Length statistics of random multicurves on closed hyperbolic surfaces

In this paper, we determine the distribution of the length partition of a random multicurve of fixed topological type on a closed hyperbolic surface using the methods of Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres. This distribution admits a polynomial density, whose coefficients can be expressed explicitly in terms of intersection numbers of psi-classes on the Deligne--Mumford compactification, and in particular it does not depend on the hyperbolic metric. This result generalizes prior work of M. Mirzakhani in the case of random pants decompositions. Results very close to ours are obtained independently and simultaneously by F. Arana-Herrera.

Our goal is to find the limiting distribution ofˆ X,R,γ as R → ∞.
The study of the length partition of random multi-geodesics is initialed by M. Mirzakhani in [Mir], where she proves the following result.
Theorem 1 ([Mir, Theorem 1.2]). If {γ 1 , . . . , γ 3g−3 } is a pants decomposition of X, then as R → ∞, the random variableˆ X,R,γ converges in law to the Dirichlet distribution of order 3g − 3 with parameters (2, . . . , 2), namely the probability distribution that has a density function (6g − 7)! · x 1 · · · x 3g−3 with respect to the Lebesgue measure on the standard simplex ∆ 3g−4 := {(x 1 , . . . , x 3g−3 ) ∈ R 3g−3 Motivations. Theorem 2 is motivated by Theorem 1 of Mirzakhani. Another motivation originates from Theorem 1.25 in the section "Statistical geometry of square-tiled surfaces" of [DGZZ21]. Namely, the statistics of perimeters of maximal cylinders of a "random" square-tiled surface associated to a given multicurve γ given by formula (1.38) from [DGZZ21] coincide with statistics of hyperbolic lengths of different components of γ given by Theorem 2 above. Though formula (1.38) from [DGZZ21] can be interpreted as a certain average of lengths statistics for individual hyperbolic surfaces X, it does not imply that such statistics do not change when X changes. The conjecture of non-varying of statistics of hyperbolic lengths for any fixed multicurve under arbitrary deformation of the hyperbolic metric, proved in Theorem 2, was one of our principal motivations.
Idea of the proof. The structure of the proof is similar to that of [Mir,Theorem 1.2]. The limiting distribution that we are after boils down to the asymptotics of multicurves counting under constraints, which can be transformed to a problem of approximating to the number of "lattice points" within a horoball in a covering space of the moduli space. By considering tiling of the covering space by translates of a fundamental domain for the action of the mapping class group, it would not be unreasonable to expect that, this number might be proportional to the volume of the horoball divided by the volume of the moduli space, and finally this is not so far from the truth. We proceed using techniques that Margulis introduced in his thesis [Mar04], and the equidistribution theorem for large horospheres initially established by Mirzakhani [Mir07a]. Similar methods were also applied in, e.g., [EM93].
Theorem 2 can be generalized to hyperbolic surfaces with cusps if Mirzakhani's work on the ergodicity of the earthquake flow can be generalized to such surfaces, which seems to be the case (see [Mir]).
Remark. While the author was finishing this paper, the paper [AH21] by F. Arana-Herrera appeared on the arXiv. Both papers are devoted to a similar circle of problems and use a similar circle of ideas, though they were written in parallel and completely independently. In particular, Arana-Herrera proves a much more general version of our Theorem 17 ([AH21, Theorem 1.3]), which is one of the key ingredients allowing to attack the counting problem and the length statistics. We learned from [AH21] that this kind of statistics was initially conjectured by S. Wolpert. Papers [AH21] and [AH22] established results closely related to Theorem 2.
Proposition 20 below is based on a theorem stated by M. Mirzakhani but presented without a detailed proof. The paper [AH21] contains a detailed proof of an even stronger estimate which implies, in particular, the statements of this theorem; see Remark 24 below.
Acknowledgment. The author is extremely grateful to Bram Petri for all his patience and support, and for reading earlier drafts of this paper. The author would also like to thank Anton Zorich for formulating the problem, Francisco Arana-Herrera for pointing out a gap in the first version of this paper, Grégoire Sergeant-Perthuis and Jieao Song for useful discussions, and the anonymous referee for a careful reading and numerous helpful suggestions.

Background
Throughout this paper, we use the symbol Σ g to denote a connected, closed, oriented, topological surface of genus g ≥ 2, the symbol Σ g,n to denote a connected closed oriented topological surface of genus g with n boundary circles labeled by {1, . . . , n} with 2g − 2 + n > 0, and the letter d to denote 3g − 3 + n.
2.1. Deformation spaces and mapping class group. Let us consider the set of orientationpreserving homeomorphisms ϕ : Σ g → X where X is an oriented complete hyperbolic surface of genus g. Two such homeomorphisms ϕ 1 : Σ g → X 1 and ϕ 2 : Σ g → X 2 are said to be equivalent if ϕ 2 • ϕ −1 1 is isotropic to an isometry. The Teichmüller space, denoted by T(Σ g ) or simply T g , is the set of such equivalence classes.
Let us denote by Homeo + (Σ g ) the group of self-homeomorphisms of Σ g that preserve the orientation, and write Homeo 0 (Σ g ) for the subgroup of Homeo + (Σ g ) consisting of homeomorphisms isotropic to the identity. The mapping class group, denoted by Mod(Σ g ) or simply Mod g , is the quotient group Homeo + (Σ g )/ Homeo 0 (Σ g ).
The group Homeo + (Σ g ) acts (properly and discontinuously) from the right on T g by precomposition, and Homeo 0 (Σ g ) acts trivially. The moduli space, denoted by M(Σ g ) or M g , is the quotient T g / Mod g .
The Teichmüller space T g,n (L 1 , . . . , L n ) and moduli space M g,n (L 1 , . . . , L n ) of oriented complete hyperbolic surfaces of genus g with n (labeled) totally geodesic boundary components of lengths L 1 , . . . , L n ≥ 0 respectively can be defined in a similar manner.

2.2.
Curves. In the introduction, the theorems are stated in terms of geodesics on a hyperbolic surface. Nevertheless, it is often more convenient to work with (the free homotopy classes of) the topological curves, and they are actually equivalent for our purposes. A curve in a topological space X is (the image of) a continuous application S 1 → X. In this paper, we are interested in curves up to free homotopy. A closed curve is said to be simple if it does not intersect itself. A multicurve is a finite multiset of disjoint simple curves, and a multicurve is ordered (or labeled ) if its underlying set is labeled. We will often write an ordered multicurve γ as an ordered list (m 1 γ 1 , . . . , m k γ k ), and its unlabeled counterpart as a formal sum γ = m 1 γ 1 + · · · + m k γ k where m i ∈ Z ≥1 , 1 ≤ i ≤ k.
The group Homeo + (Σ g ) acts on the set of closed curves on Σ g by postcomposition, and the action of the subgroup Homeo 0 (Σ g ) stabilizes sets of curves in the same free homotopy class. Thus the mapping class group acts on the set of free homotopy classes of closed curves on Σ g . We say that two closed curves α and β have the same topological type if they lie in the same mapping class group orbit. The three following subgroups, associated to γ, of the mapping class group Mod g will be useful later in the paper: • Stab(γ) which fixes the multicurve γ = m 1 γ 1 + · · · + m k γ k (but the γ i 's can be permutated), • Stab(γ) which fixes every γ i for all 1 ≤ i ≤ k, • Stab + (γ) which fixes every γ i and its orientation for all 1 ≤ i ≤ k.
Let X ∈ T g . If a closed curve α on X is not homotopic to a point, then α is freely homotopic to a unique closed geodesic on X with the minimum length over all curves in the free homotopy class of α, and we write X (α) for the length of this geodesic.
The notions of topological type and length extend naturally to multicurves.
2.3. Fenchel-Nielsen coordinates. A pair of pants is a surface that is homeomorphic to a sphere with three holes. A pants decomposition of Σ g,n is a set of disjoint simple closed curves {α 1 , . . . , α 3g−3+n } on Σ g,n such that Σ g,n {α 1 , . . . , α 3g−3+n } is a disjoint union of pairs of pants.

2.4.
Weil-Petersson volumes. The following theorem is often referred to as Wolpert's magical formula.
Theorem 4 ([Wol83]). Given a pants decomposition {α 1 , . . . , α 3g−3+n }, the formula defines a symplectic form which has the same expression in any other Fenchel-Nielsen coordinates. In particular, it is invariant under the action of the mapping class group.
The symplectic form thus defined is the so-called Weil-Petersson symplectic form, and we shall denote it by ω. See [Wol83] for a more intrinsic definition.
Every symplectic form defines a volume form. The Weil-Petersson volume of the moduli space M g,n (L 1 , . . . , L n ) is defined by The following fundamental result is due to Mirzakhani.

2.5.
Earthquakes. Multicurves can be regarded as "lattice points" in the space of measured laminations ML g . We will only need the following properties of this space. See, e.g., [Kap01, Chapter 11] for more details.
1. The space ML g is a (6g − 6)-dimensional real manifold equipped with a natural piecewise integral linear structure, i.e., ML g has an natural atlas whose transition functions are piece-wise in GL(6g − 6, Z). 2. The integral points in the coordinate charts of ML g , denoted by ML g (Z), are in natural bijection with the (free homotopy classes of) integral multicurves on Σ g . 3. The action of the mapping class group on the set of multicurves extends to ML g . 4. The group (R >0 , ×) acts on ML g , and for any multicurve γ = m 1 γ 1 + · · · + m k γ k , and any r ∈ R >0 , r · (m 1 γ 1 + · · · + m k γ k ) = r m 1 γ 1 + · · · + r m k γ k . We denote the quotient by P(ML g ). 5. Given X ∈ T g , the length function X defined on the set of multicurves extends to ML g .
Moreover, for any λ ∈ ML g , we have X·h (h −1 · λ) = X (λ) for any h ∈ Mod g , and X (r · λ) = r · X (λ) for any r ∈ R >0 . 6. The twist flow tw t γ about a multicurve γ can be extended to any measured lamination λ ∈ ML g , and we have (tw t λ (X)) · h = tw t h −1 λ (Xh) and tw t rλ (X) = tw rt λ for all t ∈ R, h ∈ Mod g , and r ∈ R >0 . 7. The space ML g carries a natural mapping class group invariant measure µ Th defined by asymptotic counting of integral points, called the Thurston measure. The Thurston measure is a Lebesgue measure in the coordinate charts of ML g , and for any open subset Let PT g := T g × ML g be the bundle of measured laminations over the Teichmüller space, and let P 1 T g := {(X, λ) ∈ PT g : X (λ) = 1} be the unit sphere bundle of PT g with respect to the length function.
The mapping class group acts on PT g from the right via (X, λ) · h := (X · h, h −1 · λ). This action is well-defined on P 1 T g since it preserves the length function (X, λ) := X (λ). Write PM g := PT g / Mod g and P 1 M g := P 1 T g / Mod g .
The earthquake flow tw t on PT n is defined by tw t (X, λ) := (tw t λ (X), λ). The earthquake flow commutes with the action of the mapping class group, and therefore descends to PM g , and to P 1 M g (since the earthquake preserves the length function).
The Thurston measure on ML g induces a measure on {λ ∈ ML g : X (λ) = 1} in the following way: let U ⊂ {λ ∈ ML g : X (λ) = 1} be an open subset. The Thurston measure of U is defined to be µ Th {s · λ ∈ ML g : λ ∈ U, s ∈ [0, 1]}. The measure ν g on P 1 T g defined by for any open subset U ⊂ P 1 T g , is invariant both under the earthquake flow (since µ WP is) and under the action of the mapping class group (since µ Th and µ WP are), and hence descends to a measure on P 1 M g that (by abuse of notation) we shall also denote by ν g . The total mass of ν g The following result is fundamental.
Theorem 6 ( [Mir08a]). The earthquake flow on P 1 M g is ergodic with respect to ν g .
We recommend [Wri] for an expository survey on this topic. .
The Thurston distance between X 1 and X 2 is defined by The Thurston distance ball centered at X ∈ T g of radius is defined to be The reason for this choice of radius is that, for small , e.g. 0 < < 1, We have therefore, for any λ ∈ ML g and any Y ∈ B X ( ), Thurston distance balls are well-defined on M γ g , and on M g , since the Thurston distance is Mod g -invariant.
2.7. Stable graphs. Given a multicurve m 1 γ 1 + · · · + m k γ k , one can associate with it a stable graph in the following way. Cut the surface along γ 1 , . . . , γ k . To each connected component S of Σ g {γ 1 , . . . , γ k }, we associate a vertex, and we decorate this vertex with the genus of S. For each component γ i of γ, we draw an edge that connects the two vertices (which could be the same) corresponding to the two connected components of Σ g {γ 1 , . . . , γ k } bounded by γ i . See Figure 1 for an example. Note that the resulting graph does not depend on m 1 , . . . , m k . More formally, a stable graph consists of the data satisfying the following properties: 1. The pair (V, E) defines a connected graph, with vertex set V and edge set E. The set H is the set of half-edges. 2. The map v assigns each half-edge to its adjacent vertex. 3. The map ι is an involution, such that the 2-cycles of ι are in bijection with E, and the fixed points of ι are in bijection with L. 4. The genus function g assigns each vertex x to its genus (the genus of the surface corresponding to x), such that the stability condition is satisfied, where n(x) denotes the number of edges and legs adjacent to x.
2.8. Graph polynomials. Given a stable graph associated to the multicurve γ = m 1 γ 1 + · · · + m k γ k , we associate to each edge e a variable x e , and define the associated graph polynomial by the formula where e runs through the edge set E, v runs through the vertex set is the edge that contains the half-edge h, and v(h) denotes the vertex incident to h. Note that P γ is of degree 2d − k. Finally, we writeP γ for the top-degree homogeneous part of P γ , andV g,n for that of V g,n . Example Example 8. Let (γ 1 , γ 2 , γ 3 ) be an ordered multicurve on Σ 3 as in Firgure 1. The Weil-Petersson where m stands for the monomial symmetric polynomial. For example, and therefore, Let γ = (m 1 γ 1 , . . . , m k γ k ) be an ordered multicurve on Σ g . Recall that Stab(γ) denotes the subgroup of Mod g that fixes each γ i , 1 ≤ i ≤ k. The quotient space M γ g := T g / Stab(γ) introduced by Mirzakhani in her thesis plays an important role in this paper.
Lemma 9. The quotient P γ / Mod g is isomorphic to M γ g as symplectic orbifolds.
Proof. Consider the map P γ → M γ g defined by (X, hγ) → π γ (Xh). This map is surjective, and descends to the quotient P γ / Mod g . The resulting map P γ / Mod g → M γ g is a local isomorphism of symplectic orbifolds. All that remains now is to show that the map P γ / Mod g → M γ g is injective. Let (X 1 , h 1 γ), (X 2 , h 2 γ) ∈ P γ such that π γ (X 1 h 1 ) = π γ (X 2 h 2 ). By definition, there exists s ∈ Stab(γ) such that X 1 h 1 s = X 2 h 2 . Therefore , which proves the injectivity.
Remark 10. Let α be a simple closed curve on Σ g . In general, X (α) is not well-defined for X ∈ M γ g . However, it is if α = γ i for some i.
The next lemma is a simple fact, but for our purposes it will be very important: it transforms the multicurves counting that we are after to a "lattice points" counting problem on M γ g . Lemma 11. Let γ = (m 1 γ 1 , . . . , m k γ k ) be an ordered multicurve, X ∈ T g , R ∈ R >0 , and Next, let us review another covering space of M g that Mirzakhani introduced. By considering the Fenchel-Nielsen coordinates associated to a pants decomposition that contains γ 1 , . . . , γ k , the Teichmüller space T g can be written as where V (resp. E; H) is the vertex (resp. edge; half-edge) set of the stable graph associated to γ. The group acts naturally on T g written in the form (3) (each copy of Z acts as the Dehn twist about a γ i ), and G γ can be identified with Stab + (γ). The quotient C γ := T g /G γ is of the form Since G γ Stab + (γ) is a subgroup of Stab(γ), T g → M γ g factors through a (ramified) covering map C γ → M γ g . The degree of this covering map is where M (γ) is the number of i such that γ i bounds a surface homeomorphic to Σ 1,1 , and Stab + (γ), Stab 0 (γ) stands for the subgroup of Stab(γ) generated by Stab + (γ) and the kernel Stab 0 (γ) of the action of Stab(γ) on T g . Note that Stab 0 (γ) is trivial when g ≥ 3, and is isomorphic to Z/2Z if g = 2 (generated by the hyperelliptic involution which fixes the free homotopy class of every simple closed curve on Σ 2 ). For more details, see the footnote on p. 369-370 of [Wri20].
Integrating functions over C γ (and M γ g ) is far less delicate than integrating function over M g . Starting from this observation Mirzakhani was able to calculate the integrals of an important class of functions defined on M g , which she called "geometric functions".
Remark 13. The function X → (m i X (γ i )) k i=1 is well-defined for X ∈ M γ g . The horosphere S A R,γ ⊂ M γ g can be written as the pre-image of R · A under this function, where R · A is defined to be {(x 1 , . . . , x k ) ∈ R k >0 : (x 1 , . . . , x k )/R ∈ A}.

Horospherical measures.
We can choose d − k simple closed curves α k+1 , . . . , α d such that {γ 1 , . . . , γ k , α k+1 , . . . , α d } is a pants decomposition. In the associated Fenchel-Nielsen coordinates, the horosphereS A R,γ is an open subset of a simplex. Let µ ∆ denote the Weil-Petersson (Lebesgue) measure on this simplex. The horospherical measure µ A R,γ , of an open subset U ⊂ T g is defined to be µ A R,γ (U ) := µ ∆ (U ∩S A R,γ ). The horospherical measure µ A R,γ is invariant under the action of the mapping class group, and hence descends to a measure on M γ g and a measure on M g ; by abuse of notation we shall denote both by µ A R,γ . Note that M γ g → M g is a (ramified) covering map of infinite degree. However, its restriction on S A R,γ is of finite degree. Thus µ A R,γ on M g is the push-forward measure of µ A R,γ by M γ g → M g . So for any open subset U of M g , µ A R,γ (π −1 γ (U )) = [Stab(γ) : Stab(γ)] · µ A R,γ (U ). In particular, the total masses of µ A R,γ on M γ g and on M g differ only by a multiplicative constant depending only on γ.
Proof. In the light of Remark 13, by taking f in Theorem 12 to be the indicator function the result desired.
where λ is the Lebesgue measure on ∆ k−1 andP γ is the top-degree homogeneous part of the graph polynomial P γ defined by (2).
Remark 16. It results from Theorem 5 that the polynomialP γ can be expressed in terms of intersections numbers of ψ-classes on the Deligne-Mumford compactification M g,n .

Horospherical measures on the unit sphere bundle. Define
PS A R,γ := {(X, γ/R) ∈ P 1 T g :ˆ X (γ) ∈ A}. Note that PS A R,γ projects via P 1 T g → T g toS A R,γ , and is invariant under the earthquake flow. Let ν ∆ denote the Lebesgue measure on PS A R,γ . The horospherical measure ν A R,γ on PT g is defined by the formula ν A R,γ (U ) := ν ∆ (U ∩ PS A R,γ ) where U is any open subset of P 1 T g . The measure ν A R,γ is Mod g -invariant, and therefore descends to a measure on P 1 M g which by abuse of notation we shall also denote by ν A R,γ . Note that µ A R,γ is the push-forward of ν A R,γ via P 1 M g → M g . Notation.

Equidistribution
In this section, we establish the equidistribution of large horospheres. The proof is adapted from that of [Mir07a, Theorem 1.1] Theorem 17. We have weak convergence of probability measures on The following immediate corollary is exceedingly useful late on.
Corollary 18. We have weak convergence of probability measures on M g Proof. This follows from the fact that µ R is the push-forward of ν R via P 1 M g → M g and Theorem 17.
The proof of Theorem 17 rests on the following series of propositions.
Let ν be a weak limit of (ν R /M R ) R>0 .
Proposition 19. The measure ν is invariant under the earthquake flow.
Proposition 20. The measure ν is absolutely continuous with respect to ν g .
Proposition 21. The measure ν is a probability measure.
Proof of Theorem 17. Proposition 19, Proposition 20, and Theorem 17 imply that ν and ν g differ by a multiplicative constant, and it follows from Proposition 21 that this constant is 1.
Proposition 19 is immediate.
Proof of Proposition 19. This follows from the fact that ν R is invariant under the earthquake flow (since ν g is).
For the rest of this section we shall prove Proposition 20 and 21, which are more technical.
5.1. Escape to infinity? In this subsection, we prove Proposition 21. The key ingredient is the following non-divergence result for the earthquake flow due to Y. Minsky and B. Weiss.

We have
where π : T g → M g is the natural projection, and M ≥ g is the compact subset of M g consisting of all surfaces whose shortest closed geodesic has length at least .
Proof of Proposition 21. It is enough to prove that for any δ > 0, we can find a compact subset The strategy is to show that there exists > 0, depending only on δ, such that the pre-image of M ≥ g under P 1 M g → M g possess the desired property. In other words, Taking c = δ/2, Theorem 22 allows us to writeS R ⊂ T g as the disjoint union ofS 1 andS 2 corresponding to the two possibilities. For convenience, we shall adapt the convention thatS * (resp. S * ) denotes the image ofS * under T g → M g (resp. T g → M γ g ), where * is a certain index. First, we show that µ R (S 1 ) ≤ µ R (S 1 ) = o(M R ) as R → ∞ even when A = ∆ k−1 (the subset of the simplex that we choose to define µ R is the whole simplex). For any point inS 1 , at least one of the following holds: 1.1. α is freely homotopic to γ i for some 1 ≤ i ≤ k. 1.2. α is disjoint from γ 1 , . . . , γ k .
ThusS 1 can be written as the union ofS 1,1 andS 1,2 corresponding to the two cases above. To simplify the notation, in the following estimates of µ R (S 1,1 ) and µ R (S 1,2 ) we assume that γ is primitive, i.e. m 1 = · · · = m k = 1 (the calculation differs from the general case only by a multiplicative constant).
For each i, the corresponding horospherical volume of S 1,1 can be estimated by taking f in Theorem 12 to be the indicator function 1{(x 1 , . . . , x k ) ∈ R k ≥0 : R ≤ x 1 + · · · + x k ≤ R + h, x i < }, and we obtain Since P γ is a polynomial of degree 2d−k and x 1 · · · x k is a factor of P γ , we have µ R (S 1,1 ) = O( 2 R 2d−3 ).

Absolute continuity.
In this subsection, we prove Proposition 20.
We use the following notation throughout the subsection. Let d denote 3g − 3. We write f = O K (g) if there exists C > 0, depending only on K, such that f ≤ Cg, and we write f = Θ K (g) if there exists C, depending only on K, such that (1/C)g ≤ f ≤ Cg.
The key to the proof are the following estimates.
Remark 24. The first part of the preceding theorem is [Mir07a, Theorem 5.5.a]. Mirzakhani proved the second part in the case when γ is a simple closed curve [Mir07a,Theorem 5.5.b], and claimed a more general version without proof [Mir,Proposition 2.1.b]. The proof of [Mir07a,Theorem 5.5.b] is concise and not easy to follow. See also the footnote on p. 390 in [Wri20]. A much stronger estimate is obtained by Arana-Herrera in a different approach [AH21, Proposition 1.5].
The rest of the proof of Proposition 20 can be adapted from Mirzakhani's original proof in the case when γ is simple. Let us sketch her arguments for the sake of self-containedness.
Corollary 25. Let U ⊂ P(ML g ) be open, K ⊂ T g be compact, x ∈ K, and p : T g × P(ML g ) → P 1 M g be the natural projection. For ∈ (0, 1), we have Proof. It is enough to prove this for A = ∆ k−1 . By (1), for any y ∈ B x ( ), we have and so Hence Theorem 23.2 implies that ν R (p(B x ( ) × U )) = O K ( 2d R 2d−1 µ Th (U x )). The result now follows from Theorem 23.1 and Corollary 15.
We need one further technical lemma.
Lemma 26. Let K be a compact subset of P 1 T g . For any N ⊂ K with ν g (N ) = 0, and any > 0, there exists an open cover

Proof. Fix a choice of Fenchel-Nielsen coordinates. There exists an open cover {B
and sup i≥1 r i can be made as small as we please (since ν g is a Lebesgue class measure). It follows from the compactness of K × [0, 1] that there exists a constant s depending only on K such that B x (r) ⊂ B x (s · r) for any x ∈ K and any r ∈ [0, 1]. By Theorem 23.1, there exists a constant s depending only on K such that µ WP (B x (s · r)) ≤ s · µ WP (B x (r)) for any x ∈ K, and any r < 1/2s. Therefore, by (5) and the lemma follows.
Proof of Proposition 20. It is sufficient, as before, to consider the case in which A is the whole simplex ∆ k−1 . Let N ⊂ P 1 M g with ν g (N ) = 0. By Proposition 21, we may assume that N is contained in a compact set K ⊂ P 1 M g . Lemma 26 implies for any > 0, there exists an open cover Hence, it follows from Corollary 25 that The proof is thus complete.

Counting
The main result of this section is the following theorem which is a refined version of [Mir08b, Theoreom 1.1].
By virtue of Lemma 11, this multicurves counting problem can be transformed to a counting problem on M γ g . Let us begin by introducing some definitions that we need to state our counting result on M γ g . The horoball (on M γ g ) is defined by S r and its associated measure µ ≤R is defined by the formula where U is any open subset of M γ g . By abuse of notation, we shall also use µ ≤R to denote the measure on M g defined by the formula for any open subset U of M g . Let X ∈ M g and let N (R) denote the number of pre-images of X under π γ : M γ g → M g which lie within the horoball B R ⊂ M γ g , i.e., N (R) := #{π −1 γ (X) ∩ B R }. We have the following counting result on M γ g . Theorem 28. Let X ∈ M g , γ = (m 1 γ 1 , . . . , m k γ k ) be an ordered multicurve, and A ⊂ ∆ k−1 be open. Then we have as R → ∞.
As an immediate corollary, we get the main result of this section: Proof of Theorem 27. This follows at once from Theorem 28 and Lemma 11.
We introduce a family of subsets A a,b of ∆ k−1 , indexed by a = (a 1 , . . . , a k−1 ) ∈ [0, 1] k−1 and b = (b 1 , . . . , b k−1 ) ∈ [0, 1] k−1 such that a i < b i for all 1 ≤ i ≤ k − 1, and defined by To prove Theorem 28, it is enough to check the case when A = A a,b for all a, b. In order to abbreviate our formulas, for the rest of this section we write The reason for the choice of A + and A − is the following elementary lemma.
Lemma 29. Choose ∈ (0, 1) small enough to ensure that A − and A + are well-defined, and let x, y ∈ M γ g with d Th (x, y) ≤ . We have Proof. Suppose that x ∈ B R . It follows from the inequality (1) that which show that y ∈ B + (1+ )R . Part (1) can be proved in a similar manner. Now we are ready to prove our main result of the section.