aa r X i v : . [ m a t h . G T ] A p r THE CURVE COMPLEX AND COVERS VIA HYPERBOLIC3-MANIFOLDS
ROBERT TANG
Abstract.
Rafi and Schleimer recently proved that the natural relation be-tween curve complexes induced by a covering map between two surfaces isa quasi-isometric embedding. We offer another proof of this result using adistance estimate via hyperbolic 3-manifolds. Introduction
Let S be a compact orientable surface of genus g with n boundary components.Define a simplicial complex C ( S ), called the curve complex , as follows. Let thevertex set be the free homotopy classes of essential non-peripheral simple closedcurves. We have a simplex for every set of homotopy classes which can be realisedsimultaneously disjointly. In particular, two classes are adjacent in the 1-skeletonof C ( S ) if and only if they have disjoint representatives on S .We will assume that the complexity ξ ( S ) = 3 g + n − C ( S ) is non-empty, connected and that all surfaces underconsideration are of hyperbolic type. (There are modified definitions for the curvecomplex in the low-complexity cases, however, we shall not deal with them here.)The distance d S ( a, b ) between vertices a and b in C ( S ) is defined to be the lengthof the shortest edge-path connecting them. This can be thought of as measuringhow “complicated” their intersection pattern is. Endowed with this distance, thecurve complex has infinite diameter and is also Gromov hyperbolic [6, 2].The curve complex has had many applications to both the study of mappingclass groups and Teichm¨uller theory. More recently, it was a key ingredient of theproof of the Ending Lamination Theorem [5, 1].We will be focussed on a map between curve complexes induced by a coveringmap.Suppose P : Σ → S is finite-degree covering map between two surfaces. Thepreimage P − ( a ) of a curve a ∈ C ( S ) is a disjoint union of simple closed curves on Date : 13 April 2011.2010
Mathematics Subject Classification.
Key words and phrases. curve complex, covering space, hyperbolic 3-manifold, quasi-isometricembedding.
Σ. This induces a one-to-many map Π : C ( S ) → C (Σ) where Π( a ) is defined to bethe set of homotopy classes of the curves in P − ( a ).To establish some notation, given non-negative real numbers A, B and K , write A ≺ K B to mean A ≤ KB + K . We shall write A ≍ K B if both A ≺ K B and B ≺ K A hold.Let ( X, d ) and ( X ′ , d ′ ) be pseudo-metric spaces and f : X → X ′ a (possiblyone-to-many) function. We call f a K -quasi-isometric embedding if there existsa positive constant K such that for all x, y ∈ X we have d ( x, y ) ≍ K d ′ ( x ′ , y ′ )whenever x ′ ∈ f ( x ) and y ′ ∈ f ( y ). In addition, if the K -neighbourhood of f ( X )equals X ′ then we call f a K -quasi-isometry and say X and X ′ are K -quasi-isometric . Furthermore, we call K the quasi-isometry constant .In this paper we give a new proof of the following theorem, originally due to Rafiand Schleimer [7]. Theorem 1.1.
Let P : Σ → S be a covering map of degree deg P < ∞ . Then themap Π : C ( S ) → C (Σ) defined above is a K -quasi-isometric embedding, where K depends only on ξ (Σ) and deg P . This result can be interpreted as saying that one cannot tangle or untangle twocurves on S by too much when passing to finite index covers.Theorem 1.1 was first proved in [7] using Teichm¨uller theory and subsurfaceprojections. Our approach uses an estimate for distance in the curve complex viaa suitable hyperbolic 3-manifold homeomorphic to S × R with a modified metric.This allows us to naturally compare distances by taking a covering map betweenthe respective 3-manifolds. The estimate (Theorem 3.1) arises from work towardsthe Ending Lamination Theorem [5] and is made explicit in [1]. More details willbe given in Section 3. Remark . A consequence of Theorem 1.1 and Gromov hyperbolicity of C (Σ) isthat the image Π( C ( S )) is quasi-convex.2. The induced map between curve complexes
Let P : Σ → S be a finite index covering of surfaces and assume ξ ( S ) ≥
2. Thecover P induces a one-to-many map Π : C ( S ) → C (Σ) defined by declaring Π( a )to be the set of homotopy classes in P − ( a ). Observe that P − ( a ) is a nonemptydisjoint union of circles and that any component of P − ( a ) must be essential andnon-peripheral. Thus Π sends vertices of C ( S ) to (non-empty) simplices of C (Σ).The preimages of disjoint curves in S must themselves be disjoint. Therefore, ifwe choose a preferred vertex in each Π( a ) then we can send any edge-path in C ( S )to an edge-path in C (Σ). (We can extend this to higher dimensional simplices if sodesired.) This gives us the following distance bound. HE CURVE COMPLEX AND COVERS VIA HYPERBOLIC 3-MANIFOLDS 3
Lemma 2.1.
Let a and b be curves in C ( S ) and suppose α ∈ Π( a ) and β ∈ Π( b ) .Then d Σ ( α, β ) ≤ d S ( a, b ) . Estimates from 3-Manifolds
We now introduce the background required to state an estimate for distances inthe curve complex using a suitable hyperbolic 3-manifold.Let S be a surface with ξ ( S ) ≥ a and b in C ( S ). By simul-taneous uniformisation, there exists a hyperbolic 3-manifold ( X, d ) ∼ = int( S ) × R with a preferred homotopy equivalence f to S such that the unique geodesic repre-sentatives of the two curves, denoted a ∗ and b ∗ , are arbitrarily short. In fact, sucha 3-manifold can be chosen so that there are no accidental cusps. For a reference,see [4].Recall that the injectivity radius at a point x ∈ X is equal to half the infimumof the lengths of all nontrivial loops in X passing through x . The ǫ -thin part (orjust thin part) of X is the set of all points whose injectivity radius is less than ǫ .By the well known Margulis lemma, the thin part comprises of Margulis tubes andcusps whenever ǫ is less than the Margulis constant. The thick part of X is thecomplement of the thin part. We shall fix such an ǫ for the rest of this paper.Let Ψ ǫ ( X ) denote the non-cuspidal part of X , that is, X with its ǫ -cusps removed.We will write Ψ( X ) for brevity. Since X has no accidental cusps we see that if S is closed then Ψ( X ) = X .Define the electrified length of a path in Ψ( X ) to be its total length occurringoutside the Margulis tubes of X . More formally, we take the one-dimensionalHausdorff measure of its intersection with the thick part of X . This induces areduced pseudometric ρ X on Ψ( X ) obtained by taking the infimum of the electrifiedlengths of all paths connecting two given points. One can show that the infimum isattained, for example, by taking a path which connects Margulis tubes by shortestgeodesic segments.The distance ρ X shall be referred to as the electrified distance on Ψ( X ) withrespect to its Margulis tubes or the reduced pseudometric obtained by electrifyingthe Margulis tubes. Theorem 3.1. [1]
Let a and b be curves in C ( S ) whose geodesic representatives a ∗ and b ∗ in X have d -length at most L ≥ . Then d S ( a, b ) ≍ K ρ X ( a ∗ , b ∗ ) where the constant K depends only on ξ ( S ) , L and ǫ . This estimate follows from the construction of geometric models for hyperbolic3-manifolds used in the proof of the Ending Lamination Theorem. We refer thereader to [1] and [5] for an in-depth discussion.
ROBERT TANG
Remark . For Theorem 3.1 to make sense we need to ensure that a ∗ and b ∗ areindeed contained in Ψ( X ). This can be done using a pleated surfaces argument,such as in [3], to show that closed geodesics of bounded length in X avoid cuspsprovided ǫ is sufficiently small.4. Proof of Theorem 1.1
Closed surfaces.
We first prove the main theorem for closed surfaces. Therequired modifications for the general case shall be dealt with in Section 4.2.Fix a length bound L and a sufficiently small value of ǫ . Let P : Σ → S bea covering map. Fix curves a, b in C ( S ) and choose α ∈ Π( a ) and β ∈ Π( b ).From Lemma 2.1, we have d Σ ( α, β ) ≤ d S ( a, b ) so it remains to prove the reverseinequality.Let ( X, d ) ∼ = int( S ) × R be a hyperbolic 3-manifold with a homotopy equivalence f to S as described in Section 3. We also assume that a ∗ and b ∗ have length at most L deg P in X . There exists a covering map Q : Ξ → X , where Ξ ∼ = int(Σ) × R , and ahomotopy equivalence ˜ f : Ξ → Σ such that P ◦ ˜ f = f ◦ Q . Note that Q ( α ∗ ) = a ∗ and Q ( β ∗ ) = b ∗ , hence α ∗ and β ∗ have length bounded above by L .Let ρ X and ρ Ξ be the respective pseudometrics on X and Ξ obtained by electri-fying their ǫ -tubes. Lemma 4.1.
The map Q is 1-Lipschitz with respect to ρ Ξ and ρ X .Proof. Using the definition of injectivity radius and the π -injectivity of coveringmaps, we see that the ǫ -thin part of Ξ is sent into that of X . It follows that theelectrified lengths of paths cannot increase under Q . (cid:3) This result, together with Theorem 3.1, proves Theorem 1.1 for closed surfaces.4.2.
Surfaces with boundary.
We now assume S has non-empty boundary. Re-call that Ψ ǫ ( X ) denotes X with its ǫ -cusps removed. Lemma 4.2.
Let X be a hyperbolic 3-manifold. Choose small constants δ > δ ′ andlet ρ and ρ ′ be the pseudometrics on Ψ δ ( X ) and Ψ δ ′ ( X ) obtained by electrifyingalong their respective ǫ -tubes. Then the natural retraction r : (Ψ δ ′ ( X ) , ρ ′ ) → (Ψ δ ( X ) , ρ ) is R -Lipschitz, where R = sinh δ sinh δ ′ .Proof. Begin with a geodesic arc γ in Ψ δ ′ ( X ). We will show that the length of γ can only increase by a bounded multiplicative factor under the retraction r .If γ is contained in Ψ δ ( X ) then we are done. So suppose γ meets some δ -cusp C of X . This cusp contains a δ ′ -cusp C ′ which does not meet γ . The cusps C and C ′ lift to nested horoballs H and H ′ in the universal cover H . We can arrangefor the horospheres ∂H and ∂H ′ to be horizontal planes at heights 1 and R > HE CURVE COMPLEX AND COVERS VIA HYPERBOLIC 3-MANIFOLDS 5 respectively in the upper half-space model. Using basic hyperbolic geometry andthe definition of injectivity radius, we can show that R = sinh δ sinh δ ′ .Now define π and π ′ to be the nearest point projections from H to H and H ′ respectively. Taking a lift ˜ γ of γ , we see thatlength(˜ γ ∩ H ) ≥ length( π ′ (˜ γ ∩ H )) = 1 R length( π (˜ γ ∩ H )) . Thus, by projecting each arc of γ inside a δ -cusp to the boundary of that cusp, wecreate a new path whose length is at most R × length( γ ). (cid:3) Let Q : Ξ → X be the covering map as described in Section 4.1. Observe thatΨ ǫ ( X ) ⊆ Q (Ψ ǫ (Ξ)) ⊆ Ψ ǫ ′ ( X )where ǫ ′ = ǫ deg P . As before, let ρ Ξ and ρ X be the pseudometrics on Ψ ǫ (Ξ) andΨ ǫ ( X ) obtained by electrifying along their respective ǫ -tubes. Combining Lemma4.2 with the proof of Lemma 4.1 gives us the following. Lemma 4.3.
Let r : Ψ ǫ ′ ( X ) → Ψ ǫ ( X ) be the natural retraction. Then the compo-sition r ◦ Q : Ψ ǫ (Ξ) → Ψ ǫ ( X ) is R -Lipschitz with respect to ρ Ξ and ρ X . Moreover,the constant R depends only on deg P . (cid:3) Finally, r ◦ Q ( α ∗ ) = a ∗ and r ◦ Q ( β ∗ ) = b ∗ and so the rest of the argument fol-lows as in Section 4.1. Acknowledgements
I am grateful to both Brian Bowditch and Saul Schleimer for many helpful dis-cussions and suggestions. This work was supported by the Warwick PostgraduateResearch Scholarship.
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