The Boundary at Infinity of the Curve Complex and the Relative Teichmüller Space
aa r X i v : . [ m a t h . G T ] M a r The Boundary at Infinity of the Curve Complex and the RelativeTeichm¨uller Space
Erica KlarreichFebruary 11, 1999
Abstract
In this paper we study the boundary at infinity of the curve complex C ( S ) of a surface S of finite type and the relative Teichm¨uller space T el ( S ) obtained from the Teichm¨uller space bycollapsing each region where a simple closed curve is short to be a set of diameter 1. C ( S ) and T el ( S ) are quasi-isometric, and Masur-Minsky have shown that C ( S ) and T el ( S ) are hyperbolicin the sense of Gromov. We show that the boundary at infinity of C ( S ) and T el ( S ) is the spaceof topological equivalence classes of minimal foliations on S . There is a strong but limited analogy between the geometry of the Teichm¨uller space T ( S ) of asurface S and that of hyperbolic spaces. Teichm¨uller space has many of the large-scale qualities ofhyperbolic space, and in fact the Teichm¨uller space of the torus is H . At one point it was generallybelieved that the Teichm¨uller metric was negatively curved; however, Masur ([13]) showed that thisis not so, apart from a few exceptional cases. Since then, Masur and Wolf ([15]) showed that T ( S )is not even hyperbolic in the sense of Gromov.One way in which T ( S ) differs from hyperbolic space is that it does not have a canonicalcompactification. A Gromov hyperbolic space has a boundary at infinity that is natural in thefollowing two senses, among others: the boundary consists of all endpoints of quasigeodesic rays upto equivalence (two rays are equivalent if they stay a bounded distance from each other), and everyisometry of the space extends continuously to a homeomorphism of the boundary. Teichm¨uller spacecannot be equipped with such a compactification but rather gives rise to several compactifications,each with advantages and drawbacks.Questions about the boundary of a hyperbolic space are interesting for many reasons; one isthat they tie in to questions of rigidity of group actions by isometry on the space. For example,in the proof of Mostow’s Rigidity Theorem, a key step in showing that two hyperbolic structureson the same compact 3-manifold are isometric is to show that a quasi-isometry between the twostructures lifts to a map of H that extends continuously to ∂ ∞ H (the Riemann sphere), and thento gain some control over the map on ∂ ∞ H . In another instance, Sullivan’s Rigidity Theoremgives geometric information about a hyperbolic 3-manifold based on quasi-conformal informationabout its associated group action on ∂ ∞ H .Although Teichm¨uller space is not hyperbolic, it is natural to be interested in boundaries ofTeichm¨uller space, since they have a strong connection to deformation spaces of hyperbolic 3-manifolds. If M is a compact 3-manifold, there is a well-known parametrization of the space ofgeometrically finite hyperbolic structures on int ( M ) by the Teichm¨uller space of the boundary1f M ; see [1]. One question is to understand the behavior of the hyperbolic structure on M asthe Riemann surface structure on ∂M “degenerates”, that is, goes to infinity in the Teichm¨ullerspace. More generally, an important problem in the theory is to describe all geometrically infinitehyperbolic structures on M ; for this purpose Thurston has introduced an invariant called the ending lamination of ∂M , intended to play a similar role to that of the Teichm¨uller space of ∂M in the geometrically finite setting. Two important boundaries of T ( S ) by Teichm¨uller andThurston involve compactifying T ( S ) by the measured foliation space, or equivalently the measuredlamination space, which is related to but not the same as the space of possible ending laminationson S .Masur and Minsky ([14]) have shown that although Teichm¨uller space is not Gromov hyperbolic,it is relatively hyperbolic with respect to a certain collection of closed subsets. In this paper wedescribe the boundary at infinity of the relative Teichm¨uller space and a closely related object,the curve complex. If α is a homotopy class of simple closed curves on S , a surface of finite type,let T hin α denote the region of T ( S ) where the extremal length of α is less than or equal to ǫ ,for some fixed small ǫ >
0. These regions play a role somewhat similar to that of horoballs inhyperbolic space; in fact for the torus, these regions are actual horoballs in H . However, Minskyhas shown ([17]) that in general the geometry of each region T hin α is not hyperbolic, but ratherhas the large-scale geometry of a product space with the sup metric. On the other hand, theseregions are in a sense the only obstacle to hyperbolicity: Masur and Minsky ([14]) have shown thatTeichm¨uller space is relatively hyperbolic with respect to the family of regions { T hin α } . In otherwords, the electric Teichm¨uller space T el ( S ) obtained from T ( S ) by collapsing each region T hin α to diameter 1 is Gromov hyperbolic (this collapsing is done by adding a point for each set T hin α that is distance from each point in T hin α ).Since T el ( S ) is Gromov hyperbolic, it can be equipped with a boundary at infinity ∂ ∞ T el ( S ).Our main result is the following: Theorem 1.1
The boundary at infinity of T el ( S ) is homeomorphic to the space of minimal topo-logical foliations on S . A foliation is minimal if no trajectory is a simple closed curve. This space of minimal foliationsis exactly the space of possible ending laminations (or foliations) on a surface S that correspondsto a geometrically infinite end of a hyperbolic manifold that has no parabolics (see [23]). Here thetopology on the space of minimal foliations is that obtained from the measured foliation space byforgetting the measures. This topology is Hausdorff (see the appendix), unlike the topology on thefull space of topological foliations; hence we may prove Theorem 1.1 using sequential arguments toestablish continuity.We will prove Theorem 1.1 by showing that the inclusion of T ( S ) in T el ( S ) extends continuouslyto a portion of the Teichm¨uller compactification of T ( S ) by the projective measured foliation space PMF ( S ): 2 heorem 1.2 The inclusion map from T ( S ) to T el ( S ) extends continuously to the portion PMF min ( S ) of PMF ( S ) consisting of minimal foliations, to give a map π : PMF min ( S ) → ∂ ∞ T el ( S ) . Themap π is surjective, and π ( F ) = π ( G ) if and only if F and G are topologically equivalent. Moreover,any sequence { x n } in T ( S ) that converges to a point in PMF ( S ) \ PMF min ( S ) cannot accumulatein the electric space onto any portion of ∂ ∞ T el ( S ) . If F min ( S ) is the space of minimal topological foliations on S , the map π : PMF min ( S ) → ∂ ∞ T el ( S ) descends to a homeomorphism from F min ( S ) to ∂ ∞ T el ( S ); hence Theorem 1.1 is a con-sequence of Theorem 1.2.Another space that we can associate to S that has a close connection to the electric Teichm¨ullerspace is the curve complex C ( S ), originally described by Harvey in [10]. C ( S ) is a simplicial complexwhose vertices are homotopy classes of non-peripheral simple closed curves on S . A collection ofcurves forms a simplex if all the curves may be simultaneously realized so that they are pairwisedisjoint (when S is the torus, once-punctured torus or four-punctured sphere it is appropriate tomake a slightly different definition; see Section 4). C ( S ) can be given a metric structure by assigningto each simplex the geometry of a regular Euclidean simplex whose edges have length 1.In the construction of the electric Teichm¨uller space, if the value of ǫ used to define the sets T hin α is sufficiently small, then T hin α and T hin β intersect exactly when α and β have disjointrealizations on S , that is, when the elements α and β of C ( S ) are connected by an edge. Hence the1-skeleton C ( S ) of C ( S ) describes the intersection pattern of the sets T hin α ; C ( S ) is the nerveof the collection { T hin α } . The relationship between T el ( S ) and C ( S ) is not purely topological.Masur and Minsky have shown ([14]) that T el ( S ) is quasi-isometric to C ( S ) and C ( S ). This impliesthat C ( S ) and C ( S ) are also Gromov hyperbolic (although in the proof if Masur and Minsky, theimplication goes in the other direction). Two Gromov hyperbolic spaces that are quasi-isometrichave the same boundary at infinity, so a consequence of Theorem 1.1 is the following: Theorem 1.3
The boundary at infinity of the curve complex C ( S ) is the space of minimal foliationson S . As with Teichm¨uller space, the curve complex is important in the study of hyperbolic 3-manifolds. Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure,and suppose S is a component of ∂M that corresponds to a geometrically infinite end e of M .Thurston, Bonahon and Canary ([25, 2, 3]) have shown that there is a sequence of simple closedcurves α n ∈ C ( S ) whose geodesic representatives in M “exit the end e ”, that is, are containedin smaller and smaller neighborhoods of S in M . Further, they showed that every such sequenceconverges to a unique geodesic lamination (equivalently, foliation) on S . In the case when thehyperbolic structure on int ( M ) has a uniform lower bound on injectivity radius, Minsky has shown([20, 22]) that the sequence { α n } is a quasigeodesic in C ( S ); a form of this was a key step in hisproof of the Ending Lamination Conjecture for such manifolds, giving quasi-isometric control ofthe ends of M . 3ince the sequence { α n } is a quasigeodesic in C ( S ), it must converge to a point F in theboundary at infinity of C ( S ), which we have described as the space of minimal foliations (orlaminations) on S . We will show that this description is natural, so that in particular when thesequence { α n } in C ( S ) arises as described above in the context of hyperbolic 3-manifolds, theboundary point F is the ending lamination. Theorem 1.4
Let { α n } be a sequence of elements of C ( S ) that converges to a foliation F in theboundary at infinity of C ( S ) . Then regarding the curves α n as elements of the projective measuredfoliation space PMF ( S ) , every accumulation point of { α n } in PMF ( S ) is topologically equivalentto F . It is interesting to note that our description of the boundary of C ( S ) ultimately does not dependon our original choice of a Teichm¨uller compactification for T ( S ), even though the Teichm¨ullerboundary of T ( S ) depends heavily on an initial choice of basepoint in T ( S ) (see Section 2 for moredetails). Kerckhoff has shown (see [12]) that the action of the modular group by isometry on T ( S )does not extend continuously to the Teichm¨uller boundary; on the other hand, the natural actionsof the modular group on T el ( S ) and C ( S ) do extend to the boundary at infinity, since this is true ofany action by isometry on a Gromov hyperbolic space. Hence the collapse used in the constructionof T el ( S ) essentially “collapses” the discontinuity of the modular group action.In Section 2 we will give an overview of some of the basic theory of Teichm¨uller space andquadratic differentials. Section 3 contains the essential ideas of Gromov hyperbolicity that we willneed. In Section 4 we discuss in more detail Masur and Minsky’s work on the electric Teichm¨ullerspace and the curve complex, and describe the quasi-isometry between them. In Section 5 weestablish some facts about convergence properties of sequences of Teichm¨uller geodesics, which areused in Section 6 to prove the main theorems. Acknowledgments.
The author would like to thank Dick Canary and Yair Minsky for interestingconversations, and Howard Masur for suggesting a portion of the argument for Proposition 5.1.
Let S be a surface of finite genus and finitely many punctures. The Teichm¨uller space T ( S ) is thespace of all equivalence classes of conformal structures of finite type on S , where two conformalstructures are equivalent if there is a conformal homeomorphism of one to the other that is isotopicto the identity on S . A conformal structure is of finite type if every puncture has a neighborhoodthat is conformally equivalent to a punctured disk. The Teichm¨uller distance between two points σ and τ ∈ T ( S ) is defined by d ( σ, τ ) = 12 log K ( σ, τ ) , K ( σ, τ ) is the minimal quasiconformal dilatation of any homeomorphism from a representa-tive of σ to a representative of τ in the correct homotopy class. The extremal map from σ to τ may be constructed explicity using quadratic differentials.A holomorphic quadratic differential q on a Riemann surface σ is a tensor of the form q ( z ) dz in local coordinates, where q ( z ) is holomorphic. We define k q k = Z Z S | q ( z ) | dxdy. Let DQ ( σ ) denote the open unit ball in the space Q ( σ ) of quadratic differentials on σ , and SQ ( σ )the unit sphere.Every q ∈ DQ ( σ ) determines a Beltrami differential k q k q | q | on σ , which in turn determines aquasiconformal map from σ to a new element τ q of T ( S ); this map is the Teichm¨uller extremal mapbetween σ and τ q . The map that sends q to τ q is a homeomorphism, giving an embedding of T ( S )in Q ( σ ); SQ ( σ ) is the boundary of T ( S ) in Q ( σ ), and T ( S ) ∪ SQ ( σ ) gives a compactification of T ( S ) which we will denote T ( S ), called the Teichm¨uller compactification of T ( S ).Any q ∈ Q ( σ ) determines a pair H q and V q of measured foliations on S called the horizontal andvertical foliations. Measured foliations are equivalence classes of foliations of S with 3- or higher-pronged saddle singularities, equipped with transverse measures; the equivalence is by measure-preserving isotopy and Whitehead moves (that collapse singularities). We will denote the measuredfoliation space by MF ( S ) and the projectivized measured foliation space (obtained by scaling themeasures) by PMF ( S ). The horizontal and vertical foliations associated to q give a metric on S in the conformal class of σ that is Euclidean away from the singularities. The map from SQ ( σ ) to PMF ( S ) defined by sending q to the projective class of its vertical foliation is a homeomorphism,so that we may think of PMF ( S ) as the boundary of T ( S ) (see [11]).A unit-norm quadratic differential q on σ determines a directed geodesic line in T ( S ) as follows:for 0 ≤ k <
1, let σ k denote the element of T ( S ) determined by the quasiconformal homeomorphismgiven by the quadratic differential k · q . Geometrically, the extremal map from σ to σ k is obtainedby contracting the transverse measure of H q by a factor of K − and expanding the transversemeasure of V q by K , where K = k − k ; note that the extremal map is K -quasiconformal. Thefamily { σ k : 0 ≤ k < } , when parametrized by Teichm¨uller arclength, gives a Teichm¨uller geodesicray; the family { σ k : − < k < } determines a complete geodesic line. Every ray and line through σ is so determined. We may think of the Teichm¨uller ray { σ k : 0 ≤ k < } as terminating at theboundary point q ∈ SQ ( σ ), or equivalently, at the projective foliation V q ∈ PMF ( S ). Similarly,every pair of foliations in PMF ( S ) that fills up S (see the next section for the definition of fillingup) determines a geodesic line in T ( S ), for which if τ ∈ L then the quadratic differential on τ thatdetermines L has the two foliations as its horizontal and vertical foliations.The compactification of T ( S ) by endpoints of geodesic rays depends in a fundamental wayon the choice of basepoint σ in T ( S ). Kerckhoff has shown (see [12]) that there exist projectivefoliations F ∈ PMF ( S ) such that there are choices of τ ∈ T ( S ) for which the Teichm¨uller rayfrom τ determined by F does not converge in T ( S ) to F , but rather accumulates onto a portion5f PMF ( S ) consisting of projective foliations that are topologically equivalent but not measureequivalent to F . Intersection number. If α is a simple closed curve on S then α determines a foliation on S (whichwe will also call α ) whose non-singular leaves are all freely homotopic to α . The non-singular leavesform a cylinder, and S is obtained by gluing the boundary curves in some preassigned manner.There is a one-to-one correspondence between transverse measures on α and positive real numbers:each measure corresponds to the height of the cylinder, that is, the minimal transverse measureof all arcs connecting the two boundary curves of the cylinder. If a measure has height c , we willdenote the measured foliation by c · α . We define the intersection number of the foliations c · α and k · β by i ( c · α, k · β ) = ck · i ( α, β )where the right-hand intersection number is just the geometric intersection number of the simpleclosed curves α and β (that is, the minimal number of crossings of any pair of representatives of α and β . Thurston has shown that the collection { c · α : α a simple closed curve, c ∈ R + } is densein MF ( S ), and that the intersection number extends continuously to a function i : MF ( S ) ×MF ( S ) → R (see for instance [7]).Note that although the intersection number of two projective measured foliations is not well-defined, it still makes sense to ask whether two projective measured foliations have zero or non-zerointersection number.A foliation F is minimal if no leaves of F are simple closed curves. We say that two measuredfoliations are topologically equivalent if the topological foliations obtained by forgetting the mea-sures are equivalent with respect to isotopy and Whitehead moves that collapse the singularities.Rees ([24]) has shown the following: Proposition 2.1 If F is minimal then i ( F , G ) = 0 if and only if F and G are topologically equiv-alent. We say that two foliations F and G fill up S if for every foliation H ∈ MF ( S ), H has non-zerointersection number with at least one of F and G . A consequence of Proposition 2.1 is that if F isminimal, then whenever G is not topologically equivalent to F , F and G fill up S .Let F min ( S ) denote the space of minimal topological foliations, with topology obtained from thespace PMF min ( S ) of minimal projective measured foliations by forgetting the measures. Our goalis to show that F min ( S ) is homeomorphic to the boundary at infinity of the electric Teichm¨ullerspace. We will use sequential arguments to show that certain maps are continuous, so it is necessaryto show the following, whose proof can be found in the appendix: Proposition 2.2
The space F min ( S ) is Hausdorff and first countable. Note: The entire space F ( S ) of topological foliations on S is not Hausdorff. If α and β are twodistinct homotopy classes of simple closed curves that can be realized disjointly on S , then regarded6s topological foliations, α and β do not have disjoint neighborhoods; every neighborhood of α or β must contain the topological foliation containing both α and β , and whose non-singular leavesare all homotopic to α or β . Extremal length. If γ is a free homotopy class of simple closed curves on S , an importantconformal invariant is the extremal length of γ , which is defined as follows: Definition 2.3
Let σ ∈ T ( S ) , and let γ be a homotopy class of simple closed curves on S . Theextremal length of γ on σ , written ext σ ( γ ) , is defined by sup ρ ( l ρ ( γ )) A ρ , where ρ ranges over all metricsin the conformal class of σ , A ρ denotes the area of S with respect to ρ , and l ρ ( γ ) is the infimum ofthe length of all representatives of γ with respect to ρ . Extremal length may be extended to scalar multiples of simple closed curves by ext ( k · γ ) = k ext ( γ ),and extends continuously to the space of measured foliations.Our goal is to describe the boundary of the relative Teichm¨uller space T el ( S ) obtained from T ( S )by collapsing each of the regions T hin γ of T ( S ) to be a set of bounded diameter, where T hin γ isthe region of T ( S ) where the simple closed curve γ has short extremal length. We will need thefollowing lemma, which gives a connection between extremal length and intersection number (seefor instance [21] Lemma 3.1-3.2 for a proof): Proposition 2.4
Let q be a quadratic differential with norm less than 1 on σ ∈ T ( S ) with hori-zontal and vertical foliations H and V , and let F be a measured foliation on S . Then ext σ ( F ) ≥ ( i ( F , H )) ; likewise ext σ ( F ) ≥ ( i ( F , V )) . In this section we will present an overview of some of the basic theory of Gromov-hyperbolic spaces.References for the material in this section are [8], [9], [5] and [4].Let (∆ , d ) be a metric space. If ∆ is equipped with a basepoint 0, define the
Gromov product h x | y i of the points x and y in ∆ to be h x | y i = h x | y i = 12 ( d ( x,
0) + d ( y, − d ( x, y )) . Definition 3.1
Let δ ≥ be a real number. The metric space ∆ is δ -hyperbolic if h x | y i ≥ min ( h x | y i , h y | z i ) − δ for every x, y, z ∈ ∆ and for every choice of basepoint.
7e say that ∆ is hyperbolic in the sense of Gromov if ∆ is δ -hyperbolic for some δ .A metric space ∆ is geodesic if any two points in ∆ can be joined by a geodesic segment (notnecessarily unique). If x and y are in ∆ we write [ x, y ], ambiguously, to denote some geodesic from x to y .Heuristically, a δ -hyperbolic space is “tree-like”; more precisely, if we define an ǫ - narrow geodesicpolygon to be one such that every point on each side of the polygon is at distance ≤ ǫ from a pointin the union of the other sides, then we have Proposition 3.2
In a geodesic δ -hyperbolic metric space, every n-sided polygon ( n ≥ is n − δ -narrow. In a geodesic hyperbolic space, the Gromov product of two points x and y is roughly the distancefrom 0 to [ x, y ]; we have Proposition 3.3
Let ∆ be a geodesic, δ -hyperbolic space and let x, y ∈ ∆ . Then d (0 , [ x, y ]) − δ ≤ h x | y i ≤ d (0 , [ x, y ]) for every geodesic segment [ x, y ] . The boundary at infinity of a hyperbolic space.
If ∆ is a hyperbolic space, ∆ can beequipped with a boundary in a natural way. We say that a sequence { x n } of points in ∆ convergesat infinity if we have lim m,n →∞ h x m | x n i = ∞ ; note that this definition is independent of the choiceof basepoint, by Proposition 3.3. Given two sequences { x m } and { y n } that converge at infinity,say that { x m } and { y n } are equivalent if lim m,n →∞ h x m | y n i = ∞ . Since ∆ is hyperbolic, it is easilychecked that this is an equivalence relation. Define the boundary at infinity ∂ ∞ ∆ of ∆ to be theset of equivalence classes of sequences that converge at infinity. If ξ ∈ ∂ ∞ ∆ then we say that asequence of points in ∆ converges to ξ if the sequence belongs to the equivalence class ξ . Write∆ = ∆ ∪ ∂ ∞ ∆. When the space ∆ is a proper metric space, the boundary at infinity may also bedescribed as the set of equivalence classes of quasigeodesic rays, where two rays are equivalent ifthey are a bounded Hausdorff distance from each other. Quasi-isometries and quasi-geodesics.
Let ∆ and ∆ be two metric spaces. Let k ≥ µ ≥ to ∆ is a relation R between elements of ∆ and ∆ that has the coarse behavior of an isometry. Specifically, let R relate every element of ∆ to some subset of ∆ (so that we allow a given point in ∆ to be related to multiple points in ∆).We say that R is a ( k, µ )- quasi-isometry if for all x and x ∈ ∆ ,1 k d ( x , x ) − µ ≤ d ( y , y ) ≤ kd ( x , x ) + µ x Ry and x Ry . Note that for a quasi-isometry, given x ∈ ∆ there is an upper boundto the diameter of the set { y ∈ ∆ | xRy } , that is independent of x .We say that R is a cobounded quasi-isometry if in addition, there is some constant L such thatif y ∈ ∆, y is within L of some point that is related by R to a point in ∆ . If R is a coboundedquasi-isometry then R has a quasi-inverse, that is, a relation R ′ that relates each element of ∆ tosome subset of ∆ , with the following property: there is some constant K for which if x and x ′ areelements of ∆ such that for some y ∈ ∆, xRy and yR ′ x ′ , then d ( x, x ′ ) ≤ K .A quasi-isometry between two δ -hyperbolic spaces extends continuously to the boundary, in thefollowing sense: Theorem 3.4
Let ∆ and ∆ be Gromov-hyperbolic, and let h : ∆ → ∆ be a quasi-isometry. Forevery sequence { x n } of points in ∆ that converges to a point ξ in ∂ ∞ ∆ , the sequence { h ( x n ) } converges to a point in ∂ ∞ ∆ that depends only on ξ , so that h defines a continuous map from ∂ ∞ ∆ to ∂ ∞ ∆ . The map h : ∂ ∞ ∆ → ∂ ∞ ∆ is injective. Theorem 3.4 is, among other things, a key step in the proof of Mostow’s Rigidity Theorem.If the metric on ∆ is a path metric, a ( k, µ ) -quasigeodesic is a rectifiable path p : I → ∆, where I is an interval in R , such that for all s and t in I ,1 k l ( p | [ s,t ] ) − µ ≤ d ( p ( s ) , p ( t )) ≤ k · l ( p | [ s,t ] ) + µ. Note that if a path p : I → ∆ is parametrized by arc length then it is a quasigeodesic if and onlyif it is a quasi-isometry.The behavior of quasigeodesics in the large is like that of actual geodesics. In particular, wehave the following analogue of Proposition 3.3: Proposition 3.5
Let s : I → ∆ be a quasigeodesic with endpoints x and y . Then there areconstants K and C that only depend on the quasigeodesic constants of s and the hyperbolicityconstant of ∆ , such that K d (0 , s ( I )) − C ≤ h x | y i ≤ Kd (0 , s ( I )) + C. The Curve Complex. If S is an oriented surface of finite type, an important related objectis a simplicial complex called the curve complex . Except in the cases when S is the torus, theonce-punctured torus or a sphere with 4 or fewer punctures, we define the curve complex C ( S ) inthe following way: the vertices of C ( S ) are homotopy classes of non-peripheral simple closed curves9n S . Two curves are connected by an edge if they may be realized disjointly on S , and in generala collection of curves spans a simplex if the curves may be realized disjointly on S .When S is a sphere with 3 or fewer punctures, there are no non-peripheral curves on S , so C ( S ) is empty. When S is the 4-punctured sphere, the torus, or the once-punctured torus, thereare non-peripheral simple closed curves on S , but every pair of curves must intersect, so C ( S ) hasno edges. For these three surfaces, a more interesting space to consider is the complex in whichtwo curves are connected by an edge if they can be realized with the smallest intersection numberpossible on S (one for the tori; 2 for the sphere); we alter the definition of C ( S ) in this way. Inthese cases, C ( S ) is the Farey graph, which is well-understood (see for example [16, 18]).We give C ( S ) a metric structure by making every simplex a regular Euclidean simplex whoseedges have length 1. The main result of [14] is the following: Theorem 4.1 (Masur-Minsky) C ( S ) is a δ -hyperbolic space, where δ depends on S . Note that C ( S ) is clearly quasi-isometric to its 1-skeleton C ( S ), so that in particular C ( S ) is alsoGromov-hyperbolic. The Relative Teichm¨uller Space.
For a fixed ǫ >
0, for each curve α ∈ C ( S ) denote T hin α = { σ ∈ T ( S ) : ext σ ( α ) ≤ ǫ } . We will assume that ǫ has been chosen sufficiently small that the collar lemma holds; in that case,a collection of sets T hin α , ...T hin α n has non-empty intersection if and only if α , ..., α n can berealized disjointly on S , that is, if α , ..., α n form a simplex in C ( S ).We form the relative or electric Teichm¨uller space T el ( S ) (following terminology of Farb [6]) byattaching a new point P α for each set T hin α and an interval of length from P α to each point in T hin α . We give T el ( S ) the electric metric d el obtained from path length.Masur and Minsky have shown the following: Theorem 4.2 [14] T el ( S ) is quasi-isometric to C ( S ) . The quasi-isometry R between C ( S ) and T el ( S ) is defined as follows: if α is a curve in C ( S ), α is related to the set T hin α (or equally well, to the “added-on” point P α ). It is not difficultto see that the relation R between C ( S ) and T el ( S ) is a quasi-isometry (see [14] for a proof). C ( S ) is -dense in C ( S ) (that is, every point in C ( S ) is within of a point in C ( S )) and thecollection { T hin α } is D -dense in T el ( S ) for some D , so the relation R may easily be extended tobe a cobounded quasi-isometry from C ( S ) to T el ( S ), making C ( S ) and T el ( S ) quasi-isometric.An immediate corollary of Theorem 4.2 and Theorem 4.1 is the following: Theorem 4.3 [14] The electric Teichm¨uller space T el ( S ) is hyperbolic in the sense of Gromov.
10e will use h·|·i el to denote the Gromov product on T el ( S ). Quasigeodesics in T el ( S ) . Since T ( S ) is contained in T el ( S ), each Teichm¨uller geodesic is a pathin T el ( S ). Because certain portions of T ( S ) are collapsed to sets of bounded diameter in T el ( S ), apath whose Teichm¨uller length is very large may be contained in a subset of T el ( S ) whose diameteris small. So to understand the geometry of these paths in T el ( S ), we introduce the notion of arclength on the scale c , after Masur-Minsky: if c > p : [ a, b ] → T el ( S ) is a path, we define l c ( p [ a, b ]) = c · n where n is the smallest number for which [ a, b ] can be subdivided into n closedsubintervals J , ..., J n such that diam T el ( S ) ( p ( J i )) ≤ c .We will say that a path p : [ a, b ] → T el ( S ) in T el ( S ) is an electric quasigeodesic if for some c > k ≥ u > k l c ( p [ s, t ]) − µ ≤ d el ( p ( s ) , p ( t )) ≤ k · l c ( p [ s, t ]) + µ for all s and t in [ a, b ] (note that the right-hand side of the inequality is automatic).Masur and Minsky have shown the following, which will be important for understanding theboundary at infinity of T el ( S ): Theorem 4.4 [14] Teichm¨uller geodesics in T ( S ) are electric quasigeodesics in T el ( S ) , with uni-form quasigeodesic constants. For the remainder of the paper we will assume that we have chosen a basepoint 0 ∈ T ( S ), giving anidentification of T ( S ) with the open unit ball of quadratic differentials on 0, and a compactification T ( S ) of T ( S ) by endpoints of Teichm¨uller geodesic rays from 0 (that is, by unit norm quadraticdifferentials or equivalently, by projective measured foliations).In view of Proposition 3.5 and the fact that Teichm¨uller geodesics are electric quasigeodesics,we can get some control over the behavior of sequences going to infinity in the electric space if weknow the behavior of the Teichm¨uller geodesic segments between elements of the sequences. Themain fact we will need is the following: Proposition 5.1
Let F and G be minimal foliations in PMF ( S ) . Suppose { x n } and { y n } aresequences in T ( S ) that converge to F and G , respectively, and let s n denote the geodesic segmentwith endpoints x n and y n . Then as n → ∞ , the sequence { s n } accumulates onto a set s in T ( S ) with the following properties:(1) s ∩ T ( S ) is a collection of geodesic lines whose horizontal and vertical foliations are topo-logically equivalent to F and G ; this collection is non-empty exactly when F and G fill up S (thatis, when F and G are not topologically equivalent).(2) s ∩ ∂ T ( S ) consists of foliations in PMF ( S ) that are topologically equivalent to F or G . roof: We will begin by showing property (2) . Let { z n } be a sequence of points lying on thesegments s n such that z n → Z ∈ PMF ( S ); we will show that Z is topologically equivalent toeither F or G .Suppose first that the z n lie over a compact region of moduli space. Then we claim that afterdropping to a subsequence there is a sequence { α n } of distinct simple closed curves on S such that ext z n ( α n ) is bounded. Since the z n lie over a compact region of moduli space, there are elements f n of the mapping class group that move the z n to some fixed compact region of Teichm¨ullerspace; since the z n are not contained in a compact region of Teichm¨uller space, we can drop toa subsequence so that the maps f n are all distinct. So, after dropping to a further subsequence,there is some curve α on S for which the curves α n = f − n ( α ) are all distinct; these curves willhave bounded extremal length on the surfaces z n , establishing the claim. Now since PMF ( S ) iscompact, after dropping to a further subsequence, the sequence { α n } converges in PMF ( S ); hencethere exist constants r n such that the sequence { r n α n } converges in MF ( S ) to a foliation Z ′ , andsince the curves α n are all distinct, we have r n →
0. If instead the z n do not lie over a compactregion of moduli space then after dropping to a subsequence there is a sequence { α n } of (possiblynon-distinct) simple closed curves such that ext z n ( α n ) →
0, and a sequence of bounded constants r n such that r n α n converges to some Z ′ ∈ MF ( S ).Let q n denote the quadratic differential on the basepoint 0 that is associated to z n by theidentification of T ( S ) with DQ (0), so that after dropping to a subsequence, q n → q ∈ SQ (0)whose vertical foliation is Z . Let F n denote the vertical foliation of q n . If we pull back q n bythe Teichm¨uller extremal map between 0 and z n to get a quadratic differential ˜ q n on z n , thevertical foliation of ˜ q n is K / n F n , where K n is the quasiconformality constant of the extremal map.By Lemma 2.4, ext z n ( r n α n ) ≥ ( i ( r n α n , K / n F n )) , so i ( r n α n , F n ) → n → ∞ . So we have i ( Z ′ , Z ) = 0.On z n , let φ n denote the quadratic differential determining the segment s n , and let H n and V n denote the horizontal and vertical foliations associated to φ n (so that as we move along s n in thedirection from x n to y n , the transverse measure of H n contracts and the transverse measure of V n grows). Since ext z n ( α n ) ≥ ( i ( α n , H n )) , we have i ( r n α n , H n ) →
0; likewise i ( r n α n , V n ) →
0. Let a n and b n be constants such that after dropping to subsequences, a n H n and b n V n converge to some H and V ∈ MF ( S ), respectively. k φ n k = i ( H n , V n ) = 1, so since i ( H , V ) must be finite, the product a n b n is bounded. So we must have at least one of the sequences { a n } and { b n } bounded (say { a n } ).Then i ( r n α n , a n H n ) → n → ∞ , so i ( Z ′ , H ) = 0.Let ˜ φ n denote the quadratic differential on x n obtained by pulling back φ n by the Teichm¨ullerextremal map from x n to z n . Let ˜ H n denote the horizontal foliation of ˜ φ n . As we move along s n from z n back to x n horizontal measure grows, so we have k n ˜ H n = H n where the constants k n are less than1. Following the argument of the first paragraph of the proof, there is a sequence { β n } of simpleclosed curves on S and a sequence of bounded positive constants t n such that ext x n ( t n β n ) → t n β n → F ′ where i ( F , F ′ ) = 0 (so that F ′ is topologically equivalent to F , by minimality of F ). This implies that i ( t n β n , ˜ H n ) →
0, so i ( t n β n , H n ) →
0. Taking limits, i ( F ′ , H ) = 0 so H is12lso topologically equivalent to F . But we have already shown that i ( H , Z ′ ) = i ( Z ′ , Z ) = 0, so byminimality Z is topologically equivalent to F , establishing property (2) .To show that s ∩ T ( S ) consists of geodesic lines determined by horizontal and vertical foliationstopologically equivalent to F and G , suppose now that { z n } is a sequence of points in the segments s n such that z n → z ∈ T ( S ). Again, let φ n denote the quadratic differential on z n that determinesthe segment s n , and let H n and V n denote the associated horizontal and vertical foliations. Afterdescending to a subsequence, we can assume that q n → q , a quadratic differential on z ; H n and V n will converge respectively to the horizontal foliation H and vertical foliation V of q . By argumentssimilar to those of the preceding paragraphs, H and V are topologically equivalent to F and G ,respectively. Now the segments s n all intersect a compact neighborhood of z , so since they forman equicontinuous family of maps a subsequence must converge uniformly on compact sets to thecomplete geodesic line containing z determined by q .When F and G are topologically equivalent, it is impossible for any point in T ( S ) to support aquadratic differential whose horizontal and vertical foliations are topologically equivalent to F and G ; hence when F and G are topologically equivalent, s ∩ T ( S ) must be empty.It remains to show that when F and G fill S , s ∩ T ( S ) is nonempty. We have shown that s ∩ ∂ T ( S ) consists of foliations in PMF ( S ) that are topologically equivalent to F or G . The setof foliations in PMF ( S ) topologically equivalent to F is closed (likewise for G ), since if F n is asequence of foliations topologically equivalent to F and F n → H ∈ PMF ( S ) then by we have i ( F , H ) = 0, so that H is topologically equivalent to F . So since F and G are not topologicallyequivalent, s ∩ ∂ T ( S ) consists of at least two connected components. The segments s n are connectedso their accumulation set s must be connected; hence s ∩ T ( S ) cannot be empty. ✷ Note that in the course of the proof we have also shown the following about sequences ofsegments whose endpoints converge to foliations that are not minimal:
Proposition 5.2
Let x n and y n be sequences in T(S) converging to F and G in PMF ( S ) , let s n be the geodesic segment with endpoints x n and y n , and let s be the set of accumulation points in T ( S ) of the segments s n . Then the only possible minimal foliations in s ∩ PMF ( S ) are those (ifany) that are topologically equivalent to F or G . Using similar arguments we can prove the following about convergence of Teichm¨uller raysemanating from a common point (not necessarily the chosen basepoint 0 in T ( S )): Proposition 5.3
Let z be a fixed point in T ( S ) , let z n be a sequence of points in T ( S ) thatconverge to Z ∈ PMF ( S ) , and let r n be the geodesic segment from z to z n . After descending to asubsequence, the segments r n converge uniformly on compact sets to a geodesic ray r with verticalfoliation V , such that i ( Z , V ) = 0 .Proof: Assume that the segments r n are paths parametrized by arclength, and extend the r n tomaps r n : R → T ( S ) by setting r n ( t ) = z n for all t ≥ d ( z, z n ). The family { r n } is equicontinuous, so13y Ascoli’s theorem, after dropping to a subsequence the maps r n converge uniformly on compactsets to a map r : R → T ( S ), which is necessarily a geodesic ray emanating from z .Let V be the vertical foliation of r . We wish to show that i ( Z , V ) = 0. Let φ n be the quadraticdifferential on z determining the segment r n , and let V n be the vertical foliation of φ n , so that V n → V . Then ext z V n → ext z V , so ext z V n is bounded. Now d ( z, z n ) = log( ext z V n ext zn V n ) (see [12]),so since d ( z, z n ) → ∞ , ext z n V n → n → ∞ . Now the argument of the third paragraph of theproof of Proposition 5.1 (changing the r n α n to V n ) shows that i ( Z , V ) = 0. ✷ As a start to proving Theorem 1.2 we will prove the following, which shows that minimal foliationsin
PMF ( S ) are an infinite electric distance from any point in T ( S ). Proposition 6.1
Let
F ∈ PMF ( S ) be minimal and let { z n } be a sequence of points in T ( S ) thatconverges to F . Then d el (0 , z n ) → ∞ as n → ∞ .Proof: Suppose that d el (0 , z n ) does not go to infinity. Then after dropping to a subsequence wemay assume that the z n lie in a bounded electric neighborhood of 0. As in the proof of Proposition5.1, we can construct a sequence of curves α n such that the values ext z n ( α n ) are bounded, andsuch that for some bounded constants r n , the sequence r n α n converges in MF ( S ) to a foliation F such that i ( F , F ) = 0. Now since α n has bounded extremal length on z n , we have that z n liesin a bounded neighborhood of T hin α n , so the values d el (0 , T hin α n ) are bounded. So the curves α n , regarded as elements of the curve complex, are a bounded distance (say M ) from some fixedcurve α . Now for each α n we can construct a chain of curves α n, , ..., α n,M such that α n, = α n and α n,M = α , and for all i , d ( α n,i , α n +1 ,i ) = 1. So α n,i and α n,i +1 are disjoint, or in other words, i ( α n,i , α n,i +1 ) = 0. After dropping to subsequences, for each fixed i , the sequence α n,i converges(after bounded rescaling) to a measured foliation F i , and for all i we have i ( F i , F i +1 ) = 0. Since F is minimal this implies that all the foliations F i are topologically equivalent to F . But F M = α ,which gives a contradition. ✷ The proof of Theorem 1.2 will be divided into the next three propositions. We begin by showingthat we have a well- defined, continuous map from
PMF min ( S ) to ∂ ∞ T el ( S ). Proposition 6.2
The inclusion map from T ( S ) to T el ( S ) extends continuously to the portion PMF min ( S ) of PMF ( S ) consisting of minimal foliations, to give a map π : PMF min ( S ) → ∂ ∞ T el ( S ) .Proof: Let
F ∈ PMF min ( S ). We must show that every sequence { z n } in T ( S ) converging to F , considered as a sequence in T el ( S ), converges to a unique point in ∂ ∞ T el ( S ). So suppose thatthere is a sequence { z n } → F that does not converge to a point in ∂ ∞ T el ( S ). Then there aresubsequences { x n } and { y n } of { z n } such that h x n | y n i el is bounded. Let s n denote the Teichm¨uller14eodesic segment between x n and y n . Since the segments s n are electric quasigeodesics with uniformquasigeodesic constants, by Proposition 3.5 there is a point p n on each s n that is a bounded electricdistance from 0. By Proposition 5.1, the points p n converge to a foliation in PMF min ( S ) that istopologically equivalent to F . But then according to Proposition 6.1, d el (0 , p n ) must go to infinityas n → ∞ . This gives a contradiction. ✷ We now show that the non-injectivity of the map π : PMF min ( S ) → ∂ ∞ T el ( S ) is limited toidentifying foliations that are topologically equivalent but not measure equivalent. Proposition 6.3
Let F and G be minimal foliations in PMF ( S ) . Then π ( F ) = π ( G ) if and onlyif F and G are topologically equivalent.Proof: Suppose first that F and G are topologically equivalent, and suppose that π ( F ) = π ( G ).Then the same argument as in the proof of Proposition 6.2 would give a sequence of points { p n } that are a bounded electric distance from 0 and that converge to a minimal foliation in PMF ( S );but this is impossible by Proposition 6.1. Hence when F and G are topologically equivalent, π ( F ) = π ( G ).Now suppose that F and G are not topologically equivalent, and let { x n } and { y n } be sequencesin T ( S ) converging to F and G , respectively. We will show that { x n } and { y n } do not convergeto the same point in ∂ ∞ T el ( S ), by showing that we can drop to subsequences so that h x n | y n i el isbounded as n → ∞ . Let s n denote the Teichm¨uller geodesic segment with endpoints x n and y n .Since F and G are not topologically equivalent, by Proposition 5.1 we can drop to a subsequenceso that the s n converge uniformly on compact sets to a Teichm¨uller geodesic line L . Choose apoint p ∈ L , and a sequence p n ∈ s n converging to p . Then as n → ∞ , d (0 , p n ) is bounded,hence d el (0 , π ( p n )) is also bounded. So by Proposition 3.5, h x n | y n i el is bounded as n → ∞ . Thus π ( F ) = π ( G ). ✷ The following proposition completes the proof of Theorem 1.2.
Proposition 6.4
The map π : PMF min ( S ) → ∂ ∞ T el ( S ) is surjective. Moreover, if { x n } is asequence in T ( S ) that converges to a non-minimal foliation in PMF ( S ) then no subsequence of { x n } converges in the electric space T el ( S ) to a point in ∂ ∞ T el ( S ) .Proof: Let
X ∈ ∂ ∞ T el ( S ), and let x n be a sequence in T el ( S ) that converges to X ; without loss ofgenerality we may assume that each x n lies in T ( S ), since if x n is one of the added-on points inthe construction of T el ( S ) then we may replace x n by a point in T ( S ) that is distance from x n ,without changing the convergence properties of the sequence { x n } . We will show that a subsequenceof { x n } converges to a minimal foliation F ∈ PMF ( S ); then π ( F ) = X .Since T ( S ) is compact, after dropping to a subsequence, { x n } converges to some F ∈ PMF ( S ).Suppose F is not minimal. We will show that for some B < ∞ , for each x n there are infinitelymany x m such that h x n | x m i el < B ; this would contradict convergence in T el ( S ) of the sequence { x n } . Fix x n , and let r mn denote the geodesic segment with endpoints x n and x m . By Proposition15.3, a subsequence of the r mn (which we will again call r mn ) converges uniformly on compact setsto a geodesic ray r n . Let H n denote the horizontal foliation of r n ; by Proposition 5.3 we have i ( F , H n ) = 0. The foliations F and H n are not minimal, so each one contains a simple closedcurve, which we will denote α and γ n , respectively. Now we have i ( α, γ n ) = 0, so that in the curvecomplex, the distance from α to γ n is at most 1; hence the electric distance from T hin α to T hin γ n is bounded independent of n , since the curve complex is quasi-isometric to T el ( S ).The simple closed curve γ n contained in H n may be chosen so that as t → ∞ , ext r n ( t ) γ n → t , r n ( t ) belongs to T hin γ n . Since the rays r mn converge to r n uniformly on compact sets, for all m sufficiently large there is a point p mn on r mn that lies in T hin γ n . Now we have d el (0 , p mn ) ≤ d el (0 , T hin γ n ) + 1 ≤ d el (0 , T hin α ) + d el ( T hin α , T hinγ n ) + 2since each thin set has diameter 1 (here the electric distance between two sets S and S means thesmallest distance between any pair of points in S and S , respectively). Note that the right-handside of the inequality does not depend on n or m since d el ( T hin α , T hin γ n ) is bounded independentof n . Now by Proposition 3.5 we have that for all m sufficiently large, h x n | x m i el is bounded, andthe bound does not depend on n or m ; this contradicts the fact that the sequence { x n } convergesto a point in the boundary at infinity of T el ( S ), so our assumption that F is not minimal must befalse. Hence F is minimal, and we have π ( F ) = X . ✷ Note that given a nonminimal foliation F , there are sequences in T ( S ) converging to F whoseelectric distance from 0 goes to infinity; however, no subsequences of these will converge to a pointin ∂ ∞ T el ( S ), so that in particular T el ( S ) ∪ ∂ ∞ T el ( S ) is not compact. It is simple to construct suchsequences: the minimal foliations are dense in PMF ( S ) (see for instance [7]), so there is a sequence {F n } of minimal foliations that converges to F . By Proposition 6.1, for every M >
0, each F n hasa neighborhood whose points are all at least M from 0 in the electric metric; hence we may easilychoose a sequence { p n } of points contained in small neighborhoods of the foliations F n , such that { p n } converges to F and d el (0 , p n ) → ∞ .If F is a foliation in PMF ( S ), let τ ( F ) denote the equivalence class of foliations in PMF ( S )that are topologically equivalent to F . We have shown that the boundary at infinity of T el ( S ) and C ( S ) can be identified with topological equivalence classes of minimal foliations. In spite of thefact that the Teichm¨uller compactification of T ( S ) by PMF ( S ) depends heavily on the choice ofbasepoint, the arguments we have given show that the description we have obtained of the boundaryof C ( S ) is natural: Theorem 1.4
Let { α n } be a sequence of elements of C ( S ) that converges to a foliation F in theboundary at infinity of C ( S ) . Then regarding the curves α n as elements of the projective measuredfoliation space PMF ( S ) , every accumulation point of { α n } in PMF ( S ) is topologically equivalentto F . Appendix
In order to use sequential arguments to prove the continuity results of the main theorems, itis necessary to understand the point-set topology of F min ( S ), the space of minimal topologicalfoliations on S . This is particularly important in light of the fact that the entire space F ( S ) oftopological foliations, with the topology induced from PMF ( S ) by forgetting the measures, is notHausdorff. We will begin with the following: Proposition 7.1
The measure-forgetting quotient map p : PMF min ( S ) → F min ( S ) is a closedmap, and the pre-image of any point of F min ( S ) is compact.Proof: To show that p is a closed map, let K ⊂ PMF min ( S ) be a closed set. Then we claimthat the set p − ( p ( K )) is closed; this will imply that p ( K ) is closed. So, let { x n } be s sequence in p − ( p ( K )) that converges to a point x in PMF min ( S ); we must show that x ∈ p − ( p ( K )). Thereis a sequence of y n ∈ K such that p ( x n ) = p ( y n ). Since PMF ( S ) is compact, after dropping to asubsequence we may assume that y n → y ∈ PMF ( S ). Now since p ( x n ) = p ( y n ), we have that x n and y n are topologically equivalent, which implies that i ( x n , y n ) = 0. Hence i ( x, y ) = 0, so since x is minimal, x and y are topologically equivalent by Proposition 2.1, so that p ( x ) = p ( y ). We nowknow y to be in PMF min ( S ), so since K is closed in PMF min ( S ), we have y ∈ K . This in turnimplies that x ∈ p − ( p ( K )), so p − ( p ( K )) is closed.To show that the pre-image of any point is compact, let z be a point in F min ( S ) and let Z = p − ( z ). Let { x n } be a sequence of points in Z ; since PMF ( S ) is compact, after dropping toa subsequence we may assume that x n converges to some x ∈ PMF ( S ). Let y be a fixed point in Z . Then the set Z is the set of all foliations in PMF ( S ) that are topologically equivalent to y .Hence i ( y, x n ) = 0 for all n , so i ( y, x ) = 0. Thus by minimality, x is topologically equivalent to y ,so x ∈ Z . So Z is compact. ✷ The space
PMF ( S ) is metrizable and normal, since it is a topological sphere; hence so is PMF min ( S ) ⊂ PMF ( S ). The following proposition will establish in particular that F min ( S )is first countable and Hausdorff, which are exactly the properties needed in order for sequentialarguments to prove continuity: Proposition 7.2
Let X be a metric space that is normal, and let p : X → ˆ X be a quotient mapthat is a closed map, and such that the pre-image of any point of ˆ X is compact. Then the quotienttopology on ˆ X is first countable and normal.Proof: We will show first that ˆ X is normal. Let S and T be disjoint closed sets in ˆ X ; we must showthat S and T have disjoint neighborhoods. The sets p − ( S ) and p − ( T ) are closed and disjoint in X ,so since X is normal there are disjoint open sets U and V such that p − ( S ) ⊂ U and p − ( T ) ⊂ V .Then X − U and X − V are closed, so p ( X − U ) and p ( X − V ) are closed since p is a closedmap. Now S has empty intersection with p ( X − U ), so ˆ X − p ( X − U ) is a neighborhood of S ;17ikewise ˆ X − p ( X − V ) is a neighborhood of T . It is easily checked that the sets ˆ X − p ( X − U ) andˆ X − p ( X − V ) are disjoint, which establishes normality.To show that ˆ X is first countable, let z ∈ ˆ X ; we must define a countable neighborhood basisaround z . Let Z = p − ( z ), and let U n be the open neighborhood around Z of radius n . Let V n = p ( U n ). If V is any neighborhood of z , then p − ( V ) is a neighborhood of the set Z , so sinceby assumption Z is compact, p − ( V ) must contain one of the sets U n ; hence V must contain oneof the sets V n . So we will be done if we can show that every V n contains a neighborhood of z . In X , let W n = p − ( V n ); note that U n ⊂ int ( W n ). Let S n = X − int ( W n ), so that S n ∩ U n = ∅ . Theset p ( S n ) is closed in ˆ X since p is a closed map, so by normality of ˆ X , there is some neighborhood T n of x disjoint from p ( S n ). Now p − ( T n ) ⊂ W n , so T n ⊂ p ( W n ) = V n . Hence the sets T n form alocal basis of neighborhoods of z . ✷ References [1] L. Bers.
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