A ROUND THE COMBINATORIAL UNIT BALL OF MEASURED FOLIATIONS ON BORDERED SURFACES

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A ROUND THE COMBINATORIAL UNIT BALL OF MEASURED FOLIATIONS ON BORDERED SURFACES

Ga¨etan Borot

^{* †}

, S´everin Charbonnier

^{* ‡}

, Vincent Delecroix

^§

, Alessandro Giacchetto

^{* ¶}

, Campbell Wheeler

^*

Abstract

The volumeB_Σ^comb(G)of the unit ball — with respect to the combinatorial length function`_G— of the space of measured foliations on a stable bordered surfaceΣappears as the prefactor of the polynomial growth of the number of multicurves onΣ. We find the range ofs∈ Rfor which (B^comb_Σ )^s, as a function over the combinatorial moduli spaces, is integrable with respect to the Kontsevich measure. The results depends on the topology ofΣ, in contrast with the situation for hyperbolic surfaces where [6] recently proved an optimal square-integrability.

*Max Planck Institut f ¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany.

†Humboldt-Universität zu Berlin, Institut f ür Mathematik und Institut f ür Physik, Rudower Chaussee 25, 10247 Berlin, Germany.

‡Universit´e de Paris, CNRS, IRIF, F-75006, Paris, France

§LaBRI, UMR 5800, Bˆatiment A30, 351 cours de la Lib´eration, 33405 Talence Cedex, France.

¶Universit´e Paris-Saclay, CNRS, CEA, Institut de Physique Th´eorique (IPhT), 91191 Gif-sur-Yvette, France

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1 Introduction

1.1 Measured foliations and Teichm ¨uller spaces

Consider a smooth connected oriented surfaceΣof genusg>0 withn >0 labelled boundaries which is stable (i.e. 2g−2+n > 0), and denote by Mod^∂Σ its pure mapping class group. A key role in this work is played by the space MFΣof measured foliations onΣ(considered up to Whitehead equivalence), where we require that∂Σis a union of singular leaves. For later convenience, we also include the empty foliation.

From the work of Thurston, MFΣis a topological space of dimension 6g−6+2nequipped with a piecewise linear integral structure. The set of integral points in MFΣis identified with the set of multicurvesMΣonΣ, and in fact MFΣis the completion of the set ofQ+-weighted multicurves. The corresponding volume form µ_Th, called the Thurston measure, can be defined by asymptotics of lattice point counting.

There are two other natural spaces attached toΣ: for a fixed L = (L₁, . . . ,Ln) ∈ R₊ⁿ, we consider the ordinary Teichm ¨uller spaceTΣ(L)and the combinatorial oneT^combΣ (L). The former is identified with the set of isotopy classes of hyperbolic structures onΣmaking the boundaries geodesics of lengthL(we may allow L₁ = · · · = Ln = 0, meaning that each boundary component is replaced by a puncture and we consider complete hyperbolic structures). The latter is the set of isotopy classes of embedded metric ribbon graphs onΣwith fixed boundary lengthL, onto whichΣretracts. By definition the associated moduli spaces are

Mg,n(L) =TΣ(L)/Mod^∂Σ, M^combg,n (L) =TΣ^comb(L)/Mod^∂Σ.

Such Teichm ¨uller spaces are equipped with a natural Mod^∂Σ-invariant symplectic form: the Weil–Petersson formω_WPin the hyperbolic setting [16], and the Kontsevich formω_Kin the combinatorial one [18]. Both measuresµ_WPandµ_Kassign a finite volume to the respective moduli spaces.

TΣ(L)andT^combΣ (L)are topologically the same space but carry different geometries; the ordinary Teichm ¨uller space has a natural smooth structure, while the combinatorial one is a polytopal complex. Nevertheless, the two geometries share many interesting properties: they posses global coordinates that are Darboux for the associated symplectic forms [4,27], and they admit a recursive partition of unity (Mirzakhani–McShane identities) that integrate to a recursion for the associated symplectic volumes [4,20]. In this article we shall examine another aspect of this parallelism regarding the asymptotic count of multicurves.

1.2 Random geometry of multicurves

Since the Weil–Petersson and the Kontsevich measures assign a finite volume to the respective moduli spaces, normalising them defines a probability measure and thus the ensemble of random hyperbolic surfaces and the ensemble of random combinatorial surfaces. We shall study the behavior of the length spec- trum of multicurves in these two ensembles. Concretely, the data of a hyperbolic metricσ∈TΣ(L)or of an embedded metric ribbon graphG∈T^combΣ (L)induces a length function

MFΣ−→R+

F7−→`σ(F), MFΣ−→R+

F7−→`_G(F).

We want to study the Thurston volume of the unit balls — with respect to these lengths functions — in the space of measured foliations:

BΣ(σ) =µ_Th { F∈MFΣ|`σ(F)61}

, B^combΣ (G) =µ_Th { F∈MFΣ|`_G(F)61} .

The functionBΣofσ∈TΣ(L)(resp.B−Σ^combofG∈TΣ^comb(L)) is mapping class group invariant, therefore descends to a functionB_g,n (resp. B^comb_g,n ) on the moduli spacesMg,n(L)(resp. M^combg,n (L)). They naturally appear in the study of the asymptotic number of multicurves with bounded length:

BΣ(σ) = lim

r→∞

#{γ∈MΣ|`σ(γ)6r}

r^6g−6+2n , B^comb_Σ (G) = lim

r→∞

#{γ∈MΣ |`_G(γ)6r}

r^6g−6+2n .

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Because the function`σon MF is not very explicit it is delicate to extract properties ofBΣ. In [22] Mirzakhani initiated the study ofBΣ(σ), and she established the following properties for punctured surfaces —i.e.over TΣ(0). Her proof can be extended to bordered surfaces and more generally to lengths measured with respect to a filling current [12].

Theorem 1.1. [22] For anyL∈Rⁿ_>₀, the functionBΣis continuous and proper onTΣ(L), and induces a function whoses-th power is integrable onMg,n(L)with respect toµ_WPfor anys <2, and not integrable fors >2.

Arana-Herrera and Athreya [6] recently proved integrability for the limit cases=2 in the case of punctured surfaces.

TheL¹-norm ofBg,nis well-understood. It is in fact the same in the hyperbolic and combinatorial setting ir- respectively of boundary lengths and coincides, up to normalisation, with the Masur–Veech volume MVg,n

of the top stratum of the moduli space of meromorphic quadratic differentials on punctured surfaces with simple poles at the punctures:

∀L∈R_>0ⁿ , MVg,n

2^4g−2+n(4g−4+n)!(6g−6+2n) = ˆ

Mg,n(L)

Bg,ndµ_WP= ˆ

M^combg,n(L)

B^comb_g,n dµ_K. (1.1) We refer to [3,4,9,21] for the justification of the various parts of this statement. Besides, the values of MVg,n

can be computed in many ways [3,8,9,17,28] and its large genus asymptotics are known [1,2].

In contrast, the computation of theL²-norm ofB_g,n is still an open problem. In this article, we study the combinatorial analogue of the above quantities. We find that the computations are much simpler, due to the polytopal nature of both MFΣandM^comb_Σ (L), that allows us to explicitly describe the functionB_Σ^comb(see Proposition2.7) and have a good understanding of its domain of integration.

Consider, for example, a torus with one boundary component. The associated moduli spaceM^comb_1,1 (L)has a single top-dimensional cell given by

(`A,`B,`C)∈R³₊`A+`B+`C= ^L₂ Z6.

HereZ₃⊂Z₆is cyclically permuting the three components, whileZ₂⊂Z₆is the elliptic involution stabilising every point. Moreover, the Kontsevich measure on such cell is dµK=d`Ad`B. We will see that

B^comb_1,1 (`A,`B,`C) = L 2

1

(`_A+`_B)(`_B+`_C)(`_C+`_A), and after integration

ˆ

M^comb1,1 (L)

B_1,1^combs

dµK= L^1−s 3

ˆ

(0,1)²

dxdy(1+y)^3(s−1)y^1−s(1−y²x²)^−s. In particular, we find integrability if and only ifs <2 and

ˆ

M^comb1,1 (L)

B_1,1^combdµ_K= π² 24, which is in agreement with the Masur–Veech volume MV1,1= ^2π₃².

More generally, the explicit description ofB_Σ^comballows us to characterise integrability, which surprisingly depends on the topology ofΣ.

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Theorem 1.2. For anyL∈R₊ⁿ, the functionB_Σ^combis continuous and proper onTΣ^comb(L). It induces onM^combg,n (L) a functionB^comb_g,n whoses-th power is integrable if and only ifs < s^∗_g,n, where assuming thatLis non-resonant according to Definition3.1:

s^∗_g,n=











+∞ ifg=0andn=3,

2 ifg=0andn∈{4, 5}, org=1andn=1, 4

3+ 2 3

1

bn/2c−2 ifg=0andn>6, 4

3 ifg=1andn>2,

1+ 1

3(2g−3) ifg>2andn=1,

1+ 1

3(2g−1) ifg>2andn>2.

Note that genericLare non-resonant. Note that the(0, 3)case is trivial, sinceM^comb_0,3 (L)is a point. The cases (0, 4),(0, 5), and(g, 1)forg>1 are also special. The general case in genus 0 and in genusg>1 are covered by the last two lines, and they constitute the central result of the article. It is proved in Section3, with three main ingredients:

• a study of the geometry of the cells in the combinatorial moduli space (Section3.1);

• an independent characterization of integrability for inverse powers of products of linear forms with positive coefficients via convex geometry (AppendixA);

• the identification of the regions of worst divergence in the integrals of (B_g,n^comb)^s, which reduce to questions involving the combinatorics of ribbon graphs and their subgraphs (Section3.4).

The origin of the difference in integrability between the two settings can be explained as follows. In the hyperbolic case,BΣis bounded from above by the product of inverse of lengths of short curves [22, Propo- sition 3.6]. By the collar lemma such curves cannot intersect each other, so we can include them in a pair of pants decomposition. This is sufficient to show thatB_g,n^s is integrable fors <2. The integrability fors=2 is proved via a finer upper bound in [6]. In the combinatorial case, there is a similar bound but no collar lemma, so there can be more short curves and this results in less integrability.

1.3 Consequences for hyperbolic surfaces with large boundaries

The two Teichm ¨uller spaces do not just sit apart from each other. From the works of Penner [26], Bowditch–

Epstein [7] and Luo [19] on the spine construction, there is a Mod^∂Σ-equivariant homeomorphism between the Teichm ¨uller spaceTΣand its combinatorial counterpart

sp:TΣ(L)−→T^combΣ (L), L∈Rⁿ₊. The rescaling flow acts forβ >0 by takingσ∈TΣ(L)and sending it to

σ^β = (sp⁻¹◦ρβ◦sp)(σ)∈TΣ(βL),

whereρ_β: T^comb_Σ (L)→ T_Σ^comb(βL)is the operation of dilating the metric on the ribbon graph by a factorβ. In many ways [4,10,19,23], the asymptotic geometry of hyperbolic surfaces with metricσ^βwhenβ→∞is described by the combinatorial geometry sp(σ)∈T_Σ^comb. In particular, [23] proves that the Weil–Petersson measure onTΣ(βL)converges to the Kontsevich measure onT_Σ^comb(L), meaning that the Jacobian

Jacβ= 1 β^6g−6+2n

(sp⁻¹◦ρβ)^∗dµ_WP dµ_K

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converges pointwise onT^combΣ (L)to 1.

The non-integrability of(B^comb_g,n )^simplies an anomalous scaling of the integral ofB_g,n^s over the moduli space of bordered Riemann surfaces when the boundary lengths tend to+∞. Indeed, the combinatorial function describes the large time limit of the hyperbolic one under the rescaling flow, that is

β→lim∞β^6g−6+2n(sp⁻¹◦ρβ)^∗BΣ =B^comb_Σ (1.2)

uniformly on compacts ofTΣ^comb. But, by change of variable, we have for anyL∈R₊ⁿ β(6g−6+2n)(1−s)

ˆ

Mg,n(βL)

B_g,n^s dµ_WP= ˆ

M^combg,n(L)

Jacβ·β^6g−6+2n (sp⁻¹◦ρβ)^∗B_g,n^s dµ_K. Then, the Fatou lemma and the pointwise convergence of the integrand asβ→+∞imply that

ˆ

M^combg,n(L)

(B^comb_g,n )^sdµK6lim inf

β→∞ β(6g−6+2n)(1−s)

ˆ

Mg,n(βL)

B^s_g,ndµWP. (1.3) Theorem1.2then implies

Corollary 1.3. Fors>s^∗_g,n, we have for anyL∈R_>0ⁿ :

β→lim∞β(6g−6+2n)(1−s)

ˆ

Mg,n(βL)

B_g,n^s dµWP= +∞.

It would be interesting to obtain an asymptotic equivalent of this integral for all values ofs. Whens <

s^∗_g,n, we cannot currently conclude whether there is equality in (1.3). This could be proved by dominated convergence only if one could describe a sufficiently integrable and uniform bound for the Jacobian Jacβ

overT_Σ^comb. This would requires careful estimates in the arguments by which the convergence of the Weil–

Petersson Poisson structure to the Kontsevich Poisson structure were proved in [23], which we do not currently have.

Fors=1, we already mentioned in (1.1) that:

β→lim∞

ˆ

Mg,n(βL)

Bg,ndµ_WP= ˆ

Mg,n(L)

Bg,ndµ_WP= ˆ

M^combg,n(L)

B^comb_g,n dµ_K.

which is shown in [4] by a direct evaluation of the integrals. It would be more satisfactory if the equality could be proved using the convergence property stated in (1.2).

In AppendixB.2, we discuss various discretisations of´

M^combg,n(L)(B_g,n^comb)^sdµKwhich can be naturally defined using the piecewise-linear integral structures on MFΣ and on T^comb_Σ . They lead to interesting arithmetic questions and give another possible way to study the behaviour of multicurve counting on surfaces with large boundaries.

1.4 Organisation of the paper

The paper is organised as follows. Section2, where we first recall definitions and facts about the combinatorial Teichm ¨uller spaceT^combΣ , and the description of the volume of the unit ball of measured foliations through the statistics of length of multicurves are recalled in Subsection2.1. Subsection2.2shows how the combinatorial structures inTΣ^comballows the parametrisation of the set of measured foliations MFΣand makes explicit the polytopal structure of the latter. Building on this parametrisation of MFΣ, Subsection2.3 is dedicated to the explicit description of the volume of the unit ballB_Σ^combin terms of rational functions.

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This is the content of Proposition2.7. As a direct application of the proposition, and as a preliminary result for the rest of the paper, the integrability of B_1,1^comb^s

is then extensively studied in Subsection2.4.

Section3is dedicated to the proof of the main result of the paper — Theorem1.2. As a preliminary study, we start with Subsection3.1by giving a precise characterisation of the vertices of the cells of the combinatorial Teichm ¨uller space. Then, in Subsection3.2, we state the propositions that lead to the main result:

Proposition3.7turns the study of integrability of B_1,1^combs

into a local integrability result; and Proposition 3.8identifies the range of integrability asgandnvary. Those propositions are proved in Subsections3.3 and3.4respectively.

The paper is supplemented with 2 appendices: the theorem of AppendixAis used in the course of the proof of Proposition3.7in subsection3.3; AppendixBdeals with the discrete approach of the integrability, coming from the integral structure ofT^comb_Σ .

Acknowledgments

We thank Don Zagier for a remark on the apparition of truncations ofζ(2)in relation with dilogarithms.

This work benefited from the support of the Max-Planck-Gesellschaft. It has been supported in part by the ERC-SyG project, Recursive and Exact New Quantum Theory (ReNewQuantum) which received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 810573. It has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. ERC-2016-STG 716083 “CombiTop”).

2 Counting multicurves

2.1 Combinatorial geometry background

Let us recall some facts about the combinatorial moduli space and the combinatorial Teichm ¨uller space (we refer to [4] for further readings).

The combinatorial moduli space. Aribbon graphis a finite graphG together with a cyclic order of the edges at each vertex. Replacing edges by oriented closed ribbons and glueing them at each vertex according to the cyclic order, we obtain a topological, oriented, compact surface|G|, called the geometric realisation of G, with the graph embedded into it and onto which the surface retracts. Thenboundary components of|G| are also calledfaces, and we always assume they are labelled as∂₁G, . . . ,∂_nG. We denote byV_G,E_G,F_Gthe set of vertices, edges and faces respectively. For connected ribbon graphs, we define the genusg>0 of the ribbon graph to be the genus of|G|, and so #VG−#EG+#FG=2−2g. The datum(g,n)is called the type ofG. A ribbon graph isreducedif its vertices have valency>3. We denote byRg,n the set of reduced and connected ribbon graphs of type(g,n), and byR^trivg,n its subset consisting of trivalent ribbon graphs only.

For 2g−2+n >0, these sets are non-empty and finite. Non-reduced or non-connected ribbon graphs will only appear in Sections3.2-3.4.

Ametric ribbon graphGis the data of a ribbon graphG, together with the assignment of a positive real number for each edge, that is`_G∈R^E₊^G. Notice that, for a pointG∈M^comb_g,n (L)and any non-trivial edgepath γ, we can define its length`_G(γ)∈R+as the sum of the length of edges (with multiplicity) whichγtravels along. In particular, we can talk about length of the boundary components`G(∂_iG)of the ribbon graph, and for a fixedL∈Rⁿ₊we define the polytope

Z_G(L) =

`∈R^E₊^G `(∂_iG) =L_i

⊂R^E₊^G.

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•

∂₁G

∂₂G ∂₃G

∂₄G

• •

•

∂₁Σ

∂₂Σ

∂₃Σ

∂₄Σ

• •

•

∂₁Σ

∂₂Σ

∂₃Σ

∂₄Σ Figure 1: A ribbon graphGof type(0, 4), and two embedded ribbon graphs[G,f]and[G,f⁰]on a sphere with 4 boundary componentsΣ, with the same underlying graphGbut different embeddings.

It has dimension #EG−n. The automorphism group ofGis acting onZG(L), and we define themoduli space of metric ribbon graphs as

M^comb_g,n (L) = [

G∈Rg,n

ZG(L) Aut(G),

where the orbicellsZG(L)/Aut(G)are glued together through degeneration of edges. This endowsM^combg,n (L) with the structure of a polytopal orbicomplex of dimension 6g−6+2n, parametrising metric ribbon graphs of genusgwithnboundary components of lengthL∈Rⁿ₊. Note that the top-dimensional cells correspond to trivalent ribbon graphs.

The combinatorial Teichm ¨uller space. Fix now a smooth connected oriented stable surfaceΣof genus g > 0 withn > 0 labelled boundaries, denoted∂₁Σ, . . . ,∂_nΣ. Anembedded ribbon graphonΣis the data [G,f]of an isotopy class of proper embeddingf: G ,→ Σof a ribbon graphG inΣonto whichΣretracts, respecting the labelling of the boundary components. As a consequence of the retraction condition,Ghas the same genus and number of boundary components asΣ. We denote byERΣthe set of embedded ribbon graph onΣ. The pure mapping class group ofΣacts onERΣ, and the quotientERΣ/Mod^∂Σ is in natural bijection withRg,n.

Anembedded metric ribbon graphGonΣis the data[G,f]of an embedded ribbon graph onΣ, together with the assignment of a positive real number for each edge:`_G∈R₊^E^G. The polytopes

Z_G(L) =

`∈R^E₊^G `(∂_iG) =L_i

⊂R₊^E^G

parametrise metrics on[G,f]with boundary perimetersL∈Rⁿ₊, and we define the combinatorial Teichmller space ofΣas

TΣ^comb(L) = [

[G,f]∈ERΣ

ZG(L),

where the cells are glued together through degeneration of embedded edges. This endowsT_Σ^comb(L)with the structure of a polytopal complex of dimension 6g−6+2n, parametrising embedded metric ribbon graphs onΣwith boundary components of lengthsL ∈ R₊ⁿ. The pure mapping class group ofΣacts on T^comb_Σ (L), and we have a natural isomorphismT^comb_Σ (L)/Mod^∂Σ=∼ M^comb_g,n (L).

Integrating functions. In [18] Kontsevich defined a 2-formω_Kon the moduli spaceM^comb_g,n (L)that is symplectic on the top-dimensional stratum. The associated symplectic volume form defines a measureµ_Kon

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M^combg,n (L). In particular, for every measurable functionf: M^combg,n (L)→R, we can consider its integral against the Kontsevich measure, defined as

ˆ

M^combg,n(L)

fdµK= X

G∈R^trivg,n

1

#Aut(G) ˆ

Z_G(L)

fdµK. (2.1)

Here, by abuse of notation, we denoted with the same symbols objects on the orbicellsZG(L)/Aut(G)and on the unfolded cellsZG(L).

Combinatorial length of curves. IfG ∈T^comb_Σ (L), the homotopy classγof a simple closed curve admits a unique non-backtracking edgepath representative on the graph underlying G, and we can define the length`_G(γ)as the length of this representative.T^comb_Σ can also be described in terms of measured foliations transverse to∂Σ, and this notion of length coincides with the intersection number ofγwith the measured foliation associated toG. More generally, we can talk about the length with respect toGof any multicurve c∈MΣby adding lengths of the components ofc. We can then introduce the function:

B_Σ^comb(G) = lim

r→∞

#{c∈MΣ|`_G(c)6r}

r^6g−6+2n .

Its basic properties have been studied in [4].

Proposition 2.1. [4] For anyL∈R₊ⁿ, the functionB_Σ^combtakes values inR+, is continuous onT^combΣ (L), and the induced functionB^comb_g,n onM^combg,n (L)is integrable with respect toµ_K.

2.2 Parametrisation of measured foliations

In this paragraph, we shall describe a parametrisation of the space of measured foliations MFΣthat depends on a chosen embedded ribbon graph[G,f]. It is dual to the parametrisation of [25] — which considers triangulations instead of ribbon graphs. This will allow us to effectively describe the functionB_g,n^combon the orbicell of the moduli spaceM^comb_g,n (L)determined by the ribbon graphG.

In what follows, it is useful to introduce a larger space MF^•Σof measured foliations, where now∂Σcan be a union of smooth and singular leaves (and we still include the empty foliation). It is a piecewise linear man- ifold of dimension 6g−6+3n, with a piecewise integral structure whose integral points are the multicurves M^•_ΣonΣwhere the components are allowed to be homotopic to boundary components. In particular, we can consider the associated Thurston measureµ^•_Thby lattice point count, and the function

B^comb,•_Σ (G) =µ^•_Th { F∈MF^•Σ|`_G(F)61}

, G∈TΣ^comb(L). We have a homeomorphism

Φ: MFΣ×Rⁿ_>₀−−→⁼^∼ MF^•_Σ,

that also respects the piecewise linear structure:Φ(MΣ×Zⁿ_>0) =M^•_Σ. Thus, it respects the measures, when MF^•Σand MFΣare equipped with their respective Thurston measures andRⁿ_>0with the Lebesgue measure.

We also notice that MFΣandRⁿ_>0naturally sit inside MF^•ΣasΦ(·, 0)andΦ(∅,·)respectively.

There is an elementary relation between the enumeration of multicurves with or without components homotopic to boundaries.

Lemma 2.2. For anyG∈T_Σ^comb(L), we have

B_Σ^comb,•(G) = (6g−6+2n)! (6g−6+3n)!

B_Σ^comb(G) Qn

i=1Li

.

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Proof. Since`_Gis homogeneous and additive under disjoint union of multicurves, we have

∀(F,x)∈MFΣ×R_>0ⁿ , `_G(F) +`_G(x) =`_G(Φ(F,x)), with`_G(x) = Xn i=1

x_iL_i. Therefore, using homogeneity of the Thurston and Lebesgue measure, we find

B_Σ^comb,•(G) = ˆ ₁

0

dt µTh { F |`_G(F)6t}

·µ_Leb {x|`_G(x)61−t}

= ˆ ₁

0

dt t^6g−6+2n(1−t)ⁿ

·µ_Th { F |`_G(F)61}

·µ_Leb {x|`_G(x)61}

= n!(6g−6+2n)!

(6g−6+3n)! ·B_Σ^comb(G)· 1 n!Qn

i=1L_i

= (6g−6+2n)!

(6g−6+3n)!

B_Σ^comb(G) Qn

i=1L_i .

Remark 2.3. The above statement can be generalised to any notion of length as follows. Letl: M^•_Σ →R+

be a locally convex function, that is additive under disjoint union of multicurves. It uniquely extends to a continuous function on MF^•_Σ, and it induces a function still denotedlon MFΣ. Furthermore, we have

µ^•_Th {l61}

= (6g−6+2n)!

(6g−6+3n)!

µ_Th {l61} Qn

i=1l(∂iΣ) .

Fix now an embedded ribbon graph[G,f]inΣ. Each edgeeof the embedded graphG ,→ Σis dual to a unique — up to homotopy of proper embeddings¹– arcαebetween two (possibly the same) boundaries of Σ, and these arcs are pairwise disjoint. To a measured foliation, we associate the set of intersection numbers² with these arcs

m_[G,f]: MF^•Σ −→R^E_>0^G F 7−→ ι(F,αe)

e∈EG

.

By definition,m_[G,f]preserves the piecewise linear integral structures of MF^•ΣandR^E_>0^G.

The mapm_[G,f]gives a description of MF^•Σand MFΣ. We will show that it in fact gives a parametrisation of MF^•Σand MFΣ, after we introduce notations to describe the image.

Definition 2.4. LetGbe a ribbon graph. Asimple loopis a non-empty, closed, non-backtracking edgepath onGthat does not pass twice through the same edge. Adumbbellis a closed, non-backtracking edgepathγ onGthat passes at most twice through each edge and such that the union of edges that are visited twice

1IfXandYare topological manifolds with boundaries, a continuous mapf:X→Yis called a proper embedding iff⁻¹(∂Y) =

∂Xand we use the natural notion of homotopies among such.

2We recall that the intersection number is defined as follows (cf.[15, Section 5.3]). For a fixed isotopy class of measured foliationF inΣ, and an arcainΣbetween two boundary components (or a simple closed curve), we have the notion of measure ofa:

µ_F(a) =sup Xk

j=1

µ_F(aj)

,

wherea₁, . . . ,a_kare arcs ofa, mutually disjoint and transverse toF, and where the sup is taken over all sums of this type. Ifαis now a homotopy class of arc inΣbetween two boundary components (or a homotopy class of simple closed curve), we set

ι(F,α) = inf

a∈αµ_F(a),

where the inf is taken over representatives ofα. Such quantity is invariant under isotopy ofFand Whitehead moves.

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forms a non-empty edgepathpfor which we have a decompositionγ = γ₁·p·γ₂·p⁻¹, where γ₁ andγ₂ are simple loops. A simple loop or a dumbbell is calledessential if it does not coincide with a boundary component ofG.

If[G,f]is an embedded ribbon graph inΣ, we call (essential) simple loop or dumbbell of[G,f]the homotopy class of the image of any (essential) simple loop or dumbbell ofGviaf.

Definition 2.5. Acornerin a trivalent ribbon graphGis an ordered triple∆= (e,e⁰,e⁰⁰)wheree,e⁰,e⁰⁰are edges incident to a vertex in the cyclic order. Equivalently, a corner consists of a vertexvtogether with the choice of an incident edgee. We say that a corner belongs to a facef∈ FGife⁰ ande⁰⁰are edges around that face. We denote C(f)the set of corners belonging tofand CGthe set of all corners ofG. If we have an assignment of real numbers(xe)_e∈E_Gand∆= (e,e⁰,e⁰⁰)is a corner, we denotex∆=xe⁰+xe⁰⁰−xe. Lemma 2.6. Fix an embedded ribbon graph[G,f]inΣ, withGtrivalent. The mapm_[G,f]is a homeomorphism onto its image, which is the convex polyhedral cone

Z^•_G=

x∈R^E_>0^G ∀∆∈CG x∆>0

.

The image ofMFΣ, denotedZG, is the union ranging over the setDG={∆:FG→CG|∆(f)∈C(f)}of the convex polyhedral cones

Z_G,∆=

x∈Z^•_G

∀f∈FG x_∆(f) =0 . (2.2)

Moreover, ZG is a fan and its rays are generated by the images of essential simple loops and essential dumbbells.

When the cell is not top-dimensional, one can obtain a similar description by resolving the non-trivalent vertices of the underlying ribbon graph (in some arbitrary way) into trivalent vertices.

Proof. Letx∈R^E_>^G₀ be in the image ofm_[G,f],i.e.there existsF∈MF^•Σsuch thatm_[G,f](F) =x. For a vertexv ofG, let us denote bye,e⁰,e⁰⁰the adjacent edges, respecting the cyclic order. Then there must be a switch at vand one should specify the weights of this switch. These are three numbersye,ye⁰,ye⁰⁰∈R_>0such that

xe=ye⁰+ye⁰⁰, xe⁰ =ye+ye⁰⁰, xe⁰⁰=ye+ye⁰.

This linear system of equations admits a solution in non-negative real numbers if and only if the three corners conditions are satisfied, namely

xe6xe⁰+xe⁰⁰, xe⁰ 6xe⁰⁰+xe, xe⁰⁰6xe+xe⁰. When the solution exists, it is unique and given by the formulas

ye= x∆

2 , x∆=xe⁰+xe⁰⁰−xefor each corner∆= (e,e⁰,e⁰⁰).

This gives the first part of the lemma. By definition, a measured foliationF ∈MF^•Σbelongs to MFΣif and only if none of its leaves is homotopic to a boundary component of Σ. This is the case when there is a stop around each facef,i.e. if and only if there exists a corner∆ = (e,e⁰,e⁰⁰)aroundfsuch thatye = 0, or equivalentlyx∆ = 0. This justifies (2.2), which is written as a finite union of convex polyhedral cones indexed by the location of the stops,i.e.maps∆:FG→CGsuch that∆(f)∈C(f), and one easily checks it is a fan.

The identification of the rays essentially follows from [24, Proof of Proposition 3.11.3]. For the reader’s convenience, we spell out the argument. Assume thatm_[G,f](F) = xbelongs to a ray ofZ_G,∆. We callσ the support ofF,i.e. the set of edges ofGwhose intersection withFis positive. By following the leaves of F, we conclude thatσis a union of closed curves onG. Moreover,σis connected, for otherwise we could writexas a non-trivial sum over the connected components contradicting thatxbelongs to a ray.

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•

• •

∂₁G

∂₂G ∂₃G

∂₄G

•

◦

•

∂₁Σ ∂₃Σ

∂₂Σ ∂₄Σ

•

• •

•

• •

•

• •

•

• •

••

◦◦

•• ••

◦◦

•• ••

◦◦

•• ••

◦◦

••

Figure 2: A ribbon graphGand an embedded ribbon graph[G,f]on a sphere with 4 boundary components Σ, and all essential simple loops and dumbbells on them.

Choose arbitrarily an orientation onσ. We claim that σpasses through each edge at most once in each direction. If this were not the case, one could choose an origin onσso that it takes the formσ= a·e·b·e whereaandbare non-empty paths. Then,σ₁=a·eandσ₂=b·eare closed curves, and there is a natural decomposition of the weights ofFinto two measured foliationsF1,F2with respective supportsσ₁,σ₂such thatx= m_[G,f](F1) + m_[G,f](F2)contradicting thatxbelongs to a ray.

e e a

b

=

e e

a + b

SinceGis trivalent, ifσpasses through each edge at most once (in any direction), it must be an essential simple loop. Now assume thatσpasses through certain edges in both directions. Ifeis an oriented edge, we use the notation ¯e for the edge with opposite orientation. Ifσwere not an essential dumbbell, there would exist oriented edgese 6= e⁰ with e 6= e¯⁰, and paths a,b,c,d such that one of the following cases holds.

• σ =a·e·b·e⁰·c·e¯·d·e¯⁰. Then, there exists a natural decompositionx= m_[G,f](F1) + m_[G,f](F2)with measured foliationsF1,F2of respective supportsσ₁=a·e·c¯·e¯⁰andσ₂=d·e¯⁰·b·e.¯

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= + e

¯ e

e⁰

¯ e⁰ b

d

a c

e e¯⁰

a c

¯

e e¯⁰

b

d

• σ = a·e·b·e¯·c·e⁰·d·e¯⁰ wherebanddare non-empty. Then, there exists a natural decomposition x= m_[G,f](F1) + m_[G,f](F2)with measured foliationsF1,F2of respective supportsσ₁=a·e·b·e·¯a·e¯ ⁰·d·e¯⁰ andσ₂ =c·e·b·¯ e·c·e¯ ⁰·d·e¯⁰.

a b

c d e¯⁰

e⁰

e

¯ e

a b

d

¯

a c¯

c

b e d

¯ e

¯

e⁰ e¯⁰

¯ e e e⁰

e⁰

= +

In both cases this contradicts the assumption thatxbelongs to a ray.

2.3 Volume of combinatorial unit balls

IfG ∈TΣ^comb, the description in Lemma2.6reduces the computation of the Thurston measure of the combinatorial unit ball{`_G61}to the computation of volumes of truncations of polyhedral cones. This can be carried out explicitly on a computer, but at a qualitative level, the result always takes the following form.

LetGbe a trivalent ribbon graph on a surfaceΣof type(g,n). We recall thatGinduces a decomposition of the space of measured foliations MFΣ into polyhedral conesZ_G,∆ where∆ : FG → CGis a choice of a corner in each face, and their union over ∆is denoted ZG. Anelementary simplex of ZG is a cone of dimension 6g−6+2ninZGwhose extremal rays are linearly independent inR^E^Gand are either essential simple loops or essential dumbbells. Asimplicial decompositionofZGis a collectionTGof simplicial cones with disjoint interior and whose union isZG. Each simplicial conet ∈ TGhas 6g−6+2nextremal rays generated by an essential simple loop or dumbbell. We denoteR(t)⊂R^E_>0^Gthis set of generators. We define det(t)to be the volume with respect to the Thurston measureµ_Thof the simplex issued from the origin and sides beingR(t). The number det(t)is a positive integer and is also the number of integral point in the semi-open simplex.

Proposition 2.7. Let G be a trivalent ribbon graph of type (g,n). For anyG ∈ ZG(L), that is any metric on the underlying graphG, B_g,n^comb(G) is a rational function of the edge lengths. More precisely, for any simplicial decompositionTGofZGwe have

B_g,n^comb(G) = 1 (6g−6+2n)!

X

t∈TG

1 det(t)·Q

ρ∈R(t)`G(ρ). Proof. By definition of a simplicial decomposition:B_Σ^comb(G) =P

t∈TGµ_Th(t∩{`G61}). From the definition of the Thurston measure

µ_Th(t∩{`G61}) = lim

r→+∞

#

x∈t∩Z^E_>0^G ^P_e∈E_Gxe`_G(e)6r

r^6g−6+2n

= 1

det(t) lim

r→+∞

#

z∈Z^R(t)_>0 ^P_ρ∈R(t)zρ`_G(ρ)6r

r^6g−6+2n

= 1

det(t)

1 (6g−6+2n)!Q

ρ∈R(t)`_G(ρ).

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•

`_B `_C

`A

•

ρ₁ •

•

ρ₂

Figure 3: The top-dimensional cell ofM^comb_1,1 (L)parametrised by edge lengths(`_A,`_B,`_C), together with two essential simple loopsρ₁andρ₂.

Remark 2.8. Proposition2.7extends to graphsGwith higher valencies by choosing any resolution into a trivalent graph with some edges of zero length.

2.4 How to use the formula: the ( 1, 1 ) case.

There is a single trivalent ribbon graphGof genus 1 with one boundary component. For a fixedL∈ R+, the associated polytope is simply

Z_G(L) =

(`_A,`_B,`_C)∈R³₊`_A+`_B+`_C= ^L₂ .

The automorphism group ofGisZ6, where the subgroupZ3⊂Z6is cyclically permuting the three edges, whileZ2 ⊂ Z6 is the elliptic involution stabilising every point and is the automorphism group ofG for which the lengths of the edges are not equal.

Ghas a unique facefand six corners; from the elliptic involution acting onG,B_1,1^combreduces to the sum of three contributions. The first one corresponds to the corner∆(f) = (A,B,C). The polytope Z_G,∆ is a simplicial cone, with extremal raysρ₁ = (1, 1, 0)and ρ₂ = (1, 0, 1)corresponding to the essential simple loops of Figure3, and with determinant 1. The two contributions are obtained by cyclic permutation of the role of(A,B,C). For a pointG= (À,`B,`C)∈ ZG(L), we find`G(ρ₁) = À+`B,`G(ρ₂) = À+`C, and det(t) =1. Similarly for the other polyhedral cones, so that

B^comb_1,1 (`A,`B,`C) = 1 2

1

(`A+`B)(`A+`C)+1 2

1

(`A+`B)(`B+`C)+1 2

1

(`A+`C)(`B+`C)

= L 2

1

(`_A+`_B)(`_B+`_C)(`_C+`_A).

(2.3)

Besides

B_1,1^comb,•(`A,`B,`C) = ˆ

R+³

dxAdxBdxC1x_A(`_B+`_C)+x_B(`_C+`_A)+x_C(`_A+`_B)61

= 1

6(`_A+`_B)(`_B+`_C)(`_C+`_A) = 2!

3!

B_1,1^comb(`A,`B,`C) L as expected from Lemma2.2.

Let us now integrate over the moduli space (see Equation (2.1)). We recall that #Aut(G) = 6, and the Kontsevich measure onZG(L)is dµ_K=d`Ad`B. Expressing`C = ^L₂ −`A−`Band performing the change

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of variable(`A,`B) = ^L₂(a,b), we can compute ˆ

M^comb1,1 (L)

B_1,1^combdµK= 1 6

ˆ

0<`A,`B<L/2

`_A+`_B<L/2

L 2

d`Ad`B

(`_A+`_B)(^L₂ −`_A)(^L₂ −`_B)

= 1 6

ˆ

0<a,b<1 a+b<1

dadb (a+b)(1−a)(1−b)

= −1 3

ˆ 1 0

ln(a) 1−a² da

= Li2(1) −Li2(−1)

6 = π²

24. As expected from (1.1), this value coincides with´

M1,1(L)B1,1dµ_WP= ^π₂₄² founde.g.in [3].

Let us look at the integral of thes-th power fors >1 ˆ

M^comb1,1 (L)

B_1,1^combdµ_K= (L/2)^1−s

6 B(s), B(s) :=

ˆ

a,b>0 a+b61

dadb

(a+b)(1−a)(1−b)^s. By elementary means we shall prove that it is finite if and only ifs <2, and more precisely Proposition 2.9. We haveB(s)∼ _2−s³ whens→2⁻.

Proof. LetD={(a,b)∈R²_>0 _|a+b61}be the 2-simplex. Ifs=2, we shall see that the non-integrability comes from the divergence of the integrand at the vertices of D, i.e. (a,b) = (0, 0), (a,b) = (1, 0) and (a,b) = (0, 1). We decompose the domain of integration, introducing

D₀₀=

(a,b)∈D

a+b6 ¹₂ , D₁₀=

(a,b)∈D

a> ¹₂ , D₀₁=

(a,b)∈D

b>¹₂ , and ˜D =D\ D₀₀∪D₁₀∪D₁₁

. We analyse separately the contributions of these domains to the integral, with obvious notations:

B(s) =B₀₀(s) +B₁₀(s) +B₀₁(s) +B˜(s).

The integrand being a continuous function on ˜D, ˜B(s)remains bounded whens → 2. For the first three contributions, the idea is to choose coordinates transforming the domain into a square and which include a coordinatec measuring the distance to the vertex, then split the integrand into a contribution coming solely from the vanishing factor in the denominator, and a remainder which will remain bounded whens approaches 2.

We start withB00(s). With the change of variable(c,u) = (a+b,_a+b^a ), we find:

B₀₀(s) = ˆ ¹

2

0

dc c^1−s ˆ ₁

0

du

(1−cu)(1−c+cu)s

= ˆ ¹₂

0

dc c^1−s+ ˆ ¹₂

0

dc c^1−s ˆ 1

0

du 1

(1−cu)(1−c+cu)s −1

!

= (1/2)^2−s 2−s +

ˆ ¹₂

0

dc c^1−sO(c)

s→2= 1

2−s+O(1),

where theO(c)is uniform forc∈[0,¹₂]ands∈(0, 2), and we observed(¹₂)^2−s=1+O(2−s)whens→2.

ForB₁₀(s), we perform the change of variable(c,u) = 1−a,_1−a^b

and get B10(s) =

ˆ ¹

2

0

dc c^1−s ˆ ₁

0

du

(1−cu)(1−c+cu)s =B00(s).

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Exchanging the role ofaandbwe also haveB₀₁(s) =B₀₀(s), hence the result.

There is no simple expression forB(s), but the expression can be transformed in various ways. For instance, with the change of variable(c,v) = (a+b,_a+b^a )sending(a,b)∈Dto(c,v)∈(0, 1)²:

B(s) = ˆ ₁

0

cdc c(1−c)s

ˆ ₁

0

dv

1+_1−c^c² v(1−v)s.

By symmetryv 7→1−v, we can restrict the integration tov∈ [0,¹₂]while multiplying the result by 2. We then sety= _2−c^c andx=1−2v, obtaining

B(s) =2^2−s ˆ

(0,1)²

dxdy(1+y)^3(s−1)y^1−s(1−y²x²)^−s as announced in the introduction.

Proposition2.9tells us that the behaviour ofB^comb_1,1 already deviates from the one ofB1,1, as the latter has a finite square-norm for the Weil–Petersson measure. This simple example shows thatB_g,n^combhas non-trivial integrability properties. The purpose of the next section is to analyse them systematically.

3 Integrability of B

_Σ^comb

3.1 Geometry of the cells in T

^combΣ

(L)

As a preparation, we study the geometry of the cellsZG(L)ofM^combΣ (L), and in particular we shall characterise the tangent cone at the vertices of the cells.

Definition 3.1. We say thatL ∈ Rⁿ₊ isnon-resonantif for any non-zero map: {1, . . . ,n} → {−1, 0, 1}, we

have Xn

i=1

_iL_i6=0.

Definition 3.2. Let Gbe a trivalent ribbon graph withnboundary components and letS ⊆ E_G. We let G^∗_S the subgraph of the dual graphG^∗in which we keep only the duals of edges fromS. We call a subset S⊆EGasupport set ofGif

• it hasnelements,

• each face ofGcontains at least an edge inS,

• each connected component ofG^∗_Scontains a unique cycle which has odd length.

Definition 3.3. LetGbe a trivalent ribbon graph. ForL ∈ R₊ⁿ andλa point of the cell closureZG(L)we define

E[λ] :={e∈EG|λe=0}. Lemma 3.4. LetGbe a trivalent ribbon graph of genusgwithnfaces.

(A) LetL∈R₊ⁿbe non-resonant andZ_G(L)be a top-dimensional cell of the combinatorial moduli spaceM^comb_g,n (L). Ifλ= (λ_e)_e∈E_G is a vertex of the cell closureZ_G(L)⊂R^E₊^G, thenE\E[λ]is a support set.

(B) Conversely, letS⊂E_Ga support set forG. Then there exists a non-resonantL∈Rⁿ₊and a vertexλof the cell closureZ_G(L)such thatS=E\E[λ].

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Lemma 3.5. LetL ∈ R₊ⁿ be non-resonant andZG(L)be a top-dimensional cell of the combinatorial moduli space M^combg,n (L). Then the tangent cones at any vertex of the cell closureZG(L)are simplicial. Furthermore, at a given vertexλthe raysr^(e)of the tangent cone are indexed by the edgese∈E[λ]in such a way that

∀e⁰ ∈E[λ], r^(e)_e0 =δ_e,e⁰.

Proof of Lemma3.4. The closure of the polytope is determined by inequalities`e>0 for eache ∈EGandn

equalities of the form X

e∈E⁽ⁱ⁾_G

a_i,e`_e=L_i, i∈{1, . . . ,n},

whereE⁽ⁱ⁾_G is the set of edges around thei-th face anda_i,e∈{1, 2}is the multiplicity of the edgeearound this face. Now, for an arbitraryS ⊆ EG, consider the inhomogeneous linear system of equations in the variables(`e)e∈E_G

`e=0 fore∈EG\S, P

e∈E⁽ⁱ⁾_G a_i,e`e=Li fori∈{1, . . . ,n}. (3.1) We claim that

1. the system (3.1) is invertible in(`e)e∈E_Gif and only ifSis a support set,

2. ifL_iis non-resonant andSis a support set then the solution of the system is such that`_e>0 fore∈S. Let us prove the first claim. The matrix associated to the family of equationsP

e∈E⁽ⁱ⁾_G a_i,e`e = Liis the incidence matrix of the graphG^∗_S. In order for the incidence matrix to be invertible there must be as many edges as vertices in each connected component of G^∗_S, hence a unique cycle. Next, degree one vertices does not play any role in the invertibility (the edge length`_eadjacent to a the vertex dual to thei-th face must be set to`_e = L_i). Hence one can get rid of the tree part of the graph. Finally the incidence matrix of a cycle is invertible if and only if it has odd length. Indeed if the cycle is even then the alternating vector (1,−1, 1,−1, . . . , 1,−1)belongs to the kernel. Whereas if the cycle is odd, the alternating vector (1,−1, 1,−1, . . . , 1)is mapped to twice a basis vector and the matrix is invertible by cyclic symmetry. This concludes the proof thatSmust be a support set.

Now let us prove the second claim. Let(Li)_i∈_{_1,...,n_} ∈ Rⁿ and (`e)e∈E_G be the corresponding solution in (3.1). Assume that fore₀ ∈Swe have`e0 =0. ThenG^∗_S_\{_e

0} contains at least one tree component. LetS⁰ be the vertices of a tree component ofG^∗_S_\{_e

0}andS⁰=S₁⁰tS₂⁰ a bipartition ofS⁰(i.e.vertices inS₁⁰are only adjacent toS₂⁰). ThenP

i∈S₁⁰Li=P

i∈S₂⁰Liand henceLiis resonant. This concludes the proof of the second claim.

We turn to the proof of the first part (A) of the lemma. Assume thatλis a vertex andS:={e∈E_G|λ_e>0}is such that the system (3.1) admits a unique solution. Necessarily #S6n. IfSis not contained in a support set then the graphG^∗_S contains an even cycle and the solution of (3.1) is not unique. Let us suppose by contradiction that #S < nand letS⁰ ⊃Sbe a support set. Thenλis a solution of the system (3.1) with the subset of edgesS⁰. It contradicts our second claim that states thatλ_ewould be positive for alle∈S⁰. For the converse — part (B) of the lemma — pick a support set and a positive vector(è)_e∈S. Because the system is bijective there is no further inequalityè>0 that can be set to an equalityè=0. In other words, completing the vector(è)_e∈Swith zeros, we obtain a vertex. Now if the positive values are generic enough the associated face lengthsLiare non-resonant.

Proof of Lemma3.5. LetLibe non-resonant. Letλ= (λe)e∈E_G be a vertex ofZG(L)andS[λ] ={e∈EG|λe>

0}. By Lemma3.4Sis a support set. The invertibility of the homogeneous linear system underlying (3.1) shows that the projection map from the tangent space

TλZG(L) =

n

\

i=1

`∈R^E^G ^X

e∈E⁽ⁱ⁾_G

a_i,e`e=0