aa r X i v : . [ m a t h . G T ] J un DISTANCE IN THE CURVE GRAPH
URSULA HAMENST ¨ADT
Abstract.
We estimate the distance in the curve graph of a surface S of finitetype up to a fixed multiplicative constant using Teichm¨uller geodesics. Introduction
The curve graph of an oriented surface S of genus g ≥ m ≥ g − m ≥ CG , d CG ) whose vertices are isotopy classesof essential (i.e. non-contractible and not puncture parallel) simple closed curveson S . Two such curves are connected by an edge of length one if and only if theycan be realized disjointly. The curve graph is a hyperbolic geodesic metric space ofinfinite diameter [MM99]. It turned out to be an important tool for understandingthe geometry of the mapping class group of S [MM00].The Teichm¨uller space T ( S ) of S is the space of all complete finite volumemarked hyperbolic metrics on S . Let P : Q ( S ) → T ( S ) be the bundle of markedarea one quadratic differentials . The bundle Q ( S ) can naturally be identified withthe unit cotangent bundle of T ( S ) for the Teichm¨uller metric . In particular, every q ∈ Q ( S ) determines the unit cotangent line of a Teichm¨uller geodesic .An area one quadratic differential q defines a singular euclidean metric on S ofarea one. Call a simple closed curve α δ -wide for q if α is the core curve of anembedded annulus of width at least δ with respect to the metric q . Here the widthof an annulus is the minimal distance between the two boundary circles. For smallenough δ , a δ -wide curve exists for all q ∈ Q ( S ) (Lemma 5.1 of [MM99]).A map Υ : Q ( S ) → CG which associates to a marked area one quadratic differ-ential q a δ -wide curve for q relates the Teichm¨uller metric on T ( S ) to the metricon CG . Namely, there is a number L > q t ) ⊂ Q ( S ) of a Teichm¨uller geodesic, the assignment t → Υ( q t ) is a coarselyLipschitz unparametrized L -quasi-geodesic (this is an immediate consequence ofTheorem 2.6 of [MM99]). This implies that there is a number κ > d CG (Υ( q s ) , Υ( q t )) ≤ κ | t − s | + κ and(1) d CG (Υ( q s ) , Υ( q t )) + d CG (Υ( q t ) , Υ( q u )) ≤ d CG (Υ( q s ) , Υ( q u )) + κ for s ≤ t ≤ u (see Lemma 2.5 of [H10] for a detailed discussion).In general these unparametrized quasi-geodesics are not quasi-geodesics withtheir proper parametrizations. More precisely, for ǫ > T ( S ) ǫ ⊂ T ( S ) be theset of all complete finite volume marked hyperbolic metrics on S whose systole (i.e.the shortest length of a simple closed geodesic) is at least ǫ . Then for a given number Date : May 17, 2012.Partially supported by the Hausdorff Center BonnAMS subject classification:30F60. L ′ > L , there exists a number ǫ = ǫ ( L ′ ) > CG defined by the image under Υ of the cotangent line of a Teichm¨ullergeodesic γ is a parametrized L ′ -quasi-geodesic only if γ entirely remains in T ( S ) ǫ [MM99, H10]. In particular, d CG (Υ( q s ) , Υ( q t )) may be uniformly bounded eventhough | s − t | is arbitrarily large.Nevertheless, the estimate (1) allows to construct for any two points ξ, ζ ∈ CG aparametrized uniform quasi-geodesic connecting ξ to ζ as follows.Note first that it is very easy to decide whether the distance in the curve graphbetween ξ and ζ is at most two. Namely, in this case there is an essential simpleclosed curve α which can be realized disjointly from ξ, ζ .If the distance between ξ, ζ is at least three then ξ, ζ define the unit cotangentline ( q t ) of a Teichm¨uller geodesic whose horizontal and vertical measured folia-tions are one-cylinder foliations with core curves homotopic to ξ, ζ , respectively.For any sequence t < · · · < t n so that d CG (Υ( q t i ) , Υ( q t i +1 )) ≥ κ for all i and d CG (Υ( q s )) , Υ( q t )) < κ for all s, t ∈ ( t i , t i +1 ), a successive application of the sec-ond part of the estimate (1) implies that d CG (Υ( q t ) , Υ( q t n )) ≥ nκ. Combined with the first part of (1), the map s ∈ [ i, i + 1) → Υ( q t i ) is a uniformquasi-geodesic in CG .Unfortunately, up to date no bounds for the number κ > tight geodesics . Unlike the problem of deciding whether or not the distancein the curve graph is at least three, determining whether or not the distance in CG between two given curves is bigger than a given number β ≥ CG up to a fixed multiplicative constant by onlydetecting pairs of curves of distance at least three. We show Theorem.
There is a number θ > with the following property. Let ( q t ) ⊂ Q ( S ) be the unit cotangent line of a Teichm¨uller geodesic. Assume that there are numbers t < · · · < t n = T such that for all i < n we have (1) t i +1 ≥ t i + 1(2) d CG (Υ( q s ) , Υ( q t )) ≥ for s ≤ t i , t ≥ t i +1 .Then d CG (Υ( q ) , Υ( q T )) ≥ n/θ − θ . Let again ( q t ) be the unit cotangent line of a Teichm¨uller geodesic. For T > t < · · · < t u = T as follows. If t i has alreadybeen determined, let t i +1 be the smallest number contained in the interval [ t i +1 , T ]so that d CG (Υ( q s ) , Υ( q t )) ≥ s ≤ t i , t ≥ t i +1 . If no such number exists put u = i . By the first part of the estimate (1), for each s ≤ t ℓ there is at least one t ≤ t ℓ − so that d CG (Υ( q s ) , Υ( q t )) ≤ κ . A successive application of the triangleinequality shows that d CG (Υ( q T ) , Υ( q )) ≤ u (3 + 2 κ ) + 3. On the other hand, wehave d CG (Υ( q T ) , Υ( q )) ≥ uθ − /θ by the Theorem.The proof of this result relies on the work of Minsky [M92] and Rafi [R10].The basic idea is that δ -wide curves for a quadratic differential q detect ”active”subsurfaces in the sense of [R10] for the Teichm¨uller geodesic defined by q . However, ISTANCE IN THE CURVE GRAPH 3 such active subsurfaces may intersect in a complicated way, and making this simpleidea rigorous and quantitative is the main contribution of this work. Unfortunatelythe proof of the theorem does not give an explicit bound for the number θ .2. Teichm¨uller geodesics and the curve graph
Let S be an oriented surface of genus g ≥ m ≥ complexity ξ ( S ) = 3 g − m ≥
1. In the sequel we always mean by a simple closedcurve on S an essential simple closed curve. Moreover, we identify curves which areisotopic unless specifically stated otherwise.We denote by CG the curve graph of S and by d CG the distance in CG . If ξ ( S ) = 1then S is a once-punctured torus or a four-punctured sphere and CG is the Fareygraph (i.e. two simple closed curves are connected by an edge if they intersect inthe smallest possible number of points).Let P : Q ( S ) → T ( S ) be the bundle of marked area one quadratic differentialsover the Teichm¨uller space T ( S ) of S . Such a quadratic differential on a surface x ∈ T ( S ) is a meromorphic section of T ′ ( x ) ⊗ T ′ ( x ) with at most simple poles at thepunctures and no other poles. It defines a singular euclidean metric on S of finitearea, and we require that the area of this metric equals one (see [S84] for details).Any two simple closed curves ξ, ζ of distance at least 3 in CG define a one-parameterfamily ( q t ) of area one quadratic differentials which form the unit cotangent line ofa Teichm¨uller geodesic. These differentials are determined by the requirement thatthe vertical and the horizontal measured foliation of q t is a one-cylinder foliationwith cylinder curves homotopic to ξ, ζ , respectively (see Proposition 3.10 of [H12]for one of the many places in the literature where such quadratic differentials areconstructed explicitly).For x ∈ T ( S ) and an essential simple closed curve α on S let Ext x ( α ) be the extremal length of α with respect to the conformal structure defined by x . If q is aquadratic differential on x then we also write Ext q ( α ) instead of Ext x ( α ). We referto [G87] for more and references.In the sequel we always mean by a quadratic differential a marked area onequadratic differential. The q -length ℓ q ( α ) of an essential simple closed curve α on S is the infimum of the lengths for the singular euclidean metric defined by q of allcurves freely homotopic to α . With this terminology, we haveExt q ( α ) ≥ ℓ q ( α ) for every simple closed curve α .There is a number δ > q admits a δ -widecurve [MM99] (see also [B06] for a more explicit and detailed discussion), i.e. asimple closed curve α which is the core curve of an embedded annulus of width atleast δ . Then the q -length of α is at most 1 /δ . Define a mapΥ : Q ( S ) → CG by associating to an area one quadratic differential q a δ -wide simple closed curvefor q .There is a number κ > q t ) of a Teichm¨ullergeodesic, the estimate (1) from the introduction holds true. In particular, if t >s, k ≥ d CG (Υ( q s ) , Υ( q t )) ≥ k + 2 κ then d (Υ( q u ) , Υ( q v )) ≥ k URSULA HAMENST¨ADT for all u ≤ s < t ≤ v .For a quadratic differential q ∈ Q ( S ) let ν ( q ) = inf { Ext q ( α ) | α } where α ranges through the essential simple closed curves. Note that the function ν : Q ( S ) → (0 , ∞ ) is continuous. Moreover, for every ǫ >
0, the set { ν ≥ ǫ } isclosed and invariant under the action of the mapping class group Mod( S ) of S , andthe quotient { ν ≥ ǫ } / Mod( S ) is compact.Denote by λ the Lebesgue measure on the real line. The following observationcan be thought of as a quantitative account on progress in the curve graph alongTeichm¨uller geodesic segments which spend a definitive proportion of time in thethick part of Teichm¨uller space. Proposition 2.1.
For every ǫ > , k > and every b ∈ (0 , there is a number R = R ( ǫ, k, b ) > with the following property. Let ( q t ) be the unit cotangent lineof a Teichm¨uller geodesic. If λ { t ∈ [0 , R ] | ν ( q t ) ≥ ǫ } ≥ bR for some R ≥ R then d CG (Υ( q ) , Υ( q R )) ≥ k. Proof.
Let ǫ > , k > , b ∈ (0 , τ = τ ( ǫ, k, b ) > q t ) there is a number σ ∈ [1 , τ ] so that either d CG (Υ( q ) , Υ( q σ )) ≥ k or λ { s ∈ [0 , σ ] | ν ( q s ) < ǫ } ≥ σ (1 − b ) . For this we argue by contradiction and we assume that the claim does not hold.Then there is a sequence ( q it ) of unit cotangent lines of Teichm¨uller geodesics sothat for each i >
0, the smallest subinterval of [0 , ∞ ) containing [0 ,
1] with theproperty that the above alternative holds true for ( q it ) is of length at least i .Since for each i ≥ q it ) and σ = 1, there isat least one s ∈ [0 ,
1] so that ν ( q is ) ≥ ǫ . This implies that the quadratic differentials q i project to a compact subset of the moduli space of quadratic differentials. Thusby invariance under the mapping class group, after passing to a subsequence wemay assume that the quadratic differentials q i converge as i → ∞ in Q ( S ) to aquadratic differential q .Let χ > z , the distancein the curve graph between any two curves on S of z -length at most 2 /δ is smallerthan χ (see Lemma 2.1 of [H10] for a detailed proof of the existence of such anumber). Let ( q t ) be the unit cotangent line of the Teichm¨uller geodesic withinitial velocity q = q .If ( q t ) is recurrent (i.e. its projection to the moduli space of quadratic differen-tials returns to a fixed compact set for arbitrarily large times) then there is some T > d CG (Υ( q ) , Υ( q T )) ≥ k + 2 χ (see Proposition 2.4 of [H10] for details). However, by continuity, for all sufficientlylarge i the q -length (or q T -length) of a curve of q i -length at most 1 /δ (or of a curveof q iT -length at most 1 /δ ) does not exceed 2 /δ . Thus by the choice of χ , d CG (Υ( q i ) , Υ( q iT )) ≥ k for all sufficiently large i which violates the assumption on the sequence ( q it ). ISTANCE IN THE CURVE GRAPH 5
On the other hand, if ( q t ) is not recurrent then there is some t > ν ( q t ) ≤ ǫ/ t ≥ t . Then the proportion of time the arc q t ( t ∈ [0 , t /b ])spends in the region { ν ≤ ǫ/ } is at least 1 − b . By continuity, for all sufficientlylarge i the proportion of time the arc q it ( t ∈ [0 , t /b ]) spends in the region { ν < ǫ } is at least 1 − b as well. Once again, this is a contradiction.Let κ > b ∈ (0 , , k > , ǫ >
0, let τ = τ ( ǫ, b/ , k + 2 κ )and let R > τ /b . Let ( q t ) be the unit cotangent line of any Teichm¨uller geodesic.There are successive numbers 0 = t < t < · · · < t u ≤ R so that R − t u ≤ τ andthat for all i < u , 1 ≤ t i +1 − t i ≤ τ and either(2) d CG (Υ( q t i ) , Υ( q t i +1 )) ≥ k + 2 κ or λ { t ∈ [ t i , t i +1 ] | ν ( q t ) < ǫ } ≥ ( t i +1 − t i )(1 − b/ . By the choice of κ , if there is some i < u such that the inequality (2) holdstrue then d CG (Υ( q ) , Υ( q R )) ≥ k . Otherwise since R − t u ≤ τ < bR/
2, we have λ { t ∈ [0 , R ] | ν ( t ) < ǫ } > R (1 − b ). The proposition follows. (cid:3) For small ǫ >
0, the (hyperbolic) length of a simple closed curve on a hyperbolicsurface x ∈ T ( S ) is roughly proportional to its extremal length. Thus by the collarlemma, there is a number ǫ < δ / x ∈ T ( S ), any two simple closedcurves on ( S, x ) of extremal length at most ǫ can be realized disjointly.For ǫ ≤ ǫ let A ǫ = A ǫ ( x )be the set of all simple closed curves on ( S, x ) of extremal length less than ǫ . Then S − A ǫ is a union Y of connected components (this is meant to be a topologicaldecomposition, i.e. we cut S open along disjoint representatives of the curves ofsmall extremal length). A component Y ∈ Y is called an ǫ -thick component for x , and ( A ǫ , Y ) is the ǫ -thin-thick decomposition of ( S, x ). Note that for χ < ǫ , a χ -thick component is a union of ǫ -thick components.If q is a quadratic differential with underlying conformal structure x then we alsocall ( A ǫ , Y ) the ǫ -thin-thick decomposition for q , and we call a component Y of Y an ǫ -thick component for q . More generally, we call a non-peripheral incompressibleopen connected subsurface Y of S ǫ -semi-thick for q if the extremal length of eachboundary circle of Y is at most ǫ . Then the result of cutting Y along all the curvesof extremal length (as curves in S ) at most ǫ is a union of ǫ -thick components.Let again q be a quadratic differential on S . For ǫ ≤ ǫ and an ǫ -semi-thicksubsurface Y ⊂ S for q let Y be the representative of Y with q -geodesic boundarywhich is disjoint from the interiors of the (possibly degenerate) flat cylinders foliatedby simple closed q -geodesics homotopic to the boundary components of Y . We call Y the geometric representative of Y . Following Rafi [R07], if Y is not a pairof pants then we define size q ( Y ) to be the shortest q -length of an essential simpleclosed curve in Y . If Y is a pair of pants then we define size q ( Y ) to be the diameterof Y .An expanding annulus for a simple closed curve α ⊂ ( S, q ) is an embedded annu-lus A ⊂ S homotopic to α with the following property. One boundary componentof A is a q -geodesic ψ , and the second boundary component is a curve which isequidistant to ψ . Moreover, A does not intersect the interior of a flat cylinderfoliated by closed q -geodesics homotopic to α (see [M92] for details). URSULA HAMENST¨ADT
By Theorem 3.1 of [R10], up to making ǫ smaller we may assume that for everyquadratic differential q and every α ∈ A ǫ = A ǫ ( q ) (this means that we calculateextremal length for the conformal structure underlying q ) the following holds true.Assume that Y, Z are the ǫ -thick components of ( S, q ) which contain α in theirboundary. Note that Y, Z are not necessarily distinct. Let
E, G be the maximalexpanding annuli in the geometric representatives Y , Z of Y, Z homotopic to α .This means that E, G are expanding annuli contained in Y , Z which are as big aspossible, i.e. which are not proper subsets of another expanding annulus. If Y = Z then these annuli are required to lie on the two different sides of α with respect toan orientation of α and the orientation of S . Then1Ext q ( α ) ≍ log size q ( Y ) ℓ q ( α ) + size q ( F q ( α )) ℓ q ( α ) + log size q ( Z ) ℓ q ( α )(3) ≍ Mod q ( E ) + Mod q ( F q ( α )) + Mod q ( G )where F q ( α ) is the (possibly degenerate) flat cylinder foliated by simple closedgeodesics homotopic to α and where Mod q is the modulus with respect to the con-formal structure defined by q . Also, ≍ means equality up to a uniform multiplicativeconstant.Let M > M . For M ≥ M and for ǫ ≤ ǫ , a boundary curve α of an ǫ -semi-thick subsurface Y of S is called M -large if Mod( E ) ≥ M where as before, E is a maximal expanding annulus homotopic to α in the geometric representative Y of Y . An ǫ -semi-thick subsurface Y is M -large if Y is not a pair of pants and ifeach of its boundary circles is M -large. We also require that Y is non-trivial, i.e.that Y = S .Since ǫ < δ / q -length of a simple closed curve α is at most p Ext q ( α ), a δ -wide simple closed curve α can not have an essential in-tersection with a boundary curve of an ǫ -thick component of S . As a consequence,a geodesic representative of α either is contained in the geometric representative Y of an ǫ -thick component Y of S , or it is homotopic to a boundary circle of such acomponent.In the case that α is contained in the geometric representative Y of an ǫ -thick component Y of S and is not homotopic to a boundary component of Y , theembedded annulus of width δ which is homotopic to α can intersect a boundarycomponent of Y only in a set of small diameter. As a consequence, up to adjustingthe number ǫ , the q -diameter of Y is bounded from below by a universal constant.If Ext q ( α ) ≤ ǫ then there are two (not mutually exclusive) possibilities. In thefirst case, α is the core curve of a flat cylinder of small circumference whose widthis uniformly bounded from below. Such a cylinder has a large modulus. The secondpossibility is that there is an expanding annulus homotopic to α whose width isuniformly bounded from below. Then this annulus is contained in the geometricrepresentative Y of an ǫ -thick component Y of S . The q -diameter of Y is boundedfrom below by a universal constant.This observation is used to show Lemma 2.2.
For every
M > , χ > there is a number ǫ ( χ, M ) < ǫ with thefollowing property. Let q ∈ Q ( S ) and assume that there is a simple closed curve β with Ext q ( β ) < ǫ ( χ, M ) . Let α be a δ -wide curve for q ; then up to isotopy, either ISTANCE IN THE CURVE GRAPH 7 α is contained in an M -large subsurface of S − β or α is contained in a flat cylinderof modulus at least χ .Proof. For χ > ν ≤ ǫ be sufficiently small that the modulus of a flat cylinderof circumference √ ν and width δ/ χ .Let α be a δ -wide curve for the quadratic differential q . By the choice of ǫ ,there is a q -geodesic representative of α which is contained in a ν -thick component Y for q .If α is homotopic to a boundary circle of Y and if moreover α is the core curveof a flat cylinder of width at least δ/ α is the core curve of a flat cylinder ofmodulus at least χ and we are done.For the remainder of this proof suppose that no such cylinder exists. The discus-sion preceding this lemma shows that in this case we may assume without loss ofgenerality that the q -diameter of Y is bounded from below by a universal constant σ = σ ( χ ) > Y of extremal length smaller than ν , size q ( Y ) is comparable to the q -diameter of Y (with a comparison factor depending on ν ). Now the q -diameter of Y is at least σ and therefore size q ( Y ) is bounded from below by a number σ ≤ σ only dependingon ν and hence only depending on χ .Let M > Y is M -large then we aredone. Otherwise let β be a boundary component of Y which is not M -large. Sincesize q ( Y ) ≥ σ , the estimate (3) shows that ℓ q ( β ) ≥ σ for a number σ > σ . In particular, we have Ext q ( β ) ≥ σ .If β is non-separating and defines two distinct free homotopy classes in Y then Y ∪ β = Y is a subsurface of S containing α . Since the length of β is boundedfrom below by a universal constant, the size of Y ∪ Y is bounded from below by auniversal constant σ > χ -thick component Y ′ of S which contains β in its boundaryand which is distinct from Y . Using again the fact that the length of β is uniformlybounded from below, the diameter of Y ′ and hence the size of Y ′ is uniformlybounded from below. Let Y = Y ∪ Y ′ and note that Y contains α , and its size isbounded from below by a universal constant σ >
0. In particular, any very shortboundary curve of Y is contained in an expanding cylinder of modulus at least M .Repeat this reasoning with Y . In at most 3 g − m steps we either find an M -large subsurface Y of S containing α whose size is bounded from below by auniversal constant, or we conclude that the shortest q -length of an essential simpleclosed curve on S is bounded from below by a universal constant. Together thisshows the lemma. (cid:3) Let X ⊂ S be a non-peripheral, incompressible, open connected subsurface whichis distinct from S , a three-holed sphere and an annulus. The arc and curve complex C ′ ( X ) of X is defined to be the complex whose vertices are isotopy classes of arcswith endpoints on ∂X or essential simple closed curves in X . Two such verticesare connected by an edge of length one if they can be realized disjointly. There is a subsurface projection π X of CG into the space of subsets of C ′ ( X ) which associatesto a simple closed curve the homotopy classes of its intersection components with X (see [MM00]). For every simple closed curve c , the diameter of π X ( c ) in C ′ ( X )is at most one. Moreover, if c can be realized disjointly from X then π X ( c ) = ∅ . URSULA HAMENST¨ADT
There also is an arc complex C ′ ( A ) for an essential annulus A ⊂ S , and thereis a subsurface projection π A of CG into the space of subsets of C ′ ( A ). We referto [MM00] for details of this construction. In the sequel we call a subsurface X of S proper if X is non-peripheral, incompressible, open and connected and differentfrom a three-holed sphere or S .As before, let δ > q ∈ Q ( S ) there is a δ -wide curvefor q . The next lemma is a version of Proposition 2.1 for Teichm¨uller geodesic arcswhich are allowed to be entirely contained in the thin part of Teichm¨uller space. Lemma 2.3.
For every χ > , k > there is a number T = T ( χ, k ) > with thefollowing property. Let q t be the cotangent line of a Teichm¨uller geodesic definedby two simple closed curves ξ, ζ . For t ∈ R let α ( t ) ∈ CG be a δ -wide curve for q t .Then one of the following (not mutually exclusive) possibilities is satisfied. (1) d CG (Υ( q ) , Υ( q T )) ≥ k . (2) There is a number t ∈ [0 , T ] and a proper subsurface Y of S containing α ( t ) such that diam( π Y ( ξ ∪ ζ )) ≥ k . (3) There is a number t ∈ [0 , T ] such that α ( t ) is the core curve of a flat cylinderof modulus at least χ .Proof. Define a shortest marking µ for a quadratic differential q ∈ Q ( S ) to consistof a pants decomposition with pants curves of the shortest extremal length for theconformal structure underlying q constructed using the greedy algorithm. Thereis a system of spanning arcs, one for each pants curve α of µ . Such an arc is ageodesic arc in the annular cover of S with fundamental group generated by α which intersects a q -geodesic representing α perpendicularly. We refer to Section 5of [R10] for details- the purpose of using such shortest markings here is for ease ofreference to the results of [R10]. A shortest marking defines a subset of CG .In the remainder of this proof we denote the diameter of a subset X of CG by diam CG ( X ). Moreover, whenever we use constants for a given surface S ofcomplexity n , we assume that they are also valid in the same context for surfacesof complexity smaller than n .For q ∈ Q ( S ) and ρ > α ρ -slim if α is not the core curve of a flatcylinder for q of modulus at least ρ . Let ρ > , k >
0, let b ∈ (0 ,
1) and let λ be theLebesgue measure on the real line. Let ( q t ) be the cotangent line of a Teichm¨ullergeodesic on S . For each t let µ t be a shortest marking for q t . Let n = ξ ( S ) ≥ S . Let δ > n thefollowing Claim:
There is a number T n = T n ( δ, ρ, k, b ) > f ( b, n ) ∈ (0 , b ]with the following property. Assume that there is a set A ⊂ [0 , T n ] with λ ( A ) ≥ bT n such that for every t ∈ A there is a δ -wide ρ -slim curve α ( t ) for q t . Then eitherdiam CG ( µ ∪ µ T n ) ≥ k or there is a set A ′ ⊂ A with λ ( A ′ ) ≥ f ( b, n ) T n so that for all t ∈ A ′ , the curve α ( t ) is contained in a proper subsurface Y of S with diam( π Y ( µ ∪ µ T n )) ≥ k .The claim easily yields the lemma. Namely, by Theorem 5.3 of [R10], for every k > ℓ = ℓ ( k ) > k with the following property. Let ( q s ) be thecotangent line of a Teichm¨uller geodesic defined by simple closed curves ξ, ζ . Let s < t , let µ s , µ t be shortest markings for q s , q t and assume that there is a subsurface Y of S such that diam( π Y ( µ s ∪ µ t )) ≥ ℓ ; thendiam( π Y ( ξ ∪ ζ )) ≥ k. ISTANCE IN THE CURVE GRAPH 9
Let T = T ( χ, k ) = T n ( δ, χ, ℓ, ) and let q t be the cotangent line of a Teichm¨ullergeodesic defined by two simple closed curves ξ, ζ . For t ∈ [0 , T ] let α ( t ) be a δ -widecurve for q t . If there is some t ∈ [0 , T ] such that α ( t ) is not χ -slim then the thirdproperty in the statement of the lemma is satisfied and we are done. Otherwiseeach of the curves α ( t ) ( t ∈ [0 , T ]) is χ -slim and hence we can apply the above claimto obtain the statement of the lemma.For the inductive proof of the claim, note that if ξ ( S ) = 1 then S either is aonce punctured torus or a four times punctured sphere. Consider first the case ofa one-punctured torus. Quadratic differentials on such a torus are just squares ofabelian differentials on the torus with the puncture filled in. As a consequence, a δ -wide ρ -slim simple closed curve in S is the core curve of a flat cyclinder (i.e. acylinder foliated by simple closed geodesics) whose modulus is at most ρ . Such flatstructures on tori project to a compact subset of the moduli space of tori.As a consequence, there is a number ǫ > δ, ρ ) with the followingproperty. If there is a δ -wide ρ -slim curve for a quadratic differential q on S then ν ( q ) > ǫ . The above claim now follows from Proposition 2.1 with f ( b,
1) = b (hereonly the first alternative in the claim occurs). The case of the four punctured sphereis completely analogous and will be omitted.Now assume that the claim holds true for all surfaces of complexity at most n − ≥
1. Let S be a surface of complexity n . Theorem 4.2 and Theorem 5.3 of[R10] and their proofs (see also [M92]) show the following.Let ( q t ) be the cotangent line of a Teichm¨uller geodesic. Assume that there isan interval [ a, b ] ⊂ R and a proper ǫ -semi-thick subsurface Y of S which is M -large for every t ∈ [ a, b ]. Here as before, M > S , chosen in such a way that the results of [R10] hold truefor this number. Let Y t be the geometric representative of Y for the quadraticdifferential q t and define q t,Y to be the quadratic differential obtained by cappingoff the boundary components with (perhaps degenerate) discs or punctured discs(see the proof of Theorem 4.2 in [R10] for details). Then ( q t,Y ) ( t ∈ [ a, b ]) is thecotangent line of a Teichm¨uller geodesic on Y (with the boundary circles of Y closed by discs or replaced by punctures). Moreover, there is a number δ ′ < δ anda number ρ ′ < ρ such that every essential simple closed curve in Y t which is δ -wideand ρ -slim for q t is δ ′ -wide and ρ ′ -slim for q t,Y .Let T n − = T n − ( δ ′ , ρ ′ , k ′ , b/ n ) be the number found for surfaces of complexityat most n − δ ′ < δ, ρ ′ < ρ and a number k ′ > k which willbe determined below. Let M > M be sufficiently large that the following holdstrue. Let s ∈ R and let Y be an ǫ -semi-thick subsurface for q s with geometricrepresentative Y s . Let β be a boundary circle of Y and let E ⊂ Y s be a maximalexpanding annulus for q s homotopic to β . If Mod q s ( E ) ≥ M then for | t − s | ≤ T n − the modulus of the maximal expanding annulus for q t homotopic to β which iscontained in the geometric representative Y t of Y for q t and lies on the same sideof β as E is not smaller than M .For this number M let ǫ ( ρ/ , M ) > ǫ < ǫ ( ρ/ , M )be sufficiently small that whenever β is a curve with Ext q s ( β ) ≤ ǫ then Ext q t ( β ) ≤ ǫ ( ρ/ , M ) for all t with | s − t | ≤ T n − .Let ℓ > q ∈ Q ( S ) the diameter in CG ofthe union of the set of curves from a shortest marking for q with the set of all δ -wide curves does not exceed ℓ (such a number exists since the extremal length and hence the hyperbolic length of a δ -wide curve is uniformly bounded from above, see[H10]). Let T n = R ( ǫ , k + 2 ℓ, b/
8) be as in Proposition 2.1. We may assume that bT n / ≥ bT n / n ≥ T n − . Let ( q t ) be the cotangent line of a Teichm¨uller geodesic on S . Proposition 2.1shows that either d CG (Υ( q ) , Υ( q T n )) ≥ k + 2 ℓ or there is a subset B of [0 , T n ] with λ ( B ) ≥ (1 − b/ T n and for every s ∈ B there is a curve of extremal length at most ǫ for q s . In the first case we conclude from the choice of ℓ that diam CG ( µ ∪ µ q Tn ) ≥ k and we are done. Thus assume that the second possibility holds true.Let us now suppose that there exists a set A ⊂ [0 , T n ] with λ ( A ) ≥ bT n suchthat for every t ∈ A there is a δ -wide ρ -slim curve α ( t ) for q t . Let χ B , χ A be thecharacteristic function of B, A , respectively. Since bT n / ≥ T n − and λ ( B ) ≥ (1 − b/ T n , for every s ∈ [0 , T n − ] we have λ { u ∈ [ T n − , T n − T n − ] | u − s ∈ B } ≥ (1 − b/ T n . Now λ ( A ) ∩ [ T n − , T n − T n − ] ≥ bT n / Z T n − T n − (cid:0) T n − Z s + T n − s χ B ( s ) χ A ( u ) du (cid:1) ds ≥ T n − Z T n − Z T n − T n − T n − χ B ( u − s ) χ A ( u ) duds ≥ bT n / . As a consequence, the Lebesgue measure of the set C = { s ∈ B ∩ [0 , T n − T n − ] | Z s + T n − s χ A ( u ) du ≥ bT n − / } is at least bT n / s ∈ C then s ∈ B ∩ [0 , T n − T n − ] and hence there is a simple closed curve β s with Ext q s ( β s ) ≤ ǫ . Moreover, there is a set D ⊂ [ s, s + T n − ]of Lebesgue measure at least bT n − / t ∈ D there is a δ -wide ρ -long curve α ( t ) for q t .By Lemma 2.2 and the choice of ǫ , for every t ∈ D the curve α ( t ) is containedin an M -large subsurface Y ⊂ S − β s for q t .A subsurface Y of S which is M -large for some t ∈ [ s, s + T n − ] is M -large forevery t ′ ∈ [ s, s + T n − ]. Now M -large subsurfaces for q t are pairwise disjoint, andtheir number is at most n . As a consequence, there is a subsurface Y ⊂ S − β s which is M -large for every t ∈ [ s, s + T n − ], and there is a subset D s ⊂ D ofLebesgue measure at least T n − b/ n so that α ( t ) ∈ Y for every t ∈ D s .Let ψ ( s ) , ψ ( s + T n − ) ⊂ Y be shortest markings for q s,Y , q s + T n − ,Y . By thechoice of T n − , we can apply the induction hypothesis to the cotangent line ( q t,Y )( t ∈ [ s, s + T n − ]) of the induced Teichm¨uller geodesic on Y .Let H s ⊂ [ s, s + T n − ] be the set of all t ∈ D s such that the curve α ( t ) iscontained in a proper subsurface Z t of Y with the additional property thatdiam( π Z t ( ψ ( s ) ∪ ψ ( s + T n − ))) ≥ k ′ . By induction assumption, either the diameter of ψ ( s ) ∪ ψ ( s + T n − ) in the curvegraph of Y is at least k ′ or λ ( H s ) ≥ f ( b/ n, n − T n − . ISTANCE IN THE CURVE GRAPH 11
Theorem 5.3 of [R10] shows that there is a number p > π Y ( ψ ( s ) ∪ ψ ( s + T n − ))) ≥ k ′ then diam( π Y ( µ ∪ µ T n )) ≥ k ′ − p .Moreover, for t ∈ H s the diameter of the subsurface projection of µ ∪ µ T n into Z t is at least k ′ − p .Define k ′ = k + p and let H ′ s = D s if diam( π Y ( ψ ( s ) ∪ ψ ( s + T n − ))) ≥ k ′ (whichimplies that diam( π Y ( µ ∪ µ T n )) ≥ k ), and define H ′ s = H s otherwise. Then λ ( H ′ s ) ≥ f ( b/ n, n − T n − for every s ∈ C , moreover for all t ∈ H ′ s the curve α ( t ) is contained in a subsurface Y s of S with diam( π Y s ( µ ∪ µ T n )) ≥ k .For s ∈ C let χ s be the characteristic function of D s , viewed as a subset of R ,and let χ ( t ) = max { χ s ( t ) | s } . Since λ ( C ) ≥ bT n / Z T n χ ( s ) ds ≥ f ( b/ n, n − bT n / f ( b, n ) = bf ( b/ n, n − / (cid:3) Now we are ready for the proof of the Theorem from the introduction. To thisend choose a number K > ξ, ζ be simpleclosed curves. If there is a proper subsurface Y of S such that diam( π Y ( ξ ∪ ζ )) ≥ K then a geodesic γ in CG connecting ξ to ζ passes through the complement of Y .The existence of such a number K > q t ) be the cotangent line of the Teichm¨uller geodesic connecting ξ to ζ . Bythe results of [MM00], there is a number χ > α = ξ, ζ is a simple closed curve so that α is the core curve of a flat cylinder for q t of modulus at least χ then the diameter of the subsurface projection of ( ξ, ζ ) intoan annulus with core curve α is at least K .Let κ > χ, κ, K let R = T ( χ, K + 4 κ ) be asin Lemma 2.3.Let t < · · · < t n = T be as in the statement of the theorem. Let u be an integerbigger than R . By Lemma 2.3 and the above discussion, for all ℓ we either havei) d CG (Υ( q t ℓ ) , Υ( q t ℓ + u )) ≥ K + 4 κ and hence d CG (Υ( q t v ) , Υ( q t w )) ≥ κ for all w ≥ ℓ + u ≥ ℓ ≥ v , orii) there is a subsurface Y of S (perhaps an annulus) so that the diameter ofthe subsurface projection of ξ, ζ into Y is at least K and that Υ( q s ) ⊂ Y for some s ∈ [ ℓ, ℓ + u ].It now suffices to show that d CG (Υ( q ℓ ) , Υ( q t ℓ + u (8 κ +8) )) ≥ κ for all ℓ . By thechoice of κ , this holds true if there is some j ≤ u (8 κ + 6) so that the alternative i)above is satisfied for q t j .Otherwise for every i ≤ κ + 6 there is some s ∈ [ ui, u ( i + 1)] such that Υ( q s ) iscontained in a subsurface Y with the property that the diameter of the subsurfaceprojection of ξ, ζ into Y is at least K . By the choice of K (see Theorem 3.1of [MM00]), there is a curve c s on γ disjoint from Υ( q s ). In particular, we have d CG ( c s , Υ( q s )) ≤ v ∈ [ uj, u ( j + 1)] for some j with | i − j | ≥ d CG ( c s , c v ) ≥ c s , c v are distinct. As a consequence, there are at least4 κ + 3 distinct points on the geodesic γ arising in this way. Then there are at leasttwo of these points, say the points c s , c v , whose distance is at least 4 κ + 2. Thisshows d CG (Υ( q t s ) , Υ( q t v )) ≥ κ which implies the required estimate. References [B06] B. Bowditch,
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