VLADIMIR SHASTIN. i=1

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1 A COMBINATORIAL MODEL OF THE LIPSCHITZ METRIC FOR SURFACES WITH PUNCTURES VLADIMIR SHASTIN Abstract. The zipped word length function introduced by Ivan Dynnikov in connection with the word problem in the mapping class groups of punctured surfaces is considered. We prove that the mapping class group with the metric determined by this function is quasi-isometric to the thick part of the Teichmüller space equipped with the Lipschitz metric Math. Subj. Class. 20F65 Key words and phrases. Mapping class group, Teichmüller space, Teichmüller metric, Thurston s asymmetric metric. 1. Introduction This paper is motivated by the recent work of Ivan Dynnikov [1] on the word problem in the mapping class groups of punctured surfaces. In this paper Dynnikov described an effective algorithm for the solution of this problem, where as the size of the algorithm s input he used a modified version of the word length function on the mapping class group. Namely for a finite generating set A of the mapping class group of a punctured surface S Dynnikov defined the zipped word length function zwl A as follows: m zwl A (ϕ) = min log 2 ( k i + 1), ϕ=a k akm m a 1,...,a m A k 1,...,k m Z where ϕ MCG(S). For special generating sets A he proved that the word problem is solvable in polynomial time with respect to zwl A : Theorem 1 (Dynnikov [1]). Let S be a compact surface, P = (P 1,..., P n ) S a non-empty collection of pairwise distinct points such that the mapping class group G = MCG(S \ P) is infinite. Let A be a finite generating set for G such that 1. every element in A is a fractional power of a Dehn twist; 2. every Dehn twist from G is conjugate to a fractional power of an element from A. Then the word problem in G is solvable in polynomial time with respect to zwl A. Remark 1. This theorem is the generalization of the main result of Dynnikov and Wiest from the paper [2]. In this work they proved Theorem 1 in the case when the group G is a braid group(the surface S is a disk) and the finite generating set A consists of Garside-like half-twists around each pair of strands. One of the main ingredients of the proof of Theorem 1 is the fact that the zipped word length function has a geometric analogue. Namely for an ordered ideal triangulation T of the surface The author is partially supported by Laboratory of Quantum Topology of Chelyabinsk State University (Russian Federation government grant 14.Z ). 1 i=1

2 2 VLADIMIR SHASTIN Dynnikov defined the complexity function c T as follows: c T (ϕ) = N log( e i, ϕ(e j ) + δ ij + 1), i,j=1 where ϕ MCG(S), {e i } N i=1 are edges of T, and δ ij is the Kroneker delta. The quantity e i, ϕ(e j ) equals 1 when e i is isotopic to ϕ(e j ); in all other cases it equals the geometric intersection number of e i and ϕ(e j ). For a generating set A as in the theorem above Dynnikov proved that zwl A is comparable with c T : there exist numbers K > 1 and C > 0, depending on A and T such that 1 K zwl A(ϕ) C c T (ϕ) K zwl A (ϕ) + C. holds for every ϕ MCG(S). The function zwl A determines the right-invariant metric ρ A on MCG(S) as follows: ρ A (ϕ, ψ) = zwl A (ψϕ 1 ), where ϕ, ψ MCG(S). Using the connection between complexity functions zwl A and c T we show that MCG(S) equipped with the metric ρ A is a combinatorial model of the Teichmüller space of the surface S. Namely we prove the following theorem, which is the main result of this paper: Theorem 2. Let S be an oriented surface with non-empty set of punctures, ɛ a positive constant, σ a hyperbolic structure on S, lying in the ɛ-thick part of the Teichmüller space T ɛ (S), and A a finite generating set of MCG(S) with the following properties: 1. every element in A is a fractional power of a Dehn twist; 2. every Dehn twist from G is conjugate to a fractional power of an element from A. Let also i σ : MCG(S) T ɛ (S) be the map that sends ϕ MCG(S) to the image of σ under ϕ. Then i σ is a quasi-isometry from MCG(S) equipped with the metric ρ A to the thick part of T (S) equipped with the Lipschitz metric. In the work [3] Young-Eun Choi and Kasra Rafi proved that the Lipschitz and the Teichmuller metrics are equal up to an additive constant in the thick part of the Teichmüller space. Using this result we obtain the following direct corollary of Theorem 2: Theorem 3. Let S be an oriented surface with non-empty set of punctures, ɛ a positive constant, σ a hyperbolic structure on S, lying in the ɛ-thick part of the Teichmüller space T ɛ (S), and A a finite generating set of MCG(S) with the following properties: 1. every element in A is a fractional power of a Dehn twist; 2. every Dehn twist from G is conjugate to a fractional power of an element from A. Let also i σ : MCG(S) T ɛ (S) be the map that sends ϕ MCG(S) to the image of σ under ϕ. Then i σ is a quasi-isometry from MCG(S) equipped with the metric ρ A to the thick part of T (S) equipped with the Teichmüller metric. This theorem is a generalization of the theorem of Dynnikov and Wiest from the work [2], where they obtained a similar result in the case when S is the sphere with punctures. Acknowledgements. I would like to thank Ivan Dynnikov for suggesting the statement of Theorem 3 and helpful discussions.

3 A COMBINATORIAL MODEL OF THE LIPSCHITZ METRIC 3 2. Main definitions Let S = S g,n be a closed orientable connected smooth surface of genus g with n marked points P 1,..., P n. Associated to S is a surface S = S g,n that is obtained from S by puncturing it at the set of marked points. In the following we will denote the resulting punctures on the surface S by the same letters as corresponding marked points on S. In this paper we ll consider only surfaces S with nonempty set of punctures, and negative Euler characteristic: the quantity χ(s) = 2 2g n must be less than zero. It is well known that any surface S of such type admits a hyperbolic structure a complete, finite-area Riemannian metric σ of constant curvature -1 (see. [4]). The surface S endowed with a fixed hyperbolic structure σ will be denoted by (S, σ) and called a hyperbolic surface. Definition 1. We say that hyperbolic structures σ 1 and σ 2 on S are equivalent if there is an isometry ϕ: (S, σ 1 ) (S, σ 2 ), isotopic to the identity map of the surface S. The Teichmüller space T (S) of S is the set of equivalence classes of hyperbolic structures on S. The ɛ-thick part T ɛ (S) of the Teichmüller space of S is the subset of T (S) consisting of hyperbolic metrics σ such that the length of each closed σ-geodesic is greater than ɛ. In the following we won t specify the constant ɛ and just write thick part of the Teichmüller space. T (S) admits a number of natural metrics: the Teichmüller metric, the Weil-Petersson metric, Thurston s assymetric metric, etc.(see [5] for the list of metrics on the Teichmüller space). All these metrics define the same topology on T (S) which makes it homeomorphic to an open ball of dimension 6g 6 + 2n. Definition 2. The mapping class group MCG(S) of a surface S is the group of isotopy classes of orientation preserving diffeomorphisms of S: MCG(S) = Diff + (S)/Diff 0 (S). Here Diff + (S) is the group of orientation preserving diffeomorphisms of S and Diff 0 (S) its normal subgroup consisting of diffeomorphisms isotopic to the identity. The group Diff + (S) acts on the space of hyperbolic structures on S by pullback. This (right) action descends to the action of MCG(S) on T (S). Definition 3. By a closed curve α we mean a differentiable map α: S 1 S. We will usually identify a closed curve with its image in the surface. A closed curve is called essential if it s not homotopic to a point or a puncture. A closed curve α is called simple if the corresponding map α: S 1 S is an embedding. A multicurve is an non-empty set of disjoint simple closed curves on S. A multicurve is called essential if all its components are essential curves. Definition 4. By an arc on a surface S we mean a differentiable map : [0, 1] S such that 1 (P) = {0, 1}. We also call an arc and denote by the corresponding map of (0, 1) to S. Again we will usually identify an arc with its image in S or S. By homotopy of an arc we mean a homotopy with fixed endpoints. An arc is called essential if it s not homotopic to a puncture. An arc is called simple if the corresponding map of (0, 1) to S is an embedding. A multiarc is a nonempty set of disjoint simple arcs on S. A multiarc is called essential if all its components are essential arcs. An ideal triangulation T on S is a maximal(by inclusion) multiarc. It s easy to see that T determines a cellular decomposition of S with two-dimensional cells corresponding to the connected components of S \ T. Each such component is a diffeomorphic image of the interior

4 4 VLADIMIR SHASTIN of a closed triangle under a smooth map ϕ that sends interior of a side of into an arc of T. It s easy to see that all ideal triangulations of S have equal numbers of edges N and triangles M: N = 6g 6 + 3n, M = 4g 4 + 2n. Proposition 1. Suppose σ is a hyperbolic metric on S, α is an essential closed curve or an essential arc on S. Then there is a unique σ-geodesic γ α among closed curves (resp. arcs) homotopic to α. Moreover 1) If α is simple, then γ α is also simple. 2) If is another essential closed curve or an arc and curves α and don t intersect, then the corresponding σ-geodesics γ α and γ don t intersect as well. 3) If α is closed, then γ α is the shortest curve among all curves homotopic to α. Proof. For a proof in the case of closed curves see [6], [7]. The proof in the case of arcs can be obtained from the arguments described in Lemma 3.2. of [6] and the formula in the proposition D of [10] describing a hyperbolic metric near a cusp. For a simple essential closed curve α and a hyperbolic structure σ we denote by l σ (α) the length of the unique σ-geodesic homotopic to α. Definition 5. Suppose that α, are simple curves. We say that α and are transverse if they coincide or intersect transversely. We say that α and are tight if they are transverse, and there are no disks among connected components of S \ (α ). α α P P 1 α P 2 Figure 1. Three types of disks in S \ (α ). Definition 6. Let α and be simple curves on S. Then the geometric intersection number i(α, ) of α and is the following quantity: i(α, ) = min α, #{x x α }, where minimum is taken over pairs of simple transverse curves α, isotopic to α and respectively. Some important properties of geometric intersection number of curves on hyperbolic surfaces are described in the following proposition and its corollary: Proposition 2. Suppose that α and are simple essential curves on S. Then for any hyperbolic metric σ on S σ-geodesics γ α and γ are tight. Moreover if α and are tight, then there is an isotopy of S that brings α and to γ α and γ respectively. Proof. For a proof in the case when S is a closed surface see [8]. The proof in the case when S is a surface with punctures is similar. Corollary 1. Let α and be simple essential curves. Then we have 1) There is a curve α isotopic to α such that α and are tight.

5 A COMBINATORIAL MODEL OF THE LIPSCHITZ METRIC 5 2) Suppose curves α and are transverse. Then they are tight if and only if the number of their intersections coincides with the geometric intersection number of the curves: #{x x α } = i(α, ). If A, B are essential multicurves, or essential multiarcs, or an essential multicurve and an essential multiarc, then we define the geometric intersection number i(a, B) of A and B as the sum of geometric intersection numbers of their components: i(a, B) = i(α, ) α A, B For a multicurve or a multiarc A and an ideal triangulation T and we denote by A T the intersection number i(a, T ). We also recall the definitions of a pseudometric space and a quasi-isometry between pseudometric spaces: Definition 7. A pseudometric space (X, d) is a set X together with a non-negative real-valued function d: X X R 0 (called a pseudometric) with the following properties: 1. d(x, x) = 0 holds for every x X, 2. d(x, y) = d(y, x) holds for all x, y X, 3. d(x, z) d(x, y) + d(y, z) holds for all x, y, z X. Unlike metric spaces the distance between two distinct points in pseudometric spaces can be 0. Definition 8. Suppose (X, d X ), (Y, d Y ) are two pseudometric spaces and K 1, C 0 are two constants. Then a map f : (X, d X ) (Y, d Y ) is called a (K, C)-quasi-isometry if two following properties hold: 1 1. For all x 1, x 2 X: d K X(x 1, x 2 ) C d Y (f(x 1 ), f(x 2 )) Kd X (x 1, x 2 ) + C. 2. For every point y Y there is a point x X such that d Y (f(x), y) C. Two pseudometrics d 1 and d 2 on a space X are called (K,C)-quasi-isometric if the identity map of X is a (K, C)-quasi-isometry between (X, d 1 ) and (X, d 2 ). 3. The Lipschitz metric Definition 9. Suppose that σ, τ are hyperbolic metrics on S and ϕ: S S is a diffeomorphism of S. Then the Lipschitz constant L(ϕ; σ, τ) of ϕ, considered as the map between metric spaces (S, σ) and (S, τ), is defined as follows: ( ) dτ (ϕ(x), ϕ(y)) L(ϕ; σ, τ) = sup x y d σ (x, y) In terms of the Lipschitz constant W.P.Thurston in the work [9] defined the following function L(σ, τ) of an ordered pair of hyperbolic metrics on S: L(σ, τ) = inf ϕ id log(l(ϕ; σ, τ)), where the infimum is taken over all diffeomorphisms ϕ of S homotopic to identity. It is obvious that this function depends only on equivalence classes of hyperbolic structures σ, τ and therefore descends to a function on the cartesian square of T (S). Thurston showed that the function L(σ, τ) has the following properties: L(σ, τ) is nonnegative L(σ, τ) equals 0 if and only if hyperbolic metrics σ and τ are equivalent

6 6 VLADIMIR SHASTIN L(σ, τ) satisfies the triangle inequality: L(σ 1, σ 3 ) L(σ 1, σ 2 ) + L(σ 2, σ 3 ), for all hyperbolic metrics σ 1, σ 2, σ 3 on S. However, in general L(σ, τ) L(τ, σ). So this function defines an asymetric metric on the Teichmüller space which is called Thurston s asymetric metric. In the following we will use another description of this assymetric metric given in the following theorem of Thurston from [9]: Theorem 4 (W. Thurston, [9]). For a suface S and hyperbolic metrics σ, τ on S the distance L(σ, τ) can be calculated by the following formula ( ( )) lτ (α) L(σ, τ) = sup log, α l σ (α) where supremum is taken over all simple essential closed curves α on S. It is easy to see directly from the definition of Thurston s assymetric metric that MCG(S) acts on T (S) by isometries of this assymetric metric. Definition 10. The Lipschitz metric d L on the Teichmüller space of the surface S is the following symmetrization of Thurston s assymetric metric: where σ, τ T (S). d L (σ, τ) = max{l(σ, τ), L(τ, σ)}, From above mentioned properties of Thurston s assymetric metric it s easy to see that d L is a metric on T (S) and that MCG(S) acts on T (S) by isometries of this metric. Now if we fix a hyperbolic structure σ on S and consider the corresponding orbit of MCG(S) in T (S), then the restriction of the Lipschitz metric on this orbit determines the right-invariant pseudometric ρ L,σ on the group MCG(S) as follows: ρ L,σ (ϕ, ψ) = d L (ϕ (σ), ψ (σ)), for all elements ϕ, ψ MCG(S). Here ϕ (σ) denotes the pullback hyperbolic metric under ϕ. Remark 2. ρ L,σ is not a genuine metric in general because some σ T (S) are fixed by non-trivial elements of MCG(S). In the sequel we will also use the complexity function on MCG(S) L,σ corresponding to the Lipschitz metric defined as follows: ϕ L,σ = L(σ, ϕ (σ)), for all ϕ MCG(S). Because of right-invariance of ρ L,σ we obtain the following equality: ρ L,σ (ϕ, ψ) = max{ ψϕ 1 L,σ, ϕψ 1 L,σ }. 4. Proofs of the main results Now we turn to the proof of the main results of this paper Proof of Theorem 2. First of all we want to mention that for all ɛ > 0 and every hyperbolic structure σ in the ɛ-thick part of the Teichmüller space of S the map i σ : (MCG(S), ρ L,σ ) (T ɛ (S), d L ) that sends ϕ MCG(S) to ϕ (σ) is a quasi-isometry. Indeed the action of MCG(S) on T ɛ (S) is cocompact (for a proof see Proposition 4.8. in [12]). So i σ is a (1, D ɛ )-quasi-isometry, where D ɛ is the diameter of the quotient space T ɛ (S)/MCG(S) in the metric d L.

7 A COMBINATORIAL MODEL OF THE LIPSCHITZ METRIC 7 So in order to prove the main result it is sufficient to show that the zipped word length metric ρ A and ρ L,σ are quasi-isometric. Moreover, because ρ A and ρ L,σ are right-invariant it s sufficient to prove that complexity functions zwl A and L,σ are comparable. Indeed let there exist K > 1 and C > 0 such that the following inequalities 1 K zwl A(ϕ) C ϕ L,σ K zwl A (ϕ) + C hold for every ϕ MCG(S). Then we have that ρ L,σ (ϕ, ψ) = max{ ψϕ 1 L,σ, ϕψ 1 L,σ } K max{zwl A (ψϕ 1 ), zwl A (ϕψ 1 )} + C K max{ρ A (ϕ, ψ), ρ A (ψ, ϕ)} + C = Kρ A (ϕ, ψ) + C ρ L,σ (ϕ, ψ) = max{ ψϕ 1 L,σ, ϕψ 1 L,σ } 1 K max{zwl A(ψϕ 1 ), zwl A (ϕψ 1 )} C 1 K max{ρ A(ϕ, ψ), ρ A (ψ, ϕ)} C = 1 K ρ A(ϕ, ψ) C. It was mentioned in the introduction that for an admissible finite generating set A of MCG(S) and an ordered ideal triangulation T of S the zipped length function zwl A is comparable with the function c T defined by the following formula: N c T (ϕ) = log( e i, ϕ(e j ) + δ ij + 1), i,j=1 where ϕ MCG(S), {e i } N i=1 are edges of T and δ ij is the Kroneker delta. The quantity e i, ϕ(e j ) equals 1 when e i is isotopic to ϕ(e j ); in all other cases it equals the geometric intersection number of e i and ϕ(e j ). We define the function T which is the analogue of the function c T and is given by the following formula: ϕ T = log( ϕ(t ) T + 1), where ϕ MCG(S). It is easy to see that c T is comparable with T : ϕ T c T (ϕ) N 2 ϕ T + N 2 log 2 holds for every ϕ MCG(S). Now our goal is to show that complexity functions L,σ and T are comparable. In order to do that we need to find a relation between σ-lengths of simple essential closed curves and their intersection numbers with the triangulation T. This relation is described in the following proposition: Proposition 3. Suppose that T is an ideal triangulation and σ is a hyperbolic metric on a surface S. Then there exist two positive numbers c and C, which depend on S, T and σ, such that for any simple essential curve α in S the following inequalities hold: c α T l σ (α) C α T. Proof. To prove this proposition we need the following lemma: Lemma 1. Let σ be a hyperbolic metric on a surface S. For every puncture P on S let us consider the set U P, consisting of horocyclic loops around P of σ-length less than one. Then every such set U P is a regular neighbourhood of the corresponding puncture, the neighbourhoods for different punctures don t intersect, and a simple closed σ-geodesic doesn t enter these neighbourhoods. Proof. For a detailed proof of this fact see Proposition in [13].

8 8 VLADIMIR SHASTIN Now we are ready to prove the proposition. Let us consider the ideal triangulation T with σ-geodesic edges isotopic to T and the σ-geodesic γ α isotopic to α. It follows from Corollary 1 that α T equals the number of intersection points of γ α and T. This number in turn equals the number of geodesic arcs α i in γ α S \ T. Because any connected component of S \ T is an interior of the ideal hyperbolic triangle, the σ-length of each arc α i lying in equals the hyperbolic length of some geodesic arc α i in the ideal hyperbolic triangle. Moreover each α i lies in the complement of the horocyclic neighbourhoods of vertices of the ideal triangle determined by horocyclic neighbourhoods U P, where U P is a horocyclic neighbourhood of the puncture P from the previous lemma. Therefore by compactness argument there are positive numbers c and C such that the length of each arc α i lying in is bounded from bellow and from above by c and C respectively. Because the number of components of S \ T is finite, there exist positive numbers c and C, depending on S, T and σ but not on α, such that any arc α i has length more than c and less than C. So we obtain that the inequalities hold for every simple closed curve α. c α T l σ (α) C α T First of all we want to mention that the proof of Theorem 2 in the case, when S is a sphere with 3 punctures is trivial because mapping class group of this surface is finite. Namely it is isomorphic to the permutation group of 3 elements. In the sequel we consider the case when S is distinct from the sphere with three punctures. For a general ϕ MCG(S) let us estimate ϕ L,σ. Using Theorem 4 we can write: ϕ L,σ = sup α ( log ( )) lϕ (σ)(α) l σ (α) = sup α ( log ( lσ (ϕ(α)) l σ (α) )). It follows from Proposition 3 that there is a positive number C, that doesn t depend on ϕ and such that ( ( )) ( ( )) lσ (ϕ(α)) ϕ(α) T sup log sup log C α l σ (α) α α T Therefore, in order to prove the theorem, it is sufficient to find positive constants C 1, C 2 such that for all ϕ MCG(S) the following inequalities are satisfied: ( ( )) ϕ(α) T (1) ϕ T C 1 sup log ϕ T + C 2. α α T At first we want to prove the left inequality. To do that we need the following proposition. Proposition 4. Let T be an ideal triangulation of S, a simple essential arc in S, and the boundary of a small neighborhood of in S. If is also connected, then it is essential and we have T 1 2 T. In the case when consists of two components 1 and 2 we have that at least one of these curves, say 1, is essential and T 3 1 T. Proof. First we need to proof the following lemma. Lemma 2. Let D be a disk with two punctures. Then there are only 3 different isotopy classes of simple essential arcs in D. Let C be a cylinder with a puncture. Then there is only one essential simple arc in C up to isotopy. Arcs representing these isotopy classes are shown in Figure 2.

9 A COMBINATORIAL MODEL OF THE LIPSCHITZ METRIC 9 Figure 2. Three types of simple arcs in the twice-punctured disk and a simple arc in the once-punctured cylinder. Proof. We only prove that there is only one(up to isotopy) simple arc connecting distinct punctures (denoted by P 1 and P 2 in the following) in a twice punctured disk. Proofs for other cases follow the same outline and are even easier. Let us consider two distinct simple essential arcs α and of given type. By small isotopy of one of these arcs we make all of their intersections transverse. If these arcs don t intersect it follows from Jordan-Schönflies theorem that they bound an embedded disk, so they are isotopic. If they intersect, then we can start from P 1 and go from it along α until the first intersection X with. We denote the parts of curves α and going from P 1 to X by α 1 and 1 respectively. The arc consisting of α 1 and 1 either bounds a disk or a once punctured disk. In the first case we can push through this disk and remove the intersection point X. In the second case we can consider arcs α 2 and 2 incident to P 2 and obtaining in the same manner as α 1, 1. These curves lie entirely (except probably their common end point) in once punctured disk bounded by α 1, 1. So they bounds a disk and we again can reduce the number of intersections points of α and by isotopy. So after a finite number of such isotopies we obtain non-intersecting arcs, which are isotopic as we ve already shown. Now we turn to the proof of the proposition. First let us consider the case when is connected (and so the ends of are distinct). Then the connected component U of S \ containing the curve is a disk with two punctures P 1, P 2 (ends of ). If is inessential, then S \ U is a disk or a once punctured disk. Therefore S is a twice or triply punctured sphere, which contradicts our assumptions on the surface. Now let us isotope such that it intersects T transversally and the number of intersections of with T equals T. By well-known isotopy extension theorem(see for example [11]) we can extend this isotopy to an ambient isotopy of S. We denote the images of curves, and of the component U by this ambient isotopy by the same letters. We also denote by U the closure of U in S. Let us consider arcs of the triangulation T which are incident to punctures P 1 and P 2. If there is an edge e of T that connects P 1 to P 2 and lies entirely in U, then by Lemma 2 e is isotopic to and therefore T = 0 T.

10 10 VLADIMIR SHASTIN If there is an edge e T that connects one of the punctures in U(e.g. P 1 ) to itself and lies entirely in U, then from Lemma 2 it follows that e bounds a once-punctured disk and so T again contains an arc isotopic to. Now if T doesn t contain edges lying entirely in U, then there exist e 1, e 2 T which intersect and are incident to P 1 and P 2 respectively. Intersecting each e i with U we obtain two curves α i e i, i {1, 2}, such that α i lies in U, is incident to P i, and intersects in one point X i (see the top left of Figure 3). By a small deformation of α 1 and α 2 we obtain two simple curves α 1 and α 2 such that they lie in U, don t intersect, don t intersect the triangulation, and intersect in points X 1 and X 2 respectively (see the top right of Figure 3). The points X 1 and X 2 divide into two segments δ 1 and δ 2. So we can combine curves α 1, δ 1, α 2 as well as α 1, δ 2, α 2 together to obtain two piecewise-smooth simple arcs 1, 2 respectively (one of them is shown in the bottom left of Figure 3). By a small deformation we can smooth these arcs without changing the number of their intersections with the triangulation(one of these arcs is shown in the bottom right of Figure 3). It follows from the construction and Lemma 2 that the resulting arcs are both isotopic to and the total number of intersection points of these arcs with T equals T. Therefore one of these curves intersects T in no more than 1 2 T points and thereby we prove the proposition in this case. X 1 X 1 α 1 P 2 P 1 α 2 P 1 α 1 α 2 P 2 X 2 X 2 X 1 α 1 P 2 P 1 α 2 P 2 P 1 δ 1 X 2 Figure 3. A picture proof of Proposition 4 (the first case). In the second case we use the similar arguments. First we mention that at least one component of is essential. Indeed, if 1 and 2 are inessential then they both bound once-punctured disks and therefore S is the sphere with 3 punctures. This contradicts our assumptions on the surface and we can assume without loss of generality that 1 is essential.

11 A COMBINATORIAL MODEL OF THE LIPSCHITZ METRIC 11 By an ambient isotopy we can move such that it intersects T transversally and the number of its intersection with T equals T. We also mention that the connected component U of S \ containing is a cylinder with a puncture P, the end of. Let us consider edges of T incident to the puncture P. If there is an edge of T that lies entirely in U, then by Lemma 2 this edge is isotopic to and therefore T = 0 1 T. Now if T doesn t contain edges lying entirely in U, then there exists an edge e T that intersects and is incident to P. If e intersects an essential component of, then there is a curve α e such that it incident to P, lies in U and intersects only in one point X (see the top left of Figure 4). Slightly deforming α we can obtain two simple curves α 1 and α 2 such that they are incident to P, lie in U, don t intersect, don t intersect T and intersect in points X 1 and X 2 respectively (see the top right of Figure 4) X α P X 1 α 1 P X 2 α 2 δ 1 γ X 1 α 1 P P X 2 α 2 Figure 4. A picture proof of Proposition 4 (the second case). The points X 1 and X 2 divide 1 into two segments δ 1 and δ 2. So we can combine the curves α 1, δ 1, α 2 as well as α 1, δ 2, α 2 together to obtain two pieceweise-smooth simple arcs γ 1, γ 2

12 12 VLADIMIR SHASTIN respectively. Just one of these arcs is essential and by a small deformation we can smooth it without changing the number of its intersection with the triangulation(see the bottom of Figure 3). It follows from the construction and Lemma 2 that the resulting arc is isotopic to and the total number of its intersection with T equals 1 T Therefore T 1 T and we prove the proposition in this case. X e 1 α 1 e 2 P P e 1 2 e 2 P P e 1 2 e e 1 γ e 2 P P e 1 2 e 2 P P e 1 2 Figure 5. A picture proof of Proposition 4 (the third case). Now we turn to the case when 2 is inessential and all edges of T incident to P don t intersect 1. In this case the component V of S \ 1 is a disk with two punctures P, P, 2 is homotopic to P and is a unique (up to isotopy) essential arc in V that connects P to itself (see Figure 5). Because edges of T that incident to P don t intersect 1, the triangulation T contains edges e 1, e 2 that both lie in V and connects P to P and P to itself respectively. These edges bound the closed triangle of T. Let us consider the closed triangle adjacent to and denote by e an edge of distinct from e 2. This edge intersects 1 because it cannot be isotopic to e 1 or e 2. Therefore there exists a point X of that lies in the interior of (see the top left of Figure 5). Now we can choose a simple arc α connecting P to X lying in V ( ) that

13 A COMBINATORIAL MODEL OF THE LIPSCHITZ METRIC 13 intersects T only in 1 point (see the top right of Figure 5). Using α we can construct (in the same manner as in the previous case) a simple essential arc γ that lies in V, connects P to P and intersects T less than in T + 2 points. These 2 additional intersections come from the intersection point of α with e 2 (see the bottom of Figure 5). It follows from Lemma 2 that γ is isotopic to and therefore T 1 T T. Here the last inequality holds because 1 is essential and so intersects T nontrivially. Now let us consider an arbitrary arc of T. Using the previous proposition we can find a simple essential closed curve 0 that lies in the boundary of a small neighbourhood of in S such that the following inequalities are satisfied: ϕ() T 3 ϕ( 0 ) T 6m ϕ( 0) T 0 T, where m is a maximal degree of a vertex in the triangulation T. The last inequality holds because 0 T T 2m. For a proof of this fact see Figure 6. 1 P 1 P 2 P 2 Figure 6. The number of intersections of with T is less than 2m. ( ) By definition ϕ T = log ϕ() T + 1, so we have: T ( ) ( ( ϕ T log 3 ϕ( 0 ) T + 1 log ) ) ϕ( 0 ) T N T T ( ( log ) ) ( ( )) 6m ϕ( 0 ) T ϕ(α) T log 6m(N + 1) sup N ( 0 ) T α α T T ( ( )) ϕ(α) T log sup + log(6m(n + 1)), α α T where N is the number of edges in T. Here the second inequality holds because for every edge of the triangulation T the curve 0 is essential and thus the quantity ϕ( 0 ) T is greater than 1. So we can take log(6m(n + 1)) as C 1 in (1). In order to prove the right inequality in (1) we need the following lemma. Lemma 3. Let T, T be a pair of ideal triangulations and α be a simple essential closed curve. Then the following inequality is satisfied: α T α T (2 T T + 1).

14 14 VLADIMIR SHASTIN Proof. First let us isotope α and T in such a way that the following is satisfied: α and T as well as T and T are tight, α and T intersect transversely, there are no triple pairwise transverse intersections of α, T and T (there can be triple intersections when some edges of T and T coincide and α intersects these edges) Now for a triangle of T we denote by α and T preimages of α and T in by the corresponding attaching map ϕ. Because α and T as well as T and T are tight, these preimages consist of embedded arcs and each arc either connects different edges of or coincides with an edge of the triangle. It is easy to see that we can isotope α in fixing the endpoints of the arcs such that α, T intersects transversely as before and each arc in α intersects any arc in T no more than in one point. Now we denote the intersection points of α and T by X 1,..., X n in such a way that α i = [X i, X i+1 ) are non-intersecting half intervals in α. Here we assume that 1 i n and X n+1 = X 1. By construction each α i lies in some triangle of T and we claim that it intersects with T no more than in 2 T T + 1 points. Indeed, the number of arcs in T which don t coincide with edges of triangle is less or equal to the number of intersection points of these arcs and edges of. In turn the number of these intersection points is less or equal to 2 T T because the preimage of an intersection point between T and T under ϕ consists of at most two points. To obtain the final estimation it sufficient to add 1 to 2 T T because α i intersects only one edge of. So we have n n α T = α i T (2 T T + 1) = α T (2 T T + 1). i=1 i=1 Now let us consider a simple essential closed curve α in S and an element ϕ MCG(S). For the triple (ϕ(α), ϕ(t ), T ) we can write the inequality from Lemma 3: ϕ(α) T ϕ(α) ϕ(t ) (2 T ϕ(t ) + 1). Because all quantities in this inequality are positive and ϕ(α) ϕ(t ) = α T, log(2 T ϕ(t ) +1) ϕ T + log 2 we obtain ( ) ϕ(α) T log ϕ T + log 2. α T Taking the supremum over all simple essential closed curves we obtain the right inequality in (1) with C 2 = log 2. This completes the proof of the theorem. Proof of Theorem 3. The Theorem 3 follows directly from the Theorem 2 and results of Choi and Rafi who proved in the work [3] that the Lipschitz metric differs from the Teichmüller metric by an additive constant in the thick part of the Teichmüller space. References [1] I. Dynnikov. Counting intersections of normal curves, unpublished preprint [2] I. Dynnikov, B. Wiest, On the complexity of braids, J. Eur. Math. Soc. (JEMS), 9:4 (2007), [3] Y.-E. Choi, K. Rafi, Comparison between Teichmüller and Lipschitz metrics, J. Lond. Math. Soc (2) 76 (2007), no. 3, [4] M. Kapovich, Hyperbolic Manifolds and Discrete Groups: Lectures on Thurston s Hyperbolization, Birkhauser s series Progress in Mathematics, 2000 [5] A. Papdopoulos, Introduction to Teichmüller Theory, Old and New, Handbook of Teichmüller spaces, Volume I, European Mathematical Society Publishing House, pp. 1-30, Zürich, 2007,

15 A COMBINATORIAL MODEL OF THE LIPSCHITZ METRIC 15 [6] A. Fathi, F. Laudenbach, V. Poenaru, Thurston s work on surfaces, translated by Djun Kim and Dan Margalit, Mathematical Notes, 48, Princeton University Press, 2012 [7] B. Farb and D. Margalit, A primer on mapping class group. Princeton Math. Ser. 49, Princeton University Press, Princeton, N.J., 2011 [8] Andrew J. Casson and Steven A. Bleiler. Automorphisms of surfaces after Nielsen and Thurston, volume 9 of London Mathematical Soci- ety Student Texts. Cambridge University Press, Cambridge, [9] W. P. Thurston, Minimal stretch maps between hyperbolic surfaces. Preprint [10] Riccardo Benedetti, Carlo Petronio, Lectures on Hyperbolic Geometry, Springer-Verlag, Berlin-Heidelberg- New York, [11] M. W. Hirsch, Differential topology, Grad. Texts in Math., vol. 33, Springer-Verlag, Berlin and New York, 1976 [12] Ursula Hamenstädt. Teichmüller theory in Moduli Spaces of Riemann Surfaces, B. Farb, R. Hain, E. Looijenga, Editors, AMS/PCMI, 2013, [13] R. C. Penner and J. L. Harer. Combinatorics of train tracks, volume 125 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, Department of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow , Russia. Laboratory of Quantum Topology, Chelyabinsk State University, Brat ev Kashirinykh street 129, Chelyabinsk , Russia. address: vashast@gmail.com