Comparison of Teichmüller geodesics and Weil-Petersson geodesics

Transcription

1 Comparison of Teichmüller geodesics and Weil-Petersson geodesics by Ali Göktürk B.Sc., Boğaziçi University; İstanbul, Turkey, 2005 M.A., Brown University; Providence, RI, 2007 Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics at Brown University PROVIDENCE, RHODE ISLAND May 2011

2 c Copyright 2011 by Ali Göktürk

3 This dissertation by Ali Göktürk is accepted in its present form by the Department of Mathematics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Jeffrey F. Brock, Ph.D., Advisor Recommended to the Graduate Council Date Richard E. Schwartz, Ph.D., Reader Date Richard Kenyon, Ph.D., Reader Approved by the Graduate Council Date Peter M. Weber, Dean of the Graduate School iii

4 Vitae Ali Göktürk was born on October 28, 1980, in İstanbul, Turkey. He pursued his undergraduate education at Boğaziçi University in mathematics major. After graduating in July 2005, he joined the Department of Mathmematics at Brown University and got his M.Sc. degree in iv

5 Acknowledgements I would like to thank my advisor, Jeff Brock, for suggesting the statements of Theorem 1.1 and Theorem 1.2, and for his guidance. v

6 Contents Vitae Acknowledgments iv v 1 Introduction 1 2 Preliminaries Quasi-geodesics and quasi-isometries Curves and pants Teichmüller spaces Weil-Petersson metric Geometry of Teichmüller geodesics via curve and pants graph Subsurface projections Combinatorics of short curves along Teichmüller geodesics Fellow-traveling theorem in small complexity Hierarchies in low complexity Teichmüller geodesics fellow-travel hierarchies Non-fellow-traveling theorem in high complexity Gluing two Teichmüller geodesics Construction of G n T (t) G n T (t) does not fellow-travel the corresponding WP geodesic vi

7 6 Questions Teichmüller geodesics and hierarchies in higher complexity The marking graph and hierarchies with annuli Comparing bi-infinite Teichmüller and Weil-Petersson geodesics Bibliography 79 vii

8 List of Figures 2.1 Elementary move An example of gluing two surfaces viii

9 Chapter One Introduction

10 2 Let S = Σ g,n be a compact surface of genus g, and with n boundary components. We define the complexity of S by ξ(s) = 3g + n. There are various natural metrics defined on the Teichmüller space T (S) of S. In this paper we study the comparison of two of them: The Teichmüller metric d T, and the Weil-Petersson metric d WP. In both metrics, T (S) is a unique geodesic space, so every pair of points is connected by a unique Teichmüller geodesic and a unique Weil-Petersson geodesic. A natural question to ask is when these two paths fellow-travel each other. In the case of bounded geometry, Brock, Masur and Minsky showed that these two paths always fellow-travel each other in the Teichmüller metric (see [3]). In this paper, we drop the bounded geometry assuption and answer the same question in full generality for the Weil-Petersson metric. Theorem 1.1. Suppose ξ(s) 5. Then there is a constant N such that for any X, Y T (S), the Teichmüller geodesic G T connecting X to Y lies in an N- neighborhood (in the Weil-Petersson metric) of the Weil-Petersson geodesic G WP connecting X to Y. In proving Theorem 1.1, instead of directly calculating distances in the Weil- Petersson metric, we look at images of G T and G WP in a combinatorial model space called the pants graph P(S) under a projection map Φ P, which is a quasiisometry from T (S) with the Weil-Petersson metric (Theorem 1.1 in [4]). The image Φ P (G WP ) of the Weil-Petersson geodesic is a quasi-geodesic in P(S). Using combinatorial machinery developed by Masur and Minsky, we will construct another quasi-geodesic P H in P(S) connecting the end points of Φ P (G WP ) and show that

11 3 this quasi-geodesic fellow-travels Φ P (G T ). Then Theorem 1.1 follows from the fact that P(S) is δ-hyperbolic when ξ(s) 5, due to Brock and Farb which guarantees quasi-geodesics with the same pair of end points to fellow-travel each other. On the other hand when ξ(s) > 5, S can be decomposed into two disjoint subsurfaces S 1 and S 2, each with complexity greater than 3. As a consequence, P(S) contains quasi-flat subspaces which violates δ-hyperbolicity. The existence of these disjoint subsurfaces leads to the construction of Teichmüller geodesics that do not fellow-travel the Weil-Petersson geodesics with the same endpoints. Theorem 1.2. Suppose ξ(s) > 5. There exist a sequence of pairs of points {X n }, {Y n } T (S) such that if G n T and Gn WP are the Teichmüller and Weil-Petersson geodesics connecting X n to Y n, then lim n Haus WP (G n T, Gn WP ) =. Here we outline the construction of G n T. Let γ be a simple closed curve that divides S into S 1 and S 2, where ξ(s 1 ) 4 and ξ(s 2 ) 4. We first construct Teichmüller geodesics X n (t) T (S 1 ) and Y n (t) T (S 2 ) of length L that satisfy d WP(S1 )(X n (0), X n (L/2)) n, d WP(S1 )(X n (L/2), X n (L)) = O(1)

12 4 and d WP(S2 )(Y n (0), Y n (L/2)) = O(1), d WP(S2 )(Y n (L/2), Y n (L)) n. Then we construct G n T (t) by gluing Xn (t) and Y n (t) along γ. More precisely, we consider the quadratic differentials on X n (t) and Y n (t), then cut open vertical slits at the punctures on X n (t) and Y n (t) corresponding to γ, and finally glue the two ends a thin flat horizontal cylinder to these vertical slits on X n (t) and Y n (t). We choose the horizontal cylinder long enough to ensure that the hyperbolic length of γ on G n T (t) is short for all t. Using the combinatorial model P(S) we will show that during t [0, L/2], G n T (t) makes definite progress in the Weil-Petersson metric restricted to S 1 and stays in a bounded set on S 2 ; whereas during t [L/2, L], G n T (t) makes definite progress S 2 and stays in a bounded set on S 1. On the other hand, G n WP is near the null stratum S(γ) = T (S 1 ) T (S 2 ) and the Weil-Petersson metric on S(γ) is the L 2 product of the Weil-Petersson metrics on each T (S i ), so G n WP travels approximately with the same speed both on S 1 and on S 2. Therefore the Weil-Petersson distance of middle point of G n T from any point on Gn WP goes to infinity as n goes to infinity. We begin in Chapter 2 with preliminaries on coarse geometry of geodesic metric spaces, curve and pants graphs, Teichmüller theory and Weil-Petersson geometry. In Chapter 3, we will introduce some important result from [15] and [18] which relates the geometry of the curve graph, the pants graph and short curves on Teichmüller

13 5 geodesics. In Chapter 4 we first introduce a special case of a combinatorial machinery called hierarchies, developed by Masur and Minsky in [15], and then prove Theorem 1.1 using this machinery. Theorem 1.2 will be proven in Chapter 5. In Chapter 6, we conclude the paper with a list of questions for further investigation.

14 Chapter Two Preliminaries

15 7 2.1 Quasi-geodesics and quasi-isometries A path-metric space (X, d) is called a geodesic metric space if every two points x, y X can be joined by a geodesic segment, an isometric image of an interval in the real line. In coarse geometry, we usualy work with set valued functions between geodesic metric spaces where the images of single points are uniformly bounded diameter sets. Such a map f is called a coarse map, and we write f(x) (with an abuse of notation) to represent an element of the image set of x. A coarse map φ : (X, d) (Y, d ) is called a (C, K)-quasi-isometry if 1. For any x, y X, we have d (φ(x), φ(y))/c K d(x, y) C d (φ(x), φ(y))+k, 2. Y is equal to the K-neighborhood of Im(φ). A coarse map which only satisfies the first condition is called a (C, K)-quasi-isometric embedding. We occasionally drop out (C, K) and just write quasi-isometry or quasiisometric embedding. For a given (C, K)-quasi-isometry φ : X Y, we can always find another (C, K)-quasi-isometry φ 1 : Y X such that φ 1 (φ(x)) has uniformly bounded diameter. Such a map is called a coarse inverse of φ. We say two spaces are quasi-isometric if there is a quasi-isometry from one space to the other. Observe that finite diameter spaces are (C, K)-quasi-isometric to a point where K is the diameter of the space, so these spaces do not have much importance in coarse geometry.

16 8 By definition, distances are coarsely preserved under a quasi-isometric embedding φ. However lengths of curves are not necessarily coarsely preserved under φ if we use the usual definition of the length function length(γ) = sup { n i=1 d(γ(t i), γ(t i 1 )) : n N and a = t 0 < t 1 < < t n = b} for a given curve γ : [a, b] X. Even if φ is a single-valued continuous function, it can take a curve of length 1 to an arbirarily long curve. For example the map φ n : [0, 1] R 2, φ n (x) = (x, sin(2πnx)) is a (1, 2)-quasi-isometry for each n, and the curve φ n ([0, 1]) has length greater than 4n. Thus we define the notion of coarse-length of a coarse path as follows (originally defined in [14], chapter 7): For s > 0 and for any coarse path γ : [a, b] (X, d), we let length s (γ) = ns where n is the smallest number for which [a, b] can be subdivided into n closed subintervals J 1,..., J n with diam X (γ(j i )) s. We call this the arclength of scale s. Lemma 2.1. Given a (C, K)-quasi-isometric embedding φ : (X, d) (Y, d ) and s > 0, there exist s > 0 C > 0 and K 0 which only depends on s, C and K such that for any path γ with length s (γ) <, we have length s (φ(γ))/c K length s (γ) C length s (φ(γ)) + K. A coarse path γ : I X (where I can be a finite interval, an infinite ray or R

17 9 itself) is called a (C, K, s)-quasi-geodesic if for any x, y I length s (γ[x, y]) C d(γ(x), γ(y)) + K where K, s 0 and C 1 and γ[x, y] is the part of γ between x and y. Clearly geodesics are (1, 0, s)-quasi-geodesics for any s > 0. As before, we occasionally drop out (C, K, s) and just write quasi-geodesic. Observe that being quasi-geodesic is a parametrization independent definition, it only uses the strict order induced from the parametrization. Now we can directly deduce from the Lemma 2.1 that (C, K, s)-quasi-geodesics are mapped to (C, K, s )-quasi-geodesics under (C, K )- quasi-isometries where C, K and s only depends on C, K, s, C and K. Quasi-isometries play an important role in δ-hyperbolic spaces. A geodesic metric space X is called δ-hyperbolic (or hyperbolic) if for a geodesic triangle, each side is contained in a δ-neighborhood of the union of the other two sides. Important examples of hyperbolic spaces are the classical hyperbolic space H n, trees, and Cayley graphs of fundamental groups of closed negatively curved manifolds. Theorem 2.2. ([2]) In a δ-hyperbolic space, the Hausdorff distance between two finite (C, K, s)-quasi-geodesics whose endpoints are distance M apart (for some M > 0) is bounded, where the bound depends only on M, δ, C, K and s. We say that two paths are M-fellow-travelers (or we just say they fellow-travel each other) if their Hausdorff distance is bounded by M. Theorem 2.3. ([2]) Spaces which are quasi-isometric to hyperbolic spaces are hy-

18 10 perbolic themselves. Observe that a path α which fellow-travels a quasi-geodesic γ is not necessarily a quasi-geodesic itself (it may backtrack). However if α and γ have parametrizations α(t) and γ(t) such that d X (α(t), γ(t)) is bounded by some M for all t, then α has to be a quasi-geodesic. In such a case we say that α and γ strongly fellow-travel each other. Notation: Throughout the rest of paper, we will be working with geodesic metric spaces that depends on a given surface S. So to simplify our notations, any constant introduced will only depend on the complexity of the surface (defined in the begining of the next section). Similarly whenever we write quasi-geodesic(s) or quasiisometr(ies), we mean (C, K, s)-quasi-geodesic(s) or (C, K)-quasi-isometr(ies) where C, K and s only depend on the complexity of the surface. Also we use the following coarse versions of equalities and inequalities: For two non-negative functions f(x) and g(x), we say f(x) is coarsely less than g(x) and write f(x) g(x) if there are constants C > 0 and K 0 (which only depend on the complexity of the surface) such that f(x) C g(x) + K ( is defined similarly). If f(x) g(x) and f(x) g(x), then we say f(x) is coarsely equal to g(x) and write f(x) g(x). Also we write f(x) 1 to indicate that there exist an M, which only depend on the complexity of the surface, such that f(x) < M. Here we should note that f(x) 1 does not necessarily mean that f(x) is bounded

19 11 for all x. For example the statement For any x X, f(x) 1 if and only if g(x) 1 could be falsely interpreted as f(x) is uniformly bounded if and only if g(x) is uniformly bounded. The correct interpretation is There exist non-negative constants M and N such that for any x X, we have f(x) < M iff g(x) < N. Remark 2.4. Certain constants in some inequalities (and coarse inequalities) will depend on the complexity of a subsurface of S rather than the complexity of S itself. But since there are only ξ(s) number of subsurfaces with different complexities, we can always choose another constant which only depends on S (and which still satisfies the inequality). For example suppose that for each subsurface W S with ξ(w ) 4, there exists function f W (x) and a constant M ξ(w ) such that f W (x) < M ξ(w ). In this case we put M = max{m 4, M 5,..., M ξ(s) } which only depends on ξ(s) and satisfies f W (x) < M for each subsurface W S with ξ(w ) 4. So now we can write this as f W (x) Curves and pants Let S = Σ g,n denote the topological surface which is obtained by removing n points from a closed, connected, orientable surface of genus g. We define the complexity

20 12 of S by ξ(s) = 3g + n. In this paper we only consider surfaces S with ξ(s) 4. Associated to S, we define the curve graph C(S) of S as follows: vertices of C(S) correspond homotopy classes of curves in S (where by a curve in S, we mean a nontrivial, non-peripheral, simple closed curve) and two vertices v and w are adjacent if they correspond to classes [α] and [β] which have representatives α and β that intersect minimally (i.e. i S (α, β) i S (α, β ) for any two curves α and β ). This minimum intersection number is zero if ξ(s) > 4, and non-zero if ξ(s) = 4 (in fact, it is equal to 1 if S is a once-punctured torus, and 2 if S is a four times punctured sphere). We endow C(S) with the graph metric by assigning length 1 to every edge which turns C(S) into a geodesic metric space. In this paper, we only consider distances between vertices of C(S), and we write α C(S) to refer to the curve class corresponding to a vertex in C(S). Also we define the geometric intersection number i S (α, β), which is the to the minimum number of intersections among all represetatives of α and β. On the other hand, d S (α, β) represents the distance in C(S) of the vertices v and w corresponding to α and β respectively. Therefore, we think of a geodesic in C(S) as a sequence of curves {α i }, such that d S (α i, α j ) = i j. Notation: For two finite subsets A and B of C(S), we write d S (A, B) := diam S (A B) & i S (A, B) := max α A,β B i S (α, β).

21 13 Lemma 2.5. (Lemma 1.2 in [5]) For two finite subsets A and B of C(S), we have d S (A, B) log(i S (A, B)). This Lemma implies that C(S) is a connected graph. A more important coarse geometric property if C(S) is the following. Theorem 2.6. ([14]) For any surface S with ξ(s) > 3, the curve graph of S is an infinite diameter δ-hyperbolic metric space. A pants decomposition P of S is a maximal collection of pairwise disjoint (and distinct) curves {α 1, α 2,..., α n } in C(S). This means that each component of S α 1 α 2 α n is a pair of pants. Using Euler characteristic arguments we can show that P has exactly ξ(s) 3 curves. We say that two pants decompositions P 1 and P 2 are related by an elementary move if P 2 can be obtained from P 1 by removing a single curve α P 1 and replacing it with another curve β such that 1. β do not intersect any of the curves in P 1 {α}, 2. i S (α, β) is minimum among all curves satisfying (1). The pants graph P(S) is the graph whose vertices correspond to pants decompositions of S and whose edges join vertices whose corresponding pants decompositions differ by an elementary move. Just like in the curve graph, we endow P(S) with

22 14 Figure 2.1: Elementary move the graph metric, denoted by d P, by assigning length 1 to every edge, and we only consider distances between vertices of P(S). With this metric P(S) is an infinite diameter metric space (this is a consequence of Theorem 3.6 in section 3.1). Theorem 2.7. ([1]) The pants graph P(S) is δ-hyperbolic if and only if ξ(s) So when ξ(s) 5, quasi-geodesics in P(S) fellow-travel each other by Theorem Notation: Depending on the context, a pants decomposition P can be thought as elements of the pants graphs P(S), or as a finite subset of the curve graph C(S). For example given two pants decompositions P 1 and P 2, the expression d P (P 1, P 2 ) stands of the pants distance between P 1 and P 2 as elements of P(S). On the other hand, d S (P 1, P 2 ) stands for diam S (P 1 P 2 ) where P 1 and P 2 are finite subset of C(S).

23 Teichmüller spaces The Teichmüller space T (S) of S is the space of all analytically finite conformal structures on S, modulo conformal isomorphism isotopic to the identity. As a consequence of the uniformization theorem, due to Klein, Poincáre, and Koebe, T (S) can also be thought as the space of all finite area hyperbolic structures on S, modulo hyperbolic isometry isotopic to the identity. The Teichmüller metric d T is originaly defined in terms of maps with minimal quasi-conformal dilatation. However, in our calculations it will be more convenient to use Kerckhoff s characterization [10] d T (X, Y ) = 1 2 sup log λ X(α) α C(S) λ Y (α), (2.1) where λ X (α) is the extremal length of α in X, which is defined as λ X (α) = sup { infα [α] l 2 σ(α ) Area σ(x) } σ is a conformal metric on X. With this metric, the Teichmüller space is a complete geodesic space, where every pair of points is connected by a unique geodesic. Geodesics in the Teichmüller space are determined by (meromorphic) quadratic differentials, which are tensors of the form ϕ(z)dz 2 in local coordinates, with ϕ meromorphic. A quadratic differential q in X T (S) determines a singular Euclidian

24 16 metric together with a pair of orthogonal singular foliations, called horizontal and vertical foliations respectively. The unit speed, parametrized Teichmüller geodesic passing through X, determined by q, is constructed by the family of quadratic differentials q(t), where q(t) is obtained by scaling the horizontal foliation of q by a factor of e t, and the vertical by e t. Another length function on T (S) that we use is the hyperbolic length function l X (α), which is defined to be the hyperbolic length of the unique geodesic representative of the curve α C(S) in X T (S) with its hyperbolic metric. Both the extremal length and the hyperbolic length of a curve α are continuous functions in T (S). Moreover, they are asymptotic in small scale. Theorem 2.8. (Maskit) Let X T (S) and α C(S). Then l 2 X (α) 2π χ(s) λ X (α) 1 2 l X(α)e l X(α)/2. Corollary 2.9. For any X T (S) and α C(S), we have l X (α) 1 iff λ X (α) 1. Using the hyperbolic length function, we define set valued functions as follows. Definition Let Φ C : T (S) C(S) assign each X the set of curves with shortest hyperbolic length on X. Similarly let Φ P : T (S) P(S) assign each X the set of pants decompostions in the form P = {α 1, α 2, α 3,..., α ξ(s) 3 } where α 1 Φ C (X)

25 17 and α i is one of the shortest curves disjoint from α 1 α 2 α i 1. The following theorem states that the shortest curves in X cannot be that long. Theorem (Bers) There exists a constant B, which only depends on the complexity of S, such that for any X T (S) and for any α P Φ P (X), we have λ X (α) B and l X (α) B. The constant B in Theorem 2.11 is called the Bers constant. Lemma For α, β C(S) and X T (S), λ X (α) λ X (β) i S (α, β) 2. For the proof, see [12], Lemma 5.1. The following Corollary of Lemma 2.12 states that curves occuring in Φ P (X) are the only possible short curves on X. Corollary There exist a constant C such that if l X (α) C or λ X (α) C, then α P for every P Φ P (X). Proof. Choose C such that C < 1/B and 1 2 CeC/2 < 1/B where B is the Bers constant. So either λ X (α) C < 1/B

26 18 or λ X (α) 1 2 l X(α)e l X(α)/2 1 2 CeC/2 < 1/B. By Theorem 2.8. Now for any P Φ P (X) and for any β P, we have λ X (β) < B by Theorem Thus by the above Lemma i S (α, β) 2 λ X (α) λ X (β) < 1 B B = 1 which implies that α is disjoint from every curve in P. This is only possible when α P. Using the same idea in the above proof, we can deduce the following Corollary. Corollary If l X (α) 1 and l X (β) 1 for α, β C(S), then i S (α, β) 1 and therefore d S (α, β) 1. This corollary, together with Theorem 2.11 implies that the map Φ C is a coarse map. Also combining these results with Lemma 3.3 and Corollary 3.7, we can conclude that Φ P is also a coarse map. Definition For any X T (S), we let P X be one of the elements of Φ P (X) and γ X be one of the elements of Φ C (X) P X. We call γ X and P X the shadow of X in C(S) and P(S) respectively. Proposition For X, Y T (S) and α, β C(S), if l X (α) 1 and l Y (β) 1,

27 19 then log(d S (α, β)) log(i S (α, β)) d T (X, Y ) Proof. By Corollary 2.9, we have λ X (α) 1 and λ Y (β) 1. Now by Kerckhoff s Theorem [10] d T (X, Y ) = 1 2 sup log λ X(α) α C(S) λ Y (α). Therefore log i S (α, β) log(i S (α, β) 2 ) log(λ X (β) λ X (α)) (Lemma 2.12) ( ) = log λx (β) λ λ Y (β) Y (β) λ X (α) ) log (λ X (α) 1 and λ Y (β) 1) ( λx (β) λ Y (β) d T (X, Y ). (Equation 2.1)

28 Weil-Petersson metric The Weil-Petersson metric g WP on T (S) arises from the L 2 inner product φ, ϕ WP = X φ ϕ ρ 2 on the cotangent space Q(X) = TX (T (S)) to Teichmüller space, naturally the holomorphic quadratic differentials on X, where ρ(z) dz is the hyperbolic metric on X. An important difference between the Teichmüller metric and the Weil-Petersson metric is that the Weil-Petersson metric is not complete, due to Wolpert [20] and Chu [6]. The lack of completeness is precisely due to the existence of paths in T (S) along which a simple closed geodesic is pinched to a cusp. The completion T (S) of the Teichmüller space is identified with the augmented Teichmüller space, which is constructed by extending the range of the Fenchel-Nielsen parameters: given a pants decomposition P = {α 1, α 2,..., α ξ(s) 3 } and the Fenchel- Nielsen coordinates (l α1, θ α1, l α2, θ α2,..., l αξ(s) 3, θ αξ(s) 3 ) (R + R) ξ(s) 3 corresponding to P, we allow the length parameters to have value 0 and take the quotient by identifying (0, t 1 ) (0, t 2 ) on each R + R. This way we obtain T (S)

29 21 from T (S) by adding strata consisting of spaces of the form S(σ) = {X l α (X) = 0 iff α σ}, where σ is a non-empty subset of a pants decomposition. We call S(σ) the σ-null stratum, which consists of nodal Riemann surfaces where paired cusps are introduced along the curves in σ. Therefore each S(σ) are identified with the product of Teichmüller spaces T (S 1 ) T (S 2 ) T (S n ) where S i s are connected components of S σ (see [11] for details). The extension of the Weil-Petersson metric in T (S) to each strata S(σ) = T (S 1 ) T (S 2 ) T (S n ) is simply the L 2 product of the Weil-Petersson metrics on each T (S i ). With the extended Weil-petersson metric, T (S) has the structure of a CAT(0) space where T (S) and each strata S(σ) is geodesically convex in T (S) (see [19]). A unique geodesic space X is called CAT(0) if for any geodesic triangle, there exists a comparison triangle in R 2 with sides of the same length as the sides of such that distances between points on are less than or equal to the distances between corresponding points on (see [2] for details). In a CAT(0) space, every pair of finite geodesics, whose endpoints are a bounded distance away from each other, are strong fellow-travelers. So a Weil-Petersson geodesic G in T (S) whose endpoints are close to a given stratum strongly fellowtravels a geodesic G in that stratum. The following theorem and its corollary provides a useful criterion for when the above situation can happen.

30 22 Theorem (Corollary 21 in [19]) Given σ = {α 1, α 2,..., α k } C(S) and a point X T (S), the minimum Weil-Petersson distance of X to the stratum S(σ) is estimated by min Z S(σ) d WP (X, Z) = 2πl + O(l 2 ), where l = l X (α 1 ) + l X (α 2 ) + + l X (α k ). Corollary ([4]) For any X T (S) and for any non-empty subset σ of P X, there exists X σ S(σ) such that i. P Xσ = P X, ii. d WP (X, X σ ) 1. Proof. Let X S(P X ) be the unique point of the maximally noded stratum S(P X ). By Theorem 2.11, l X (α) 1, α P X and so l X = α P X l X (α) 1. Thus by Theorem 2.17, d WP (X, X) = 2πl X + O(lX 2 ) 1, Now since σ P X, S(P ) is also a stratum of S(σ), which is a product of Teichmüller spaces, we can also apply Theorem 2.17 to bound distances between points in S(σ) and X. Let X σ S(σ) with P Xσ = P X (which can be constructed by using the extended Fenchel-Nielsen coordinates), so again by Theorem 2.11, α P X l Xσ (α) 1 and thus again by Theorem 2.17, d WP (X σ, X) 1,

31 23 and by triangle inequality, d WP (X, X σ ) 1. It is also proven in [4] that the regions of the form V L (P ) = {X T (S) l X (α) L, α P } have bounded diameter in the Weil-Petersson metric, where the bound only depends on L and ξ(s). Also by Theorem 2.11 we have V B (P ) = T (S). P P(S) This means that the set valued map Q : P(S) (T (S), d WP ) P V 2B (P ) is a coarse map which is coarsely onto. The following Theorem by J. Brock is the main tool that we use in this paper to estimate Weil-Petersson distances. Theorem (Thm 1.1 in [4]) The map Q : P(S) (T (S), d WP ) P V 2B (P ) is a quasi-isometry.

32 24 Outline of the Proof. For two pants decompositions P 1 and P 2 which differ by an elementary move, the regions V 2B (P 1 ) and V 2B (P 2 ) have non-empty intersection. This shows that Q is a Lipschitz map. On the other hand if V 2B (P ) V 2B (P ) for some P and P, then P and P are both short on any X in V 2B (P ) V 2B (P ), which implies i S (P, P ) 1. Therefore d P (P, P ) by Lemma 3.3 and Corollary 3.7. Finally, A compactness argument shows that points in V B (P ) lies a uniformly definite distance from points outside V 2B (P ). This implies that a unit length Weil-Petersson geodesic segment can be covered by a uniform number of regions of the form V 2B (P ). Therefore Q does not contract distances too much and the theorem follows. Remark Observe that the map Φ P : (T (S), d WP ) P(S) X P X is a coarse inverse of Q, and so is also a quasi-isometry.

33 Chapter Three Geometry of Teichmüller geodesics via curve and pants graph

34 26 In the previous chapter, we defined two coarse maps Φ C and Φ P from the Teichmüller space T (S) into C(S) and P(S) respectively. Theorem 2.19 states that Φ P is a quasi-isometry when T (S) is endowed with the Weil-Petersson metric. So given a Teichmüller geodesic and a Weil-Petersson geodesic with the same pair of endpoints, instead of calculating the Weil-Petersson Hausdorff distance between them, we can look at their shadows in P(S) and calculate their Hausdorff distance there. The geometry of the pants graph has strong connections with the geometry of the curve graphs of subsurfaces via subsurface projections. In the first section, we will define these subsurface projection maps and introduce important results from [15] relating the geometry of P(S) and C(S) to the geometry of the curve graphs of subsurfaces via these projection maps. 3.1 Subsurface projections By a subsurface W of S, we mean an isotopy class of an incompressible, nonperipheral, connected open subsurface. Suppose ξ(w ) > 3. The arc graph C (W ) of W is a graph defined as follows: The vertices of C (W ) are isotopy classes of essential, non-peripheral simple closed curves or arcs with end points on W. Two vertices v and w are connected by an edge if they have disjoint representatives. For a surface W with ξ(w ) 4 we define a set valued map ψ W : C (W ) C(W )

35 27 as follows: 1. For a simple closed curve α C (W ), we let ψ W (α) = α. 2. For an arc β C (W ), let γ 1 and γ 2 be the boundary component(s) of W on which the end points of β lie. If γ 1 = γ 2, then the end points of β divide γ 1 into two arcs α 1 and α 2. Let ψ W (β) be the union of two homotopy classes of simple closed curves obtained by gluing β to α 1 and α 2. If γ 1 γ 2, we first orient β in such a way that β + goes from γ 1 to γ 2, and endow γ 1 and γ 2 with the boundary orientation of W coming from a fixed orientation of W. In this case, ψ W (β) be the simple closed curve obtained by the concatenation β + γ 2 + β γ 1 +. Now let W S be a proper subsurface of S with ξ(w ) 4. Given [α] C(S), let α be a representative of [α] such that the intersection of α with each component of W is minimum, and let π W ([α]) C (W ) be the union of nontrivial, nonperipheral components of α W. Definition 3.1. Given a proper subsurface W of S with ξ(w ) 4 and α C(S), we define the projection of α onto W by π W (α) = ψ W (π W (α)). Also for sets or elements A and B in C(S), we define the subsurface distance and

36 28 diameter as d W (A, B) := diam C(W ) (π W (A) π W (B)) and diam W (A) := diam C(W ) (π W (A)). Lemma 3.2. (Lemma 2.3 in [15]) Let W S be a proper subsurface of S with ξ(w ) 4. Then for α, β C(S) with d S (α, β) = 1, if π W (α β) is non-empty then diam W (α β) 2. This Lemma implies that π W is a well defined coarse map. On the other hand, it does not imply that π W is a Lipschitz map. In fact, given any N N +, let α, β C(W ) such that d W (α, β) = N (which is possible since C(W ) is infinite diameter). Now if we let α, β C(S) denote the curves α and β as seen in the curve graph C(S), then we have π W (α ) = α and π W (β ) = β. On the other hand W does not intersect α and β in S, which implies that d S (α, β ) = 2. So π W can project pair of points which are distance 2 apart to pair of points which are distance N apart. However the intesection number of α and β in S coarsely bounds the subsurface distance. Lemma 3.3. For two finite sets A, B C(S) and W S with ξ(w ) 4, if π W (A B) then we have d W (A, B) log(i S (A, B)).

37 29 Proof. First observe that d W (A, B) = diam C(W ) (π W (A) π W (B)) = max{d C(W ) (α, β) α, β A B} 2 max α A,β B d C(W ) (α, β), and log(i S (A, B)) = max α A,β B log(i S (α, β)). Therefore it is enough to show that d W (α, β) log(i S (α, β)) for α, β C(S). Now If W = S, then the result follows from Lemma 2.5. If W S, then by the definition of subsurface distance, d W (α, β) = max{d W (α, β ) α π W (α), β π W (β)}, so let α π W (α) and β π W (β). By Lemma 2.5 applied to the subsurface W, we have d W (α, β ) log(i W (α, β )). (see Remark 2.4) So if we show that i W (α, β ) i S (α, β), the theorem follows. By the definitions of π W (α) and π W (β), there exists α and β, components of α W and β W such that α ψ W ( α) and β ψ W ( β). Clearly i W ( α, β) i S (α, β). Now let γ 1, γ 2, γ 3 and γ 4 be the components of W on which the end points of α and β lie.

38 30 So for 1 j 4, i W ( α, γ i ) 1 and i W ( β, γ i ) 1. By the definition of ψ W ( α), α is composed of α (possibly used twice), and parts of γ i s (similar for β ). Therefore by looking at all possible intersections of parts of α and β, we get i W (α, β ) 4 i W ( α, β) + 2 ( 4 i=1 i W ( α, γ i ) + i W ( β, ) γ i ) + 1 i,j 4 i W (γ i, γ j ) 4 i S (α, β) i S (α, β) and the proof is complete. The following Theorem by Masur and Minsky gives strong contraction properties for the subsurface projection maps π W. Theorem 3.4. (Theorem 3.1 in [15]) Let W be a proper subsurface of S with ξ(w ) 4 and let g be a geodesic segment, ray, or biinfinite line in C(S). Then there exists a constant M such that if π W (α) for every α g, then diam W (g) < M. Corollary 3.5. Let M be the constant in Theorem 3.4. Let W be proper subsurface of S with ξ(w ) 4 and let α, β C(S) such that d W (α, β) > M.

39 31 Then for any geodesic g C(S) connecting α to β, we have min γ g d S(γ, W ) 1. In other words, g has to go through the vicinity of W. Proof. Assume by contradiction that min γ g d S(γ, W ) 2. This implies that every vertex of g intersects with W. In particular, every vertex of g has non-empty projection in C(S). Therefore by Theorem 3.4, we have d W (α, β) < M, which contradicts with our assumption. So one can have an idea of what curves can appear near a geodesic g C(S) by looking at the subsurface projection distances of its end points. Large subsurface projection distances also plays an important role in estimating distances in the pants graph. Theorem 3.6. (Theorem 6.12 in [15]) Given a surface S, there is a constant M 2 such that for any M M 2 there exists constants C M and K M such that for any

40 32 two pants decompositions P 1 and P 2 of S, we have 1 d K M P(P 1, P 2 ) C M W S [d W (P 1, P 2 )] M K M d P (P 1, P 2 ) + C M, where the sum is over all subsurfaces W of S with ξ(w ) 4 (including S itself), and the function [ ] M is given by [x] M = x if x > M 0 if. x M Corollary 3.7. Given a surface S and two pants decompositions P 1 and P 2 of S, if there exists an M > 0 such that d W (P 1, P 2 ) < M for every subsurface W of S with ξ(w ) 4 (including S itself), then d P (P 1, P 2 ) < M where M only depends on M and ξ(s).

41 3.2 Combinatorics of short curves along Teichmüller geodesics 33 We know by Theorem 2.19 that the shadow of a Weil-Petersson geodesic in the pants graph is a quasi-geodesic. The following Theorem, due to Masur and Minsky, states the same fact for the shadows of Teichmüller geodesics in the curve graph. Theorem 3.8. ([14]) Let G(t) be a finite, infinite or bi-infinite Teichmüller geodesic. Then the path γ G(t), the shadow of G(t) on C(S), is a quasi-geodesic. Proof. In [14] Masur and Minsky showed that the curve complex C(S) is δ-hyperbolic by constructing a family of paths Γ in C(S) and showing that this family is coarsely transitive and has the contraction property. Also they showed that if a family of paths Γ satisfies these properties, then each path in Γ is a quasi-geodesic. This family Γ is constructed using Teichmüller geodesics in the following way: For any Teichmüller geodesic G(t), define F G : R C(S) as F G (t) = α G(t) where α G(t) is a curve in C(S) with shortest extremal length on G(t). Thus α G(t) is a quasi-geodesics. Also for any t, we have l G(t) (γ G(t) ) 1 and λ G(t) (α G(t) ) 1,

42 34 which implies d S (α G(t), γ G(t) ) 1 by Corollary 2.9 and Corollary Therefore the path γ G(t) strongly fellow-travels α G(t), and so is itself a quasi-geodesic. For any proper subsurface W of S with ξ(w ) 4, let FN W : T (S) T (W ) be the map defined by restricting of Fenchel-Nielsen coordinates to W. More precisely, choose a pants decomposition of S that contains W. FN W is the forgetful map which sends Fenchel-Nielsen coordinates on T (S), with respect to this pants decomposition, to Fenchel-Nielsen coordinates on T (W ) (See [Min95] chapter 6). Theorem 3.9. (Theorem 5.3 in [18]) Let G(t) be a bi-infinite Teichmüller geodesic in T (S). For any proper subsurface W of S with ξ(w ) 4, there exists an interval I W = [a, b] (possibly empty) and a geodesic G W : I W T (W ) such that 1. If t < a (resp. b > t), then d W (P G(t), P G(a) ) 1 (resp. d W (P G(t), P G(b) ) 1) 2. For any t I W, we have W P G(t) and d T (W ) ((FN W G(t)), G W (t)) 1. The interval I W is called the interval of isolation for W. In particular, the

43 35 boundary curves of W are short on G(t) for all t I W. Here we remark the resemblance of the first part of Theorem 3.9 and Corollary 3.5. Given a Teichmüller geodesic segment G(t), t [a, b] and a proper subsurface W of S with ξ(w ) 4, if d W (P G(a), P G(b) ) is sufficiently large then the interval I W cannot be empty and has to intersect with the interval [a, b]. This means that the boundary curves of W has to be short somewhere on the geodesic G(t), t [a, b]. Therefore we can have an idea of what curves can appear on the shadow of G(t) in C(S) by looking at the subsurface projection distances of its end points. An important consequence of the second statement of Theorem 3.9 is the following. Corollary The path π W (P G(t) ) C(W ), t I W is a quasi geodesic. Proof. By the second part of the above Theorem, W P G(t), and so the set of curves P G(t) W = {α 1 (t), α 2 (t),..., α k (t)} is equal to π W (P G(t) ). We know from [Min95] that the extremal lengths of each α i (t) in G(t) and in FN W (G(t)) are comparable λ G(t) (α i (t)) λ FNW (G(t))(α i (t)).

44 36 On the other hand l G(t) (α i (t)) 1 since α i (t) P G(t), and so applying Corrolary 2.9, we get l FNW (G(t))(α i (t)) 1. By Corollary 2.14 applied to the subsurface W, we deduce that π W (P G(t) ) is close to the shortest pants decomposition on FN W (G(t)) in C(W ) d W (π W (P G(t) ), P FNW (G(t))) 1. (see Remark 2.4) Now combining the second statement of Theorem 3.9 and Proposition 2.16, we have d W (P FNW (G(t)), P GW (t)) 1 for t I γ, and so by triangle inequality we get d W (π W (P G(t) ), P GW (t)) 1. This means that the path {π W (P G(t) )} t IW strongly fellow-travels the path {P GW (t)} t IW in C(W ) which is a quasi-geodesic by Theorem 3.8. Therefore {π W (P G(t) )} t IW is also a quasi-geodesic.

45 Chapter Four Fellow-traveling theorem in small complexity

46 38 Let S be surface with ξ(s) 5. In this chapter we will prove Theorem 1.1. The proof of Theorem 1.1 for complexity 4 is a direct consequence of δ-hyperbolicity of P(S), Theorem 3.8 and Theorem Proof. of Theorem 1.1 for ξ(s) = 4: Let G(t) and G(s) be Teichmüller and Weil- Petersson geodesics respectively, with the same end points. By Theorem 2.19, it is enough to show that P G(t) and P G(s) are fellow-travellers in P(S). Since G(s) is a geodesic in the Weil-Petersson metric, P G(s) is a quasi-geodesic in P (S), again by Theorem On the other hand, P(S) is the same as C(S) in complexity 4, and by Theorem 3.8 P G(t) = γ G(t) is also a quasi-geodesic in P(S) = C(S). Thus P G(t) and P G(s) are both quasi-geodesics in P(S) with the same end points and therefore they fellow-travel by the δ-hyperbolicity of P(S). In higher complexity, P(S) is not equal to C(S). Thus in order to prove Theorem 1.1 in complexity 5, we will use a combinatorial device called hierarchies of geodesics, developed by Masur and Minsky in [15], which connects the geometry of P(S) to the geometry of the curve graphs of subsurfaces of S. For simplicity, we will only give the definition of hierarchies without annuli and just for complexity 5.

47 Hierarchies in low complexity Here S is either twice punctured torus or five-times punctured sphere. Observe that in this complexity, pants decompositions consist of 2 disjoint curves. Also for any γ C(S), S γ has a unique component, denoted by X γ, which has complexity 4 (which is either once punctured torus or four-times punctured sphere). On the other hand, any proper subsurface W of S with ξ(w ) = 4 is equal to X γ for some γ C(S). Definition 4.1. (Hierarchies in complexity 5) let P 1 = {α 1, β 1 } and P 2 = {α 2, β 2 } be two pants decompositions in P(S). A hierarchy H connecting P 1 to P 2 consists of the following: 1. A geodesic g H in C(S), called the main geodesic of H, connecting a vertex of P 1 (say α 1 ) to a vertex of P 2 (say α 2 ). We call P 1 and P 2 the initial and terminal pants decompositions and denote them by I(H) and T (H) respectively. 2. A geodesic g γ in C(X γ ) for each γ g H connecting the predecessor of γ to the successor of γ on g H (with the convension that the predecessor of α 1 is β 1 and the successor of α 2 is β 2 ). We denote the first and the last vertex of g γ by I(g γ ) and T (g γ ) respectively. It is clear, from the connectedness of the curve graphs, that any pair of pants decompositions can be connected with hierarchy.

48 40 Lemma 4.2. (Lemma 6.2 in [15]) Let H be a hierarchy with initial and terminal pants decompositions I(H) and T (H). Then there exists a constant M such that for any proper subsurface W of S with ξ(w ) 4, if d W (I(H), T (H)) > M, then W is the domain of some geodesic in H. In the case of complexity 5, any proper subsurface W of S with ξ(w ) 4 is equal to X γ for some γ C(S). Also X γ is the domain of some geodesic in H if and only if γ g H. The proof of Lemma 4.2 for complexity 5 surfaces is a straightforward application of Theorem 3.4 Any hierarchy H has a resolution into a sequence of pants decompositions {P i } P(S) connecting I(H) to T (H). Here is the construction: Each pants decomposition P i consists of a vertex α in g H and a vertex β in g α which are chosen inductively as follows i. P 1 consists of the initial vertex α 1 of g H and the initial vertex β 1 of g α1, which is nothing but I(H) itself. ii. Suppose P i = {α i, β i } where α i g H and β i g αi. If β i T (g αi ), we let P i+1 = {α i, γ} where γ is the successor of β i on g αi. If β i = T (g αi ), then by the construction of H, β i is the successor of α i on g H and α i = I(g βi ). In this case we let P i+1 = {β i, δ} where δ is the successor of α i on g βi.

49 41 It is clear that P i and P i+1 are adjacent in P(S). Moreover the sequence eventually terminates, say at P N = {α N, β N }, where α N is the last vertex of g H and β N the last vertex of g αn, which is nothing but T (H). We call this sequence the resolution of H (Here we should remark that for higher complexity surfaces, hierarchies may have more than one resolution). The following theorem is one of the most important uses of hierarchies. Theorem 4.3. (Theorem 6.10 in [15]) Let H be a hierarchy and let P 1, P 2,..., P N be its resolution. We have d P (I(H), T (H)) N. Corollary 4.4. Resolutions of hierarchies are quasi-geodesics in the pants graph. Proof. of the case ξ(s) = 5: The above theorem says that the length of the resolution of a hierarchy is coarsely equal to the distance between its endpoints. So given a hierarchy H and its resolution {P i } 1 i N, if we show that any subpath of {P i } 1 i N is also a resolution of some hierarchy, we are done. In fact, we simply choose the part of the original hierarchy which intersects with the vertices of this subpath. It is easy to check that this is a hierarchy and its resolution is this subpath.

50 4.2 Teichmüller geodesics fellow-travel hierarchies 42 Let S be a surface with ξ(s) = 5, G(t) : [0, l] T (S) be a finite Teichmüller geodesic and let X = G(0), Y = G(l). Also let H be a hierarchy connecting I(H) = P X and T (H) = P Y. We assume this setup in the rest of this section and introduce some notations, Notation: 1. For any t, we write P t := P G(t) & γ t := γ G(t) 2. For any γ C(S) and A, B C(S), we write d γ (A, B) := d Xγ (A, B) & diam γ (A) := diam Xγ (A). We know by Theorem 2.19 that the Weil-Petersson geodesic G(s) connecting X and Y induces a quasi-geodesic P G(s) connecting P X and P Y in P(S). On the other hand, the resolution of H is also a quasi-geodesic connecting P X and P Y. So they fellow-travel in P(S) which is δ-hyperbolic by Theorem 2.7. Therefore Theorem 1.1 follows from the following theorem. Theorem 4.5. The path P t fellow-travels the resolution of the hierarchy H in the pants graph P(S). More precisely, for any point Z on G, there exists P Z in the

51 43 resolution of H such that d P (P Z, P Z ) 1 To bound the pants distance between P Z and P Z, we will use Corollary 3.7 and bound d W (P Z, P Z ) for every subsurface W of S with ξ(w ) 4 (including S itself). So we need to have some control over subsurface distances of the path P t. Recall from Theorem 3.9 that for each subsurface X γ, there exists an interval I Xγ (possibly empty) called the interval of isolation for X γ. In our setting, we only consider the part of G(t) for t [0, l]. So for any γ C(S), we let Ĩ γ = I Xγ [0, l] if I Xγ [0, l] {0} otherwise Definition 4.6. The combinatorial length of Ĩγ = [a, b] is defined as Ĩγ c = d γ (P a, P b ) In [15], Masur and Minsky prove that two hierarchies whose end pants decompositions are close to each other have fellow-traveling long geodesics (Lemma 6.6, Common Links). Combining Theorem 3.9 and its Corollary with Theorem 3.4, we can prove a similar argument between G and H. Lemma 4.7. For any γ C(S).

52 44 i. Ĩγ c d γ (P X, P Y ). ii. If γ g H, then Ĩγ c d γ (I(g γ ), T (g γ )) and g γ fellow-travels the quasi-geodesic π Xγ (P t ) min Ĩγ t max Ĩγ in C(X γ ). iii. For any α, β {P t t [0, l]} H where α, β γ, we have d γ (α, β) Ĩγ c Proof. Let Ĩγ = [a, b]. i. By definition Ĩγ c = d γ (P a, P b ). Now by the first part of Theorem 3.9, there exists a constant A such that d γ (P a, P X ) < A and d γ (P b, P Y ) < A, so by triangle inequality d γ (P X, P Y ) 2A d γ (P a, P b ) d γ (P X, P Y ) + 2A which implies the result.

53 45 ii. Suppose γ g H. Let α the vertex preceeding γ and β be the vertex following γ on g H. By the construction of H, α = I(g γ ) and β = T (g γ ) and since the segments on g H connecting α to P X and β to P Y have bounded diameter projections onto C(X γ ) by Theorem 3.4, there exists another constant B such that d γ (I(g γ ), P X ) < B and d γ (T (g γ ), P Y ) < B. Combining with part 1, we have d γ (I(g γ ), P a ) < A + B and d γ (T (g γ ), P b ) < A + B, again by triangle inequality d γ (I(g γ ), T (g γ )) 2A 2B d γ (P a, P b ) d γ (I(g γ ), T (g γ )) + 2A + 2B which implies the first part. Also this means that the end points of the geodesic g γ are bounded distance away from the end points of the path π Xγ (P t ) a t b, which is a quasi-geodesic by Corollary Therefore by hyperbolicity of C(X γ ), g γ and π Xγ (P t ) are fellow-travelers. iii. Given α {P t t [0, l]} H other than γ, it is enough to show that d γ (α, P X ) Ĩγ c. First suppose that α {P t t [0, l]}, say α P t0 for some t 0 [0, l]. Let t 1 Ĩγ

54 46 be the closest point to t 0 on Ĩγ. By the first part of Theorem 3.9, we have d γ (P t1, α) 1. Now by Corollary 3.10, π Xγ (P t ), t Ĩγ is a quasi-geodesic in C(X γ ), so d γ (P a, P t1 ) length{π Xγ (P t ), t [a, t 1 ]} length{π Xγ (P t ), t Ĩγ} d γ (P a, P b ) = Ĩγ c. Finally we have d γ (P a, P X ) 1 again by the first part of Theorem 3.9. Therefore combining these with triangle inequality, we get the result d γ (α, P X ) Ĩγ c. Now suppose α H. If we show that α has bounded d γ distance from a point on π Xγ (P t ), then we can use the above argument to get the result. If α g γ, then by the second part, α is bounded d γ distance away from a point on {π Xγ (P t ), t Ĩγ}. On the other hand if α g γ, then it has distance 0 or 1 from a point on g H {γ}. So by Theorem 3.4, either d γ (α, P X ) 1 or d γ (α, P Y ) 1 depending on whether α < γ or α > γ on g H.

55 47 Lemma 4.8. (Order of subsurfaces) Let α, β be two vertices of g H. There is a constant M such that if Ĩα c, Ĩβ c > M and α < β on g H, then we have min Ĩα < min Ĩβ and max Ĩα < max Ĩβ. Proof. If we just prove min Ĩα < min Ĩβ, then max Ĩα < max Ĩβ easily follows by changing the roles of α and β. Let Ĩα = [a, b] and Ĩβ = [c, d]. By the first part of Theorem 3.9, there is a constant M 1 such that for any 0 t a and b s l, we have d α (P a, P t ) < M 1 & d α (P b, P s ) < M 1, in particular d α (P b, P Y ) < M 1. Now since α < β, any vertex between β and T (g H ) P Y on g H intersect with X α, so by Theorem 3.4 there is another constant M 2 such that d α (β, P Y ) < M 2. Let M = 2M 1 + M and assume by contradiction that min Ĩα min Ĩβ (i.e 0 c a). Thus d α (P a, P c ) < M 1.

56 48 On the other hand β P c by the second part of Theorem 3.9, so d α (P c, β) 1. Combining these inequalities with d α (β, P Y ) < M 2 and d α (P Y, P b ) < M 1 and using triangle inequality, we get d α (P a, P b ) < 2M 1 + M = M which contradicts with our assumption Ĩα c = d α (P a, P b ) > M. Proof. of Theorem 4.5: Given γ C(S) where Ĩγ c 0 and 0 t l, we define ρ t (γ) = min{ t x, x Ĩγ}. Now let Z be a point on G and let z [0, l] be the position of Z on G (i.e G z = Z). Also let M be the constant given in Lemma 4.8. We choose P Z = {α, β} in the resolution of H as follows: Let A H = {v g H d S (P Z, v) is minimal} and B H = {v A H Ĩv c > M}. If B H, we choose α B H so that ρ z (α) is minimal, otherwise we choose α to be any point in A H. Choose β g α so that d α (P Z, β) is minimal.

57 49 To bound d P (P Z, P Z ), we will use Corollary 3.7 and bound d W (P Z, P Z ) for every subsurface W of S with ξ(w ) 4 (including S itself). Case 1 (W = S): This is a direct consequence of Theorem 3.8 which states that the path γ t of shortest curves forms a quasi-geodesic in C(S), and so it fellowtravels the main geodesic g H of H. Therefore by the construction of P Z, d S (P Z, P Z ) d S (P Z, α) = min v gh d S (P Z, v) min v gh d S (γ Z, v) 1 where γ Z P Z is the shortest curve on Z. Case 2 (W S) Let γ C(S) be such that W = X γ. We will further divide into 2 subcases. (γ = α): Let t 0 Ĩα be the closest point to z. By the first part of Theorem 3.9 d α (P Z, P t0 ) 1. Also by the second part of Lemma 4.7, π α (P t ) for t Ĩα fellowtravels g α in C(X α ), so min v gα d α (P t0, v) 1. Therefore d α (P Z, P Z ) = d α (P Z, β) = min v gα d α (P Z, v) min v gα (d α (P Z, P t0 ) + d α (P t0, v)) 1

58 50 (γ α): By the first and the third parts of Lemma 4.7, we have d γ (P Z, P Z ) Ĩγ c d γ (P X, P Y ). So if the right hand side is bounded, we are done. Thus we can assume that Ĩγ c > M and d γ (P X, P Y ) > M where M is the constant given in Lemma 4.2. This implies that γ g H. Without loss of generality assume that γ < α on g H. Let M = M 1 + M 2 where M 1, M 2 are the constants given in Theorem 3.4 and first part of Theorem 3.9 respectively. Claim: d γ (P Z, α) M Arguing by contradiction, assume that d γ (P Z, α) > M. Let P Z = {α, β } and suppose that d γ (P Z, α) = d γ (α, α). Let g = {v 0 = α, v 1, v 2,..., v n = α} be a geodesic in C(S) connecting α to α. Since d γ (α, α) > M > M 1, by Theorem 3.4, γ occurs somewhere on g, say γ = v i where i n 1 since γ α. So d γ (P Z, γ) = max{d γ (α, γ), d γ (β, γ)} d γ (α, γ) + 1 n = n = d γ (P Z, α). On the other hand α A H which implies that d γ (P Z, α) d γ (P Z, γ). Therefore γ is also in A H and since we assumed that Ĩγ c > M, γ is in B H. So B H therefore by the construction of PZ, α is also in B H (i.e. Ĩα c > M) and ρ z (α) ρ z (γ). At this point, we have Ĩα c > M, Ĩγ c > M and γ < α, so we