that is a complete 3-dimensional Riemannian manifold of constan t curv ature − 1 such that the fundamen tal group π 1 ( M ) is finitely generated. W e exclude the somewhat degenerate case π 1 ( M ) has an ab elian subgroup of finite index, that is π 1 ( M ) is an elementary Kleinian group. An imp ortant subset of M is its c onvex c or e C M whic h is the smallest, non-empty , closed, con v ex subset of M . The b oundary ∂ C M of this con v ex core is a surface of finite top ological type whose geometry was describ ed b y W. Thurston [23]. Note that giv en a hy- p erb olic 3–manifold M = H 3 /G , w e can also define the conv ex core as the quotient CH(Λ) /G where CH(Λ) is the con vex hull of the limit set Λ = Λ( G ) of G , see [7] for a detailed discussion on the pleated struc- ture of the b oundary of the conv ex core. If M is geometrically finite, then there is a natural homeomorphism betw een eac h comp onent of ∂ C M and each comp onent of the conformal b oundary Ω /G of M . Eac h comp onen t F of ∂ C M inherits an induced h yperb olic structure from M . Th urston also pro v ed such each comp onen t is a pleated surface, that is a hyperb olic surfaces whic h is totally geo desic almost everywhere and such that the lo cus of p oints where it fails to b e totally geo desic is a geo desic lamination. F ormally a pleated surface is defined in the follo wing w a y . Definition 2.14. A ple ate d surfac e with top ological t yp e S in a hy- p erb olic 3–dimensional manifold M is a map f : S − → M such that: 18 SARA MALONI • the path metric obtained by pulling back the hyperb olic metric of M b y f is a hyperb olic metric m on S ; • there is an m -geo desic lamination λ suc h that f sends eac h leaf of λ to a geo desic of M and is totally geo desic on S − λ . In this case, w e say that the pleated surface f admits the geo desic lamination λ as a ple ate d lo cus and λ is called the b ending lamination and the images of the complemen tary comp onen ts of λ are called the flat pie c es (of the pleated surface). The b ending lamination of each comp onent of ∂ C M carries a natural transv erse measure, called the b ending me asur e (or ple ating me asur e ). In the case M = Σ × R , there are tw o comp onen ts ∂ + C M and ∂ − C M of ∂ C M and w e will denote pl ± ∈ M L (Σ) the resp ectiv e pleating measure on eac h one of them. W e will deal with manifolds for whic h the b ending lamination is r ational , that is, supp orted on closed curves. The subset of rational measured laminations is denoted M L Q (Σ) ⊂ M L (Σ) and consists of measured laminations of the form P k i =1 a i δ γ i , where the curv es γ i ∈ S (Σ) are disjoin t and non-homotopic, a i ≥ 0, and δ γ i denotes the transv erse measure which giv es weigh t 1 to each intersection with γ i . If P k i =1 a i δ γ i is the b ending measure of a pleated surface Σ, then a i is the angle b et w een the flat pieces adjacen t to γ i , also denoted θ γ i . In particular, θ γ i = 0 if and only if the flat pieces adjacent to γ i are in a common totally geo desic subset of ∂ C /G . W e tak e the term pleated surface to include the case in whic h a closed leaf γ of the b ending lamination maps to the fixed p oint of a rank one parab olic cusp of M . In this case, the image pleated surface is cut along γ and th us ma y b e disconnected. Moreo v er the b ending angle b et w een the flat pieces adjacen t to γ is π . See discussion in [22] or [4]. An imp ortant result, due to Bonahon and Otal, ab out the existence of hyperb olic manifolds with prescrib ed b ending laminations is the fol- lo wing. Recall that a set of curves { γ 1 , . . . , γ n } in a surface Σ fil ls the surface if for an y γ ∈ S (Σ) there exist j ∈ { 1 , . . . , n } suc h that i ( γ , γ j ) 6 = 0. Theorem 2.15 (Theorem 1 of [3]) . Supp ose that M is 3 –manifold home omorphic to Σ × (0 , 1) , and that ξ ± = P i a ± i γ ± i ∈ M L Q (Σ) . Then ther e exists a ge ometric al ly finite gr oup G such that M = H 3 /G and such that the b ending me asur es on the two c omp onents ∂ C ± ( G ) of ∂ C ( G ) e qual ξ ± r esp e ctively, if and only if a ± i ∈ (0 , π ] for al l i and { γ ± i , i = 1 , . . . , n } fil l up Σ (i.e. if i ( ξ + , γ ) + i ( ξ − , γ ) > 0 for every γ ∈ S ). If such a structur e exists, it is unique. THE ASYMPTOTIC DIRECTIONS 19 Sp ecialising now to the Maskit embedding M = M (Σ), let ρ = ρ τ where τ = ( τ 1 , . . . , τ ξ ) ∈ C ξ b e a representation ρ : π 1 (Σ) − → S L (2 , C ) suc h that the image G = G ( τ 1 , . . . , τ ξ ) ∈ M . The b oundary ∂ C ( G ) of the conv ex core has ξ + 1 comp onen ts, one ∂ + C facing Ω + /G and homeomorphic to Σ, and ξ triply punctured spheres whose union w e denote ∂ − C . The induced hyperb olic structures on the comp onen ts of ∂ − C are rigid, while the structure on ∂ + C v aries. W e recall that w e de- noted pl + ( G ) ∈ M L (Σ) the b ending lamination of ∂ + C . F ollowing the discussion ab ov e, w e view ∂ − C as a single pleated surface with b ending lamination π ( σ 1 + . . . + σ ξ ), indicating that the triply punctured spheres are glued across the annuli whose core curve s σ 1 , . . . , ξ σ 2 corresp ond to the parab olics S i ∈ G . Corollary 2.16. A lamination η ∈ M L Q (Σ) is the b ending me asur e of a gr oup G ∈ M if and only if i ( η , σ 1 ) , . . . , i ( ξ , σ ξ ) > 0 . If such a structur e exists, it is unique. W e call η ∈ M L Q (Σ) admissible if i ( η , σ 1 ) , . . . , i ( ξ , σ ξ ) > 0. 2.5.1. Ple ating r ays. Denote the set of pro jective measured laminations on Σ b y P M L = P M L (Σ) and the pro jectiv e class of η = a 1 γ 1 + . . . + a k γ k ∈ M L by [ η ]. The ple ating r ay P = P [ η ] of η ∈ M L is the set of groups G ∈ M for which pl + ( G ) ∈ [ η ]. T o simplify notation we write P η for P [ η ] and note that P η dep ends only on the pro jectiv e class of η , also that P η is non-empt y if and only if η is admissible. In particular, we write P γ for the ray P [ δ γ ] . As pl + ( G ) increases, P η limits on the unique geometrically finite group G cusp ( η ) in the algebraic closure M of M at whic h at least one of the supp ort curv es to η is parab olic, equiv alen tly so that pl + ( G ) = θ ( a 1 γ 1 + . . . + a k γ k ) with max { θ a 1 , . . . , θ α k } = π . W e write P η = P η ∪ G cusp ( η ). The follo wing k ey lemma is pro v ed in Proposition 4.1 of Choi and Se- ries [4], see also Lemma 4.6 of Keen and Series [11]. The essence is that, b ecause the tw o flat pieces of ∂ C ( G ) on either side of a b ending line are in v arian t under translation along the b ending line, the translation can ha v e no rotational part. Lemma 2.17. If the axis of g ∈ G is a b ending line of ∂ C ( G ) , then T r( g ) ∈ R . Notice that the lemma applies even when the bending angle θ γ along γ v anishes. Th us if G ∈ P η γ 1 ,...,γ k , where η γ 1 ,...,γ k = P k i =1 a i δ γ i , w e hav e T r g ∈ R for an y g ∈ G whose axis pro jects to a curv e γ i , i = 1 , . . . , k . In order to compute pleating ra ys, w e need the following result whic h is a sp ecial case of Theorems B and C of [4], see also [11]. Recall that a 20 SARA MALONI co dimension- p submanifold N  → C n is called total ly r e al if it is defined lo cally by equations = f i = 0 , i = 1 , . . . , p , where f i , i = 1 , . . . , n are lo cal holomorphic co ordinates for C n . As usual, if γ is a b ending line w e denote its bending angle b y θ γ . Recall that the c omplex length λ ( A ) of a loxodromic element A ∈ S L (2 , C ) is defined b y T r A = 2 cosh λ ( A ) 2 , see e.g. [4] for details. By construction, P γ 1 ,...,γ k ⊂ M ⊂ R (Σ). Theorem 2.18. The c omplex lengths λ ( γ 1 ) , . . . , λ ( γ k ) ar e lo c al holo- morphic c o or dinates for R (Σ) in a neighb ourho o d of P η γ 1 ,...,γ k . Mor e- over P η γ 1 ,...,γ k is c onne cte d and is lo c al ly define d as the total ly r e al sub- manifold = T r γ i = 0 , i = 1 , 2 of R . A ny k –tuple ( f 1 , f 2 , . . . , f k ) , wher e f i is either the hyp erb olic length < λ ( γ i ) or the b ending angle θ γ i , ar e glob al c o or dinates on P η γ 1 ,...,γ k . This result extends to P η γ 1 ,...,γ k , except that one has to replace < λ ( γ i ) b y T r γ i in a neigh b ourho o d of a p oint for which γ i is parab olic. In fact, as discussed in [4, Section 3.1], complex length and traces are in terc hangeable except at cusps (where traces must b e used) and p oin ts where a b ending angle v anishes (where complex length must b e used). The parameterisation b y lengths or angles extends to P γ 1 ,...,γ k . Notice that the ab o v e theorem gives a local c haracterisation of P η γ 1 ,...,γ k as a subset of the representation v ariet y R and not just of M . In other w ords, to lo cate P , one do es not need to chec k whether nearb y p oints lie a priori in M ; it is enough to chec k that the traces remain real and a w a y from 2 and that the b ending angle on one or other of θ γ i do es not v anish. As w e shall see, this last condition can easily b e c hec k ed b y requiring that further traces b e real v alued. 3. Main theorems In this section w e will prov e our main results. As explained in the In tro duction, we will extend to a general hyperb olic surface Σ g ,n the results pro v ed b y Series [22] for the case of a t wice punctured torus Σ 1 , 2 . As already observ ed by Series, almost all the results of Section 6 [22] generalise straigh tforw ardy , but for Section 7 [22] some non-trivial extensions are needed. So w e will only restate the most imp ortan t theorems of Section 6 without pro of and refer to the original pap er for a more detailed discussion. Almost all the results of Section 7 still remain true, but we will discuss how to generalise them more deeply . In addition, w e find how to include the case of ‘exceptional curves’ in the pro of of the main theorems (so we will not need to discuss that case separately). W e will also correct some misprin ts in [22]. All these remarks will b e explained in detail later on. THE ASYMPTOTIC DIRECTIONS 21 The key idea for proving these theorems is to understand the ge- ometry of the top comp onent ∂ + C ( G ) of the conv ex core for groups G = G η ( θ ) ∈ P η ⊂ M as θ − → 0 . Recall that the definition of M de- p ends on the c hoice of a pan ts decomp osition P C = { σ 1 , . . . , σ ξ } , whic h tells us the curv es which will b e pinched in the b ottom surface of the asso ciated manifold. Before stating the results, w e need to fix some notation. W e will use Series’ notation, so that the interested reader can refer to the pap er [22] more easily . Notation 3.1. Giv en a quantit y X = X ( σ i ) whic h dep end on the pants curv e σ i ∈ P C , we will write X ( σ i ) = O ( θ e ) , meaning that X 6 cθ e as θ − → 0 for some constan t c > 0, where e is an exp onent (usually e = 0 , 1). R emark 3.2 . Note that the estimates b elo w all dep ends on the lami- nation η . So, more precisely , one has X 6 c ( η ) θ e . Ho w ever it is easily seen, by following through the arguments, that the dep endence on η is alwa ys of the form X ( σ i ) 6 cq e θ e , where q = i ( σ i , η ) and where, no w, c is a universal constan t indep enden t of η . The dep endence of the constan ts on η is not imp ortant for our argumen t, but it may b e useful elsewhere. The main theorem in Section 6 of [22] is Prop osition 6.1. The pro of of this result relies on three other main lemmas prov ed in the same section, namely Prop osition 6.6 and 6.11 for the asymptotic b eha viour of the imaginary part of the parameters τ i and Prop osition 6.14 for the real part. (See Series’ article for the pro ofs.) The only remark is that the role pla y ed in Σ 1 , 2 b y the curv e γ T for the pan ts curves σ 1 and σ 2 should b e replaced b y the curves D i , dual to the pan ts curv e σ i . In fact, the imp ortant prop erty of γ T is that it in tersects σ i minimally . In particular, for the second part of the pro of of Corollary 6.5 instead of using T r[ T , S − 1 i ] you should use T r D i , and for Prop osition 6.18 instead of calculating i σ i ( γ , T ) y ou should deal with i σ i ( γ , D i ). The ideas for the pro ofs remain how ever the same. Finally , w e remind the reader that the twist parameters p i used in this article are twice the v alue of the ‘old’ parameters (again called p i ) used by Series in [22]. The parameters p i w e are using in this article are the twist parameters using D. Thurston’s standard p osition (as defined in [13]). A generalisation of Prop osition 6.1 of [22] is the following. Theorem 3.3. L et η = P k i =1 a i δ γ i b e an admissible r ational me asur e d lamination on the surfac e Σ = Σ g ,b and let G = G η ( θ ) b e the unique 22 SARA MALONI gr oup in M with pl + ( G ) = θ η . Then, as θ − → 0 , we have: < τ i = − p i ( η ) q i ( η ) + O (1) and = τ i = 4 + O ( θ ) θ q i ( η ) , wher e O (1) denotes a universal b ound indep endent of η . Corollary 3.4. With the same hyp othesis as The or em 3.3, as θ − → 0 , we have: = τ i < τ i < τ i = τ i = p i p j + O ( θ ) and = τ i = τ j = q j q i + O ( θ ) . This result is enough in order to prov e Theorem B. W e will follo w Series’ pro of v ery closely . Pr o of of The or em B. Let η = P ξ 1 a i δ γ i b e admissible and let G = G η ( θ ) b e the unique group for whic h pl + ( G ) = θ η . Let h ( θ ) denote the h yp erb olic structure of ∂ C + ( G ). Let l + σ i b e the hyperb olic length of the geo desic represen tativ e of σ i on the hyperb olic surface ∂ + C ( G ). Since l + σ i − → 0, for all i = 1 , . . . , ξ , the limit of the structures h ( θ ) in P M L (Σ) is in the linear span of δ σ 1 , . . . , δ σ ξ . W e w an t to prov e that the limit is the barycentre P ξ 1 δ σ i . Let δ, δ 0 ∈ S . Since σ 1 , . . . , σ ξ are a maximal set of simple curves on Σ, the thin part of h ( θ ) is even tually con tained in collars A i around σ i of appro ximate width log( 1 l + σ i ) and the lengths of δ, δ 0 outside the collars A i are b ounded (with a b ound dep ending only on the combinatorics of δ, δ 0 and hence the canonical co ordinates i ( δ ) , i ( δ 0 )). By the results of Section 6.4 of [22], the t wisting around A i is b ounded. W e deduce that for an y curv e transv erse to σ i w e ha v e (3.1) l + δ = 2 ξ X i =1 q i ( δ ) log ( 1 l + σ i ) + O (1) , see for example Prop osition 4.2 of Diaz and Series [6]. By Theorem 3.3 w e ha ve l + σ i l + σ j − → q j ( η ) q i ( η ) , and since η is admissible, q i ( ξ ) > 0 for i = 1 , . . . , ξ . Thus log l + σ i log l + σ j − → 1. Hence l + δ l + δ 0 − → P ξ i =1 q i ( δ ) P ξ i =1 q i ( δ 0 ) = i ( δ, P ξ 1 δ σ i ) i ( δ 0 , P ξ 1 δ σ i ) . The result follo ws from the definition of con v ergence to a p oint in P M L (Σ).  THE ASYMPTOTIC DIRECTIONS 23 The next results are the k ey to ols for the pro ofs of Theorems A. W e need to fix more notation. Supp ose that γ is a b ending line of ∂ C + ( G ) for a group G ( τ ) ∈ P η . The T op T erms’ Relationship 2.11, together with the condition T r γ ∈ R of Lemma 2.17, gives asymptotic conditions for τ ∈ P ξ , in terms of the canonical co ordinates i ( γ ) of γ . In particular, for τ = ( τ 1 , . . . , τ ξ ) ∈ C ξ set τ i = x i + iy i , ρ = k ( y 1 , . . . , y ξ ) k = ( y 2 1 + . . . + y 2 ξ ) 1 2 , and η i = y i ρ . Define E γ ( τ 1 , . . . , τ ξ ) = η 2 · · · η ξ ( q 1 x 1 + p 1 ) + . . . + η 1 · · · η ξ − 1 ( q ξ x ξ + p ξ ) = η 1 · · · η ξ ξ X i =1 ( q i x i + p i ) η i , where as usual i ( γ ) = ( q 1 ( γ ) , p 1 ( γ ) , . . . , q ξ ( γ ) , p ξ ( γ )) and y i > 0 , i = 1 , . . . , ξ . The reason why we introduced this notation is the follo wing result, whic h generalises Prop osition 7.1 of [22]. Again Series’ pro of extends clearly to our case. Prop osition 3.5. Supp ose that η ∈ M L Q is an admissible lamination, that G ( τ 1 , . . . , τ ξ ) ∈ P η has b ending me asur e pl + ( G ) = θ η , and that γ is a b ending line of η . Then, as θ − → 0 , we have E γ ( τ 1 , . . . , τ ξ ) = O ( θ ) . No w we wan t to lo cate the pleating ray P η where η = P k i =1 a i γ i . If G ∈ P γ 1 ,...,γ k , then ∂ C + ( G ) − { γ 1 , . . . , γ k } is flat, so that not only γ 1 , . . . , γ k , but also any curve δ ∈ lk( γ 1 , . . . , γ k ), is a b ending line for G , where lk( γ 1 , . . . , γ k ) denotes the link of the simplex ( γ 1 , . . . , γ k ) in the complex of curves C (Σ). One can think of it as the set of all curv es δ ∈ S = S (Σ) disjoint from γ 1 , . . . , γ k . Thus τ = ( τ 1 , . . . , τ ξ ) is constrained b y the equations = T r γ i = = T r δ = 0 ∀ i = 1 , . . . , k , ∀ δ ∈ lk( γ 1 , . . . , γ k ) and hence, using the Prop osition 3.5, it is constrained by the following equations E γ i ( τ 1 , . . . , τ ξ ) + O ( θ ) = 0 , and E δ ( τ 1 , . . . , τ ξ ) + O ( θ ) = 0 for all δ ∈ lk( γ 1 , . . . , γ k ) and for i = 1 , . . . , k . No w w e w ould lik e to describ e ho w to solv e these equations sim ultaneously for τ 1 , . . . , τ ξ . F ollowing the analysis in Section 7 of [22], we recall that for an y curv e ω ∈ S we hav e E ω ( τ 1 , . . . , τ ξ ) = i ( ω ) · u , 24 SARA MALONI where i ( ω ) = ( q 1 ( ω ) , p 1 ( ω ) , . . . , q ξ ( ω ) , p ξ ( ω )) and u = ( u 11 , u 12 , . . . , u ξ 1 , u ξ 2 ) = η 1 · · · η ξ ( x 1 η 1 , 1 η 1 , . . . , x ξ η ξ , 1 η ξ ) = ( η 2 · · · η ξ x 1 , η 2 · · · η ξ , . . . , η 1 · · · η ξ − 1 x ξ , η 1 · · · η ξ − 1 ) with x i = < τ i , η i = = τ i ρ as ab o v e. W e will use linear algebra and Th urston’s symplectic form Ω Th to solv e the equations i ( γ i ) · u = 0 , i ( δ ) · u = 0 for all δ ∈ lk( γ 1 , . . . , γ k ) and for i = 1 , . . . , k . As already noted in Section 2.1.2, this symplectic form induces a map R 2 ξ − → R 2 ξ defined b y x = ( x 1 , y 1 , . . . , x ξ , y ξ ) − → x ∗ = ( y 1 , − x 1 , . . . , y ξ , − x ξ ) suc h that Ω Th ( i ( γ ) , i ( δ )) = i ( γ ) · i ( δ ) ∗ where · is the usual inner pro duct on R 2 ξ . W e need the follo wing Lemma, which generalise Lemma 7.2 of [22]. See Section 2.6 of Penner [19] for a definition of standar d train trac ks. Note that, although not necessary , w e will use the language of the curv e and marking complexes, since many readers may find it useful. See Section 2.1.3 for the basic definitions. Lemma 3.6. (i) Supp ose that g = ( γ 1 , . . . , γ k ) is a simplex in the c omplex of curves C (Σ) . Then γ i ar e supp orte d on a c ommon standar d tr ain tr ack and i ( γ i ) ar e indep endent ve ctors in i ( M L Q (Σ)) ⊂ ( Z + × Z ) ξ . (ii) Given any simplex g = ( γ 1 , . . . , γ k ) in the c omplex of curves C (Σ) , we c an find curves γ k +1 , . . . , γ ξ , D k +1 , . . . , D ξ ∈ lk C (Σ) ( g ) such that the elements ( γ 1 , . . . , γ ξ ) and ( γ 1 , . . . , γ j , D j +1 , . . . , D ξ ) with j = k , . . . , ξ − 1 , ar e simplic es in C (Σ) and such that the ve ctors i ( γ 1 ) , . . . , i ( γ ξ ) , i ( D k +1 ) . . . , i ( D ξ ) sp an a subsp ac e of r e al dimension 2 ξ − k in i ( M L Q (Σ)) ⊂ ( Z + × Z ) ξ . Pr o of. ( i ): F ollowing Series’ proof, the disjointness of the curves γ 1 , . . . , γ k tells us they are supp orted on a common standard train trac k. The sec- ond part of ( ii ) is prov ed, as a particular case, in the pro of of ( ii ). ( ii ): The idea is to complete g to a pants decomp osition of Σ and to consider the dual curves of the pants curves added. In detail let γ k +1 , . . . , γ ξ b e such that { γ 1 , . . . , γ ξ } is a pan ts decomp osition of Σ and let D i b e the dual curv e of γ i . (Note that D i is disjoint from any pan ts curv e γ j when j 6 = i and intersects γ i t wice.) Using the language of Masur and Minsky [17], we can say we ha v e chosen a complete, clean marking µ = ( γ 1 , . . . , γ ξ ; D 1 , . . . , D ξ ) (that is a vertex in the marking THE ASYMPTOTIC DIRECTIONS 25 complex where γ 1 , . . . , γ k are curves in the base of µ ) and we define a path µ = µ 0 , µ 1 , . . . , µ 2 ξ − k b y the requirement µ i is obtained from µ i − 1 b y flipping γ k + i and D k + i for i = 1 , . . . , ξ − k . The simplices in the statemen t of the theorem are then the bases of the markings µ i for i = 0 , . . . , 2 ξ − k . W e wan t to show that the v ectors i ( γ 1 ) , . . . , i ( γ ξ ), i ( D k +1 ) . . . , i ( D ξ ) are linear indep endent. Without loss of generalit y , w e can assume the map i is defined with resp ect to the marking µ 0 . Indeed, if that it is not the case, the c hange of co ordinates betw een the map i and a new map i 0 defined with respect to a new marking µ 0 is a linear map, which doesn’t c hange our conclusion about the linear indep endence of the v ectors. No w the vector i ( γ i ) = ( q 1 , p 1 , . . . , q ξ , p ξ ) is defined b y p i = 1 and q j = p j = q i = 0 for all j 6 = i and the vector i ( D i ) = ( q 1 , p 1 , . . . , q ξ , p ξ ) is defined b y q i = 2 and q j = p j = p i = 0 for all j 6 = i . (See Remark 2.1 for a description of the conv en tion on dual curves that we are using.) This pro v es that the v ectors i ( γ 1 ) , . . . , i ( γ ξ ) , i ( δ k +1 ) . . . , i ( δ ξ ) are linearly indep enden t.  No w w e can state precisely Theorem A of the Introduction. Theorem 3.7 (Theorem A) . Supp ose that η = P k i =1 a i γ i is admis- sible (and k 6 ξ ). L et i ( η ) = ( q 1 ( η ) , p 1 ( η ) , . . . , q ξ ( η ) , p ξ ( η )) . L et L η : [0 , ∞ ) − → C ξ b e the line t 7→ ( w 1 ( t ) , . . . , w ξ ( t )) wher e w i ( t ) = − p i q i + it q 1 q i . L et ( τ 1 ( θ ) , . . . , τ ξ ( θ )) ∈ C ξ b e the p oint c orr esp onding to the gr oup G η ( θ ) with pl + ( G ) = θ η , so that the ple ating r ay P η is the image of the map p η : θ − → ( τ 1 ( θ ) , . . . , τ ξ ( θ )) for a suitable r ange of θ > 0 . Then P η appr o aches L η as θ − → 0 in the sense that if t ( θ ) = 4 θq 1 , then |< τ i ( θ ) −< w i ( t ( θ )) | = O ( θ ) and |= τ i ( θ ) −= w i ( t ( θ )) | = O (1) , i = 1 , . . . , ξ . R emark 3.8 . Note that here, in con trast to the approac h follow ed by Series in [22], we do not need to exclude from our statemen ts the case of ‘exceptional curves’ and to b e dealt with separately . F or completeness, w e include a definition of exceptional curves, but the in terested reader should see [22] for a deep er discussion. Definition 3.9. A geo desic lamination η = P k i =1 a i δ γ i is exc eptional if the matrix ( q i ( γ j )) i =1 ,...,ξ j =1 ,...,k has no maximal rank. W e are now ready to pro v e the theorem. 26 SARA MALONI Pr o of of The or em 3.7. W e will use the previous notation, that is we will write τ i ( θ ) = τ i = x i + iy i , ρ = k ( y 1 , . . . , y ξ ) k , and η i = y i ρ , where the dep endence on θ is clear. By Theorem 3.3, we ha v e y i − 4 θq i = O (1). On the other hand, with t = t ( θ ) as in the statement of the theorem, w e find = w i ( t ) = t q 1 q i = 4 θq i . Thus for i = 1 , . . . , ξ we hav e |= τ i ( θ ) − = w i ( t ( θ )) | = O (1) , as θ − → 0, as w e w an ted to pro v e. No w, let’s deal with the co ordinates x i = < τ i ( θ ). Given γ 1 , . . . , γ k , let γ k +1 , . . . , γ ξ , D k +1 , . . . , D ξ the curv es defined b y Lemma 3.6. If ( τ 1 , . . . , τ ξ ) ∈ P η , then the curves γ 1 , . . . , γ k , γ k +1 , . . . , γ ξ , D k +1 , . . . , D ξ are all b ending lines of G ( τ 1 , . . . , τ ξ ). It follows, that = T r( γ i ) = = T r( D j ) = 0 for i = 1 , . . . , ξ and j = k + 1 , . . . , ξ . So, b y Prop osition 3.5, it follo ws that E ζ ( τ 1 , . . . , τ ξ ) = O ( θ ) as θ − → 0 for ζ ∈ { γ 1 , . . . , γ ξ , D k +1 , . . . , D ξ } . Defining η = η 1 · · · η ξ and regarding these as equations in R 2 ξ for a parameter u ∈ R 2 ξ , where u = ( u 11 , u 12 , . . . , u ξ 1 , u ξ 2 ) = η ( x 1 η 1 , 1 η 1 , . . . , x ξ η ξ , 1 η ξ ) , w e ha v e, for ζ ∈ { γ 1 , . . . , γ ξ , D k +1 , . . . , D ξ } , (3.2) i ( ζ ) · u = O ( θ ) . By Theorem 2.4, we hav e Ω Th ( γ i , ζ ) = 0 for i = 1 , . . . , k for an y ζ ∈ lk C ( γ i ) ∪ { γ 1 , . . . , γ k } . Hence i ( ζ ) · i ( γ i ) ∗ = 0 for i = 1 , . . . , k and for all ζ ∈ { γ 1 , . . . , γ ξ , D k +1 , . . . , D ξ } . Since i ( γ 1 ) , . . . , i ( γ ξ ), i ( D k +1 ) . . . , i ( D ξ ) are indep enden t, it follo ws that w e can write (3.3) u ( θ ) = λ 1 ( θ ) i ( γ 1 ) ∗ + . . . + λ k ( θ ) i ( γ k ) ∗ + η ( θ ) v ( θ ) where v = v ( θ ) is in the linear span of i ( γ 1 ) , . . . , i ( γ ξ ), i ( D k +1 ) . . . , i ( D ξ ) and || v || = 1. Using (3.2) w e find that u · v = O ( θ ) (where the constan ts dep end on i ( γ 1 ) , . . . , i ( γ ξ ), i ( D k +1 ) . . . , i ( D ξ )). Then v · i ( γ i ) ∗ = 0 for i = 1 , . . . , k giv es η ( θ ) = O ( θ ). Equating the tw o sides of (3.3) gives u i 1 = η x i η i = λ 1 p i ( γ 1 ) + · · · + λ k p i ( γ k ) + O ( θ ) , u i 2 = η η i = − λ 1 q i ( γ 1 ) − · · · − λ k q i ( γ k ) + O ( θ ) . (3.4) So we prov ed u b elongs to the k –dimensional subspace Π generated by i ( γ 1 ) ∗ , . . . , i ( γ k ) ∗ . No w we wan t to prov e u is approximately parallel THE ASYMPTOTIC DIRECTIONS 27 to the v ector i ( η ) ∗ , that is ( λ 1 , . . . , λ k ) is prop ortional to ( a 1 , . . . , a k ). T o do this, and to av oid the restriction to non exceptional curves, we mo dify sligh tly Series’ approac h. By Corollary 3.4, w e ha v e     y i y j − a 1 q j ( γ 1 ) + · · · + a k q j ( γ k ) a 1 q i ( γ 1 ) + · · · + a k q i ( γ k )     = O ( θ )     x j y i y j x i − a 1 p j ( γ 1 ) + · · · + a k p j ( γ k ) a 1 p i ( γ 1 ) + · · · + a k p i ( γ k )     = O ( θ ) . (3.5) W e can now put this information together as:     ( y i y j + i x j y i y j x i ) − a 1 Q j ( γ 1 ) + · · · + a k Q j ( γ k ) a 1 Q i ( γ 1 ) + · · · + a k Q i ( γ k )     = O ( θ ) , where we defined Q i ( γ ) = q i ( γ ) + ip i ( γ ) in order to keep the notation more neat. Defining new v ariables W i = λ 1 ( q i ( γ 1 ) + ip i ( γ 1 )) + · · · + λ k ( q i ( γ k ) + ip i ( γ k )), w e hav e, by (3.4), < W i = − u i 2 and = W i = u i 1 . So w e ha v e     < W j < W i − y i y j     = O ( θ ) and     = W j = W i − x j y i y j x i     = O ( θ ) . Hence w e get (3.6)     ( < W j < W i + i = W j = W i ) − ( y i y j + i x j y i y j x i )     = O ( θ ) . No w using equations 3.5, 3.6 and the definition of the v ariables W i , w e get     λ 1 Q j ( γ 1 ) + · · · + λ k Q j ( γ k ) λ 1 Q i ( γ 1 ) + · · · + λ k Q i ( γ k ) − a 1 Q j ( γ 1 ) + · · · + a k Q j ( γ k ) a 1 Q i ( γ 1 ) + · · · + a k Q i ( γ k )     = O ( θ ) . Since this is true for all i, j = 1 , . . . , ξ , i 6 = j , and since the matrix ( Q r ( γ s )) r =1 ,...,ξ s =1 ,...,k has maximal rank (b ecause, since the curves γ 1 , . . . , γ k are distinct , the lines of that matrix are linearly indep endent) and since the norm of the vector ( λ 1 − a 1 , . . . , λ k − a k ) is one, then w e can conclude the follo wing:     λ i λ j − a i a j     = O ( θ ) , ∀ i, j = 1 , . . . , k , i 6 = j, that is u = α i ( η ) ∗ for some α > 0, as w e w an ted to pro v e.  R emark 3.10 . W e were able to get rid of the h yp othesis of non-exceptionality , since we lo oked sim ultaneously at b oth the length and the t wist of the Dehn–Th urston co ordinates for the distinct curv es γ 1 , . . . , γ k . 28 SARA MALONI References 1. F. Bonahon Ge o desic laminations on surfac es , Contemp Math 269 (2001) 1– 37. 2. F. Bonahon Ge o desic laminations with tr ansverse H¨ older distributions , Ann Sci Ecole Norm Sup 30 (1997) 205–240. 3. F. Bonahon, J.-P . 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