GEOMETRY OF PERIODIC REGIONS ON FLAT SURFACES AND ASSOCIATED SIEGEL-VEECH CONSTANTS MAX

Keywords: flat surfaces, moduli spaces, abelian differentials, configurations, saddle connections, Siegel-Veech constants

An Abelian differential gives rise to a flat structure (translation surface) on the underlying Riemann surface. In some directions the directional flow on the flat surface may contain a periodic region that is made up of maximal cylinders filled by parallel geodesics of the same length. The growth rate of the number of such regions counted with weights, as a function of the length, is quadratic with a coefficient, called Siegel-Veech constant, that is shared by almost all translation surfaces in the ambient stratum.

We evaluate various Siegel-Veech constants associated to the geometry of configurations of periodic cylinders and their area, and study extremal properties of such configurations in a fixed stratum and in all strata of a fixed genus.

INTRODUCTION

1.1. Statement of some known results. Suppose that M g,m is a closed connected oriented surface M g of genus g with a set of m labelled marked points Σ = {P 1 , . . . , P m }.

By a translation surface S we mean a flat Riemannian metric and a parallel vector field on M g,m . The metric has cone type singularities at all of the points P i of Σ where the total angle is of the form 2π(d i + 1) for some integer d i ≥ 1.

Each geodesic on a translation surface moves in a constant direction, so geodesics do not have self intersections and a regular geodesic that connects a non-singular point to itself comes back with the same angle, so it is a periodic geodesic. A periodic geodesic is always part of a maximal connected periodic region: a maximal cylinder of parallel periodic geodesics of the same length. We refer to such a maximal cylinder as a periodic cylinder or, for short, a cylinder, and we say that the common length of the periodic geodesics that make up the cylinder is the width of the cylinder. The number N cyl (S, C , L) of cylinders of width less than L grows like cπL 2 : a first fundamental result of Masur [START_REF] Masur | Lower Bounds for the number of saddle connections and closed trajectories of a quadratic differential[END_REF][START_REF] Masur | The growth rate of trajectories of a quadratic differential[END_REF] states that there exist two positive constants c 1 and c 2 such that c 1 πL 2 ≤ N cyl (S, C , L) ≤ c 2 πL 2 . Using this fact, Eskin and Masur [START_REF] Eskin | Asymptotic formulas on flat surfaces[END_REF] prove the deep theorem that for almost every surface in a stratum there is an exact asymptotics N cyl (S, C , L) ∼ cπL 2 . The constant c is called a Siegel-Veech constant. The main tool to study Siegel-Veech constants is the method of Veech [START_REF] Veech | Siegel Measures[END_REF] that also showed the exact asymptotics in a more general setting, but for a weaker form of convergence.

The main object of this paper is the computation of various Siegel-Veech constants related to counting periodic regions in different ways. Siegel-Veech constants are particularly interesting because they appear in various fields. In arithmetic, they give the asymptotics of the number of primitive points in certain lattices. In dynamics they are related to the sum of Lyapunov exponents for the Teichmüller geodesic flow in any invariant suborbifold of a stratum of Abelian differentials by the main formula of [START_REF] Eskin | Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow[END_REF]. For a Teichmüller curve, they have an algebro-geometric interpretation involving degrees of certain line bundles given by a formula of Bouw-Möller [START_REF] Bouw | Teichmüller curves, triangle groups, and Lyapunov exponents[END_REF]. Furthermore they are related to slopes of effective divisors and intersection theory in the moduli space of curves: this aspect is studied by Chen (see [START_REF] Chen | Square-tiled surfaces and rigid curves on moduli spaces[END_REF] for example) and Chen-Möller (see [START_REF] Chen | Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus[END_REF] for example).

By identifying R 2 and C, a translation surface inherits from C a complex structure on M g and a holomorphic one form (Abelian differential) ω. A zero of ω of order d corresponds to a conical singularity of angle 2π(d + 1). There is a one to one correspondence between translation surfaces endowed with a distinguished direction and Abelian differentials. If we denote by α = (d 1 , . . . , d m ) the orders of the zeros of ω then we have

∑ d i = 2g -2.
For a given α = (d 1 , . . . , d m ) such that ∑ d i = 2g -2 and d i ≥ 1, for i = 1, . . . , m, we consider the stratum H (α) of the moduli space of Abelian differentials on M g,m that have zeros at the points of Σ of orders (d 1 , . . . , d m ), or equivalently the moduli space of translation surfaces S with singularities at the points of Σ of angles (2π(d 1 + 1), . . . , 2π(d m + 1)). The dimension of H (α) is dim C H (α) = 2g + m -1. The stratum H (α) may be non connected [START_REF] Veech | Moduli spaces of quadratic differentials[END_REF] but contains at most three connected components [START_REF] Kontsevich | Connected components of the moduli space of Abelian differentials with prescribed singularities[END_REF]. The stratum H (α) admits a volume element [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF][START_REF] Veech | Gauss measures for transformations on the space of interval exchange maps[END_REF] that induces a finite SL(2, R)-invariant measure on the hyperspace H 1 (α) of area one surfaces in H (α). The volumes of the connected components of the strata of Abelian differentials were effectively computed by A. Eskin and A. Okounkov [START_REF] Eskin | Asymptotics of number of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials[END_REF].

The boundary of a periodic cylinder contains singularities. Generically, each boundary component contains exactly one singularity so it is a closed saddle connection, i.e. a geodesic that joins a singularity to itself (and contains no other singularity).

Consider a translation surface S. For each positive real number L > 0, denote by N cyl (S, L) the number of periodic cylinders in S of width at most L and by N area (S, L) the total area of these cylinders. It was shown in [START_REF] Eskin | Asymptotic formulas on flat surfaces[END_REF] that Theorem (Eskin-Masur). Let H (α) be a stratum, where α = (d 1 , . . . , d m ) with d i ≥ 1, for i = 1, . . . , m. For every connected component K of H 1 (α), there exist constants c cyl (K) and c area (K) such that for almost every translation surface S in K one has lim L→∞ N cyl (S, L) πL 2 = c cyl (K) lim L→∞ N area (S, L) πL 2 = c area (K).

The Siegel-Veech constants c cyl (K) and c area (K) only depend on K. An earlier version of this result (in a more general setting) where convergence is replaced by convergence in L 1 was proved in [START_REF] Veech | Siegel Measures[END_REF].

The results in [START_REF] Eskin | Asymptotic formulas on flat surfaces[END_REF][START_REF] Veech | Siegel Measures[END_REF] assure the existence of quadratic asymptotics (Siegel-Veech constants) if one counts cylinders with weights (under certain conditions). The existence of all the Siegel-Veech constants we consider in this paper is justified by these results (see section 1.3). The function N area (S, L) for example counts cylinders with weight the area of the cylinder.

One can also only count cylinders with sufficiently big area: for x ∈ [0, 1), denote by N cyl,A≥x (S, L) the number of cylinders in S of width at most L and of area at least x. We denote the corresponding Siegel-Veech constant by c cyl,A≥x (K). It is shown in [START_REF] Vorobets | Periodic geodesics on generic translation surfaces Algebraic and Topological Dynamics[END_REF] that Theorem (Vorobets [START_REF] Vorobets | Periodic geodesics on generic translation surfaces Algebraic and Topological Dynamics[END_REF]). Let H (α) be a stratum, where α = (d 1 , . . . , d m ) with d i ≥ 1, for i = 1, . . . , m. Then for any connected component K of H 1 (α) (a) c mean area (K) = c area (K)

c cyl (K) = 1 2g + m -2 = 1 dim C H (α) -1 (b) c cyl,A≥x (K) c cyl (K) = (1 -x) 2g+m-3 = (1 -x) dim C H (α)-2 .
The ratio c mean area (K) of part a) of the theorem can be interpreted as the (asymptotic) mean area of a cylinder on a generic surface in K in the following sense: for almost any surface M in K one has

c area (K) c cyl (K) = lim L→∞ N area (M, L) N cyl (M, L) .
Part b) of the theorem is an answer to a question of Veech in [START_REF] Veech | Siegel Measures[END_REF] where the author asks if there is a simple formula for

c cyl,A≥x (K) c cyl,A≥0 (K) (which is the same as c cyl,A≥x (K) c cyl (K) ).
As a by-product of our results using the methods from [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] we get an alternative proof of the theorem of Vorobets by evaluating an explicit integral that is a simplified version of the integral used in [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF].

The methods from [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] allow for the computation of various other Siegel-Veech constants associated to counting cylinders for more specific data that we allude to next.

It might happen that the geodesic flow in a given direction on M contains several (maximal) periodic cylinders in that direction. Generically, this only happens if the boundary saddle connections of the cylinders are homologous. The boundary saddle connections might be part of a larger family of homologous saddle connections, where the extra saddle connections do not bound a cylinder. We refer to such a family as a configuration of homologous saddle connections or simply as a configuration. A topological representation of such configurations can be obtained by taking blocks of surfaces as in figures 6, 7, and 8 (the surfaces are drawn as tori, but might have arbitrary genus), arranging them in a cyclic order and then identifying the boundary components.

Note that the fact that the saddle connections are homologous implies that they are all of the same length. Call this the length of the configuration. It also implies that a configuration persists under small deformations of the translation structure.

One says that two configurations of homologous saddle connections correspond to the same topological type if the saddle connections are based at the same singularities, if the complementary regions are of the same topological type and have the same number and type of singularities, e.t.c. In particular, they always have the same number of complementary cylinders. The length of the configuration is the common width of the cylinders coming from the configuration.

For each connected component K of a stratum there are only a finite number of admissible topological types of configurations, i.e. topological types of configurations that are realized on at least one surface in K. In fact almost all surfaces share the same admissible topological types. We will consider only configurations of this special type. Their distinguishing feature is that they persist under any small deformation. For related counting problems in the case of general periodic components (so not configurations), see [START_REF] Naveh | Tight upper bounds on the number of invariant components on translation surfaces[END_REF][START_REF] Lindsey | Counting invariant components of hyperelliptic translation surfaces[END_REF].

For S in K, denote by N conf (S, C , L) the number of configurations of homologous saddle connections on S of type C and whose length is at most L.

Theorem (Eskin-Masur-Zorich, [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF]). For a given connected component K of a stratum H 1 (α) and an admissible topological type C of configurations there exists a Siegel-Veech constant c conf (K, C ) such that for almost any surface S in K one has

lim L→∞ N conf (S, C , L) πL 2 = c conf (K, C ).
The authors of [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] give a method to compute these Siegel-Veech constants.

1.2. Statement of results. We denote by N cyl (S, C , L) the number of cylinders of length less than L coming from a configuration of type C . For a real number p ≥ 0 we denote by N area p (S, C , L) the sum of the p-th power of the area of each of these cylinders. We denote the corresponding Siegel-Veech constants by c cyl (K, C ), resp. c area p (K, C ). For p = 1 we write c area (K, C ). Note that if C comes with q cylinders then c cyl (K, C ) = qc conf (K, C ).

It follows from the general result of [START_REF] Eskin | Asymptotic formulas on flat surfaces[END_REF] that there is a Siegel-Veech constant c area p (K, C ) such that for almost any surface S in K one has

lim L→∞ N area p (S, C , L) πL 2 = c area p (K, C ).
The methods from [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] can be applied to compute c area p (K, C ) in a way similar to the computation of c conf (K, C ). The expression for c area p (K, C ) contains a constant M that depends only on combinatorial data such as the dimension of the stratum, the order of the singularities and the possible symmetries. It is given by an explicit formula in § 13.3. of [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF]. It also contains the "principal boundary stratum" H 1 (α ′ ) determined by K and C (the stratum of possibly disconnected surfaces we get by contracting the closed saddle connections of the configuration). The constant n always denotes the complex dimension of H 1 (α ′ ). If C comes with q cylinders then we have

n = dim C H (α) -q -1.
With this notation we show in section 2.2, Theorem 1. Given a real number p ≥ 0. Let H (α) be a stratum, where α = (d 1 , . . . , d m ) and d i ≥ 1, for i = 1, . . . , m. Let K be a connected component of H 1 (α) and C an admissible topological type of configuration containing q ≥ 1 cylinders. Then

c area p (K, C ) = M • Vol(H 1 (α ′ )) Vol(K) • (n -1)! (p + 1) • (p + 2) • • •(p + q + n -1) • q c cyl (K, C ) = M • Vol(H 1 (α ′ )) Vol(K) • (n -1)! (n + q -1)! • q c area (K, C ) = M • Vol(H 1 (α ′ )) Vol(H 1 (α)) • (n -1)! (n + q)! • q
where n = dim C H (α)q -1. M denotes the combinatorial constant given in § 13.3. of [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] and H 1 (α ′ ) denotes the principal boundary stratum.

Note that c area (K, C ) = c area 1 (K, C ) and c cyl (K, C ) = c area 0 (K, C
), so the second and third equation of the previous theorem follow from the first.

Remark.

(a) Evaluation of c area p is motivated by the question of M. Möller related to the study of quasimodular properties of the related counting function. (b) We recall that the saddle connections in a configuration of given type C can be named. Choose and fix one of the saddle connections that bounds a cylinder. In the proof of theorem 1 we show that if we only consider the area of this cylinder then we get the same formulas for c area p (K, C ) and c area (K, C ) except that the factor q is missing.

We get as a corollary:

c mean area p (K, C ) = c area p (K, C ) c cyl (K, C ) = (d -2)! (p + 1) • (p + 2) • • •(p + d -2)
,

where d = dim C H (α) = 2g + m -1.
Remark. (a) The quotient of the corollary can be interpreted as the (asymptotic) mean area of a cylinder coming from a configuration of type C , where the area is counted with a power p.

(b) For a natural number p ≥ 1 we obtain

c mean area p (K, C ) = 1 p+d-2 p .
Define N area p (S, L) in the same way as we defined N area (S, L) in section 1.1, except that the area of each cylinders is counted with a power p. If c area p (K) denotes the corresponding Siegel-Veech constant, then we have

c cyl (K) = ∑ C c cyl (K, C ) and c area p (K) = ∑ C c area p (K, C ),
where the sum is taken over all admissible topological types of configurations for K with at least one cylinder. This implies Corollary 3. Given a real number p ≥ 0 and let H (α) be a stratum, where α = (d 1 , . . . , d m ) with d i ≥ 1, for i = 1, . . . , m. Then for any connected component K of H 1 (α)

c mean area p (K) = c area p (K) c cyl (K) = (d -2)! (p + 1) • (p + 2) • • •(p + d -2)
,

where d = dim C H (α) = 2g + m -1.
For p = 1, corollary 3 becomes part (a) of the theorem of Vorobets as stated in section 1.1. Corollary 2 gives more detailed information as corollary 3. For example, the mean area c mean area p (K, C ) of a cylinder coming from a configuration of type C does not depend on the number of cylinders. This means in particular that the mean area of a cylinder is the same if the cylinder makes up the whole periodic region of a configuration or if the periodic region is made up of several cylinders.

As a variation of the above, we denote by N area p ,conf (S, C , L) the p-th power of the total area of the periodic region (union of the cylinders) on S coming from a configuration of topological type C whose length is at most L. The corresponding Siegel-Veech constant is denoted by c area p ,conf (K, C ). We show in section 2.3 Theorem 4. Given a real number p ≥ 0 and let H (α) be a stratum, where α = (d 1 , . . . , d m ) and d i ≥ 1, for i = 1, . . . , m. Let K be a connected component of H 1 (α) and C an admissible topological type of configuration containing q ≥ 1 cylinders. Then

(a) c area p ,conf (K, C ) = M • Vol(H 1 (α ′ )) Vol(K) • (n -1)! (q -1)! 1 (p + q) • • •(p + q + n -1) (b) c area p ,conf (K, C ) c conf (K, C ) = q(q + 1) • • •(q + n -1) (p + q)(p + q + 1) • • •(p + q + n -1)
.

M denotes the combinatorial constant given in § 13.3. of [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] and H 1 (α ′ ) denotes the principal boundary stratum.

Remark. (a) For a natural number p ≥ 1 we obtain

c area p ,conf (K, C ) c conf (K, C ) = q • (q + 1) • • •(q + p -1) (d -1) • d • • •(d + p -2)
,

where d = dim C H (α) -1. (b)
The quotient in part (b) of the preceeding theorem can be interpreted as the asymptotic mean area of the periodic part (taking the p-th power of the area). For p = 1 we get qc mean area (K, C ), which is consistent, as c mean area (K, C ) is the mean area of a cylinder.

(c) For q = 1 we have c area p ,con f (K, C ) = c area p (K, C ).

We next count configurations with cylinders of large area. We recall that the saddle connections in a configuration of given type C can be named. Choose and fix one of the saddle connections that bound a cylinder and call this the first cylinder. Given x ∈ [0, 1). Denote by N conf,A 1 ≥x (S, C , L) the number of configurations of type C of length at most L and such that the area of the first cylinder is at least x (of the area one surface S). We denote by c conf,A 1 ≥x (K, C ) the corresponding Siegel-Veech constant. Note that we have c conf,A 1 ≥0 (K, C ) = c conf (K, C ). We show in section 2.5 Theorem 5. Given x ∈ [0, 1). Let H (α) be a stratum of Abelian differentials on a surface M g,m , where α = (d 1 , . . . , d m ) with d i ≥ 1, for i = 1, . . . , m. Then for any connected component K of H 1 (α) and any admissible topological type C of configuration for K containing at least one cylinder,

c conf,A 1 ≥x (K, C ) c conf (K, C ) = (1 -x) 2g+m-3 = (1 -x) dim C H (α)-2 .
Summing over all configurations we get as a corollary part (b) of the theorem of Vorobets but our result contains more detailed information.

We next count configurations with periodic regions of large area. Let C be a topological type of configuration that comes with q ≥ 1 cylinders. For x ∈ [0, 1), denote by N conf,A≥x (S, C , L) the number of configurations on S of type C of length at most L and such that the total area of the q cylinders is at least x (of the area one surface S). We denote the corresponding Siegel-Veech constant by c conf,A≥x (K, C ).

The incomplete Beta function B(t; a, b) and the regularized incomplete Beta function I(t; a, b) are defined for t ∈ [0, 1] by

B(t; a, b) = t 0 u a-1 (1 -u) b-1 du, I(t; a, b) = B(t; a, b) B(1; a, b) = B(t; a, b) B(a, b) .
For more details about the incomplete Beta function see section 4: Figure 9 represents the density function d dt I(t; a, b) for various values for a and b. With this notation we show in section 2.4: Theorem 6. Given x ∈ [0, 1), and let H (α) be a stratum, where α = (d 1 , . . . , d m ) with d i ≥ 1, for i = 1, . . . , m. Suppose that K is a connected component of H 1 (α) and that C is an admissible topological type of configurations for K containing exactly q cylinders.

c conf,A≥x (K, C ) c conf (K, C ) = I(1 -x; n, q) = (1 -x) n q-1 ∑ k=0 n -1 + k k x k , where n = dim C H (α) -q -1 = dim C H (α ′ ).
n =20 q =2 q =4 q =6 n =20 q =2 q =4 q =6 FIGURE 1. Graphs of the function f (x) = c conf,A>x (K, C ) c conf (K, C )
See figure 1 for the graph of I(1x; n, q) for various values for n and q.

Remark.

(a) The fraction

c conf,A≥x (K, C ) c conf (K, C
) can be interpreted as the mean proportion of configurations of type C whose periodic complementary region is big, that is, the total area of the cylinders is at least x of the area of the surface. (b) I(1; n, q) = 1 and

lim x→1 I(0; n, q) (1 -x) n = q-1 ∑ l=0 n + l -1 l = n + q -1 n , so c conf,A≥x (K, C ) c conf (K, C ) ∼ (1 -x) n n + q -1 n as x → 1.
In this form we can compare the result with the previous one for one cylinder, given in Theorem 5.

In section 2.6 we consider the problem of correlation between the area of two cylinders. Let C be an admissible configuration for a connected component K that comes with at least two cylinders. Choose (and fix) two cylinders and let x, x 1 ∈ [0, 1). We denote by N A 2 ≥x,A 1 ≥x 1 (S, C , L) the number of configurations of width length at least L such that the area A 1 of the first cylinder is at least x 1 and such that the area A 2 of the second cylinder is at least x of the remaining surface, i.e. it is at least x(1 -A 1 ). We denote by c A 2 ≥x,A 1 ≥x 1 (K, C ) the corresponding Siegel-Veech constant. To simplify notation we will write

c A 1 ≥x 1 (K, C ) instead of c conf,A 1 ≥x 1 (K, C ). We show in section 2.6, Theorem 7. For any connected component K of a stratum H 1 (α), where α = (d 1 , . . . , d m ) with d i ≥ 1, for i = 1, . . .

, m, and any admissible topological type C of configurations containing at least two cylinders,

c A 2 ≥x,A 1 ≥x 1 (K, C ) c A 1 ≥x 1 (K, C ) = (1 -x) dim C H (α)-3 .
The result does not depend on x 1 .

Morally, we compute the asymptotic probability that, among configurations whose first cylinder has area x 1 , we have a second cylinder with area at least x(1x 1 ). Comparing Theorems 7 and 5 we see that this is the probability that, among all configurations, the area of the second cylinder is at least x, except that the parameter space has one fewer dimension. So, in some sense, except for the fact that the area of the first cylinder gives a restriction on the range for the area of the second cylinder, the area of the second cylinder is independent of the area of the first cylinder.

In the results presented above we studied individual configurations. In the remaining part of the paper we study extremal properties of configurations among all configurations in a given stratum or even among all strata for a fixed genus.

In section 3.1 we address the question of finding topological types of configurations C (admissible for some connected component K) that maximizes

c mean area conf (K, C ) = c area (K, C ) c conf (K, C ) .
The constant c mean area conf (K, C ) can be interpreted as the asymptotic mean area of the periodic part (union of the cylinders) of the complementary region of a configuration of topological type C .

Each stratum has at most three connected components that are classified by the invariants "hyperellipticity", and "parity of spin structure". (We recall in section 3.2 the classification of connected components from [START_REF] Kontsevich | Connected components of the moduli space of Abelian differentials with prescribed singularities[END_REF].)

The quantity c mean area conf (K, C ) varies considerably among strata. For the connected stratum H (1, 1, . . . , 1), the maximal value of c mean area conf (K, C ) over all the configurations is 1 4 . For the connected component H hyp (g -1, g -1) it is equal to 1 2g . The following proposition gives an uniform bound on the ratio c mean area conf (K, C ).

Theorem 8. Let K be any connected component of a stratum H (α) and C be any admissible topological type of configuration for K, then the asymptotic mean area of the periodic complementary regions satisfies

c mean area conf (K, C ) ≤ 1 3 .
The maximum is attained for any genus g ≥ 2: for each g ≥ 2 there is a topological type of configuration C g that is admissible for the component H odd (2, 2, . . . , 2) (g -1 zeros of order 2) of Abelian differentials on a surface of genus g such that the corresponding constant c mean area conf is 1 3 .

To prove this result we need to determine the maximal number of cylinders that can come from any configuration which is admissible for a fixed stratum H (α). We insist on the fact that we compute here the number of cylinders in rigid collections of saddle connections, which is different from the studies of Naveh [START_REF] Naveh | Tight upper bounds on the number of invariant components on translation surfaces[END_REF] and Lindsey [START_REF] Lindsey | Counting invariant components of hyperelliptic translation surfaces[END_REF] where they count the number of parallel cylinders.

In section 3.2 we answer a question of A. Eskin and A. Wright: is it possible to find in each connected component of each stratum a topological type of configuration whose complementary regions are tori with boundary and cylinders.

The answer depends on the connected component. We show (Proposition 11): for hyperelliptic components this is not possible; for the components with even spin structure when the genus is even this is not possible unless we allow one of the complementary regions to be a genus two surface; in all other cases this is possible. 1.3. Notation. We introduce here most of the notation we need for the computation of Siegel-Veech constants. For survey material on Abelian differentials and translation surfaces see [START_REF] Gutkin | Affine mappings of translation surfaces: Geometry and arithmetic[END_REF][START_REF] Zorich | On random matrices, zeta functions and dynamical systems[END_REF][START_REF] Masur | Rational billiards and flat structures[END_REF]. For the exact definition of configurations and related results see [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF].

Let M g,n denote a closed oriented surface M g of genus g on which there are m marked points Σ = {P 1 , . . . , P m }. By (R, ω) we denote a Riemann surface structure R on M g together with an Abelian differential ω. If ω is not identically zero then Σ is the set of zeros of ω. We usually only write ω for (R, ω). If we denote by (d 1 , . . . , d m ) the orders of the zeros of ω then we have

∑ d i = 2g -2.
The form ω can be used to define an atlas of adapted coordinates on R. In these adapted coordinates the Abelian differential ω becomes dz in a neighborhood of any point of R \ Σ and is

(d i + 1)w d i dw = d(w d i +1
) in a neighborhood of a point P i ∈ Σ, where d i is the order of the zero P i . Transition functions away from the zeros for these adapted coordinates are translations. We refer to a surface together with an atlas whose transition functions are translations as translation surfaces.

Using such an atlas, R \ Σ inherits from the complex plane a flat (zero curvature) Riemannian metric. A zero of order d i of the Abelian differential (that is a regular point of the Riemann surface structure) corresponds to a conical singularity of the flat metric of total angle 2π(d i + 1). These points are also called saddles.

The horizontal unit vector field on C pulls back by adapted coordinates to a horizontal unit vector field on R \ Σ. Away from the singularities, the leaves of the corresponding foliation are geodesics with respect to the flat metric. In fact, for each θ ∈ [0, 2π[ we have a unit vector field and so a foliation in that direction that comes from the unit vector field in direction θ of the complex plane.

The converse construction is also possible: suppose that S is a translation surface structure on M g,m , i.e. an atlas on M g,m whose transition functions are translations. A translation surface inherits from R 2 a flat Riemannian metric and a parallel vector field on M g,m . We assume that M g is the metric completion of M g,m . The points of Σ that are not regular points for the metric, are cone type singularities where the total angle is of the form 2π(d i + 1) for some d i ∈ N.

By identifying R 2 and C, a translation surface inherits from C a complex (Riemann surface) structure on M g and a holomorphic one form (abelian differential) ω. The zeros of ω are contained in Σ.

Each translation surfaces can be represented by a polygon in the plane whose edges come in pairs of parallel sides of the same length. Identifying each pair by a translation one obtains a translation surface. The vertices give rise to singularities (or regular points if the total angle is 2π.)

We identify two translation surface structures if there is a bijection of the underlying topological surface that is in local coordinates a translation. We identify two Abelian differentials (for some complex structures) if they are biholomorphically equivalent. There is then a one to one correspondance between Abelian differentials and translation surfaces. We denote by H the moduli space of Abelian differentials ω or equivalently of translation surfaces S. The moduli space H is an algebraic variety.

Given

α = (d 1 , . . . , d m ) such that 2g -2 = ∑ i d i .
The set H (α) of Abelian differentials that share the same zero structure α is called a stratum. The stratum H (α) is an algebraic subvariety that admits a natural affine structure and a natural "Lebesgue" volume element induced by this affine structure [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF][START_REF] Veech | Gauss measures for transformations on the space of interval exchange maps[END_REF]. The dimension of

H (α) is dim C H (α) = 2g + m -1.
The area of the translation surface S defined by the Abelian differential ω = φ (z)dz is S |φ (z)| 2 dxdy. We denote by H 1 (α) the hyperspace of H (α) of area one surfaces. Masur [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF] and Veech [START_REF] Veech | Gauss measures for transformations on the space of interval exchange maps[END_REF] showed that H 1 (α) with the measure induced by the measure on H (α) is of finite volume.

There is a natural SL(2, R) action on H (α). An element g ∈ SL(2, R) acts on local coordinates by postcomposition of g. The measure on H (α) and H 1 (α) is SL(2, R) invariant. If we represent a translation surfaces by a polygon in R 2 , then the action of g is the usual action on R 2 .

Geodesic segments in the flat metric have constant angle with respect to the flat metric, so a geodesic segment γ can be represented by a holonomy vector hol(γ) in R 2 . The angle and length of the vector is given by the direction and length of the geodesic segment. We will often use γ both for the geodesic segment and for the holonomy vector.

As we said above, a closed geodesic is always contained in a maximal cylinder of closed geodesics and each boundary component of the cylinder is generically a closed saddle connection. As all of the geodesics in the cylinder are represented by the same vector we say that this vector represents the cylinder. Its length is the width of the cylinder.

Suppose that on a translation surface we find in some direction a configuration of homologous saddle connections, meaning a maximal family of saddle connections, that are homologous relative to the singularities. All of the saddle connections are parallel and of the same length, so they share the same holonomy vector. Its length is the length of the configuration.

A configuration defines the following data: the named singularities the saddle connections are based at; the topological type of the complementary regions; the knowledge of which saddle connection bounds which complementary region, so in particular the cyclic order of the complementary regions; the order of the singularities (if there are any) in the interior of each complementary region; the singularity structure on the boundary (the type of boundary and angles at the boundary singularities to be described below).

We say that two configurations are of the same topological type if they define the same data. Almost all surfaces in a given connected component K share the same topological types of configurations that can be realized on the surface. We talk about an admissible topological type for K.

Denote by V (S, C ) the (discrete) set of holonomy vectors associated to configurations on S of type C . We are allowed to associate weights to the holonomy vectors. Denote by B(L) ⊂ R 2 the disk of radius L centered at the origin and by N(S, C , L) the cardinality of V (S, C ) ∩ B(L), where the elements of V (S, C ) ∩ B(L) are counted with their weights. So if we write an element of V (S, C ) as (v, w(v)) where w(v) is the weight of the holonomy vector v, then N(S,

C , L) = ∑ v∈V (S,C )∩B(L) w(v).
By using appropriate weights on the holonomy vectors the counting function N(S, C , L) becomes the counting functions N conf (S, C , L), N cyl (S, C , L), N area (S, C , L) e.t.c. introduced in section 1.2. If for example we count holonomy vectors associated to a configuration with weight one (resp. the number of cylinders, total area of the cylinders) then

N(S, C , L) equals N conf (S, C , L) (resp. N cyl (S, C , L), N area (S, C , L)).
Given a connected component K of some stratum H 1 (α) and an admissible topological type C for K. It follows from [START_REF] Eskin | Asymptotic formulas on flat surfaces[END_REF] that the set of (weighted) holonomy vectors V (S, C ) and the associated counting functions N(S, C , L) we consider in this paper verify the following conditions :

(A) for every g ∈ SL(2, R), V (gS, C ) = gV (S, C ).

(B) for every S ∈ K there exists a constant c(S) > 0 such that N(S, L, C ) ≤ c(S)L 2 . The constant c(S) can be chosen uniformly on compact sets of K. (C) there exist constants L > 0 and ε > 0 such that N(S, L, C ) is L 1+ε (K, µ) as a function of S. In fact, the authors of [START_REF] Eskin | Asymptotic formulas on flat surfaces[END_REF] show that the above conditions are verified for the set V (S) of holonomy vectors of closed saddle connections on S where the weight associated to a holonomy vector is the number of saddle connections on S that share this vector. The weights we use in this paper are "invariant under SL(2, R)", so the sets V (S, C ) we use verify condition (A). The number of configurations is bounded by the number of saddle connections, and as we use bounded weights, the counting functions N(S, C , L) we consider also satisfy conditions (B) and (C).

For

f ∈ C ∞ 0 (R 2 ) one defines the function f : K → R by f (S) = ∑ v∈V (S,C ) w(v) f (v).
The fact that the sets V (S, C ) and associated counting functions N(S, C , L) we consider in this paper satisfy the conditions (A), (B), (C), implies, using [START_REF] Eskin | Asymptotic formulas on flat surfaces[END_REF], that:

Theorem ( [START_REF] Eskin | Asymptotic formulas on flat surfaces[END_REF][START_REF] Veech | Siegel Measures[END_REF]). Let K be a connected component of some stratum H 1 (α) of Abelian differentials and C an admissible topological type of saddle connections for K. Then for any of the sets V = V (S, C , L) and associated counting functions N V (S, C , L) we consider in this paper there is a constant c V (K, C ) such that the following holds :

(a) For almost any translation surface S in K,

lim L→∞ N V (S, C , L) πL 2 = c V (K, C ). (b) For any f ∈ C ∞ 0 (R 2 ), 1 vol(K) K f (S)dvol(S) = c V (K, C ) R 2 f (x, y)dxdy.
If the convergence in (a) is replaced by convergence in L 1 then this theorem follows from a more general theorem under similar hypotheses in [START_REF] Veech | Siegel Measures[END_REF]. In that paper, (b) is proved for integrable functions f of compact support. We will only use (b) for the characteristic function on a disk.

The paper [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] explains how this last theorem can be used to compute the Siegel-Veech constant c conf (K, C ). The same method can be used to compute the other Siegel-Veech constants we consider in this paper.

The strategy used in [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] is as follows: if we apply (b) for the characteristic function f ε of the disc B(ε) of radius ε centered at the origin then the integral of the right hand side becomes πε 2 and we have

fε (S) = ∑ v∈V (S,C )∩B(L) w(v) = N(S, C , ε). So c V (K, C ) = 1 vol(K) 1 πε 2 K N V (S, C , ε)d vol(S).
Denote by K(ε, C ) the subset of translation surfaces in K that contain at least one configuration of type C of length smaller than ε. So N V (S, C , ε) is zero outside K(ε, C ). Denote by K thick (ε, C ) the subset of K(ε, C ) of translation surfaces that contain exactly one configuration of type C of length smaller than ε but contain no other closed saddle connection of length smaller than ε. Using a result from [START_REF] Eskin | Asymptotic formulas on flat surfaces[END_REF] the autors of [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] show that

vol(K(ε, C )) = vol(K thick (ε, C )) + o(ε 2 ). So c V (K, C ) = 1 vol(K) 1 πε 2 K(ε,C ) N V (S, C , ε)d vol(S) = lim ε→0 1 vol(K) 1 πε 2 K thick (ε,C ) N V (S, C , ε)d vol(S). ( 1
)
Consider a translation surface S in K thick (ε, C ). On S we have a configuration of closed saddle connections of topological type C of length smaller than ε and no other short closed saddle connection. Cutting along the closed saddle connections we decompose S into several pieces. There will be some, say p ≥ 1, surfaces S 1 , . . . , S p with boundary and some, say q ≥ 0, periodic cylinders C 1 , . . . ,C q , all having the same width. The pieces are arranged in a cyclic order. The boundary of each S i is made up of two closed saddle connections.

For each connected component S i , by taking out the boundary and then taking the compactification we get a surface with either one boundary component, a figure eight, as in the right part of figure 2, or we get a surface with two boundary components, a pair of holes, as in the right part of figure 3.

In the first case, the figure eight boundary describes two interior sectors of the surface of angles 2π(a ′ + 1) and 2π(a ′′ + 1), for some integers a ′ , a ′′ ≥ 0. (Figure 2 illustrates the case a ′ = 1 and a ′′ = 0.) By shrinking the figure eight boundary to a point we produce a singularity of order a ′ + a ′′ if a ′ + a ′′ ≥ 1, or a regular point if a ′ + a ′′ = 0. (See the left part of figure 2.) We get in this way closed surfaces S ′ i that belong to some stratum H (α ′ i ), for i = 1, . . . , p. We write α ′ = ⊔ p i=1 α ′ i and H (α ′ ) = Π p i=1 H (α ′ i ). We will say that the surface S ′ whose connected components are S ′ 1 , . . . , S ′ p belongs to H (α ′ ). We say that S ′ belongs to the principal boundary of H (α) determined by C and that H (α ′ ) is the corresponding principal boundary stratum. All this data (topological type of complementary regions, cyclic order, boundary stratum, e.t.c.) are the same for any configuration of the same topological type C . In fact the topological type is characterized by this data.

2π(a ′ + a ′′ + 1) 2π(a ′ + 1) 2π(a ′′ + 1) FIGURE 2.
By shrinking a closed saddle connection, the singular point it is based at might become a regular point with total angle 2π. In this case the regular point will be considered as a marked point of order 0. So α ′ i can contain one or two 0. The procedure of shrinking saddle connections can be reversed. We summarize the description form [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] in case where H (α) has only one connected component K. Start with a (maybe disconnected) surface T ′ in H (α ′ ) and call the connected components T ′ i . Choose some holonomy vector γ in B(ε). There are two types of surgery. A figure eight surgery where we start with a singularity or a marked point of order a ≥ 0, choose a ′ , a ′′ ≥ 0 such that a = a ′ + a ′′ and then metrically create a figure eight boundary that consists of two saddle connections in direction γ that are of length |γ| and are based at the same singularity. There will be two sectors of angles 2π(a ′ + 1) and 2π(a ′′ + 1) (see figure 2). A pair of holes surgery where we start with two points that are either a singularity or a marked point of orders b ′ ≥ 0 and b ′′ ≥ 0 and then create a boundary saddle connection at each singularity. The pair of holes boundary has two boundary singularities of angles π(2b ′ + 3) and π(2b ′′ + 3) (see figure 3).

By performing an appropriate surgery on each T ′ i we obtain surfaces T i that are homeomorphic to the surfaces S i and have the same type of singularities in the interior and on the boundary. We then take q cylinders C j with a marked point on each boundary. We finally combine the surfaces T i and the cylinders C j in the way prescribed by the topological type C to produce a surface T in K thick (ε, C ). We do this by identifying pairs of boundary components by an isometry that identifies boundary singularities. The boundary components give rise to a configuration of closed saddle connections of topological type C . In fact each surface in K thick (ε, C ) can be produced in this way.

The parameters used to produce surfaces in K thick (ε, C ) are the following:

• a maybe disconnected surface in H (α ′ ).

• a holonomy vector γ in B(ε).

• a combinatorial constant M that only depends on the configuration. There is a M : 1 correspondence between the surfaces in K thick (ε, C ) and the surfaces in H 1 (α ′ ). This is mainly due to the facts that at a zero of order k there are k + 1 sectors of angle 2π where we can produce a saddle connection in the direction of γ and to possible symmetries of the surface in K thick (ε, C ) (see § 13.3. of [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF]). • the heights h i of the q cylinders C i (the width is given by |γ|). • for each cylinder C i a twist parameter t i ∈ [0, |γ|) that describes the relative position of the marked points on the two boundary components.

Remark. If H (α) has more than one connected component than the correspondence between the thick part K thick (ε, C ) of a connected component K of H 1 (α) and the principal boundary H (α ′ ) is slightly more complicated. For example, to construct a surface T in K thick (ε, C ), where K is a hyperelliptic component, we must start with a surface T ′ in the principal boundary such that all connected components T ′ i of T ′ are hyperelliptic surfaces. So only the hyperelliptic components of some strata are in the principal boundary of K. It might even happen that only part of a connected component of a stratum is in the principal boundary. We still denote the principal boundary of K thick (ε, C ) by H (α ′ ), although H (α ′ ) might be the union of (parts of) connected components of some strata.

We remark in passing that there are also some parameters used to describe the surgeries but this does not affect our computations.

If q > 1, then to parametrize the q tori we will always replace h q by h = h 1 + • • • + h q . So the heights (h 1 , . . . , h q-1 ) are in the cone ∆ q-1 (h) given by the conditions h i > 0, for 1 ≤ i ≤ q -1, and

h 1 + • • • + h q-1 < h.
For a given ε > 0 we define

H ε (0 q ) = {(γ, h, h 1 , . . . , h q-1 ,t 1 , . . . ,t q ) | γ ∈ B(ε), (h 1 , . . . , h q-1 ) ∈ ∆ q-1 (h), (t 1 , . . . ,t q ) ∈ [0, |γ|] q }.
In what follows, |γ| will always be small, but if no specific restriction on |γ| is needed we will write H (0 q ). We write dν(T ) for the measure

dν(T ) = dγ dh q-1 ∏ i=1 dh i q ∏ i=1 dt i .
We refer to the elements of H ε (0 q ) as tori as we obtain a torus with q marked points by joining the q cylinders. We write H ε 1 (0 q ) for the subset of H ε (0 q ) of area 1 tori, meaning that they satisfy the condition h|γ| = 1.

We remark in passing that H ε (0 q ) can be interpreted as the ε -neighborhood of the "cusp" of the moduli space of flat tori with q marked points except that the marked points are already named by the way we parametrize them. This is in contrast to the cusp of the usual moduli space H ε (0, . . . , 0 q times ) where the marked points can be arbitrarily named.

We denote by dν(S) (resp. dν(S ′ )) the measure on H (α) (resp. H (α ′ )). It is shown in [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] that dν(S) = dν(S ′ ) • dν(T ).

Let S be a translation surface in H 1 (α). For r a positive real number we denote by rS ∈ H (α) the surface we get by multiplying the flat metric on S by r. Equivalently, if we represent S by an Abelian differential ω with respect to some complex structure, then rS corresponds to the Abelian differential rω with respect to the same complex structure. Note that we have in particular area(rS) = r 2 area(S).

If X is a subset of H 1 (α) then the cone C(X) is defined to be

C(X) = {rS | 0 < r < 1, S ∈ X} ⊂ H (α).
If we denote by d vol(S) the measure on H 1 (α) induced by the measure dν(S) then we have dν(S) = r dim R H (α)-1 dr d vol(S).

So in particular one has

vol(H 1 (α)) = dim R H (α)ν(H (α)).
One has analogous statements for H 1 (α ′ ) and H 1 (0 q ) with their induced measures denoted by d vol(S ′ ) and d vol(T ).

Note that there are q+1 complex parameters for H ε (0 q ) and that one has dim R H

(α) = dim R H (α ′ ) + dim R H (0 q ). So by writing n = dim C H (α ′ ) we have dim R H (α ′ ) = 2n dim R H (0 q ) = 2(q + 1) dim R H (α) = 2(n + q + 1) We recall for completeness that if α = (d 1 , . . . , d m ) with d i ≥ 1 for i = 1, . . . , m, then dim R H (α) = 2(2g + m -1).

SIEGEL-VEECH CONSTANTS.

2.1. General method. We will describe in this section the method used to compute the Siegel-Veech constants introduced in section 1.2. Consider a stratum H (α) of Abelian differentials where α = (d 1 , . . . , d m ) and d i ≥ 1 for i = 1, . . . , m. Suppose that K is a connected component of H 1 (α) and that C is an admissible topological type of configurations for K containing exactly q ≥ 1 cylinders. We denote the corresponding principal boundary stratum by H (α ′ ). Recall that K thick (ε, C ) is the set of translation surfaces in K that contain exactly one configuration of type C of length smaller than ε but do not contain another closed saddle connection of length smaller than ε. To simplify notation we will write from now on K ε for K thick (ε, C ).

Suppose that N(S, C , ε) is one of the above mentioned counting functions that counts configurations on S of length smaller than ε and of topological type C. We want to evaluate

K ε N(S, C , ε)dν(S).
The decomposition of a surface S ∈ H (α) as q tori with marked points on the boundary and a surface S ′ in H (α ′ ) on which one performs surgeries (M choices) gives the following parametrisation of the cone C(K ε ): before performing the surgeries, a surface S in C(K ε ) is made up of sS ′ 1 and tT 1 for some scalars s,t where S ′ 1 ∈ H 1 (α ′ ) and T 1 ∈ H 1 (0 q ). The conditions are the following : (i) the total area s 2 + t 2 of S satisfies s 2 + t 2 ≤ 1; The area 1 surface 1 √ s 2 +t 2 S is in K ε so the waist curve of t √ s 2 +t 2 T 1 is smaller than ε which means that the waist curve of T 1 is smaller than ε √

s 2 +t 2 t
. So the second condition is:

(ii) T 1 has to lie in H ε ′ 1 (0 q ), where ε ′ = ε √ s 2 +t 2 t . We have (2) C(K ε ) N(S, C , ε)dν(S) = = M vol(H 1 (α ′ )) 1 0 s 2n-1 ds √ 1-s 2 0 t 2q+1 dt H ε ′ 1 (0 q ) N(S, C , ε)d vol(T ) + o(ε 2 ), where ε ′ = ε ′ (s,t) = ε• √ s 2 +t 2 t
. Note that if we construct S ∈ C(K ε ) as described above then S contains exactly one configuration of type C and of width less than ε, so N(S, C , ε) can be replaced by the weight with which we count configurations.

This integral is a simplification of the integral from [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF], page 133. In [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] the statement is about the volume of C(K ε ) (in our notaton) as the authors only count configurations with weight one, so N(S, C , ε) = 1 on K ε . 2.2. Mean area of a cylinder. Fix an admissible configuration C in a connected component K of a stratum H 1 (α) that comes with q ≥ 1 cylinders and let p be a real number p ≥ 0.

Proof of Theorem 1. We choose and fix one of the named closed saddle connection of C that bounds a cylinder and call this cylinder the first one. We denote by N area p 1 (S, C , ε) the number of configurations on S of type C of length at most ε, counted with weight the p-th power of the area of the first cylinder. We denote the corresponding Siegel-Veech constant by c area p 1 (K, C ).

As described above, we decompose a surface S in C(K ε ) as sS ′ 1 and tT 1 . We use the usual parameters γ, h, h 1 , . . . , h q-1 ,t 1 , . . . ,t q to parametrize T 1 ∈ H ε 1 (0 q ). We assume the notation chosen so that the first cylinder of S corresponds to the first cylinder of T 1 . So the area of the first cylinder of S is t 2 h 1 w, where w = |γ|. We use the following weight:

t 2 h 1 w s 2 + t 2 p .
This weight is invariant under scaling of S and if the area s 2 + t 2 of S equals 1 then it reduces to (t 2 h 1 w) p which is the p-th power of the area of the first cylinder of S. A surface S ∈ C(K ε ) contains exactly one configuration of type C of width at most ε, so we need to evaluate the integral of equation [START_REF] Chen | Square-tiled surfaces and rigid curves on moduli spaces[END_REF] for

N(S, C , ε) = N area p 1 (S, C , ε) = t 2 s 2 + t 2 p (h 1 w) p .
The first step to do this is to show that Lemma 9.

H ε 1 (0 q ) (h 1 w) p d vol(T ) = 2πε 2 (p + 1) • (p + 2) • • •(p + q -1)
,

where for q = 1 the denominator is by convention equal to 1.

Proof. For q = 1 we have h 1 w = hw = 1, so the computation is a simplified version of the general case. Assume that q > 1. We will first integrate over the domain C(H ε 1 (0 q )), still using the parameters γ, h, h 1 , . . . , h q-1 ,t 1 , . . . ,t q for T ∈ C(H ε 1 (0 q )). To have a weight that is invariant under scaling of T and that becomes (h 1 w) p for an area one torus T 1 , we use the weight

h 1 w hw p = h 1 h p .
The domain of integration for T ∈ C(H ε 1 (0 q )) is as follows: The area wh of T satisfies wh ≤ 1. We have 1

√ wh T ∈ H ε 1 (0 q ), so the length w √ wh of the waist curve of 1 √ wh T is smaller than ε, which gives h ≥ w ε 2 . (See figure 4.) (3)
C(H ε 1 (0 q )) h 1 h p d vol(T ) = 2π 0 dθ ε 0 w dw 1/w w/ε 2 dh h 0 h 1 h p dh 1 • • ∆ q-2 (h-h 1 ) dh 2 . . . dh q-1 [0,w] q dt 1 . . . dt q . h w = |γ| ε h = 1 w h = w ε 2 FIGURE 4. Domain of integration for C(H ε 1 (0))
The volume of the cone

∆ q-2 (h -h 1 ) is (h -h 1 ) q-2 (q -2)!
and the volume of the cube [0, w] q is w q . Using the change of variables u = h 1 /h, the right hand side of equation ( 3) then evaluates to

πε 2 (q -2)! B(p + 1, q -1) q + 1 = πε 2 q + 1 1 (p + 1) • • •(p + q -1)
,

where B(•, •) is the Beta function defined by [START_REF] Kontsevich | Connected components of the moduli space of Abelian differentials with prescribed singularities[END_REF]. We get the right hand side using relations [START_REF] Masur | Interval exchange transformations and measured foliations[END_REF] and ( 9). The weight

f (S) = (h 1 /h) p satisfies f (rS) = f (S) and, if S 1 ∈ H ε 1 (0 q ), f (S 1 ) = (h 1 w) p , so C(H ε 1 (0 q )) h 1 h p d vol(T ) = = 1 0 r 2(q+1)-1 dr H ε 1 (0 q ) f (rS 1 vol(S 1 ) = 1 2(q + 1) H ε 1 (0 q ) (h 1 w) p d vol(S 1 ),
which completes the proof of the lemma.

To continue the proof of Theorem 1, we evaluate equation [START_REF] Chen | Square-tiled surfaces and rigid curves on moduli spaces[END_REF]. We use lemma 9

with ε ′ = ε • √ s 2 + t 2 t
and integrate the function

t 2 s 2 + t 2 p that is the remaining part of N area p 1 (S, C , ε). So equation (2) becomes C(K ε ) N area p 1 (S, C , ε)dν(S) = M • Vol(H 1 (α ′ )) • 2πε 2 (p + 1) • (p + 2) • • •(p + q -1) J p + o(ε 2 ),
where

J p = 1 0 s 2n-1 ds √ 1-s 2 0 t 2q+1 • t 2 s 2 + t 2 p-1
dt Using polar coordinates s = r cos θ , t = r sin θ we get

J p (C ) = 1 2(n + q + 1) π 2 0 (cos θ ) 2n-1 (sin θ ) 2p+2q-1 dθ . Using u = cos 2 θ we get (4) J p (C ) = B(n, q + p) 4(n + q + 1) = 1 4(n + q + 1) (n -1)! (p + q) • • •(p + q + n -1)
.

Using the fact that for

S ∈ C(K ε ), N area p (S, C , ε) = t 2 s 2 +t 2 p
is invariant under scaling of S and reduces to the initial definition of N area p (S 1 , C , ε) if S 1 ∈ K ε , we can show as in the proof of lemma 9 that

K ε N area p 1 (S 1 , C , ε)d vol(S 1 ) = 2(n + q + 1) C(K ε ) N area p 1 (S, C , ε)dν(S),
where 2(n + q + 1) = dim R H (α). And so

K ε N area p 1 (S, C , ε)d vol(S) = = M • πε 2 • Vol(H 1 (α ′ )) • (n -1)! (p + 1) • • •(p + q + n -1) + o(ε 2 ).
Using equation ( 1) we conclude that

c area p 1 (K, C ) = M • Vol(H 1 (α ′ )) Vol(K) • (n -1)! (p + 1) • • •(p + q + n -1)
.

Note that if we define N area p i (S, C , ε) in the same way as N area p 1 (S, C , ε), except that we use the area of the i-th cylinder, for i = 1, • • • , q, then we have

N area p (S, C , ε) = ∑ i N area p i (S, C , ε). It follows that c area p (K, C ) = qc area p 1 (K, C
), which completes the proof of Theorem 1.

Remark. (a)

We note that the volume of the stratum H 1 (α ′ ) of disconnected surfaces was computed in [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF], equation [START_REF] Masur | Lower Bounds for the number of saddle connections and closed trajectories of a quadratic differential[END_REF]; writing

H (α ′ ) = Π p i=1 H (α ′ i ) and n i = dim C H (α ′ i ), we have Vol(H 1 (α ′ )) = 1 2 p-1 • ∏ p i=1 (n i -1)! (n -1)! • p ∏ i=1 Vol(H 1 (α ′ i )).
(b) As an example we consider the moduli space of tori. To have a configuration we need to mark one regular point. The only possible topological type C of configuration is a closed saddle connection based at this regular point. We get, using Lemma 9 for p = 0 and q = 1,

c conf (H 1 (0), C ) = lim ε→0 1 πε 2 • Vol(H ε 1 (0)) Vol(H 1 (0)) = 1 πε 2 • 2πε 2 π 2 /3 = 6 π 2 = 1 ζ (2)
, which is the well-known factor for the proportion of coprime lattice points in Z ⊕ Z.

2.3.

Mean area of the periodic region. Fix an admissible topological type of configuration C for a connected component K of a stratum H 1 (α) that comes with q ≥ 1 cylinders. Recall that N area p ,conf (S, C , L) denotes the p-th power of the total area of the periodic region (union of the cylinders) on S coming from a configuration of topological type C whose length is at most L.

Proof of theorem 4. The argument is a special case of the argument in section 2.2. By decomposing as before a surface S in C(K ε ) as sS ′ 1 and tT 1 we use the weight

N area p ,conf (S, C , ε) = t 2 s 2 + t 2 p .
So by taking p = 0 in lemma 9, (5)

H ε 1 (0 q ) d vol(T ) = 2πε 2 (q -1)! .
We then have

C(K ε ) N area p ,conf (S, C , ε)dν(S) = M • Vol(H 1 (α ′ )) • 2πε 2 (q -1)! J p + o(ε 2 ),
where J p satisfies relation [START_REF] Dutka | The incomplete beta function -a historical profile[END_REF]. We conclude as in section 2.2 that

K ε N area p ,conf (S, C , ε)d vol(S) = M • πε 2 • Vol(H 1 (α ′ )) • (n -1)! (q -1)! • (p + q) • • •(p + q + n -1) + o(ε 2 ).
It suffices to apply relation (1).

2.4.

Configurations with periodic regions of large area. Fix an admissible configuration C in a connected component K of a stratum H 1 (α) that comes with q ≥ 1 cylinders and let x ∈ [0, 1) be a real parameter. Recall that N conf,A≥x (S, C , ε) denotes the number of configurations on S of type C of length smaller than ε and such that the total area of the q cylinders is at least x (of the area one surface S).

Proof of Theorem 6. We use a modification of the argument from section 2.2. Here we consider the subset of K ε consisting of surfaces S whose area of the periodic part is at least x (of the area 1 surface S). Construct S in the cone of this set using sS ′ 1 and tT 1 for some scalars s,t and S ′ 1 ∈ H 1 (α ′ ) and T 1 ∈ H ε ′ 1 (0 q ). We need to evaluate the integral in equation ( 2) with an additional condition : when scaling S by 1 s 2 +t 2 we get a surface of area one that satisfies

area t √ s 2 + t 2 T 1 = t 2 s 2 + t 2 > x ⇐⇒ t ≥ x 1 -x s.
We count a configuration that satisfies this additional constraint with weight 1, so equation (2) becomes [START_REF] Eskin | Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow[END_REF], the right hand side becomes

C(K ε ) N conf,A>x (S, C , ε)dν(S) = M vol(H 1 (α ′ )) 1 0 s 2n-1 ds √ 1-s 2 √ x 1-x s t 2q+1 dt H ε ′ 1 (0 q ) d vol(T ) + o(ε 2 ), where ε ′ = ε • √ s 2 + t 2 t . Using relation
M • 2πε 2 (q -1)! • Vol(H 1 (α ′ )) • I x + o(ε 2 )
where

I x = √ 1-x 0 s 2n-1 √ 1-s 2 √ x 1-x s t 2q+1 s 2 + t 2 t 2
dtds. The domain of integration for s and t is described in Figure 5.

s t 1 1 0 √ 1 -x √ x FIGURE 5. Domain of integration.
Using polar coordinates s = r cos θ , t = r sin θ and setting α = arccos √ 1x we get

I x = 1 2(n + q + 1) π 2 α (cos θ ) 2n-1 (sin θ ) 2q-1 dθ .
Using the change of variables u = cos 2 θ we get

I x = 1 4(n + q + 1) 1-x 0 u n-1 (1 -u) q-1 du = B(1 -x; n, q) 4(n + q + 1) ,
where B(•; •, •) is the incomplete Beta function as defined in equation [START_REF] Masur | Lower Bounds for the number of saddle connections and closed trajectories of a quadratic differential[END_REF]. Note that we have c conf (K, C ) = c conf,A≥0 (K, C ), so up to the same constant, the integral used to compute c conf,A≥x (K, C ), resp c conf (K, C ) is given by I x , resp I 0 (or equivalently J 0 , see equation ( 4)), so

c conf,A≥x (K, C ) c conf (K, C ) = I x I 0 = I(1 -x; n, q).
We proved Theorem 6 of section 1.2, where, to expand I(1x; n, q), we use Lemma 14.

2.5. Configurations with a cylinder of large area. Fix an admissible configuration C in a connected component K of a stratum H 1 (α) that comes with q ≥ 1 cylinders and let x be a parameter that satisfies 0 ≤ x < 1.

We choose and fix one of the named closed saddle connection of C that bounds a cylinder and call this cylinder the first one. Recall that N conf,A 1 ≥x (S, C , L) is the number of configurations of type C of length at most L and such that the area of the first cylinder is at least x (of the area one surface S).

Proof of theorem 5. Suppose that we have q ≥ 1 cylinders. The argument is as in section 2.4 with the following modification: We replace "area" by "area of the first cylinder" in the condition area

t √ s 2 +t 2 T 1 = t 2 s 2 +t 2 > x. Denote by Cusp a (ε ′ ) the subset of H ε ′ 1 (0 q )
of tori T 1 such that the area of the first cylinder of T 1 is at least a = x s 2 +t 2 t 2 . Using the usual parameters (γ, h, h i ,t j ), a torus T is in the cone of Cusp a (ε ′ ) if the area one torus 1 √ wh T is in Cusp a (ε ′ ), so the parameters for T have the additional constraint wh 1 wh > a. To compute the volume of Cusp a (ε ′ ) we proceed as in the proof of lemma 9 for x = 0. The only difference is that we replace the condition 0 ≤ h 1 ≤ h by ah ≤ h 1 ≤ h. So (h 1 , . . . , h q-1 ) is in a cone whose volume is ((1-a)h) q-1 (q-1)! . So the volume is as in equation ( 5) except that we have an extra factor (1a) q-1 . So we need to integrate (6)

I ′ x = √ 1-x 0 s 2n-1 √ 1-s 2 √ x 1-x s t 2q+1 (1 -a) q-1 s 2 + t 2 t 2 dtds.
Using polar coordinates s = r cos θ , t = r sin θ followed by the change of variables w = cos 2 θ

1-x we get (7)

I ′ x = (1 -x) n+q-1 4(n + q + 1) B(n, q). Note that c conf (K, C ) = c conf,A 1 ≥0 (K, C ), so c conf,A 1 ≥x (K,C ) c conf (K,C ) = I ′ x I ′ 0 as claimed. (Recall that dim C H (α) = n + q + 1.)
Remark. For q = 1 we have c conf,A 1 ≥x (K, C ) = c conf,A≥x (K, C ) (see Theorem 6).

2.6. Correlation between the area of two cylinders. Let C be an admissible configuration for a connected component K that comes with at least two cylinders. Choose (and fix) two cylinders and let x, x 1 ∈ [0, 1). Recall that we denote by N A 2 ≥x,A 1 ≥x 1 (S, C , L) the number of configurations of length at least L such that the area A 1 of the first cylinder is at least x 1 and such that the area A 2 of the second cylinder is at least x(1 -A 1 ). We denote by c A 2 ≥x,A 1 ≥x 1 (K, C ) the corresponding Siegel-Veech constant. To simplify notation we will write

c A 1 ≥x 1 (K, C ) instead of c conf,A 1 ≥x 1 (K, C ).
Proof of theorem 7. We proceed as in the proof of Theorem 5 (see section 2.5). Using the same notation, we have the condition that the area Ã1 of the first cylinder of t √ s 2 +t 2 T 1 is at least x 1 and the area Ã2 of the second cylinder of t √ s 2 +t 2 T 1 is at least x(1 -Ã1 ). This means that the area A 1 of the first cylinder of T 1 satisfies A 1 ≥ a 1 = x 1 s 2 +t 2 t 2 and the area A 2 of the second cylinder is at least x(1 -Ã1 ) s 2 +t 2 t 2 which gives A 2 ≥ x s 2 +t 2 t 2 -A 1 = a -xA 1 , where a = x s 2 +t 2 t 2 . Denote by Cusp a 1 ,a (ε ′ ) the subset of tori in H ε ′ 1 (0 q ) that satisfies these conditions. Using the usual parameters (γ, h, h i ,t j ), a torus T is in the cone C(Cusp a 1 ,a (ε ′ )) if and only if T 1 = 1 √ wh T is in Cusp a 1 ,a (ε ′ ). So the areas A 1 and A 2 of the first and second cylinders of T 1 must satisfy A 1 = wh 1 wh ≥ a 1 and A 2 = wh 2 wh ≥ ax wh 1 wh . So we get

a 1 h ≤ h 1 ≤ h, ( * ) ah -xh 1 ≤ h 2 ≤ h -h 1 . ( * * )
Equation ( * * ) has a solution if and only if ahxh 1 ≤ hh 1 which can be written as h 1 ≤ 1-a 1-x h so we need to modify ( * ):

a 1 h ≤ h 1 ≤ 1 -a 1 -x h. ( * ′ )
Equation ( * ′ ) has a solution if and only if a 1 ≤ 1-a 1-x . This translates into x 2 s 2 +t 2 t 2 ≤ 1, where

x 2 = x + x 1 (1 -x), which in turn becomes t ≥ x 2 1 -x 2 s.
We also have 0 ≤ h 1 ≤ h and 0 ≤ h 2 ≤ hh 1 . But for all possible t, s we have a 1 ≥ 0, ahxh 1 ≥ 0, and 1-a 1-x ≤ 1, so h 1 can take all values between a 1 h and 1-a 1-x h and h 2 can take all values between ahxh 1 and hh 1 .

The computation of the volume of Cusp a 1 ,a (ε ′ ) is as in the proof of Lemma 9 for x = 0, except that we replace

∆ q-1 (h) dh 1 . . . dh q-1 = h q-1 (q -1)! by 1-a 1-x h a 1 h dh 1 h-h 1 ah-ph 1 dh 2 ∆ q-3 (h-h 1 -h 2 ) dh 3 . . . dh q-1 = [(1 -a) -(1 -x)a 1 ] q-1 1 -x h q-1 (q -1)! = (1 -a 2 ) q-1 1 -x h q-1 (q -1)! ,
where a 2 = x 2 s 2 +t 2 t 2 . So the volume is as in equation ( 5) except that we have an extra factor

(1-a 2 ) q-1 1-x
. So we need to integrate

1 1 -x √ 1-x 2 0 s 2n-1 √ 1-s 2 x 2 1-x 2 s t 2q+1 (1 -a 2 ) q-1 s 2 + t 2 t 2 dtds.
Note that up to the factor 1/(1x) this is I ′ x 2 as defined in [START_REF] Eskin | Asymptotic formulas on flat surfaces[END_REF]. The integral used to compute

c X 1 ≥x 1 (K, C ) = c conf,X 1 ≥x 1 (K, C ) is I ′ x 1 , so we conclude c A 2 ≥x,A 1 ≥x 1 (K, C ) c A 1 ≥x 1 (K, C ) = I ′ x 2 1 -x • 1 I ′ x 1
.

Using equation ( 7) and 1 -

x 2 = (1 -x)(1 -x 1 ) we find c A 2 ≥x,A 1 ≥x 1 (K, C ) c A 1 ≥x 1 (K, C ) = (1 -x 2 ) n+q-1 (1 -x)(1 -x 1 ) n+q-1 = (1 -x) n+q-2 .

EXTREMAL PROPERTIES OF CONFIGURATIONS

3.1. Maximal total mean area of a configuration. Consider an admissible topological type C of configuration for some connected component K of some stratum H (α), where α = (d 1 , . . . , d m ) satisfies d i ≥ 1, for i = 1, . . . , m. We denote by q(C ) the number of cylinders that come with C . In this section we prove Theorem 8, so we look for a topological type of configuration C (that is admissible for some connected component K) that maximizes

c mean area conf (K, C ) = c area (K, C ) c conf (K, C ) = q(C ) 2g + m -2 ,
where for the last equality we used Corollary 2 and c cyl (K, C ) = q(C )c conf (K, C ).

For a given stratum H (α), we start by determining the maximal possible number of cylinders q max (α) that can come from an admissible topological type of configuration for H (α) :

q max (α) = max C in H (α) q(C ).
It is shown in [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] that topological types of configurations can be constructed by creating singular points of the following three types:

(a) a cylinder, followed by k ≥ 1 surfaces S i of genus g i ≥ 1 with figure eight boundary, followed by a cylinder. See figure 6 for k = 3 and g i = 1, i = 1, 2, 3. We say that the newborn singularity is of type I. (b) a cylinder, followed by k ≥ 0 surfaces S i of genus g i ≥ 1 with figure eight boundary, followed by a surface S k+1 of genus g k+1 ≥ 1 with a pair of holes boundary. See Figure 7 for k = 2 and g i = 1, i = 0, 1, 2. We say that the newborn singularity is of type II. For k = 0 we just have a cylinder followed by a surface with a pair of holes boundary. We might also reverse the order: a pair of holes torus followed by k ≥ 0 figure eight tori followed by a cylinder. (c) a singularity of type III, which is obtained from a singularity of type II by replacing the cylinder by a surface with a pair of holes boundary. As for surfaces of type II there might be no surface with a figure eight boundary. See Figure 8 where all of the surfaces are of genus 1 and where we have two surfaces with figure eight boundary.

′ + 1) + ∑ k i=1 (a i + 2) + (b ′′ 2 + 1)
. All orders bigger or equal to 2 are possible. For a given small γ one can create all of the boundaries of the surfaces involved in the above construction by an appropriate figure eight surgery or a pair of holes surgery (see Figures 2 and3) such that the boundaries all have holonomy vector γ.

By arranging the blocks in a cyclic order and identify boundary components we create an admissible topological type of configuration of homologous saddle connections where each saddle connection is based at a newborn singularity of one of the three types. There is only the following obstruction: one needs to either use at least one surface with a pair of holes boundary or, if one only uses surfaces with figure eight boundaries, then one needs to use at least one cylinder.

Each cylinder is bounded by two saddle connections, so to have a one to one correspondence we think of the cylinder as being cut into two parts by the central waist curve. Each half cylinder then has a saddle connection γ on the boundary (that is not the waist curve) that joins a saddle P to itself. We say that P accounts for this half-cylinder. A zero of type I (see Figure 6) accounts for two half-cylinders (so for one cylinder), a zero of type II (see Figure 7) accounts for one half of a cylinder, and a zero of type III (see Figure 8) does not account for any half cylinder. Note that only singularities of type II can have order 1. It follows that if for n ∈ N we define

χ(n) = 1/2, if n = 1 1, if n > 1 then we have q max (d 1 , . . . , d k ) ≤ ∑ m i=1 χ(d i ).
To maximize the number of cylinders we construct the zeros of orders greater than or equal to 2 by zeros of type I. The other zeros of order 1 must be created by zeros of type II (coming from a cylinder followed by a torus with a pair of holes boundary). This works well if there is an even number of zeros of order 1 in which case we constructed a configuration with ∑ n i=1 χ(d i ) cylinders. If there is an odd number of zeros of order 1 then the construction of the zeros of order 1 ends with a surface with a pair of holes boundary. In this case we are obliged to construct one of the zeros of order greater than 1 by a surface of type II, so we constructed a configuration with ∑ m i=1 χ(d i ) -1/2 cylinders. We showed Proposition 10. Consider a stratum H (α), where α = (d 1 , . . . , d m ) satisfies d i ≥ 1, for i = 1, . . . , m. We have

(8) q max (α) = n ∑ i=1 χ(d i ) ,
where the square brackets denote the integer part of a number.

Remark. The proposition says that it is possible to find in each stratum H (α) a topological type of configuration C such that q(C ) = q max (α) is as stated. It is not possible to find a topological type C in each connected component of H (α) with q max (α) cylinders.

We next show that given any connected component K of a stratum H (α) and any admissible topological type C of configuration for K,

c mean area conf (K, C ) ≤ 1 3 . Proof. Let g ≥ 2.
Denoting by ℓ(α) the length of α we have

c mean area conf (K, C ) = q(C ) 2g -2 + ℓ(α) ≤ q max (α) 2g -2 + ℓ(α) .
We determine max α∈Π(2g-2)

q max (α) 2g -2 + ℓ(α)
,

where Π(2g -2) denotes the set of permutations of 2g -2. First let us prove that a partition α containing at least one entry 1 is not maximizing. Indeed, if there is at least one pair of entries 1 we can modify the initial partition α by replacing the elements 1, 1 with a single entry 2. This does not change the value (8) of q max (α), but decreases the denominator in the ratio q max /(2g -2 + ℓ(α)).

If there is a single entry 1 in the partition α, then ∑ n i=1 χ(d i ) is not an integer. We can modify α by deleting the entry 1 and increasing some other entry by 1. This operation does not change the value (8) of q max (α), but decreases the denominator in the ratio q max /(2g -2 + ℓ(α)).

Thus we have proved that all of the entries of the partition maximizing the ratio q max (α)/(2g-2 + ℓ(α)) are greater then or equal to 2. The formula [START_REF] Eskin | Asymptotics of number of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials[END_REF] for such partitions simplifies to q max (α) = ℓ(α). Now note that for any g ≥ 2 the function

f g (x) = x 2g -2 + x = 1 - 2g -2 2g -2 + x
is strictly decreasing. Thus, among all partitions of 2g -2 with entries strictly greater than one we have to chose the one maximizing ℓ(α). This is the partition (2, . . . , 2) where the order 2 appears g -1 times. For this partition we get q max (2, . . . , 2) = g -1 = ℓ(α) and so c mean area conf (K, C ) is bounded from above by 1/3 as claimed.

Proof of Theorem 8. It suffices to show that for each genus g there is an admissible topological type of configuration C for a connected component K of the stratum H (2, . . . , 2) (g -1 zeros of order 1) such that c mean area conf (K, C ) attains the upper bound 1/3.

Note that for the upper bound 1/3 for c mean area conf (K, C ) we used q max . As we explained in the proof of the relation [START_REF] Eskin | Asymptotics of number of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials[END_REF], the only way to obtain q max (2, . . . , 2) is to only use zeros of type I (a cylinder followed by a torus with a figure eight boundary that was created at a regular point followed by a cylinder). Doing this we obtain a surface in H (2, . . . , 2) with a configuration that comes with g -1 cylinders.

By Lemma 14.2 in [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] the surface in H (2, . . . , 2) we constructed that comes with a maximizing configuration has odd parity of spin structure, so this surface is in H odd (2, . . . , 2). (We recall in the next section the classification of connected components.) Proposition 8 is proved.

3.2. Configurations with simple complementary regions. This section provides an answer to the following question of Alex Eskin and Alex Wright: is it possible to find in each connected component of a stratum an admissible topological type of configuration whose complementary regions are tori (with boundary) and cylinders. Motivations for this problem can be found in [START_REF] Wright | Cylinder deformations in orbit closures of translation surfaces[END_REF].

The answer depends on the connected component, so we need to recall the classification of connected components for strata H (α) of Abelian differentials from [START_REF] Kontsevich | Connected components of the moduli space of Abelian differentials with prescribed singularities[END_REF]. Some connected components are characterized by the fact that they only contain hyperelliptic surfaces. For a surface S in a stratum H (d 1 , . . . , d n ) where all d i are even one has the notion of parity of spin structure that is either 0 or 1. We then have Theorem (M. Kontsevich, A. Zorich [START_REF] Kontsevich | Connected components of the moduli space of Abelian differentials with prescribed singularities[END_REF]). Let H (d 1 , . . . , d m ) be a stratum of Abelian differentials on a surface of genus g ≥ 4. The strata H (2g -2) and H (g -1, g -1) are the only strata to have a hyperelliptic component H hyp (2g -2), resp. H hyp (g -1, g -1). Apart from these hyperelliptic components we have:

(a) If at least one of the d i is odd then there is only one non-hyperelliptic component. (b) If all of the α i are even then there are two non-hyperelliptic components, H even with even and H odd with odd parity of spin structure.

Remark. The classification in [START_REF] Kontsevich | Connected components of the moduli space of Abelian differentials with prescribed singularities[END_REF] also covers g = 2, 3 but we will not need this.

Proposition 11. Let H comp (α) denote a connected component of a stratum of Abelian differentials on a surface of genus g ≥ 5.

Then:

(a) If H comp (α) is hyperelliptic (so H hyp (2g -2) or, if g -1 is even, H hyp (g -1, g -1)) then it is not possible to find an admissible topological type of configuration whose complementary regions are only tori (with boundary) and cylinders. (b) If g is even and H comp (α) = H even (α) then we can find a topological type of configuration whose complementary regions are tori, cylinders and one surface of genus two. But it is not possible to only have tori and cylinders. (c) In all remaining connected components one can explicitly construct an admissible topological type of configuration whose complementary regions are tori and cylinders.

Proof. First note that a configuration containing only tori and cylinders, or tori, cylinders, and at most one genus two surface does not occur in hyperelliptic strata, since a hyperelliptic surface can contain at most two closed homologous saddle connections, which rules out g ≥ 5. Lemma 14.5 in [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF] describes precisely configurations in hyperelliptic components of strata.

We suppose for what follows that H comp (α) is not a hyperelliptic component.

Recall the description of singularities of types I, II, and III from the previous section. If apart from cylinders, we only use surfaces of genus 1, then we must do figure eight surgeries and pair of holes surgeries at regular marked points. So we have: (a) A zero of type I has order 2k, where k ≥ 1 is the number of tori with figure eight boundaries (see Figure 6). (b) A zero of type II has order 2k + 1, where k ≥ 0 is the number of tori with figure eight boundaries (see figure 7). (c) A zero of type III has order 2k + 2, where k ≥ 0 is the number of tori with figure eight boundaries (see figure 8).

Lemma 14. The incomplete Beta function satisfies

B(1x, n, q) = (1x) n B(n, q) q-1 ∑ l=0 n + l -1 l x l .

Proof.

n+q-1

∑ k=n n + q -1 k (1 -x) k x n+q-1-k = (1 -x) n q-1 ∑ k=0 n + q -1 n + k (1 -x) n+k x q-1-k = (1 -x) n q-1 ∑ k=0 k ∑ j=0 n + q -1 n + k k j (-1) k-j x q-1-j = (1 -x) n q-1 ∑ l=0 q-1 ∑ k=0 n + q -1 n + k k q -1 -l (-1) k+l+q+1 x l
It suffices to apply Lemma 13

  FIGURE 2. Figure eight construction

FIGURE 3 .

 3 FIGURE 3. Creating a pair of holes

FIGURE 6 .FIGURE 7 .FIGURE 8 .

 678 FIGURE 6. Block of surfaces creating a zero of type I

  Corollary 2. Given a real number p ≥ 0 and let H (α) be a stratum of Abelian differentials on a surface M g,m , where α = (d 1 , . . . , d m ) with d i ≥ 1, for i = 1, . . . , m.

	Then for any
	connected component K of H 1 (α) and any admissible type C of configuration containing
	at least one cylinder,

Acknowledgements. We would like to thank Anton Zorich for having initiated the present work. He asked most of the questions and initiated some answers. We thank Alex Wright for the formulation of the problem of section 3.2, and for some comments on typos. We thank the anonymous referee(s) for the careful reading of the manuscript. Both authors thank ANR "GeoDyM" for financial support.

partially supported by ANR "GeoDyM".

Assume that α contains at least one odd d i , so α = (2a 1 , • • • , 2a p , 2b 1 + 1, • • • , 2b r + 1) with p ≥ 0 and r ≥ 1 (p = 0 corresponds to the case when all α i are odd).

We construct blocs of surfaces that contain zeros of type II of orders 2b 1 +1,. . . , 2b r +1. Note that since ∑ i d i is even, the number r of odd d i is even, so in our construction the first and last surface is a cylinder. If there is at least one zero of even order then we construct in addition blocs of surfaces that contain zeros of type I of orders 2a 1 ,. . . ,2a p . For this construction the first and last surface is also a cylinder. Arranging the blocs in a cyclic order and identifying cylinders we get in each case an admissible topological type of configuration for H (α) whose complementary regions are cylinders and tori.

Suppose now that all d i in H (α) are even, so we can not have a zero of type II. We then either have only zeros of type I or only zeros of type III as having zeros of types I and III necessarily implies that we have a zero of type II.

It is easy to verify ( §14.1 in [START_REF] Eskin | Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems, and the Siegel-Veech Constants[END_REF]) that if we only have zeros of type I then the corresponding surface has an odd parity of spin structure and if we only only have zeros of type III then the parity of the resulting surface is the parity of g -1. So for g odd we are done, but for even g the parity of the spin structure is 1 in both cases.

If we only have zeros of type III but replace one of the pair of holes boundary tori with a genus 2 surface with a pair of holes boundary then the parity of the spin structure of the resulting surface is the parity of g. So for even g we constructed a surface of parity 0. This completes the proof.

TOOLBOX

We recall some well known facts about the Beta function and incomplete Beta function. The (real) Gamma function is defined for each t > 0 by

It satisfies

The Incomplete Beta function defined for real positive numbers t, a, b by

The regularized incomplete Beta function is defined as

See figure 9 for the density function f (t) = d dt I(t; a, b) for various values for a and b. See [START_REF] Dutka | The incomplete beta function -a historical profile[END_REF] for a historical development of the incomplete Beta function.

Lemma 12.

Taking x = 1 we get [START_REF] Masur | The growth rate of trajectories of a quadratic differential[END_REF].

Taking the derivative of the recurrence relation with respect to x we get Ĩx (A, B) = xI ′

x (A, B) and hence Ĩx (A, B + 1) = -xI x (A, B) + (1x) Ĩx (A, B). Taking x = 1 we get [START_REF] Masur | Rational billiards and flat structures[END_REF].

Lemma 13. For q ≥ 0 and l ≤ q the value of

Proof. Using the recurrence relation on binomial coefficients one obtains Î(n, q + 1, l) = Î(n, q, l). So we have Î(n, q + 1, l) = Î(n, l, l). Noting that Î(n, l, l) = Ĩ(n, l + 1) we get the desired result.