Thomas Morzadec

Geometric compactification of moduli spaces of half-translation structures on surfaces.

1 Introduction.

The goal of this paper is to construct and to describe a geometric compactification, natural under the action of the mapping class group, of the space of homothety classes of halftranslation stuctures on a compact surface, endowed with the equivariant Gromov topology. It is part of the wide field of study of deformations of geometric structures on surfaces (see for instance [Gol]). Let Σ be a compact, connected, orientable surface of genus g > 2, without boundary for simplicity (see [START_REF] Morzadec | Compactification géométrique de l'espace des modules des structures de demi-translation sur une surface[END_REF]) for the general case). After the founding fathers Gauss and Riemann who have studied conformal geometry on surfaces, the Teichmüller spaces T (Σ) of isotopy classes of hyperbolic metrics on Σ, have been studied by for instance Fricke, Klein, Fenchel, Nielsen, and the moduli spaces of real projective structures by for instance Goldman and Choi [CG]. The analysis of the space Flat(Σ) of half-translation structures on a Σ is currently blooming, with the works notably of Calta, Eskin, Hubert, Lanneau, Masur, McMullen, Möller, Myrzakhani, Schmidt, Smillie, Veech, Weiss, Yoccoz and Zorich. When these deformations spaces are non compact, it is important and useful to consider the asymptotic behavior of the sequences of geometric structures that leave all compact subsets. Only few results are known about the compactifications of spaces of geometric structures, except for the Teichmüller space, for which several compactifications have been built, notably by Thurston (see [FLP]), and also by Bestvina, Morgan, Paulin, Shalen. A compactification of Flat(Σ) has been recently proposed in [DLR]. This article aims at proposing a new one.

Let Σ be a connected, orientable surface. A half-translation structure (or flat structure with conical singularities and holonomies in {± Id}) on Σ is the data consisting of a (possibly empty) discrete subset Z of Σ and of a Euclidean metric on Σ -Z with conical singular points of angles of the form kπ, with k ∈ N and k 3, at each point of Z, such that the holonomy of every piecewise C 1 loop of Σ -Z is contained in {± Id}. As the set Flat(Σ) of isotopy classes of half-translation structures on Σ identifies with the quotient by SO(2) of the set of isotopy classes of holomorphic quadratic differentials on Σ, we will denote by [q], with q a holomorphic quadratic differential on Σ, a half-translation structure on Σ. We refer for instance to [Wri, Zor] and Section 2.1 for background on half-translation structures.

Assume that Σ is compact and that χ(Σ) < 0. Let us define the mixed structures on Σ, which are CAT(0) tree-graded spaces in the sense of Drutu-Sapir, arising, as we shall see, as geometric degenerations of elements of Flat(Σ). Recall that a tree-graded space (see [START_REF] Druţu | Tree-graded spaces and asymptotic cones of groups[END_REF]Def. 1.10]) is the data of a complete geodesic metric space X and of a covering P of X by closed geodesic subsets of X, whose elements are called pieces, such that :

• two distinct pieces have at most one common point ;

• any simple geodesic triangle of X is contained in a single piece.

We will need the following definitions. Let p : Σ → Σ be a universal cover of Σ with covering group Γ. Let Σ 0 be a tight subsurface of Σ, that is a proper closed subsurface with smooth boundary, such that no connected component of Σ 0 is a disk or a pair of pants, no connected component of Σ-Σ 0 is a disk, and no cylinder component of Σ 0 can be homotoped in another connected component of Σ 0 . Let W be a connected component of Σ 0 or of Σ -Σ 0 and W be a connected component of the preimage of W in Σ. We associate to W a complete geodesic metric space X W of the following type :

• if W is a strip, then X W is empty ;

• if W is a connected component of Σ 0 which is not a strip or if W is a connected component of Σ -Σ 0 and W is a pair of pants, then X W is a point ;

• if W is a connected component of Σ -Σ 0 and W is neither an annulus nor a pair of pants, -either X W is the completion of the lift to W of a half-translation structure [q W ] on W which extends to the filled in surface, possibly with singularities of angle π.

-or X W is the dual R-tree to a filling measured hyperbolic lamination on W (endowed with any complete hyperbolic metric, see for instance [START_REF] Bonahon | Geodesic laminations on surfaces[END_REF] and [START_REF] Otal | Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3[END_REF]§ 2.3] and Subsection 6.1 for the definitions).

Finally, to every proper homotopy class c of boundary components of Σ 0 , we associate a compact interval X c of R, called an edge. We refer to the picture of Section 7 for a graphical understanding of the following definition. Definition 1.1 A mixed structure on Σ is a metric space X such that there exists a (possibly empty) tight, proper subsurface Σ 0 , as above, such that X is obtained by gluing some complete metric spaces X W and X c , where W is a connected components of Σ 0 or of Σ -Σ 0 , and c is a proper homotopy class of boundary components of Σ 0 , as above (see Section 7 for the precisions), such that :

• if W is a strip of Σ-Σ 0 , and if c is the proper homotopy class of the boundary components of W , then the length of X c is positive ;

• if W is a connected component of Σ -Σ 0 whose image in Σ is a pair of pants, there exists at least a proper homotopy class c of boundary components of W such that the length of X c is positive.

We will see (see Section 7) that a mixed structure is a CAT(0) tree-graded space, and that the actions of the stabilizers of W in Γ on the pieces X W and X c as above glue together to give an isometric action of Γ on X. We will see that the action of Γ on X defines the subsurface Σ 0 , up to isotopy. Moreover, a mixed structure is defined (up to Γ-equivariant isometry) by the half-translation structures and the measured hyperbolic laminations on the connected components of Σ -Σ 0 , and by the lengths of the edges, that define its pieces (see Section 7).

Let Mix(Σ) be the space of Γ-equivariant isometry classes of mixed structures on Σ. We endow it with the equivariant Gromov topology (see [START_REF] Paulin | The Gromov topology on R-trees[END_REF][START_REF] Paulin | Sur la compactification de Thurston de l'espace de Teichmüller. dans "Géométries à courbure négative ou nulle, groupes discrets et rigidités[END_REF] and Section 7) which is the topology such that two elements are close if they contain large finite subsets which are almost isometric in an equivariant way under a large finite subset of Γ. Let P Mix(Σ) be the space of homothety classes of mixed structures on Σ, endowed with the quotient topology. We identify Flat(Σ) with the space of Γ-equivariant isometry classes of Γ-invariant half-translation structures on Σ, which is a subset of Mix(Σ). Let P Flat(Σ) be its image in P Mix(Σ). The two main results of this paper are the following ones.

Theorem 1.1 The space P Flat(Σ) is an open and dense subset of P Mix(Σ), and P Mix(Σ) is compact. The action of the mapping class group of Σ on P Flat(Σ) extends continuously to P Mix(Σ) Let (X, d) be a metric space with an isometric action of a group Γ. For all γ ∈ Γ, the translation distance of γ in X is X (γ) = inf x∈X d(x, γx). Let PR Γ = ((R + ) Γ -{0})/R + * . We denote by [X] the image of X ∈ Mix(Σ) in P Mix(Σ) and by [x γ ] γ∈Γ the image of (x γ ) γ∈Γ ∈ (R + ) Γ -{0} in PR Γ .

Theorem 1.2 The map [X] → [ X (γ)] γ∈Γ is an embedding of P Mix(Σ) onto its image.

In Section 2, we recall some definition and basic facts about half-translation structures on a surface. In Section 3, we give some results about the action of the covering group on the universal cover of a surface endowed with a half-translation structure. In Section 4, we recall two tools that we will need in this paper, which are the subsurfaces filled by a family of homotopy classes of simple closed curves and the generalized subsurfaces with geodesic boundary, for a half-translation structure. In Section 5, we recall what are the ultralimits of sequences of metric spaces in Gromov's sense. In Section 6, we study the ultralimits of universal covers of sequences of half-translation structures on a surface. Finally, in Section 7, we define the mixed structures, we recall the definition of the equivariant Gromov topology, and we prove the above two theorems. Using Theorem 1.2, we construct a mapping class group-equivariant homeomorphism between our compactification of P Flat(Σ) and the one constructed by Duchin-Leininger-Rafi in [DLR].

Our approach is fundamental in understanding in a geometric way the degenerations of half-translation structures, and allow an extension (work in progress) to flat structures with conical singularities and finite holonomy groups (for instance the quotient by SO(2) of the set of isotopy classes of cubic holomorphic differentials on Σ, whose moduli spaces have started to be studied for instance by , by Labourie [Lab] and Loftin [Lof]).

Acknoledgement : I want to thank Frédéric Paulin for many advices and corrections that have deeply improved the writting of this paper.

2 Notation and background.

Half-translation structures.

In the whole paper, we will use the definitions and notation of [START_REF] Bridson | Metric spaces of non-positive curvature[END_REF] for a surface endowed with a distance (Σ, d) : (locally) CAT(0), δ-hyperbolic,... Notably, a geodesic (resp. a local geodesic) of (Σ, d) is an isometric (resp. locally isometric) map : I → Σ, where I is an interval of R. It will be called a segment, a ray or a geodesic line of (Σ, d) if I is respectively a compact interval, a closed half line (generally [0, +∞[) or R. If there is no precision, a geodesic is a geodesic line. Let Σ be a connected, orientable surface, with empty boundary. Definition 2.1 A half-translation structure (or flat structure with conical singularities and holonomies in {± Id}) on a surface Σ is the data of a (possibly empty) discrete subset Z of Σ and a Euclidean metric on Σ -Z with conical singularity of angle k z π at each z ∈ Z, with k z ∈ N, k z 3, such that the holonomy of every piecewise

C 1 loop in Σ -Z is contained in {± Id}.
As the set Flat(Σ) of isotopy classes of half-translation structures on Σ identifies with the quotient by SO(2) of the set of isotopy classes of holomorphic quadratic differentials on Σ, we will denote by [q], with q a holomorphic quadratic differential on Σ, a half-translation structure on Σ (see [START_REF] Morzadec | Laminations géodésiques plates. À paraître dans Ann. Institut Fourier[END_REF]§ 2.5] and [START_REF] Strebel | Quadratic Differentials[END_REF]Def. 1.2 p. 2] for the definition of a holomorphic quadratic differential). A half-translation structure defines a geodesic distance d on Σ that is locally CAT(0). We will call local flat geodesic a local geodesic of a half-translation structure. A continuous map : R → Σ is a local flat geodesic if and only if it satisfies (see [START_REF] Strebel | Quadratic Differentials[END_REF]Th. 5.4 p.24] and [START_REF] Strebel | Quadratic Differentials[END_REF]Th. 8.1 p. 35]) : for every t ∈ R,

• if (t) does not belong to Z, there exists a neighborhood V of t in R such that |V is a Euclidean segment (hence, |V has constant direction) ; • if (t) belongs to Z, then the two angles defined by ([t, t + ε[) and (]t -ε, t]), with ε > 0 small enough, measured in the two connected components of U -(]t -ε, t + ε[), with U a small enough neighborhood of (t), are at least π. 00 00 11 11 π π Let C (Σ) and S (Σ) be the sets of free homotopy classes of essential closed and simple closed curves on Σ. Let α ∈ C (Σ). If m is a hyperbolic metric on Σ, then α has a unique m-geodesic representative. However, if [q] is a half-translation structure on Σ, then α has at least a [q]-geodesic representative, but it may not be unique. In that case, the set of geodesic representatives foliates a maximal flat cylinder (with singular points on its boundary), whose interior is embedded into (Σ, [q]). We denote by m (α) and [q] (α) the lengths of the geodesic representatives of α.

In this article, we will consider (complete) hyperbolic metrics and half-translation structures on Σ. Whereas a hyperbolic (local) geodesic is uniquely determined by its image (up to changing the origin), a flat (local) geodesic is not. However, we will sometimes still denote by the image of a flat (local) geodesic , if there is no confusion. Let d be the distance defined by a half-translation structure or a hyperbolic metric on Σ, and let p : ( Σ, d) → (Σ, d) be a universal cover of covering group Γ. Since (Σ, d) is complete and locally CAT(0), according to the theorem of Cartan-Hadamard, the space ( Σ, d) is complete and CAT(0). We will denote by ∂ ∞ Σ the boundary at infinity of ( Σ, d) and

∂ 2 ∞ Σ = ∂ ∞ Σ × ∂ ∞ Σ -∆ (with ∆ = {(x, x), x ∈ ∂ ∞ Σ}).
If Σ is a (possibly trivial) cover of a compact surface, which will always be the case in this article, the boundary at infinity ∂ ∞ Σ does not depend on the complete locally CAT(0) distance on Σ, up to a unique Γ-equivariant homeomorphism. If g is a geodesic of ( Σ, d), we denote by E(g) = (g(-∞), g(+∞)) ∈ ∂ 2 ∞ Σ its pair of points at infinity.

3 Isometric action of the covering group on the universal cover of a surface endowed with a half-translation structure.

Let (X, d) be a complete CAT(0) metric space. If (X, d) is endowed with an isometric action of a group Γ, for every γ ∈ Γ, the translation distance of γ is

X (γ) = inf x∈X d(x, γx).
The element γ is said to be elliptic if it has a fixed point in X, parabolic if X (γ) = 0 but γ has no fixed point in X, and hyperbolic if X (γ) > 0. Then, if (X, d) is complete (that will always be the case), there exists at least one geodesic Ax(γ) called a translation axis of γ in (X, d), which is invariant under γ and such that d(x, γx) = X (γ) if x ∈ Ax(γ) (see [START_REF] Bridson | Metric spaces of non-positive curvature[END_REF]Chap. 2.6]). Since the translation axes of a hyperbolic element are pairwise at finite Hausdorff distance, according to [START_REF] Bridson | Metric spaces of non-positive curvature[END_REF]Th. 2.13 p.182], if X is a surface, the union of all the translation axes of a hyperbolic element is a flat strip, possibly reduced to a single geodesic, or is isometric to R × [0, +∞[ or to R 2 . Let (Σ, [q]) be a compact, connected surface endowed with a half-translation structure, and let p : ( Σ, [ q]) → (Σ, [q]) be a universal cover with covering group Γ, as in Section 2.1. We will need the Gauss-Bonnet formula for a half-translation structure on a surface. Let P be a compact subsurface of (Σ, [q]) with piecewise geodesic boundary. For every point x ∈ ∂P , the interior angle θ(x) > 0 is the flat angle between the two germs at x of geodesic segments contained in ∂P , measured in the angular sector inside of P . For every point x ∈ • P , the angle θ(x) is the total flat angle at x (i.e. nπ with n ∈ N and n 3 if x is a singular point, and 2π otherwise).

Lemma 3.1 [START_REF] Dankwart | On the large-scale geometry of flat surfaces[END_REF]Prop. 3.6] We have 2πχ(P ) =

x∈ • P (2π -θ(x)) + x∈∂P (π -θ(x)).
Since the boundary of P is piecewise geodesic, the angles θ(x) at the points x ∈ ∂P are at least π except at the points where ∂P is not locally geodesic. If we denote by θ 1 , . . . , θ n the angles at these points, the Gauss-Bonnet formula implies the following result. Lemma 3.2 We have 2πχ(P )

x∈ • P (2π -θ(x)) + n i=1 (π -θ i ).
Let be a geodesic of ( Σ, [ q]).

Definition 3.1 The side + andof are the connected components of Σ -.

Let be a periodic geodesic of (Σ, [q]) and let be a lift of in Σ. Let γ ∈ Γ -{e} be an element of the stabilizer of , and let x ∈ . Then, the total curvature of [x, γx[ at side + is the sum t∈[x,γx[ (π -θ + (t)), where θ + (t) is the angle of at t measured in the side +.

Lemma 3.3 (see [START_REF] Rafi | A characterization of short curves of a Teichmüller geodesic[END_REF]Rem. 3.2]) The total curvature of [x, γx[ at side + is an integral multiple of π. It is zero, or it is at most -π.

Let γ ∈ Γ be a hyperbolic element of translation length (γ) and let F (γ) be the (possibly degenerated) flat strip, union of all the translation axes of γ. Let x ∈ Σ and let x ⊥ be the orthogonal projection of x onto F (γ). Let Ax(γ) be the translation axis of γ in Σ that bounds F (γ) and contains x ⊥ .

Lemma 3.4 The segment [x, γx] meets Ax(γ) and d(x, γ x) 2 (γ) 2 + 2 d(x, x ⊥ ) 2 .
Proof.

[x 1 , x 2 ] does not meet [x ⊥ , γx ⊥ ]. Then, the closed curve [x 1 , x 2 ] • [x 2 , γx ⊥ ] • [γx ⊥ , x ⊥ ] • [x ⊥ , x 1 ]
is simple and bounds a topological disk P . For all t ∈ [x ⊥ , γx ⊥ ], we denote by θ(t) the angle made by the rays of Ax(γ) starting at t, measured in the side of Ax(γ) containing x, and we denote by θ 1 , θ 2 the interior angles of P at x 1 and x 2 , and by θ -and θ + the interior angles of P at x ⊥ and γx ⊥ . The segment [γx, γx ⊥ ] is the image of [x, x ⊥ ] by γ, hence θ -+ θ + = θ(x ⊥ ). Since χ(P ) = 1, according to the Gauss-Bonnet formula (Lemma 3.2), we should have

2 π z∈ • P (2π -θ(z)) + π -θ 1 + π -θ 2 + π -θ -+ π -θ + + t∈]x ⊥ ,γx ⊥ [ (π -θ(t)) 3π -(θ 1 + θ 2 ) + π -(θ -+ θ + ) + t∈]x ⊥ ,γx ⊥ [ (π -θ(t)) < 3π + t∈[x ⊥ ,γx ⊥ [ (π -θ(t)),
whereas, according to Lemma 3.3, we have t∈

[x ⊥ ,γx ⊥ [ (π -θ(t)) -π. Hence, the segment [x, γx] meets [x ⊥ , γx ⊥ ] at least at a point y.
00 00 11 11

0 1 0 0 1 1 0 0 1 1 0 1 00 11 00 11 x 1 x 2 x θ 1 θ 2 γx y F (γ) x ⊥ γx ⊥ θ + θ - Ax(γ) Since ( Σ, [ q]) is CAT(0), we have d(x, y) 2 d(x, x ⊥ ) 2 + d(x ⊥ , y) 2 and d(y, γx) 2 d(γx, γx ⊥ ) 2 + d(y, γx ⊥ ) 2 , hence d(x, γ x) 2 (γ) 2 + 2 d(x, x ⊥ ) 2 .
Let Ax(α) and Ax(β) be two translation axes of two hyperbolic elements α, β ∈ Γ that are not some powers of a common element. Then, their translation axes have no common point at infinity. Let w be a point of Ax(α) and z be a point of Ax(β). We assume that Ax(α) and Ax(β) are in the boundaries of the (possibly degenerated) flat strips union of all the translation axes of α and β, and that the segment [w, z] does not meet a translation axis of α or β other than Ax(α) and Ax(β). Since ( Σ, [ q]) is proper, there exists x ∈ Ax(α) and y ∈ Ax(β) such that d(x, y) minimizes the distance between Ax(α) and Ax(β). It may happen that x = y, then [x, y] = {x}.

Lemma 3.5 We have d(z, w) d(w, x) + d(z, y) + d(x, y) -2( (α) + (β)).
Proof. We denote by w and z the endpoints of the intersection [w, z] ∩ Ax(α) and [z, w] ∩ Ax(β) (possibly equal to w and z).

w z T t z w x Ax(α) θ2 θ1 θ3 Ax(β) T Ax(β) t2 y x Ax(α) θ1 π 2 z w z t1 θ2 w Ax(β) y z π 2 x w w z θ1 θ2 Ax(α) Ax(β) θ1 T x w z Ax(α) θ3 z t θ2 w
Assume first that x = y. Since Ax(α) and Ax(β) are distinct and respectively α and βinvariant, they can share a segment of length at most max{ (α), (β)}. Let t be the endpoint of the intersection

[x, w] ∩ [x, z] (possibly equal to x). Note that t ∈ [w , x]. If w = t, then z = t and d(z, w) = d(z, t) + d(t, w) d(w, x) + d(z, x) -2 max{ (α), (β)},
which proves the result. Hence, we assume for now on that w = z . Assume for a contradiction that d(w, t) > (α) or d(z, t) > (β). Let T be the triangle with vertices t, w and z . According to Lemma 3.3, by the assumption on Ax(α), Ax(β), and on [w, z], the sum of the total curvatures of [t, w] and of [t, z] at theirs sides containing T is at most -π, and according to the Gauss-Bonnet formula applied to T , if we denote by θ 1 , θ 2 and θ 3 the interior angles of y], with t 1 the closest to x. If t 1 is distinct from x, let T be the triangle with vertices w , t 1 and x. Assume for a contradiction that d(w , x) > (α). Then, by the assumption on Ax(α), Ax(β), and since ]x, y[ meets no translation axis of α nor β, and according to Lemma 3.3, the total curvature of [w , x] at the side of Ax(α) containing T is at most -π, and, if we denote by θ 1 and θ 2 the interior angles of T at w and t 1 , according to the Gauss-Bonnet formula applied to T , we should have

2π π -θ 1 + π -θ 2 + π - π 2 -π < 3π

Subsurface with geodesic boundary or filled up by simple closed curves

In this Section 4, we assume that Σ is a (possibly trivial) cover of a compact surface whose Euler characteristic is negative. Let p : Σ → Σ be a universal cover with covering group Γ. The results of Subsections 4.1 and 4.2 are folklore. We refer to [START_REF] Morzadec | Compactification géométrique de l'espace des modules des structures de demi-translation sur une surface[END_REF] for complete proofs.

4.1 Subsurface filled up by a set of (free) homotopy classes of simple closed curves or by a measured hyperbolic lamination.

Assume in this Subsection 4.1 that Σ is compact. A tight subsurface of Σ is a closed subsurface Σ of Σ with smooth boundary, such that :

• no connected component of Σ is a disk or a pair of pants ;

• no connected component of Σ -Σ is a disk.

• no cylinder connected component of Σ can be homotoped in another connected component of Σ .

If C is a closed subset of Σ, an essential closed curve α of Σ topologically cuts C if it is not freely homotopic to a curve disjoint from C. Let S be a subset of S (Σ).

Lemma 4.1 (see [START_REF] Morzadec | Compactification géométrique de l'espace des modules des structures de demi-translation sur une surface[END_REF]Lem. 4.1]) There exists a (non unique) tight subsurface W of Σ such that :

• every essential closed curve of Σ that topologically cuts W has a positive intersection number with at least one element of S ; • W contains a representative of each element of S.

Lemma 4.2 (see [START_REF] Morzadec | Compactification géométrique de l'espace des modules des structures de demi-translation sur une surface[END_REF]Lem. 4.2]) Let S and S be two subsets of S (Σ) such that S ⊆ S. If S and S fill up respectively W and W then, up to isotopy, we have W ⊆ W .

Corollary 4.3 (see [START_REF] Morzadec | Compactification géométrique de l'espace des modules des structures de demi-translation sur une surface[END_REF]Coro. 4.3]) Assume that S fills up two subsurfaces W and W . Then W and W are isotopic.

The subsurface W , which is unique up to isotopy, will be called the subsurface filled up by S. It may be non connected. Let [q] be a half-translation structure and let m be a (complete) Riemannian metric on Σ. If S is finite and {c[s]} [s]∈S is a set of [q] or m-geodesic representatives of the elements of S, contained in W , then the complement of the union

[s]∈S c[s] in W is a finite union of disks and (possibly) cylinders that can be homotoped to a boundary component of W . Otherwise, there would exist an essential closed curve α contained in a connected component of Σ -( [s]∈S c[s]), and we would have i(s, α) = 0 for all s ∈ S.

Let g and b be the genus and the number of boundary components of

W . Let K = (g + 1)(2g + b)(3g + b) (this constant is not optimal).
Lemma 4.4 (see [START_REF] Morzadec | Compactification géométrique de l'espace des modules des structures de demi-translation sur une surface[END_REF]Lem. 4.4]) Let W be a tight subsurface filled up by a subset S of S (Σ). Then there exists a subset S ⊆ S with at most K elements that fills up W .

Let m be a hyperbolic metric on Σ. A measured hyperbolic lamination (Λ, µ) of (Σ, m) is a closed subset of Σ which is a disjoint union of simple geodesics, called leaves, endowed with a transverse measure. We refer for instance to [START_REF] Bonahon | Geodesic laminations on surfaces[END_REF] for the definition of a transverse measure and the definition of the intersection number with (Λ, µ). Lemma 4.5 (see [START_REF] Morzadec | Compactification géométrique de l'espace des modules des structures de demi-translation sur une surface[END_REF]Lem. 4.5]) There exists a unique tight subsurface W (Λ) of Σ (up to isotopy) such that Λ is contained in W and for every essential closed curve β of Σ that topologically cuts W , we have i((Λ, µ), β) > 0.

The subsurface W (Λ) is said to be filled up by (Λ, µ). It is connected if and only if Λ is minimal.

4.2 Subsurfaces with flat geodesic boundary.

We now discuss some geometric properties of subsurfaces filled up by some set of (homotopy classes of) simple closed curves. Let [q] be a half-translation structure on Σ. Let D be an open disk embedded in Σ with piecewise smooth boundary. The perimeter peri(D) of D is the length of the boundary of the completion of D for the induced metric.

Lemma 4.6 (see [START_REF] Morzadec | Compactification géométrique de l'espace des modules des structures de demi-translation sur une surface[END_REF]Lem. 4.6]) The diameter of D is at most peri(D) and the area of

D is at most 1 4π peri(D) 2 .
If α is an isotopy class of simple closed curves, let C(α) be the (possibly degenerated) flat cylinder union of all the geodesic representatives of α. If c is a piecewise smooth simple closed curve in α, disjoint from C(α), let b(α, c) be the boundary component of C(α) such that b(α, c) and c bound a cylinder disjoint from the interior of C(α) (b(α, c) is unique since Σ is not a torus). The following notion of geometric realization of surfaces is due to [START_REF] Rafi | A characterization of short curves of a Teichmüller geodesic[END_REF]§2]. Let W be a non trivial π 1 -injective connected subsurface of Σ. The fundamental group π 1 (W ) (with respect to any choice of basepoint in W ) is a subgroup of Γ, that is determined by the homotopy class of W , up to conjugation.

Let p : Σ W = Σ/π 1 (W ) → Σ be the W -cover of Σ, and let p : Σ → Σ W be the unique universal cover such that p = p • p . Let [ q] and [ q] be the pullbacks of [q] on Σ W and on Σ. The surface Σ W is homotopically equivalent to W , and the preimage p -1 (W ) has a unique connected component W that is not simply connected. The map p | W : W → W is a homeomorphism and the complement of W in Σ W is a finite union of open annuli that can be homotoped to boundary components of W . A generalized subsurface is a closed connected union of a (possibly empty) subsurface with some finite connected (metric) graphs, glued at some points of the boundary of the subsurface. Its boundary is the union of the boundary of this subsurface with these graphs. If W is not a cylinder, the [ q]-geometric realization of W is the unique generalized subsurface W [ q] of Σ W homotopic to W within Σ W , whose boundary is the union of some [ q]-geodesic representatives of the boundary components of W (see [START_REF] Rafi | A characterization of short curves of a Teichmüller geodesic[END_REF]p. 188]), that contains a unique [ q]-geodesic representative of each boundary component of W . We call boundary components of W [ q] the [ q]-geodesic representatives of the boundary components of W contained in W [ q] , and we say that a boundary component of W [ q] corresponds to a boundary component of W if they are freely homotopic. If W is a cylinder, the [ q]-geometric realization of W is the (possibly degenerated) flat cylinder, union of all the [ q]-geodesic representatives of the boundary components of W . For every essential homotopy class of closed curves α of W , the [ q]-geodesic representatives of α are contained in W

[ q] .
Similarly, if W is a connected component of the preimage of W in Σ, the [ q]-geodesics having the same pair of points at infinity than the boundary components of W may not be pairwise disjoint. However, there exists a (possibly non unique) generalized subsurface properly homotopic to W within Σ whose boundary is the union of some geodesics having the same pairs of points at infinity than the boundary components of W . If W is not a strip, we call [ q]-geometric realization of W the unique such generalized subsurface W [ q] that contains a unique [ q]-geodesic representative of each boundary component of W , and we call boundary components of W [ q] the [ q]-geodesic representatives of the boundary components of W contained in W [ q] . We say that a boundary component of W [ q] corresponds to a boundary component of W if they have the same ordered pair of points at infinity. If W is a strip, we call [ q]-geometric realization of W the (possibly degenerated) flat strip, union of all the [ q]-geodesics having the same (unordered) pair of points at infinity than the boundary components of W .

We denote by Sing(

W [ q] ) the set of singular points of [ q] contained in W [ q] . Lemma 4.8 The space W [ q] is convex. Moreover, for all ε > 0, if the union Sing( W [ q] )∪∂ W [ q] is ε-dense into W [ q] , then W [ q] is 2ε-hyperbolic.
Proof. The proof is essentialy the same as the one of [START_REF] Dankwart | On the large-scale geometry of flat surfaces[END_REF]Prop. 3.7]. It suffices to replace

ρ = sup x∈Σ d(x, Sing(Σ)) by ρ = sup x∈ W [ q] d(x, Sing( W [ q] ) ∪ ∂ W [ q] ). Remark. If the diameter Diam(Σ, [q]) is finite, the universal cover ( Σ, [ q]) is 2 Diam(Σ, [q])- hyperbolic.
Assume that W is neither a cylinder nor a pair of pants, and that W is filled up by a finite set of isotopy classes of simple closed curves S( W ) of cardinality at most K. We can always assume that S( W ) does not contain the homotopy class of any boundary component of W . Let ε = max α∈S( W ) [ q] (α) and let b be the number of boundary components of W . Lemma 4.9 (see [START_REF] Morzadec | Compactification géométrique de l'espace des modules des structures de demi-translation sur une surface[END_REF]Lem. 4.9])The length of any boundary component of W [ q] is at most Kε, the diameter of W [ q] is at most 11Kε and its area is at most

1 π (1 + 2b) 2 (Kε) 2 .
5 Ultralimits of sequences of metric spaces.

The ultralimit of a sequence of metric spaces is a notion introduced by M. Gromov (see [Gro], we refer for example to [Dru] for background and precisions on the content of this Section 5). It uses ultrafilters (introduced by H. Cartan) that are a way of picking an accumulation value of a sequence in a compact metrizable space, avoiding extraction arguments. Let ω be a non principal ultrafilter on N (see [START_REF] Bourbaki | Topologie générale[END_REF]§ 6.4]). We say that a sentence A(n) is true for ω-almost all n ∈ N if there exists I ∈ ω such that A(n) is true for all n ∈ I. If (a n ) n∈N is a sequence of a topological space E and a ∈ E, we say that (a n ) n∈N ω-converges to a, and write lim ω a n = a, if for every neighborhood V of a, the element a n belongs to V for ω-almost all n ∈ N. The lim ω satisfies the usual properties of limits. If E is compact, for example if E = [-∞, +∞], for every non principal ultrafilter ω and for every sequence (a n ) n∈N in E, there exists a unique accumulation value a of (a n ) n∈N in E such that lim ω a n = a.

Let (X n , d n , n ) n∈N be a sequence of pointed metric spaces endowed with an isometric action of a group Γ. Let

X ω = {(x n ) n∈N ∈ n∈N X n : lim ω d n (x n , n ) < +∞}.
Then, the function

d ω : X ω × X ω → R + defined by d ω ((x n ) n∈N , (y n ) n∈N ) = lim ω d n (x n , y n )
is a pseudo-distance on X ω . We denote by X ω the quotient of X ω by the equivalence relation

(x n ) n∈N ∼ (y n ) n∈N if d ω ((x n ) n∈N , (y n ) n∈N ) = 0,
and the equivalence class of an element

(x n ) n∈N of X ω is denoted by [x n ] n∈N . Then, the pseudo-distance d ω induces a distance d ω on X ω . If for ω-almost all n ∈ N, F n is a subset of X n , let [F n ] n∈N = {[x n ] n∈N : x n ∈ F n for ω-almost all n ∈ N}. The diagonal action of Γ on n∈N X n is said to be admissible if lim ω d n ( n , γ n ) < +∞ for every γ ∈ Γ. Then, the action Γ × X ω → X ω defined by γ[x n ] n∈N = [γx n ] n∈N is an isometric action on X ω . Let ω = [ n ] n∈N . If the action of Γ is admissible, the pointed metric space (X ω , d ω , ω ) is called the ultralimit of (X n , d n , n ) n∈N
for ω and is denoted by lim ω (X n , d n , n ). We recall a few properties of ultralimits of metric spaces : Gromov, and

• If for all n ∈ N, (X n , d n , n ) is a geodesic metric space, so is (X ω , d ω , ω ) ; • If for all n ∈ N, (X n , d n , n ) is CAT(0), so is (X ω , d ω , ω ) ; • If for all n ∈ N, (X n , d n , n ) is δ n -hyperbolic in the sense of
lim ω δ n = δ, then (X ω , d ω , ω ) is δ-hyperbolic. Let (X n , d n , n ) n∈N
be a sequence of CAT(0) pointed metric spaces with an admissible isometric action of Γ, and let γ ∈ Γ. If γ is hyperbolic in X n , and Ax Xn (γ) is a translation axis of γ in X n for ω-almost all n ∈ N, and if

lim ω d n ( n , Ax Xn (γ)) < +∞, then [Ax Xn (γ)] n∈N is a translation axis of γ in X ω if Xω (γ) > 0, and it is contained in the set of fixed points of γ in X ω if Xω (γ) = 0. If γ is elliptic in X n , and x n (γ) is a fixed point of γ in X n for ω-almost all n ∈ N, and if lim ω d n ( n , x n (γ)) < +∞, then [x n (γ)] n∈N is a fixed point of γ in X ω .
In the two cases, we have Xω (γ) = lim ω Xn (γ).

Remark. The ultralimit X ω does not change if we choose another sequence of base points ( n ) n∈N , as long as lim ω d n ( n , n ) < +∞. The choice of the base points will not always be specified.

6 Ultralimits of sequences of half-translation structures.

Let Σ be a compact, connected, orientable surface such that χ(Σ) < 0, and let p : Σ → Σ be a universal cover with covering group Γ. Recall that the boundary of Σ is empty, to simplify the writting of the article, but the results can be extended very easily to a surface with nonempty boundary (see [START_REF] Morzadec | Compactification géométrique de l'espace des modules des structures de demi-translation sur une surface[END_REF]). Let Flat(Σ) be the set of isotopy classes of halftranslation structures on Σ. Let ([q n ]) n∈N be a sequence in Flat(Σ) and, for all n ∈ N, let [ q n ] be the pullback of [q n ] on Σ. Let ω be a non principal ultrafilter on N as in Section 5.

Let S be a finite generating set of Γ. For all n ∈ N, define

f n : Σ → R + by f n (x) = max s∈S d n (x, sx). Let λ n = inf x∈ Σ f n (x) and let us choose a point n ∈ Σ such that f n ( n ) λ n + 1. Let [ q n ] = 1 λn [ q n ],
and let d n be the distance defined by

[ q n ] on Σ. For all γ ∈ Γ, we have lim ω d n ( n , γ n ) < +∞, hence lim ω ( Σ, [ q n ], n ) is endowed with an isometric action of Γ. Moreover, if [x n ] n∈N ∈ lim ω ( Σ, [ q n ], n ), there exists s ∈ S such that d n (x n , sx n ) 1 for ω-almost all n ∈ N,
hence this action has no global fixed point. Remark 6.1 For all γ ∈ Γ -{e} and n ∈ N, according to Lemma 3.4, if F n (γ) is the flat strip, union of all the translation axes of γ in ( Σ, [ q n ]), we have

d n ( n , F n (γ)) d n ( n , γ n ).
Since the action of Γ on ( Σ, [ q n ]) n∈N is admissible, we have lim ω d n ( n , F n (γ)) < +∞. Hence, the ultralimit [F n (γ)] n∈N exists, and limω( Σ,[ qn]) (γ) = lim ω ( Σ,[ qn]) (γ). Moreover, according to Lemma 3.4, the set [F n (γ)] n∈N is exactly the union of all the translation axes or the set of fixed points of γ in lim ω ( Σ, [ q n ]). Moreover, according to Lemma 3.5, if γ 1 and γ 2 are not powers of a common element, then they have at most one common fixed point in lim ω ( Σ, [ q n ]).

We will see (Section 6.6) that the ultralimit lim ω ( Σ, [ q n ]) is a surface endowed with a half-translation structure if and only if there exists ε > 0 such that for every α ∈ S (Σ) and for ω-almost all n ∈ N, we have [qn] (α) ε. In Section 6.1, we consider the case where there does not exist such a uniform lower bound on the [q n ]-lengths of the elements of S (Σ). We will notably consider the case where there exist subsurfaces with fixed homotopy type whose boundary components (if any) have their lengths that ω-converge to 0, and whose areas ω-converge to 0. Let us introduce some general definition. Let m be a hyperbolic metric on Σ, and let m be its pullback on Σ.

For i ∈ {1, 2, 3}, let (x i , y i ) be an element of ∂ 2 ∞ Σ and let λ i be the geodesic of ( Σ, m) whose ordered pair of points at infinity is (x i , y i ). We assume that the corresponding unordered pairs of points are pairwise distinct. The pairs (x 1 , y 1 ) and (x 2 , y 2 ) are interlaced if the geodesics λ 1 and λ 2 intersect each other. If (x 1 , y 1 ) and (x 2 , y 2 ) are not interlaced, the pair (x 3 , y 3 ) is contained between (x 1 , y 1 ) and (x 2 , y 2 ) if λ 3 is contained in the connected component of Σ -λ 1 ∪ λ 2 bounded by λ 1 and λ 2 , and the pair (x 3 , y 3 ) is caught between (x 1 , y 1 ) and (x 2 , y 2 ) if it is contained between (x 1 , y 1 ) and (x 2 , y 2 ) and if λ 3 intersects any geodesic segment joining λ 1 and λ 2 . These definitions do not depend on the choice of m.

If X is an oriented geodesic of Σ (for any complete, CAT(0) metric on Σ) or an element of Γ -{e}, we denote by E(X) ∈ ∂ 2 ∞ Σ its ordered pair of points at infinity or of fixed points at infinity. If X, Y and Z are three such elements, we say that X and Y are interlaced if E(X) and E(Y ) are interlaced and that Z is caught or contained between X and Y if E(Z) is caught or contained between E(X) and E(Y ).

In the remainder of this section, we will consider a tight, connected subsurface W of Σ (possibly equal to Σ). Let p : Σ W → Σ be a W -cover of Σ and let W be the unique connected component of the preimage of W in Σ W which is not simply connected. Let W be a connected component of the preimage of W in Σ. Let ([ q n ]) n∈N and ([ q n ]) n∈N be the pullbacks of ([q n ]) n∈N on Σ W and on Σ. For all n ∈ N, we denote by W n and W n the [ q n ] and [ q n ]-geometric realizations of W and of W .

Typical degenerations.

In this subsection, we assume that :

• lim ω [qn] (α) = α < +∞, for every α ∈ C (Σ) ; • lim ω [qn] (b) = 0, for every boundary component b of W .
Let c be a boundary component of W . We denote by Γ c the stabilizer of c in Γ and by γ c a primitive generator of Γ c . For all n, let F n ( c) be the (possibly degenerated) flat strip, union of all the geodesic representatives of c. Let γ be a primitive element of Γ -{e} whose translation axes in any ( Σ, [ q n ]) are interlaced with c, and let (Ax n (γ)) n∈N be a sequence of translation axes of γ in ( Σ, [ q n ]). Lemma 6.2 The geodesic Ax ω (γ) = [Ax n (γ)] n∈N is the only translation axis or set of fixed points of γ in lim ω ( Σ, [ q n ]), and the intersection Ax

ω (γ) ∩ [F n ( c)] n∈N is a geodesic segment orthogonal to the boundary of [F n ( c)] n∈N if [F n ( c)]
n∈N is not reduced to a geodesic line, and is reduced to a point otherwise. Moreover, if γ is another primitive element of Γ -{e} whose translation axes in any ( Σ, [ q n ]) are interlaced with c, and if Ax ω (γ ) is the set of fixed points or the translation axis of γ in lim ω ( Σ, [

q n ]), then Ax ω (γ ) ∩ [F n ( c)] n∈N = Ax ω (γ) ∩ [F n ( c)] n∈N .
We denote by c the unique point of Ax ω (γ) ∩ [F n ( c)] n∈N that belongs to [ W n ] n∈N . We will need the following lemma (which is not surprising for specialist) in order to prove Lemma 6.2. Let c be the image of c in Σ. Let α γ be the free homotopy class of closed curves defined by γ in Σ. Let S be a filling finite set of simple closed curves in Σ, such that c does not belong to S. For all n ∈ N, we denote by I n and J n the geodesic segments in F n ( c) perpendicular to its boundary components c n and c n (reduced to two points if F n ( c) is reduced to a single geodesic) such that one endpoint of I n (resp. J n ) is the first (resp. last) intersection point between Ax n (γ) and F n ( c).

Ax n (γ)

I n J n F n Lemma 6.3 (see [Mor1, Lem. 6.3]) We have d n (I n , J n ) [qn] (c) s∈S i(α γ , s).
Proof of Lemma 6.2. According to Remark 6.

1 the ultralimit [Ax n (γ)] n∈N exists. Let us first prove that the intersection Ax ω (γ) ∩ [F n ( c)] n∈N is a geodesic segment ortho- gonal to the boundary components of [F n ( c)] n∈N , if [F n ( c)]
n∈N is a flat strip, and is reduced to a point otherwise.

Let I n and J n be the segments (possibly reduced to a point) orthogonal to F n ( c), having as an endpoint the first and the last intersection point between Ax n (γ) and F n ( c), as in Lemma 6.3. Since lim ω [qn] (c) = 0, according to Lemma 6.3, we have lim ω d n (I n , J n ) = 0. For all n ∈ N, let x n ∈ J n be the last intersection point between F n ( c) and the oriented geodesic Ax n (γ). Let (z n ) n∈N be a sequence of points such that, for all n, z n belongs to [x n , Ax n (γ)(+∞)[, and let z n⊥ be the orthogonal projection of z n onto F n ( c). Let c n be the boundary component of F n ( c) containing z n⊥ and let + be the side of c n containing z n . The interior of F n ( c), if not empty, is not contained in the side + of c n . Hence, according to Lemma 3.3, the total curvature of a fundamental domain of c n for the action of γ Z c , at the side +, is at most -π. Let us prove that d n (x n , z n⊥ )

[qn] (c). Otherwise, z n⊥ is not equal to x n and the open segment ]x n , z n⊥ [ fully contains an entire fundamental domain of c n for the action of γ Z c . Let a n be the last intersection point between [z n , x n ] and

[z n , z n⊥ ]. The curve [x n , a n ] • [a n , z n⊥ ] • [z n⊥ , x n ] bounds a topological disk T . z n⊥ cn zn θ1 T Axn(γ)
x n an θ2

The interior angle at z n⊥ is at least π 2 . Let θ 1 and θ 2 be the two other interior angles of the boundary of T which are less than π. By the Gauss-Bonnet formula applied to T , we should have

2πχ(T ) π - π 2 + π -θ 1 + π -θ 2 -π 3 2 π,
a contradiction since χ(T ) = 1. Hence, we have d n (x n , z n⊥ )

[qn] (c), and

d n (z n , F n ( c)) = d n (z n , z n⊥ ) d n (z n , x n ) -[qn] (c). Since lim ω [qn] (c) = 0, we deduce that [z n ] n∈N belongs to [F n ( c)] n∈N if
z n⊥ βn xn an γ -k c ( βn) γ k c ( βn) Fn( c) In Jn
Let ( β n ) n∈N be a sequence of translation axes of another element of Γ -{e} which is interlaced with c, or another sequence of translation axes of γ, and let n ∈ N. Since the action of γ Z c on ∂ ∞ Σ is a South-North dynamic with fixed points c(+∞) and c(-∞), there exists k ∈ N such that Ax n (γ) and β n are caught between γ k c ( β n ) and γ -k c ( β n ). Moreover, the integer k only depends on E(Ax n (γ)) and E( β n ), hence does not depend on n ∈ N. Hence, we have

lim ω d n (Ax n (γ)∩F n ( c), β n ∩F n ( c)) lim ω 2k [qn] (c) = 0, and [Ax n (γ)] n∈N ∩[F n ( c)] n∈N = [ β n ] n∈N ∩ [F n ( c)] n∈N .
According to Remark 6.1, any translation axis or any set of fixed points of an element of Γ -{e} in lim ω ( Σ, [ q n ]) is the ultralimit of a sequence of translation axes of γ in ( Σ, [ q n ]). Hence, the element γ has no other translation axis or set of fixed points than Ax ω (γ), and if

γ ∈ Γ -{e} is interlaced with c, we have Ax ω (γ) ∩ [F n ( c)] n∈N = Ax ω (γ ) ∩ [F n ( c)] n∈N .
Let Γ W be the stabilizer of W in Γ, let γ 1 , γ 2 be two (not necessarily distinct) elements of Γ W that do not preserve any boundary component of W and let c be a boundary component of W . Let (Ax n (γ 1 )) n∈N and (Ax n (γ 2 )) n∈N be two sequences of translation axes of γ 1 and γ 2 , and let

Ax ω (γ 1 ) = [Ax n (γ 1 )] n∈N and Ax ω (γ 2 ) = [Ax n (γ 2 )] n∈N . Recall that γ c is a primitive generator of Γ c . For all n ∈ N, let c n be the boundary component of W n corresponding to c. Lemma 6.4 If [x 1 , x 2 ] is a geodesic segment joining Ax ω (γ 1 ) to Ax ω (γ 2 ) in [ W n ] n∈N , then the segment [x 1 , x 2 ] can intersect [ c n ] n∈N at most at c . Notably, Ax ω (γ 1 ) and Ax ω (γ 2 ) can intersect [ c n ] n∈N at most at c . Proof. There exists a sequence ([x 1,n , x 2,n ]) n∈N of geodesic segments joining Ax n (γ 1 ) to Ax n (γ 2 ) such that [x 1 , x 2 ] = [[x 1,n , x 2,n ]] n∈N .
Let 0 be a geodesic of ( Σ, [ q 0 ]) that is interlaced with c, and for all n ∈ N, let n be a geodesic of ( Σ, [ q n ]) having the same pair of points at infinity than 0 . There exists k ∈ N such that Ax n (γ 1 ) and Ax n (γ 2 ) are contained between γ -k c ( n ) and γ k c ( n ), with k independent of n as in the proof of Lemma 6.2.

cn γ -k c ( n ) γ k c ( n ) Axn(γ 2 ) Axn(γ 1 ) x 1,n x 2,n For all n ∈ N, the union Ax n (γ 1 ) ∪ [x 1,n , x 2,n ] ∪ Ax n (γ 2 ) is contained in a connected generalized subsurface of Σ whose boundary is contained in γ k c ( n ) ∪ γ -k c ( n ) ∪ c n . Since we have [γ k c ( n )] n∈N ∩ [ c n ] n∈N = [γ -k c ( n )] n∈N ∩ [ c n ] n∈N = c , we see that [[x 1,n , x 2,n ]] n∈N can intersect [ c n ] n∈N at most at c . Let x be a point of Ax ω (γ 1 ). Lemma 6.5 The geodesic segment [x, γ c x] intersects [ c n ] n∈N at c .
Proof. This follows from Lemma 6.4 using γ 2 = γ c γ 1 γ -1 c and from Remark 6.1.

In the remainder of this Section 6.1, we assume moreover that lim ω Diam

[ qn] ( W n ) = 0. Lemma 6.6 The ultralimit [ W n ] n∈N is an R-tree T W . Proof. Since W n is convex for all n ∈ N, the ultralimit [ W n ] n∈N is convex. Since lim ω Diam [ qn] ( W n ), for all n ∈ N, the union of ∂ W n with the set of singular points of [ q n ] is ε n -dense in W n , with lim ω ε n = 0. According to Lemma 4.8, the convex set [ W n ] n∈N is 0-hyperbolic, hence it is an R-tree.
Let Γ W be the stabilizer of W in Γ. If Γ W does not have a global fixed point in T W , let T min be the minimal subtree for the isometric action of Γ W on T W , i.e. the smallest non empty subtree of T W that is invariant by Γ W . It is the union of the translation axes of the hyperbolic elements of Γ W . The isometric action of Γ W on T min is said to have small edge stabilizers if Γ W has no global fixed point, all the elements defined by the boundary components of M are elliptic elements, and the stabilizers of the non trivial segments are trivial or cyclic. Lemma 6.7 Either Γ W has a global fixed point in T W or the isometric action of Γ W on T min has small edge stabilizers.

Proof. Assume that Γ W has no global fixed point in T W so that T min is well defined. Let γ ∈ Γ W be an element defined by a boundary component b of W . We have lim ω [ qn] (b) = 0, and according to Remark 6.1, we have T min (γ) = lim ω Wn (γ) = 0, hence γ is elliptic in T min .

Let α, β be two elements of Γ W -{e} which are not powers of a common element. Let us prove that α and β have at most one common fixed point. For all n ∈ N, let Ax n (α) and Ax n (β) be two translation axes of α and β in W n , that minimize the distance between the translation axes of α and β. Let [x n ] n∈N and [y n ] n∈N be fixed points of α and β in T min . For all n ∈ N, let F n (α) be the flat strip union of all the translation axes of α. According to Remark 6.1, we have

d n (x n , F n (α)) d n (x n , αx n ), hence d n (x n , Ax n (α)) d n (x n , αx n ) + h n (α), where h n (α) is the height of F n (α). Since h n (α) 2 Diam [ qn] ( W n ) ω-converges to zero, we have lim ω d n (x n , Ax n (α)) lim ω d n (x n , αx n ) + lim ω h n (α) = 0.
Hence, we can always assume that x n belongs to Ax n (α) for all n. Similarly, we can always assume that y n belongs to Ax n (β) for all n.

Assume that [x n ] n∈N = [y n ] n∈N . Let (z n ) n∈N and (w n ) n∈N be two sequences of points such that [z n ] n∈N is a fixed point of α and [w n ] n∈N is a fixed point of β. As above, we can assume that z n belongs to Ax n (α) and w n belongs to Ax n (β) for all n ∈ N. Since lim ω h n (α) = lim ω h n (β) = 0, up to replacing Ax n (α) and Ax n (β) by others translation axes of α and β, we can always assume that the segment [w n , z n ] does not intersect a translation axis of α or β other than Ax n (α) and Ax n (β). Since lim ω d n (x n , y n ) = 0 and lim ω max{ Wn (α), Wn (β)} = 0, according to Lemma 3.5, we have

lim ω d n (w n , z n ) lim ω d n (w n , x n ) + lim ω d n (z n , y n ). Hence [w n ] n∈N is equal to [z n ] n∈N if and only if [w n ] n∈N = [z n ] n∈N = [y n ] n∈N = [x n ] n∈N . This proves the result.
Let m be a hyperbolic metric on Σ, and let m be its pullback on Σ. If (Λ, µ) is a measured hyperbolic lamination on (Σ, m), it lifts to a measured hyperbolic lamination ( Λ, µ) on ( Σ, m). If λ is an isolated leaf of Λ, the measure on any arc transverse to λ, that is disjoint of the rest of the lamination, is a Dirac measure at the intersection point between the arc and the leaf, of mass δ λ > 0. We replace the leaf λ by a flat strip of width δ λ foliated by parallel leaves. Thus, we get a surface Σ and a measured lamination, with a transverse measure, such that the measures on the arcs transverse to the lamination have no atom. The map

d : Σ × Σ → R + defined by d (x, y) = inf c || µ c ||,
where c is an arc transverse to Λ between

x and y and || µ c || is the total mass of µ c , is a pseudo-distance on Σ . The quotient space T (Λ,µ) = Σ / ∼, with x ∼ y if d (x, y) = 0, endowed with the distance d Λ defined by d , is an R-tree, called the R-tree dual to (Λ, µ) (see for example [START_REF] Morgan | Free actions of surface groups on R-trees[END_REF][START_REF])| = d(w n , γx n ) + d(z n , γx n ) -d(w n , z n ). Moreover, since the space is CAT(0), we have d(w n , γx n ) w n , γx n⊥ ) 2 + d(γx n , γx n⊥ ) d(z n , γx n ) 2 d(z n , γx n⊥ ) 2 + d(γx n , γx n⊥ ) 2 . Hence d(w n , γx n ) -d(w n , γx n⊥ ) d(γx n , γx n⊥ ) 2[END_REF]), and the action of Γ on Σ defines an isometric action of Γ on T (Λ,µ) . For every element γ ∈ Γ, we denote by α γ the associated free homotopy class of closed curves of Σ. Recall that for every γ ∈ Γ, we have

T (Λ,µ) (γ) = i(α γ , (Λ, µ)) (see [MS, § 1]).
Lemma 6.8 If Γ W has no global fixed point in T W , the minimal R-tree T min endowed with the isometric action of Γ W is dual to a measured hyperbolic lamination (Λ, µ) on W (for any hyperbolic metric).

Proof. It is a consequence of Lemma 6.7 and of [Sko].

Decomposition of Σ into pieces.

In this short subsection, we introduce the notation which will be used in the next three subsections, that aim at decomposing the surface Σ into revelant pieces. For all α ∈ C (Σ), let α = lim ω [qn] (α). Recall that by the definition of (λ n ) n∈N and ([q n ]) n∈N (see the beginning of Section 6), we have α < +∞ for every α ∈ C (Σ). Let S 0 = {α ∈ S (Σ) : α 0 = 0}. If S 0 = ∅, we denote by Σ 0 the tight subsurface filled up by S 0 (defined up to isotopy, see Lemma 4.3).

Let W be a tight connected subsurface of Σ. Let p : Σ W → Σ be a W -cover of Σ, let W be the connected component of the preimage of W in Σ W that is not simply connected, and let W be a connected component of the preimage of W in Σ. For all n ∈ N, let [ q n ] and [ q n ] be the pullbacks of [q n ] on Σ W and on Σ, and let W n and W n be the [ q n ] and [ q n ]-geometric realizations of W and W . Finally, let Γ W be the stabilizer of W in Γ.

Let us introduce the following example of typical degenerating sequence of half-translation structures on a surface.

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 n 1 W 2 1 Σ 1 W 1 β 1 1 n 1 W 0 W 1 W 0 W 2 β 2 1 0 0 1 n 1 n 3 n 3 n 1
Let n ∈ N. The half-translation structure [q n ] on Σ is defined piece by piece. It is obtained by gluing by two flat cylinders of height 1 and girth 1 n along the slits, the unit square flat torus with a vertical slit of perimeter 1 n , a square flat torus with side lengths 3 n and two vertical slits of perimeters 1 n , and a square torus whose both vertical and horizontal measured foliations are minimal, whose transverse measure to the horizontal foliation is multiplied by 1 n , and whose transverse measure to the vertical foliation is constant. The singularities are at the 8 marked points, and are of angle 3π.

Recall that for all n ∈ N, if (F v n , µ v n ) and (F h n , µ h n ) are the vertical and the horizontal foliations of a representative

q n ∈ Q(Σ) of [q n ], for all α ∈ C (Σ), we have max{i((F v n , µ v n ), α), i((F h n , µ h n ), α)} [qn] (α) i((F v n , µ v n ), α) + i((F h n , µ h n ), α). Hence, • lim ω [qn] (β 1 ) = lim ω [qn] (β 2 ) = 0 ; • for every α ∈ S (W 0 ), we have lim ω [qn] (α) = 0 ; • for every α ∈ S (W 1 ), we have lim ω [qn] (α) = [q] (
α), with [q] the metric on the square flat torus of area 1, minus a point ;

• for every α ∈ S (W 2 ), we have lim ω [qn] (α) = i(α, (F v , µ v ))
, where (F v , µ v ) is the vertical foliation of q n for all n ∈ N. In this example, we have Σ 0 = W 0 .

6.3

The subsurface Σ 0 .

If Σ 0 is not a union of cylinders, assume in this subsection 6.3 that W is a connected component of Σ 0 that is not a cylinder (the cylinder components will be considered in Subsection 6.7).

There exists a subset S 0 (W ) of S 0 with at most K ∈ N elements (see Lemma 4.4) that fills up W . We can always assume that S 0 (W ) does not contain the isotopy class of any boundary component of W . Let S 0 ( W ) be the set of lifts of the elements of S 0 (W ) in W . Lemma 6.9 We have lim

ω Diam [ qn] ( W n ) = 0, lim ω Area [ qn] ( W n ) = 0 and for every boundary component β of W , we have lim ω [ qn] (β) = 0. Hence, the ultralimit [ W n ] n∈N is an R-tree T W .
Proof. The set S 0 ( W ) has at most K < +∞ elements and, for every α ∈ S 0 ( W ), we have lim ω [ qn] (α) = 0. Hence, for every ε > 0, there exists I ∈ ω such that for all n ∈ I and α ∈ S 0 ( W ), we have [ qn] (α) < ε. Then, according to Lemma 4.9, we have

Diam [ qn] ( W n ) 11Kε and Area [ qn] ( W n ) 1 π (1 + 2b) 2 (Kε) 2 (where b is the number of boundary components of W ). Hence lim ω Diam [ qn] ( W n ) = 0 and lim ω Area [ qn] ( W n ) = 0. Hence, according to Lemma 6.6, the ultralimit [ W n ] n∈N is an R-tree.
Lemma 6.10 The action of Γ W on T W has a unique global fixed point W , and for every boundary component c of W , the point c (see Lemma 6.2) is equal to W .

We will need the following well-known lemma, obtained by smoothing the Dirac masses of negative curvature at the singularities, see for instance [Min] for a similar result. Lemma 6.11 Let [ q] be a half-translation structure on Σ W , and let E be a finite set of homotopy classes of essential closed curves in W . For every ε > 0, there exists a Riemannian metric g with nonpositive curvature on the smooth surface

Σ W such that | [ q] (α) -g (α)| < ε for all α ∈ E.
Proof of Lemma 6.10. Let γ be an element of Γ W -{e}, and let α γ be the free homotopy class of closed curves defined by γ in W . For every ε > 0, there exists I ∈ ω such that for all n ∈ I and s 0 ∈ S 0 ( W ), we have

[ qn] (s 0 ) ε. Let E = S 0 ( W ) ∪ { α γ }.
According to Lemma 6.11, there exists a Riemannian metric g with nonpositive curvature on Σ W such that for every α ∈ E, we have | [ qn] (α)g (α)| < ε. Let W g be the g-geometric realization of W . Since S 0 ( W ) fills up W , the union G of some g-geodesic representatives of the elements of S 0 ( W ) (one by element of S 0 ( W )) is a graph whose complementary components in W g are disks and half-open cylinders that can be homotoped to some boundary components of W g . Moreover, since the lengths of these g-geodesics are at most 2ε, and S 0 ( W ) has at most K elements, the sum of the lengths of the edges of G is at most 2Kε. Hence, the perimeters of the complementary disks are at most 4Kε and the circonferences of the complementary cylinders are at most 2Kε. Since g is a Riemannian metric with nonpositive curvature, the length of an intersection segment of a g-geodesic representative of α γ with a complementary component of G is at most 4Kε, and since the intersection number between two g-geodesics is minimal, we have g ( α γ ) 2Kε s 0 ∈S 0 ( W ) i( α γ , s 0 ), and [ qn] ( α γ ) 2Kε s 0 ∈S 0 ( W ) i( c γ , s 0 )+ε. Hence, we have T W (γ) = lim ω [ qn] ( c γ ) = 0, so that γ has a fixed point in T W . Hence, all the elements of Γ W are elliptic in T W , and according to a lemma of Serre (see [START_REF] Shalen | Dendrology of groups : an introduction[END_REF]p. 271]), Γ W has a global fixed point. Moreover, according to Lemma 3.5 used as in the proof of Lemma 6.7, two elements of Γ W which are not powers of a common element have at most one common fixed point. Hence, the global fixed point is unique . Remark. According to Lemmas 6.9 and 6.10, in both cases (lim ω λ n = +∞ or lim ω λ n < +∞), the subsurface Σ 0 cannot be equal to Σ, since then Γ would have a global fixed point in lim ω ( Σ, [ q n ]).

Moreover, if γ 1 ∈ Γ W -

Complementary connected components of Σ 0 in Σ.

In this Subsection 6.4, we assume that W is the closure of a connected component of Σ -Σ 0 . Then W is non trivial, compact, connected and π 1 -injective. We assume first that W is neither a cylinder nor a pair of pants. Lemma 6.12 Either there exists ε > 0 and I ∈ ω such that for all n ∈ I and α ∈ S (W ), we have [qn] (α) ε, or there exists a sequence (α k ) k∈N in S (W ) such that ( α k ) k∈N (strictly) decreases to zero.

Remark. In the first case, for all n ∈ I and α ∈ C (W ), we have [qn] (α) ε. We will see that this case cannot happen if lim ω λ n = +∞.

We will need the following two lemmas. Let m be a hyperbolic metric on Σ and let W m be the m-geodesic realization of W . Let (Λ, µ) be a measured hyperbolic lamination of W m . Lemma 6.13 There exists a possibly constant sequence (α k ) k∈N in S (W ) such that the sequence (i((Λ, µ), α k )) k∈N decreases to zero.

Proof. If Λ contains a simple closed geodesic β, we take α k equal to the homotopy class of β for all k ∈ N. Otherwise, let L 1 be a minimal component of Λ. For every k ∈ N, let c be a transverse arc to L 1 , that meets L 1 and does not meet any other minimal component of Λ, such that the total µ-mass of c is at most 1 k . Since the intersection of any leaf of L 1 with c is dense in the Cantor set L 1 ∩ c, there exist two distinct points in c belonging to the same leaf. Then, if α k is the concatenation of the segment of this leaf and of the segment of c between these two points, we have i(α k , (Λ, µ)) 1 k . Moreover, the image of α k contains the image of an essential simple closed curve α k ∈ S (W ), and i(α k , (Λ, µ)) i(α k , (Λ, µ)) 1 k . We recall that (Λ, µ) fills up W if i(α, (Λ, µ)) > 0 for all α ∈ C (Σ) that topologically cuts W (or equivalently for all α ∈ C (W ), since W is not a cylinder). If (Λ , µ ) is another measured hyperbolic lamination of W m , recall that (Λ, µ) and (Λ , µ ) jointly fill up W if i(α, (Λ, µ)) + i(α, (Λ , µ )) > 0 for all α ∈ C (W ), and i((Λ, µ), (Λ , µ )) > 0.

Lemma 6.14 Assume that (Λ, µ) and (Λ , µ ) jointly fill up W , and let (Λ n , µ n ) n∈N and (Λ n , µ n ) n∈N be two sequences of measured hyperbolic laminations of Σ that respectively ωconverge to two measured hyperbolic laminations of Σ whose intersections with W m are equal to (Λ, µ) and (Λ , µ ). There exist I ∈ ω and ε > 0 such that i(α, (Λ n , µ n ))+i(α, (Λ n , µ n )) ε for all n ∈ I and α ∈ S (W ).

Proof. Assume for a contradiction that there exists a sequence

(α n ) n∈N in S (W ) such that lim ω i(α n , (Λ n , µ n )) = lim ω i(α n , (Λ n , µ n )) = 0.
Then, there exists a positive real sequence (t n ) n∈N such that (t n α n ) n∈N ω-converges to a measured hyperbolic lamination (L, ν) of W m . First, L is not a boundary component of W m since then α n would spiral around this boundary component, for n large enough, which is impossible for an essential simple closed geodesic of W m . Moreover, we have

lim ω t n < +∞, hence i((L, ν), (Λ, µ)) + i((L, ν), (Λ , µ )) = 0
If L contains a simple closed geodesic α, then i(α, (Λ, µ)) + i(α, (Λ , µ )) = 0, which is impossible since (Λ, µ) and (Λ , µ ) jointly fill up W . Let L 1 be a minimal component of L, and let Σ(L 1 ) ⊆ W m be the connected subsurface filled up by L 1 (see Definition 4.5). If Σ(L 1 ) were not equal to W m up to isotopy, there would exist a boundary component α of Σ(L 1 ) which would be essential in W , and then i(α, (Λ, µ))+i(α, (Λ , µ )) = 0, a contradiction. Hence, the lamination L 1 fills up W . By the condition i((L, ν), (Λ, µ))+i((L, ν), (Λ , µ )) = 0, we have Λ ⊆ L 1 and Λ ⊆ L 1 , and since L 1 is minimal, we have L 1 = Λ = Λ , a contradiction since i((Λ, µ), (Λ , µ )) > 0.

Proof of Lemma 6.12. For all n ∈ N, we choose a quadratic differential q n representing [q n ], and we denote by (F h n , µ h n ) and (F v n , µ v n ) (the equivalence classes of) its horizontal and vertical measured foliations. There exist two positive real sequences

(x n ) n∈N and (y n ) n∈N such that x n (F h n , µ h n ) n∈N and y n (F v n , µ v n ) n∈N ω-converge to two (equivalence classes of) measured foliations (F h , µ h ) and (F v , µ v ). For all n ∈ N and α ∈ C (Σ), we have max{i(α, (F h n , µ h n )), i(α, (F v n , µ v n ))} [qn] (α) i(α, (F h n , µ h n )) + i(α, (F v n , µ v n )). (1)
Assume for a contradiction that lim ω x n = 0. Since µ h is nonzero, there exists α ∈ S (Σ)

such that lim ω i(α, (F h n , x n µ h n )) = i(α, (F h , µ h )) > 0. However, lim ω i(α, (F h n , x n µ h n )) lim ω x n [qn] (α) = 0, since lim ω [qn] (α) < +∞.
Hence, lim ω x n is nonzero, and similarly lim ω y n is nonzero.

Assume for a contradiction that lim ω x n = lim ω y n = +∞. Then, for all α ∈ S (W ), we have

lim ω [qn] (α) lim ω (i(α, (F h n , µ h n )) + i(α, (F v n , µ v n ))) lim ω ( 1 x n i(α, (F h n , x n µ h n )) + 1 y n i(α, (F v n , y n µ v n ))) = 0,
a contradiction since α is essential in W and hence does not belong to S 0 . Hence, up to permuting (F h n , µ h n ) n∈N and (F v n , µ v n ) n∈N , we can assume that lim ω x n < +∞. This means that the sequence (F h n , µ h n ) n∈N , without renormalization, ω-converges to a (nonzero equivalence class of) measured foliation, still denoted by (F h , µ h ).

Assume first that lim ω y n < +∞. Then, the sequence (F v n , µ v n ) n∈N also ω-converges, without renormalization, to a (nonzero, equivalence class of) measured foliation, still denoted by (F v , µ v ). If α is a boundary component of W , it is also a boundary component of Σ 0 and we have lim ω [qn] (α) = 0. Hence, by Equation ( 1

) i(α, (F h , µ h )) = i(α, (F v , µ v )) = 0.
Consequently, if (Λ h , ν h ) and (Λ v , ν v ) are the measured hyperbolic laminations of (Σ, m) associated with (F h , µ h ) and (F v , µ v ) (see for instance [Lev]), then no leaf of Λ v or Λ h intersects the boundary of W m , and the intersections

Λ h W = Λ h ∩ W m and Λ v W = Λ v ∩ W m
, endowed with the induced transverse measures (if not empty), are measured hyperbolic laminations.

If (Λ h W , µ h ) and (Λ v W , µ v ) jointly fill up W , according to Lemma 6.14, there exist I ∈ ω and ε > 0 such that, for all n ∈ I and α ∈ S (W ), we have i(α,

(F h n , µ h n )) + i(α, (F v n , µ v n )) ε, hence [qn] (α) max{i(α, (F h n , µ h n )), i(α, (F v n , µ v n ))} ε 2
, so that the first conclusion of Lemma 6.12 holds.

Recall that there exists no α ∈ S (W ) such that i(α, (Λ h , µ h )) + i(α, (Λ v , µ v )) = 0, since then α would belong to S 0 by Equation ( 1), and would have a representative contained in

Σ 0 . Hence, if (Λ h W , µ h ) and (Λ v W , µ v ) do not jointly fill up W , then i((Λ h W , µ h ), (Λ v W , µ v )) = 0. Assume first that i((Λ h W , µ h ), (Λ v W , µ v
)) = 0 and that the two sets Λ h W and Λ v W are not empty. Let Σ(Λ h W ) ⊆ W m be the subsurface with m-geodesic boundary filled up by

Λ h W . Assume for a contradiction that Σ(Λ h W ) is strictly contained in W m . Let β be a boundary component of Σ(Λ h W ). Since Λ h W fills up Σ(Λ h W ) and i((Λ h W , µ h ), (Λ v W , µ v )) = 0, we have i(β, (Λ v W , µ v W )) = 0 and i(β, (Λ h W , µ h W )) = 0 since β is a boundary component of Σ(Λ h W ). Hence i(β, (Λ h , µ h )) + i(β, (Λ v , µ v )) = 0, a contradiction since β is essential in W . Hence, Λ h W fills W and similarly Λ v W fills W . Since i((Λ h , µ h ), (Λ v , µ v
)) = 0, one of these hyperbolic laminations is contained in the otherone. Moreover, the two laminations are minimal since W is connected, and hence they are equal. Hence, either one of the sets Λ h W , Λ v W is empty, or the laminations Λ h W and Λ v W are equal. In both cases, according to Lemma 6.13, there exists a sequence

(α k ) k∈N of S (W ) such that (i(α k , (Λ h W , µ h )) + i(α k , (Λ v W , µ v ))
) k∈N strictly decreases to zero, hence the second conclusion of Lemma 6.12 holds by Equation [START_REF])| = d(w n , γx n ) + d(z n , γx n ) -d(w n , z n ). Moreover, since the space is CAT(0), we have d(w n , γx n ) w n , γx n⊥ ) 2 + d(γx n , γx n⊥ ) d(z n , γx n ) 2 d(z n , γx n⊥ ) 2 + d(γx n , γx n⊥ ) 2 . Hence d(w n , γx n ) -d(w n , γx n⊥ ) d(γx n , γx n⊥ ) 2[END_REF].

Assume next that lim ω y n = +∞. If β is a boundary component of W , as above, by Equation ( 1), we have i(β, (F h , µ h ))

lim ω [qn] (β) = 0. Consequently, if we denote by (Λ h , ν h ) the measured hyperbolic lamination of (Σ, m) associated with (F h , µ h ), then no leaf of Λ h intersects the boundary of W m . Moreover, the intersection Λ h W = Λ h ∩ W m is not empty, otherwise all the elements of S (W ) would belong to S 0 by Equation ( 1). Hence, the set Λ h W , endowed with the induced transverse measure, is a measured hyperbolic lamination. For all α ∈ S (W ) and n ∈ N, we have

max{i(α, (F h n , µ h n )), 1 y n i(α, (F v n , y n µ v n ))} [qn] (α) i(α, (F h n , µ h n )) + 1 y n i(α, (F v n , y n µ v n )).
Hence, lim ω [qn] (α) = i(α, (Λ h , ν h )). As above, there exists a sequence (α k ) k∈N of S (W ) such that i(α k , (Λ h , ν h )) strictly decreases to zero, hence the second conclusion of Lemma 6.12 holds by Equation ( 1).

Case of degeneration.

In this Subsection 6.5, we assume as in the previous one that W is the closure of a complementary connected component of Σ 0 which is neither a cylinder nor a pair of pants. We will study the asymptotic behavior of the restriction to W of the sequence of halftranslation structures ([q n ]) n∈N under the hypothesis that the second conclusion of Lemma 6.12 holds, i.e. there exists a sequence (α k ) k∈N of S (W ) such that ( α k ) k∈N strictly decreases to zero. Remark. This case corresponds to the subsurface W 2 of the example of Subsection 6.2, since for every α ∈ S (W 2 ) we have lim ω [qn] (α) = i(α, (F v , µ v )), and (F v , µ v ) is filling. Lemma 6.15 If there exists a sequence (α k ) k∈N such that ( α k ) k∈N strictly decreases to zero, then we have lim ω Diam [ qn] ( W n ) = 0, lim ω Area [ qn] ( W n ) = 0 and for every boundary component β of W , we have lim ω [ qn] (β) = 0. And the ultralimit [ W n ] n∈N is an R-tree, on which the isometric action of Γ W has no global fixed point. Moreover, the minimal subtree of [ W n ] n∈N for the action of Γ W is dual to a measured hyperbolic lamination (Λ, µ) (for any hyperbolic metric on Σ), which fills up W .

Remark. Since W is connected, if Λ is filling, it is also minimal.

Proof. For all p ∈ N, we denote by W p the subsurface of Σ filled up by {α k : k p} (defined up to isotopy). According to Lemma 4.2, if p 2 p 1 , up to isotopy, the subsurface

W p 2 is contained in the interior of W p 1 . Moreover, if W p 2 is not isotopic to W p 1 , then at least one connected component of W p 1 -W p 2 has a negative Euler characteristic, hence χ(W p 2 ) > χ(W p 1 )
. Since the Euler characteristic of W p is non positive for all p ∈ N, there exists P ∈ N such that W p is isotopic to W P for all p P . We denote by W P a (isotopy class of a) connected component of W P . Since W P is filled up by some essential simple closed curves, it is not a pair of pants. Assume for a contradiction that it is a cylinder. Let α be a isotopy class of simple closed curves contained in W P . For every ε > 0, there exists p P such that αp ε, and α p fills up W P . Since W P is assumed to be a cylinder, we have α = α ±1 p , and α = αp < ε. Hence α belongs to S 0 , which is impossible since α is essential in W . Hence, the subsurface W P is not a cylinder.

Let ε > 0. There exists p 0 > P such that α k < ε 2 for all k > p 0 , and according to Lemma 4.4, there exists a subset of {α k } k>p 0 , with at most K ∈ N elements (where K only depends on the topology of W ), that fills up W P . Since K < +∞, there exists I ∈ ω such that [qn] (α) ε for all α in this subset and n ∈ I. Let Σ W P be a W P -cover of Σ, let W P be the connected component of the preimage of W P in Σ W P which is not simply connected, and for all n ∈ I, let [ q n ] be the pullback of [q n ] to Σ W P . We denote by W P,n the [ q n ]-geometric realization of W P . According to Lemma 4.9, the length of any boundary component of W P,n is at most Kε, its diameter is at most 11Kε and its area is at most 1 π (1 + 2b) 2 (Kε) 2 (where b is the number of boundary components of W P ). Hence, if β is (the free homotopy class of) a boundary component of W P , then β = 0 and β belongs to S 0 . Therefore β is a boundary component of Σ 0 and of W . Hence, W P = W P = W (we will then replace W P and W P by W ). Since the diameter of W n ω-converges to zero, according to Lemma 6.6, the ultralimit [ W n ] n∈N is an R-tree. Recall that by the definition of Σ 0 , for all α ∈ S (W ), we have α > 0, hence the action of Γ W on [ W n ] n∈N has no elliptic element, except the stabilizers of the boundary components of W . Hence, according to Lemmas 6.7 and 6.8, the minimal subtree for the action of Γ W is dual to a measured hyperbolic lamination (Λ, µ) (for any hyperbolic metric m on Σ), of (the m-geometric realization of) W . Moreover, for every α ∈ C (W ), we have i(α, (Λ, µ)) = α > 0, hence (Λ, µ) fills up W .

Let T min be the minimal subtree of [ W n ] n∈N for the action of Γ W , and let (Λ, µ) be the measured hyperbolic lamination of (the m-geometric realization of) W dual to T min .

By the compactness of the space of projective measured hyperbolic laminations, there exists a positive real sequence (t k ) k∈N such that the sequence (t k α k ) k∈N ω-converges to a measured hyperbolic lamination (Λ α , µ α ) of W . Lemma 6.16 The laminations Λ and Λ α are equal.

Proof. Since lim ω t k < +∞, we have

i((Λ, µ), (Λ α , µ α )) = lim ω t k i((Λ, µ), α k ) = lim ω t k α k = 0.
Since Λ is minimal and fills up W , we have Λ = Λ α . Lemma 6.17 all n ∈ N, let c n be the boundary component of W n corresponding to c. Then, according to Remark 6.1, the set of fixed points of γ in T W is [ c n ] n∈N . Moreover, the minimal subtree T min is the union of the translation axes of the hyperbolic elements of Γ W in T W , and according to Lemma 6.4, this union intersects [ c n ] n∈N only at c .

Case of non degeneration.

In this Subsection 6.6, we assume as in the previous one that W is the closure of a complementary connected component of Σ 0 which is neither a cylinder nor a pair of pants. We consider now the case where the first conclusion of Lemma 6.12 holds, i.e. there exists ε > 0 such that for all α ∈ S (W ) and ω-almost all n ∈ N,s we have [qn] (α)

ε. The boundary components of W (if any) are also some boundary components of Σ 0 , hence their [q n ]-lengths ω-converge to zero. Since W has finitely many boundary components, there exists I ∈ ω such that for every boundary component c of W and all n ∈ I, we have [qn] (c) ε 5 . We replace the sequence ([q n ]) n∈N by ([q n ]) n∈I . Hence, in the remainder of this Subsection 6.6, we can assume that ε > 0 is given so that :

• for all α ∈ C (W ), we have lim ω [qn] (α) < +∞ ;

• for all n ∈ N and α ∈ C (W ), we have [qn] (α) ε ;

• for every boundary component c of W and n ∈ N, we have [qn] (c) ε 5 , and lim ω [qn] (c) = 0. Remark. This case corresponds to the subsurface W 1 of the example of Subsection 6.2, since for every α ∈ S (W 1 ) we have lim ω [qn] (α) = [q] (α) 1. Lemma 6.18 For all n ∈ N, the boundary components of W n (if any) are simple and pairwise disjoint. Moreover, there exists no essential arc of W n between two (possibly equal) boundary components of W n , of length less than ε 4 .

Proof. For all n ∈ N, in the construction of W n (see [Raf]), the boundary components of W n are simple. Assume for a contradiction that there exist two distinct boundary components that intersect each other, or that there exists an essential arc of length less than ε 4 between two (possibly equal) boundary components of W n . Since W is not a cylinder, the union of the image(s) of this (or these) boundary component(s) with (possibly) the essential arc, contains a closed curve of length less than 2 5 ε + 2 4 ε < ε, which is not freely homotopic to a point nor to a boundary component of W , and hence is (freely homotopic to) an essential closed curve of W , a contradiction.

Remark. This does not necessarily mean that the geodesic representatives of the boundary components of W in (Σ, [q n ]) are simple and pairwise disjoint for all n ∈ N.

According to Lemma 6.18, the boundary components of W n are pairwise at distance at least ε 4 . Let H W be the subset of non trivial and non peripheral elements of Γ W : their action on [ W n ] n∈N is hyperbolic. For every γ ∈ H W , let F (γ) be the (possibly degenerated) flat Proof. Let I ∈ ω be such that for any sequence of singular points (y n ) n∈N , we have lim ω d n (y n , z n ) = 0 if and only if there exist i ∈ I such that y n = y i,n for all n ∈ I. Since, for all n ∈ I, the ball B n (x n , δ) contains k < +∞ singular points, there exist η > η > 0 small enough such that B n (z n , η ) is contained in B n (x n , δ) and for ω-almost all n ∈ I, the only singular points contained in B n (z n , η ) belongs to B n (z n , η 2 ). Let n ∈ I. The boundary C n of B n (z n , η) is naturally endowed with a cyclic order. Let x 0 be a point of C n and let (x i ) i∈{0,...,pn} , with p n ∈ N, be a finite sequence, increasing for the cyclic order, defined by d n (x i , x i+1 ) = r, with r ∈ [ η 4 , η 3 ], for all i ∈ {0, . . . , p n -1}, and such that x pn is contained between x 0 and x 1 . There exists r ∈ [ η 4 , η 3 ] such that x kn = x 0 . Moreover, the ball B n (z n , η) contains at most one point of any Γ W -orbit, hence it embeds into W n , and according to [START_REF] Minsky | Harmonic maps, length, and energy in Teichmüller space[END_REF]Lem. 4.1], for all n ∈ I, the length of C n is at most Lη, where L depends on χ( W ). Hence, the integer p n is bounded by Lη/ η 3 = 3L, and there exists p ∈ N * such that it is equal to p ∈ N for ω-almost all n ∈ I. Up to changing I, we can assume that for all n ∈ I, there exists a topological disk P n whose boundary is a finite union of p Euclidean segments of equal length between η 4 and η 3 , whose endpoints belong to C n . Moreover, for all n ∈ I, the boundary

C n is contained in B n (z n , η) -B n (z n , 2η
3 ), and by the choice of η, the distance d ω is locally Euclidean on [B n (z n , η) -B n (z n , 2η

3 )] n∈N , hence [P n ] n∈N is a topological disk whose boundary is a union of p Euclidean segments.

zn xi η 2 η x1 = x k Pn θ(xi)
The interior Euclidean angle θ n (x i ) at a vertex x i is determined by the distance between the midpoints of the segments adjacent to x i , and it ω-converges to an angle θ ω (x i ) determined by the distance between the midpoints of the segments adjacent to the corresponding vertex. According to the Gauss-Bonnet formula applied to P n , for all n ∈ I, we have

2 π = i∈I (2π -θ i ) + p i=1 (π -θ n (x i )). Hence p i=1 (π -θ n (x i )) = 2 π -i∈I (2π -θ i ). The sequence ( p i=1 (π -θ n (x i ))) n∈N ω-converges to p i=1 (π -θ ω (x i )). Hence, p i=1 (π -θ ω (x i )) = 2 π -i∈I (2π -θ i ).
Moreover, according to the remark preceding Lemma 6.22, the distance

d ω is locally Euclidean on [B n (z n , η)] n∈N , except at [z n ] n∈N . Hence, the point [z n ] n∈N
is a conical singularity of angle θ 0, and by the Gauss-Bonnet formula applied to

[P n ] n∈N , we have 2 π = 2 π -θ -p i=1 (π -θ ω (x i )), hence θ = 2π + i∈I (θ i -2π). Hence, the set [ W n ] n∈N -[∂ W n ]
n∈N is a surface endowed with a locally Euclidean metric with conical singularities of angles kπ, k ∈ N and k 3, which is Γ W -invariant by naturality. This ends the proof of Lemma 6.20. Lemma 6.23 The convex hull of γ∈H W F (γ), minus the points c , where c is a boundary component of W (if any), is a surface without boundary, endowed with a locally Euclidean metric with conical singularities of angles of the form kπ, k ∈ N and k 3, which is CAT(0) and Γ W -equivariant. Moreover, the intersection of any translation axis of any hyperbolic element of Γ with [ W n ] n∈N is contained in the convex hull of γ∈H W F (γ), if it is not empty.

Proof. There exists a finite set of free homotopy classes (relative to the boundary of W ) of simple arcs of W joining some boundary components of W , having pairwise disjoint representatives, such that no arc is freely homotopic to an arc contained in a boundary component of W , and the union of some simple and pairwise disjoint representatives of these homotopy classes cuts W into a finite set of disks. Then, the union of all the lifts of these arcs in W cuts W into disks. We extend them into a set S of pairwise disjoint proper biinfinite simple paths of Σ (possibly not in a Γ-equivariant way), that intersect the boundary of W in exactly two points, such that every path cuts Σ into two connected components.

Let m be a hyperbolic metric on Σ and let m be its pullback on Σ. We consider the m-geodesic representatives of the elements of S, and we still denote by S the corresponding set of pairwise disjoint biinfinite m-geodesics. The elements of S are pairwise disjoint, each geodesic s ∈ S divides W m into two connected components, and the intersection of the union of the geodesics of S with W m cuts W m into disks. Since the geodesics are pairwise disjoint, the disks can be labelled by the (unordered) sets of boundary components of W m that meet theirs boundaries. We denote by (D k ) k∈N the closures of the connected complementary components of ( S) ∩ W m in W m , that are (topological) disks. Let n ∈ N. For every s ∈ S, let s n be a [ q n ]-geodesic having the same (ordered) pair of points at infinity than s. Since the geodesics s n are properly homotopic to the pairwise disjoint elements of S, to each disk D k , with k ∈ N, corresponds a unique convex generalized disk D k (n) (i.e. a (possibly empty) disk with a finite number of spikes of finite lengths (possibly zero)), whose boundary is contained in

( s∈ S s n ) ∪ ∂ W n . It may happen that D k (n) is a graph.
Let k 1 , k 2 ∈ N and let (x 1,n ) n∈N and (x 2,n ) n∈N be two sequences of points such that for all n ∈ N, x 1,n belongs to D k 1 (n) and x 2,n belongs to D k 2 (n). Since every geodesic of S cuts W m into two connected components, if the point x 1 , x 2 belong respectively to D k 1 and D k 2 , there exists a finite sequence ( s i ) i∈I in S, such that the m-geodesic segment joining x 1 to x 2 cuts the geodesics ( s i ) i∈I . Then, for all n ∈ N, the geodesics ( s i,n ) i∈I cut the geodesic segment joining x 1,n to x 2,n in W n into a bounded number of geodesic segments of finite lengths (several geodesics can be cut at the same time). Hence, the geodesic segment

[[x 1,n , x 2,n ]] n∈N in [ W n ] n∈N is the concatenation of a finite number of geodesic segments whose endpoints belong to i∈I ([ s i,n ] n∈N ), hence it is contained in k∈N [D k (n)] n∈N . Therefore k∈N [D k (n)] n∈N is convex. Let us show that the union k∈N [D k (n)] n∈N intersects [∂ W n
] n∈N only at the points c , where c is a boundary component of W . For all n ∈ N, if s n is a geodesic corresponding to an element s ∈ S, it is interlaced with two boundary components c and c of W , and its pair of points at infinity does not depend on n. Hence, there exists γ ∈ Γ -{e} and k ∈ Z such that for all n ∈ N, a translation axis Ax n (γ) of γ is interlaced with c n , and s n is caught between γ -k c (Ax n (γ)) and γ k c (Ax n (γ)), where γ c ∈ Γ W -{e} preserves c. Similarly, there exists γ ∈ Γ -{e} and k ∈ Z such that for all n ∈ N, a translation axis Ax n (γ ) of γ is interlaced with c n , and s n is caught between γ -k c (Ax n (γ )) and γ k c (Ax n (γ )), where γ c ∈ Γ W -{e} preserves c . According to Lemma 6.2, the geodesic [ s n ] n∈N can intersect [∂ W n ] n∈N only at the points c and c . Hence, the convex union k∈N

[D k (n)] n∈N , minus the points c , is contained in [ W n ] n∈N -[∂ W n ] n∈N ,
and according to Lemma 6.20, it is a surface without boundary. Since it is convex, it is CAT(0) as

[ W n ] n∈N . Finally, it is naturally Γ W -invariant.
Let γ ∈ H W . By the definition of S, there exists a sequence ( s i ) i∈Z of S that cuts the translation axis Ax m (γ) of γ in W m into bounded intervals. Then, for all n ∈ N, if Ax n (γ) is a translation axis of γ in W n , it is cut by the sequence of geodesics ( s i,n ) i∈N into bounded intervals, and the order of the ( s i ) i∈N does not depend on n ∈ N (it may happen that several geodesics are intersected at the same time). Hence, the geodesic

[Ax n (γ)] n∈N is contained in k∈N [D k (n)] n∈N . Hence, all the (possibly degenerated) flat strips F (γ), with γ ∈ H W , are contained in k∈N [D k (n)] n∈N , and since k∈N [D k (n)] n∈N is convex, the convex hull of γ∈H W F (γ) is contained in k∈N [D k (n)] n∈N . Moreover, since the convex hull of γ∈H W F (γ) is Γ W -equivariant
, according to Lemma 3.4, it contains all the points c , where c is a boundary component of W , hence it contains all the geodesic segments joining them, and notably the segments

[ s n ] n∈N ∩ [ W n ] n∈N , with s ∈ S. Hence, the convex hull of s∈ S [ s n ] n∈N ∩ [ W n ] n∈N , which is equal to k∈N [D k (n)] n∈N , is equal to the convex hull of γ∈H W F (γ). Finally, if γ is a hyperbolic element of Γ -Γ W whose translation axis Ax ω (γ) in lim ω ( Σ, [ q n ]) intersects [ W n ] n∈N ,
according to Lemma 6.2, there exists two boundary components c and c of W such that the intersection Ax ω (γ)

∩ [ W n ] n∈N is the segment [ c , c ], hence is contained in the convex hull of γ∈H W F (γ).
Lemma 6.24 The locally Euclidean metric with conical singularities on W ω is a halftranslation structure [ q ω ] on W ω , that is the pullback of a half-translation structure [ q ω ] on a finite type surface W ω homeomorphic to W -∂ W . Moreover, if W = Σ, [ q ω ] can be extended to the compact surface obtained from W ω by filling in the punctures, with possibly some singular points of angle π at the added points.

Proof. Let x = [x n ] n∈N be a point of W ω such that there exists r > 0 small enough, such that the (Euclidean) disk D ω (x, 2r) does not contain any singular point. Then, there exists I in ω such that for all n ∈ I, the disk

D n (x n , r) in W n does not contain any singular point of [ q n ]. Let x 1 = [x 1,n ] n∈N and x 2 = [x 2,n
] n∈N be two points of D ω (x, r), and let r 1 , r 2 > 0 be such that the disks D ω (x 1 , r 1 ) and D ω (x 2 , r 2 ) are contained in D ω (x, r). We can always assume that the distance

d n (x 1,n , x 2,n ) is constant for all n ∈ N. For all n ∈ I, let ψ i,n : D n (x i,n , r i ) → D(0, r i ) (with i = 1, 2, where D(0, r i ) = {z ∈ C : |z| < r i }) be the inverse of an exponential map at x i,n . Then, if r 1 + r 2 < d ω (x 1 , x 2 ), for ω-almost all n ∈ N, there exists c n ∈ C such that the restriction of ψ 2,n •ψ -1 1,n to ψ 1,n (D n (x 1,n , r 1 )∩D n (x 2,n , r 2 )) is either z → z +c n or z → -z +c n . Moreover, we have |c n | = d n (x 1,n , x 2,n ), hence the sequence (c n ) n∈N ω-converges to c ∈ C. Let us define the isometry ψ i,ω : D ω (x i , r i ) → D(0, r i ) by ψ i,ω ([w n ] n∈N ) = lim ω ψ i,n (w n ) (i ∈ {1, 2}). Then, the restriction of ψ 2,ω • ψ -1 1,ω to ψ 1,ω (D ω (x 1 , r 1 ) ∩ D ω (x 2 , r 2 )) is either z → z + c or z → -z + c.
Hence, the set of maps thus defined spans a maximal atlas of charts on the complement of the singular points in W ω , such that the exchange maps are of the form z → ±z + c, with c ∈ C. Since the singular points are of angles kπ, with k ∈ N and k 3, it defines a half-translation structure [ q ω ] on W ω . This half-translation structure is naturally Γ Winvariant, and since W ω is simply connected and the action of Γ W is properly discontinuous, it defines a half-translation structure [ q ω ] on the surface W ω = Γ W \ W ω . Moreover, since Γ W is finitely generated, the surface W ω is of finite type. Let us prove that W ω is homeomorphic to W -∂ W . Since Γ W has b conjugation classes of parabolic elements (b is the number of boundary components of W ), the surface W ω has b punctures. Moreover, the surface W ω is orientable, and two translation axes of two elements γ 1 and γ 2 of H W in W ω are properly homotopic to two disjoint biinfinite paths if and only if any two translation axes of γ 1 and γ 2 in W n are properly homotopic to two disjoint biinfinite paths, for any n ∈ N (see the proof of [START_REF] Morzadec | Compactification géométrique de l'espace des modules des structures de demi-translation sur une surface[END_REF]Lem. 6.26]). Hence, the cardinality of a maximal set of isotopy classes of essential simple closed curves of W ω having pairwise disjoint representatives is equal to the cardinality of a maximal set of isotopy classes of simple closed curves having pairwise disjoint representatives of W . Hence, the Euler characteristics of W ω and of W are equal, and since W ω has b punctures, the surfaces W ω and W have the same genus. Hence, W ω is homeomorphic to W -∂ W .

The complex structure defined by [ q ω ] on W ω can be extended to the compact surface without puncture obtained from W ω by filling in the punctures. Let a be a puncture of W ω , and let (U, z) be a chart of the extended complex structure, with a ∈ U . Let q ω ∈ Q( W ω ) be a representative of [ q ω ]. We define the holomorphic map ϕ on U -{a} by ϕ(x) = q ω,1 (x)( dz 1 dz ) 2 , for any chart (U 1 , z 1 ) of W ω such that x belongs to U 1 , where q ω,1 is the representative of q ω in (U 1 , z 1 ). Let a be the fixed point of an elliptic element γ a ∈ Γ W -{e} in the completion of W ω , such that the free homotopy class of closed curves defined by γ a in W ω as some representatives contained in any neighborhood of a. Let x be a point of W ω , close enough to a (for the distance d ω ). Then, by the isoperimetric inequality (see for example [START_REF] Bridson | Metric spaces of non-positive curvature[END_REF]Thm. 2.17 p. 426]), the angle sector, bounded by [ a, x], and [ a, γ a x] has a finite area. Moreover, it projects to a closed neighborhood of a in W ω , whose [ q ω ]-area is finite since the cover projection does not increase the area. Hence, the integral of |ϕ| on a closed neighborhood of a is finite, and ϕ can be extended to a meromorphic function, with at most a simple pole in a. Hence, the half-translation structure [ q ω ] on the compact surface with punctures W ω can be extended to a half-translation structure on the compact surface, possibly with singularities of angle π at the punctures.

6.7 Complementary components of Σ 0 which are cylinders or pair of pants.

It remains to study the ultralimits of the geometric realizations of the connected components of the preimages in Σ of the connected components of Σ 0 and of Σ -Σ 0 which are pair of pants or cylinders, and of the cylinders that can be homotoped to a boundary component of Σ 0 .

Let α be the free homotopy class of closed curves of a boundary component of Σ 0 , and let α be a lift of α in Σ. For all n ∈ N, we denote by F n ( α) the (possibly degenerated) flat strip, union of all the geodesics of ( Σ, [ q n ]) having the same pair of points at infinity than α.

The ultralimit [F n ( α)] n∈N exists and can be neither a plane nor a half-plane, since the height of F n ( α) is bounded, hence it is either a flat strip or a single geodesic. According to Remark 6.1, it is the set of fixed points in lim ω ( Σ, [ q n ]) n∈N of the elements of the stabilizer of α in Γ, and according to Lemma 6.2, the intersection of any translation axis of any hyperbolic element of Γ with [F n ( α)] n∈N is either empty, or a point, or a geodesic segment orthogonal to the boundary components of the flat strip [F n ( α)] n∈N .

Assume until the end of this section that W is the closure of a connected component of Σ -Σ 0 which is a pair of pants. According to [START_REF] Rafi | A characterization of short curves of a Teichmüller geodesic[END_REF]Rem. 3.2], the interior curvature of any boundary component of W n is at most -π. Hence, according to the Gauss-Bonnet formula, we should have 2πχ( W ) -3π, which is impossible since χ( W ) = -1. Hence, up to changing the notation, for ω-almost all n ∈ N, we can assume that the boundary components c 1,n and c 2,n of W n , corresponding to c 1 and c 2 , intersect each other. The geodesic c 3,n either intersects the union of the other ones, or is disjoint from the union of the other ones, and the interior of W n is a cylinder. In the second case, the union c 1,n ∪ c 2,n contained a simple closed curve isotopic to c 3,n , and according to Lemma 4.7, there exists a point in c 1,n ∪ c 2,n at distance of c 3,n at most

( [ qn] ( c 1,n )+ [ qn] ( c 2,n )),
and W n can be cut into a disk of perimeter at most 2(

[ qn] ( c 1,n ) + [ qn] ( c 2,n )) + ( [ qn] ( c 1,n ) + [ qn] ( c 2,n )) + [ qn] ( c 3,n ) 4( [ qn] ( c 1,n ) + [ qn] ( c 2,n )). In both cases, the diameter of W n is at most 4( [ qn] ( c 1,n ) + [ qn] ( c 2,n )).
According to Lemma 6.6, the ultralimit [ W n ] n∈N is an R-tree. Moreover, there exist some homotopy classes of arcs joining the boundary components of W , whose [ q n ]-lengths ω-converge to zero. As in the proof of Lemma 6.10, the action of Γ W has a global fixed point W which is equal to c for every boundary component c of W .

Mixed structures.

Let Σ be a compact, connected, orientable surface, such that χ(Σ) < 0, and let p : Σ → Σ be a universal cover with covering group Γ. Let Σ 0 be a tight proper subsurface of Σ, and let Σ 0 be its preimage in Σ. If W is a connected component of Σ 0 or of Σ -Σ 0 , we denote by Γ W the stabilizer of W in Γ.

Let W be a connected component of Σ -Σ 0 and let W be a connected component of the preimage of W in Σ. Assume first that W is neither a cylinder nor a pair of pants. Let T Λ be the R-tree dual to a measured hyperbolic lamination (Λ, µ) on W (for any complete hyperbolic metric).

Lemma 7.1 (see [START_REF] Morzadec | Compactification géométrique de l'espace des modules des structures de demi-translation sur une surface[END_REF]Lem. 7.1]) If Λ is filling, the stabilizers of the boundary components of W have a unique fixed point in T Λ . The other elements have no fixed point.

The surface W is a finite type surface. Let [q W ] be a half-translation structure on W , that can be extended to the compact surface obtained from W by filling in the punctures, with possibly some singularities of angle π at the added point. Let p W : ( W , [ q W ]) → (W, [q W ]) be a universal cover. Then ( W , [ q W ]) is not complete.

Lemma 7.2 The completion W c of ( W , [ q W ]) is the union of W and of countably many isolated points which are exactly the fixed points of the stabilizers of the boundary components of W in Γ W .

Let Σ 0 be as above. Let W be a connected component of Σ 0 or of Σ -Σ 0 . Let X W be a complete geodesic metric space endowed with an isometric action of the stabilizer Γ W of W in Γ, such that :

• if W is a strip, X W is empty ; • if W is a connected component of Σ 0 ,
which is not a strip, X W is a point, and the action of Γ W on X W is trivial ;

• if W is a connected component of Σ -Σ 0 such that W = p( W )
is neither a cylinder nor a pair of pants, X W is -either the R-tree dual to a filling measured hyperbolic lamination (Λ, µ) of W (for any complete hyperbolic metric), and then the action of Γ W on X W is dual to (Λ, µ), -or X W is the completion of a universal cover of a half-translation structure on W , that can be extended to a half-translation structure on the compact surface obtained from W by filling in the punctures, with possibly some singularities of angle π at the added points, and then the action of Γ W on X W is the covering action, extended at the added points ;

• if W is a connected component of Σ -Σ 0 such that W = p( W ) is a pair of pants, X W is a point, and the action of Γ W on X W is trivial.

Finally, for every proper homotopy class c of boundary components of Σ 0 (two boundary components of Σ 0 can be properly homotopic because of the strips), let X c be an edge (i.e. a compact interval of R, possibly reduced to a point). We assume that if c bounds a connected component of Σ -Σ 0 which is a strip, the length of this edge is nonzero and if W is a connected component of Σ -Σ 0 whose image in Σ is a pair of pants, then there exists at least one boundary component c of W such that the length of X c is nonzero. We endow X c with the trivial action of the stabilizer Γ c of c in Γ. Moreover, if W and W are the connected components of Σ 0 or of Σ -Σ 0 bounded by c, we identify the fixed points of Γ c in X W and X W (which exist and are unique, according to Lemmas 7.1 and 7.2) with the (possibly non distinct) endpoints of X c .

W 1 W 2 W Σ γ W ∂ Σ 0 X W0 X W1 X W2 X W X γ W W 0
Let (X, d) be the topological space obtained by this gluing, endowed with the length distance induced by the distances on the different metric spaces. We identify the metric spaces X W , and X c with their images in X. The actions of the stabilizers Γ W above extend uniquely in an isometric action of Γ by setting γ(X ) = X γ for every γ ∈ Γ, where is a connected component of Σ 0 or of Σ -Σ 0 , or a boundary component of Σ 0 . If Γ acts on a CAT(0) geodesic space X, and if Γ is a subgroup of Γ, if there exists a unique minimal non empty closed convex subset of X which is preserved by Γ , it is called the convex core of Γ . By definition of the action of Γ on X, we see that if W is a connected component of Σ 0 or of Σ -Σ 0 , then the convex core of Γ W in X is X W , and if c is a boundary component of Σ 0 , the set of fixed points of Γ c in X is X c . The elements of Γ that have a fixed point in X are those that preserve a connected component of Σ 0 or a connected component of Σ -Σ 0 whose image in Σ is a pair of pants. The other elements are hyperbolic. Moreover, if γ ∈ Γ is a hyperbolic element in X, it can have more than one translation axis only if it preserves a piece which is the completion of a surface endowed with a half-translation structure.

Lemma 7.3 The space (X, d) just defined is a CAT(0) tree-graded space, whose set of pieces are the spaces X W and X c as above, which is uniquely determined by the half-translation structures and by the measured hyperbolic laminations on the connected components of Σ-Σ 0 , and by the lengths of the edges corresponding to the boundary components of Σ 0 , up to Γequivariant isometry. Moreover, the action of Γ on X determines uniquely the subsurface Σ 0 (up to isotopy), the measured hyperbolic laminations (up to isotopy) and the half-translation structures (up to isometry), on the connected components of Σ -Σ 0 .

Remark. Recall that Σ 0 is proper in Σ, and by construction, the space X is not reduced to a point, and the action of Γ on X has no global fixed point.

Proof. According to the construction, two distinct pieces Y and Y have at most one common point x which is then a fixed point of the stabilizer Γ c of a boundary component c of Σ 0 in Γ. Moreover, the boundary component c cuts Σ into two connected components, and if W and W are two connected components of Σ 0 or of Σ -Σ 0 that are not contained in the same connected component of Σ -c, then there exists no common point between the pieces X W and X W , except possibly x. Hence, any simple loop of X is contained in a unique piece, and since all the pieces are simply connected, the space X is simply connected. Moreover, it is clearly locally CAT(0) and hence globally CAT(0). And any simple geodesic triangle is contained in a unique piece, hence (X, d) is a tree-graded space.

Conversely, if Z is a connected component of Σ 0 which is not a cylinder, then Γ Z is neither trivial nor cyclic, it fixes a point Z in X, either Γ Z is a free group of rank at least 3 or a free group of rank 2 and the action of Γ Z on X -{ Z } has one orbit, and Z is maximal (for the inclusion) for these properties. Hence, the connected components of Σ 0 which are not cylinder are exactly the connected closed subsurface of Σ whose stabilizers of the connected components of the preimage in Σ satisfy these properties, and the action of Γ on (X, d) determines the connected components of Σ 0 which are not cylinders, up to isotopy.

Let us show that the action of Γ on X determines the cylinder components of Σ 0 . No cylinder connected component of Σ 0 can be homotoped to a boundary component of another connected component of Σ 0 . Hence, if Z is a cylinder component of Σ -Σ 0 , it is not bounded by a cylinder component of Σ 0 but by two connected components of Σ 0 whose Euler characteristics are negative. And Z is a cylinder component of Σ 0 if and only if Γ Z is cyclic and pointwise preserves an edge (possibly reduced to a point) in X and Z is not a connected component of Σ -Σ 0 . Hence, the action of Γ on (X, d) determines the cylinder components of Σ 0 , up to isotopy.

Let us show that the action of Γ on X determines the half-translation structures and the measured hyperbolic laminations, on the connected components of Σ -Σ 0 that are neither a cylinder nor a pair of pants. Let W be such a connected component of Σ -Σ 0 . If the piece X W is the R-tree dual to a measured hyperbolic lamination (Λ, µ), then (Λ, µ) is determined (up to isotopy) by the set of translation distances of the elements of Γ W (see [START_REF] Bonahon | Geodesic laminations on surfaces[END_REF]). If X W is the completion of ( W , [ q W ]) as above, then

(W, [q W ]) = ( W , [ q W ])/Γ W is determined by the action of Γ W on ( W , [ q W ]).
Let us show that the space (X, d) and the action of Γ are determined by the data of Σ 0 , of the half-translation structures and the measured hyperbolic laminations on the connected components of Σ -Σ 0 that are neither a cylinder nor a pair of pants, and by the lengths of the edges, up to Γ-equivariant isometry. Assume that (X, d) and (X , d ) correspond to two constructions as above, with the same subsurface Σ 0 , the same half-translation structures and measured hyperbolic laminations, and the same lengths of the edges. Let W be a connected component of Σ -Σ 0 that is not a pair of pants. Then, if the piece X W is the completion of the universal cover of (W, [q W ]), it is determined by (W, [q W ]) up to a Γ W -equivariant isometry. Similarly, if X W is the R-tree dual to a measured hyperbolic lamination (Λ, µ), it is determined by (Λ, µ) up to a Γ W -equivariant isometry. In both cases, the map that associates to the fixed point of a non trivial elliptic element γ ∈ Γ W in the piece X W of X, the fixed point of γ in the piece X W of X (the fixed points exist and are unique according to Lemmas 7.1 and 7.2), extends in a unique way into a Γ W -equivariant isometry. Hence, the map Φ which associates to the set of fixed point(s) of a non trivial elliptic element γ ∈ Γ-{e} in X, the set of fixed point(s) of γ in X , extends in a unique way in an isometry, that defines an Γ W -equivariant isometry between X W and X W , for every connected component W of Σ 0 or of Σ -Σ 0 , and since γ(X ) = X γ , and γ(X ) = X γ for every γ ∈ Γ, the global isometry is Γ-equivariant.

so that |d(x n , γ 2 x n ) -d(x n , γx n ) -d(γx n , γ 2 x n )| < 3ε.
Let P n be the set of pieces covering X n , let Σ 0n be the tight subsurface associated with X n as in Lemma 7.3, and let Σ 0n be its preimage in Σ. Assume for a contradiction that γ is elliptic in X n . Then γ belongs to the stabilizer Γ W of a connected component W of Σ 0n or of Σ -Σ 0n whose image in Σ is a pair of pants. Either γ does not preserve any boundary component of W , and it fixes a unique point W in X n , or γ preserves a boundary component of W . In the former case, it fixes an edge (possibly reduced to a point), and it preserves another piece Y n of X n . According to Lemma 7.1 and 7.2, any element of Γ -{e} has at most one fixed point in any piece of X n that is not an edge. Hence, up to (possibly) replacing the fixed point W of γ by the good endpoint of the edge, the point W is the unique common point between the segments [x n , W ] and [ W , γx n ], and by construction of X n , the union

[x n , W ] ∪ [ W , γx n ] is a geodesic segment, equal to [x n , γx n ]. Similarly, the segment [x n , γ 2 x n ] is the union [x n , W ] ∪ [ W , γ 2 x n ]. Hence, we have |d(x n , γ 2 x n ) -d(x n , γx n ) -d(γx n , γ 2 x n )| = 2 d(x n , W ), hence d(x n , W ) < 3 2 ε and d(x n , γx n ) < 3ε, a contradiction. Hence γ is hyperbolic in X n .
If x n belongs to a translation axis of γ in X n , then d(x n , γx n ) = Xn (γ), and | X (γ)-Xn (γ)| < ε. Otherwise, let x n⊥ be the orthogonal projection of x n onto the (possibly degenerated) flat strip union of all the translation axes of γ in X n , and let Ax n (γ) be the translation axis containing x n⊥ . The segment [x n , x n⊥ ] and [x n , γx n ] may share a initial segment. Let a be the last intersection point between [x n , x n⊥ ] and [x n , γx n ]. Similarly, let b be the last intersection point between [γx n , γx n⊥ ] and [γx n , x n ]. Assume for a contradiction that the segment

[x n , γx n ] is disjoint from Ax n (γ). Then, the curve [a, x n⊥ ]•[x n⊥ , γx n⊥ ]•[γx n⊥ , b]•[b, a]
is a simple geodesic quadrangle, hence it is contained in a piece Y n ∈ P n (see [START_REF] Druţu | Tree-graded spaces and asymptotic cones of groups[END_REF]Lem. 2.5]).

Let y n be in [x n , x n⊥ ] ∩ Y n . Then γy n belongs to Y n ∩ [γx n , γx n⊥ ]. Since γx n⊥ is not equal to x n⊥ , the piece Y n is neither a point nor an edge. If Y n is the R-tree dual to a filling lamination or if Y n is the completion of a surface endowed with a half-translation structure, according to Lemma 3.4, the segment [y n , γy n ] intersects Ax n (γ), a contradiction. Hence, the segment [x n , γx n ] meets Ax n (γ), and similarly the segments [γx n , γ 2 x n ] and [x n , γ 2 x n ] meet Ax n (γ).

Moreover, since there exists a unique segment joining two points in a CAT(0) geodesic space, the first points of intersection between [x n , γx n ] and Ax n (γ), and [x n , γ 2 x n ] and Ax n (γ) are equal. Similarly, the last points of intersection between [γx n , γ 2 x n ] and Ax n (γ), and [x n , γ 2 x n ] and Ax n (γ) are equal. Let w n (resp. z n ) be the last (resp. first) point of intersection between [x n , γx n ] and Ax n (γ) (resp. [γx n , γ 2 x n ] and Ax n (γ)).

x n γx n γ 2 x n x n⊥ w n z n Ax n (γ) γx n⊥ γ 2 x n⊥ |d(x n , γx n ) -Xn (γ)| 2 3ε( X (γ) + ε), so | X (γ) -Xn (γ)| 2 3ε( X (γ) + ε) + ε.
Let W be a (open) non trivial, π 1 -injective, connected subsurface of Σ, and let Flat(W ) be the set of (isotopy classes) of half-translation structures on W , that can be extended to halftranslation structures on the compact surface obtained from W by filling in the punctures, with possibly some singularities of angle π at the added points. We identify Flat(W ) with the Γ-equivariant isometry classes of universal covers of half-translation structures on W and we endow Flat(W ) with the Gromov equivariant topology. Let W → W be a universal cover with covering group Γ W . For every element [q W ] of Flat(W ), we denote by [ q W ] its lift on W . Let H W be the set of non trivial and non peripheral elements of Γ W .

Lemma 7.6 The map π : (W, [q W ]) → ( [ q W ] (γ)) γ∈Γ W from Flat(W ) to (R + ) Γ W -{0} is a homeomorphism onto its image.

Proof. As in Lemma 7.5, this map is injective and continuous. Let ((x γ,n ) γ∈Γ W ) n∈N be a sequence of the image of Flat(W ) in (R + ) Γ W -{0} that converges to (x γ ) γ∈Γ W in the image of Flat(W ). Let (W, [q n ]) n∈N be the sequence of preimages of ((x γ,n ) γ∈Γ W ) n∈N in Flat(W ).

For all n ∈ N, let λ n be the minimal displacement of generator of ( W , [ q n ]), for a finite generating set of Γ W , as in Section 6. Assume for a contradiction that lim ω λ n = +∞. Then, there exists a generator s ∈ Γ such that lim ω ( W ,[ qn]) (s) = +∞, and x s,n does not ω-converge to x s ∈ R + . Hence, we have lim ω λ n < +∞, and the asymptotic limit lim ω ( W , [ q n ]) n∈N , without renormalization, exists and is endowed with an isometric action of Γ W . Since the point (x γ ) γ∈Γ W is in the image of Flat(W ), there exists ε > 0 such that for all γ ∈ H W , we have x γ > ε . Hence, according to Lemma 6.12 (that immediately extends to the universal cover of a finite type surface), there exist ε > 0 and I ∈ ω such that for all γ ∈ H W and n ∈ I, we have ( W ,[ qn]) (γ) > ε. Hence, according to Section 6.6 (that immediately extends to the universal cover of a finite type surface), the space lim ω ( W , [ q n ]) n∈N , minus the fixed points of the peripheral elements of Γ W , is the universal cover of W endowed with a half-translation structure, that extends to the punctures. Moreover, the ultralimit lim ω ( W , [ q n ]) n∈N is the limit of a subsequence of ( W , [ q n ]) n∈N for the Gromov equivariant topology (see [START_REF] Paulin | The Gromov topology on R-trees[END_REF]). Hence, every subsequence of (W, [q n ]) n∈N has a subsequence that converges to an element of Flat(W ), equal to (W, [q W ]) by injectivity of π. Hence the sequence (W, [q n ]) n∈N converges to (W, [q W ]), and the inverse map is continuous.

Lemma 7.7 The equivariant Gromov topology on Mix(Σ) is metrizable.

Proof. According to Lemma 7.5, the space Mix(Σ) is Hausdorff and with countable bases of neighborhoods of points. Let us prove that it is separable. Let Σ 0 be a tight subsurface of Σ, let Σ 0 be its preimage in Σ, let A be the set of connected components of Σ -Σ 0 whose image in Σ is not a pair of pants, and let B be the finite set of Γ-orbits of boundary components of Σ 0 . Let E(Σ 0 ) be the subset of Mix(Σ) whose underlying tight subsurface is Σ 0 . For every connected component W of Σ -Σ 0 , let W = p( W ) and let F 1 ( W ) be the set of completions of universal covers of (isotopy classes of) half-translation structures on W , and F 2 ( W ) be the set of R-trees dual to a filling measured hyperbolic lamination on W . We endow F 1 ( W ) and F 2 ( W ) with the Gromov equivariant topology. Let Φ be the map from W ∈A (F 1 ( W ) F 2 ( W )) × (R + ) B , endowed with the product topology, to Mix(Σ), that associates to ((X W ) W ∈A , (t b ) b∈B ) the Γ-equivariant isometry classes of mixed structures such that the piece preserved by Γ W , with W ∈ A, is X W , and the length of the edges pointwise preserved by the stabilizers of the elements of b ∈ B is t b (that exists and is unique under the few conditions before Lemma 7.3).

Moreover, for every element W ∈ A, according to [START_REF] Paulin | The Gromov topology on R-trees[END_REF]Thm. 5.2], the space F 2 ( W ) is separable, and according to Lemma 7.6, the space F 1 ( W ) is separable. Since (R + ) B is separable and A is countable, the space W ∈A (F 1 ( W ) F 2 ( W )) × (R + ) B is separable, and since the map Φ is continuous and surjective, the space E(Σ 0 ) is separable. Finally, the space Mix(Σ) is the countable union of the sets E(Σ 0 ), with Σ 0 a isotopy class of tight subsurface of Σ, hence Mix(Σ) is separable. Since Mix(Σ) has countable bases of neighborhoods of points, it has a countable basis. Moreover, the space (R + ) Γ -{0} is normal, hence according to Lemma 7.5, the space Mix(Σ) is also normal. According to a theorem of Urysohn (see for instance [START_REF] Dugundji | Topology[END_REF]Ch. 9,9.2]), it is metrizable.

The group R + * acts on Mix(Σ) by multiplications of the distances. Let P Mix(Σ) and P Flat(Σ) be the quotients of Mix(Σ) and of Flat(Σ) by these actions, endowed with the quotient topology of the equivariant Gromov topology.

Theorem 7.8 The space P Flat(Σ) is an open and dense subset of P Mix(Σ), which is compact.

Proof. Let ( Σ, [ q n ]) n∈N be a sequence of Flat(Σ). Let S be a finite generating set of Γ, and for all n ∈ N, let λ n be the minimal displacement of the generators on ( Σ, [ q n ]) and let n be a point such that max{d n ( n , s n ) : s ∈ S} is less than λ n + 1, as in Section 6. Let [ q n ] = 1 λn [ q n ], and let [q n ] be the half-translation structure on Σ defined by [ q n ]. Let ω be a non principal ultrafilter on N as in Section 5. Then, the ultralimit lim ω ( Σ, [ q n ], n ) n∈N is a CAT(0), geodesic, complete space, endowed with an isometric action of Γ.

Let us show that lim ω ( Σ, [ q n ], n ) n∈N has a Γ-invariant subset which is a mixed structure. Let S 0 = {α ∈ S (Σ) : lim ω [qn] (α) = 0} and let Σ 0 be the tight subsurface of Σ filled up by S 0 , and let Σ 0 be its preimage in Σ. No connected component of Σ 0 is a pair of pants. For all connected component W of Σ 0 or of Σ -Σ 0 and n ∈ N, we denote by W n the [ q n ]-geometric realization of W . If W is a connected component of Σ -Σ 0 , let W be the finite type surface, image of W in Σ. According to Lemmas 6.10,6.15,6.19,6.25, the convex core of Γ W in [ W n ] n∈N is a unique point W if W is a connected component of Σ 0 or if W is a pair of pants in Σ -Σ 0 , and it is either the completion of a universal cover of W endowed with a half-translation structure that can be extended (as in Section 6.6), or the R-tree dual to a filling measured hyperbolic lamination on W (for any complete hyperbolic metric). In the two cases, we denote it by X W . Finally, if Γ c is the stabilizer in Γ of a boundary component c of Σ 0 , we have seen in Section 6.7, that Γ c pointwise preserves a (maximal) flat strip (possibly reduced to a geodesic), and there exists a geodesic segment e c , possibly reduced to a point, orthogonal to the boundary components of the flat strip (if any), such that the intersection of the translation axis of any hyperbolic element of Γ which is interlaced with c (see Section 6 for the definition), is e c . Then, the union X of the points W , of the convex subsets X W and of the geodesic segments e c , where W is a connected component of Σ 0 or of Σ -Σ 0 and c is a boundary component of Σ 0 , is a convex tree-graded subspace of lim ω ( Σ, [ q n ], n ) n∈N , notably it is CAT(0). Moreover, it is naturally Γ-invariant, and hence endowed with an isometric action of Γ. Moreover, if W is a connected component of the preimage of a cylinder component W of Σ -Σ 0 , and if for all n ∈ N we denote by h n ( W ) the height of the [ q n ]-geometric realization of W , then lim ω h n ( W ) > 0, otherwise there would exist an element of S 0 that would cut W and that would be contained in the union of W and of the two connected components of Σ 0 that bound W , and W would be contained in Σ 0 . Similarly, if W is a connected component of Σ -Σ 0 whose image in Σ is a pair of pants, there exists at least a boundary component c of W such that lim ω h n ( c) > 0, with h n ( c) the height of the flat strip union of all the [ q n ]-geometric representatives of c, , otherwise W would be contained in Σ 0 . Hence, the subspace X, endowed with the action of Γ, satisfies all the properties of Definition 7.1.

Moreover, by the definition of the ultralimits, the ultralimit lim ω ( Σ, [ q n ], n ) n∈N is a limit of a subsequence of ( Σ, [ q n ]) n∈N for the equivariant Gromov topology, in the space of Γequivariant isometry classes of metric spaces endowed with an isometric action of Γ. Since X is Γ-invariant in lim ω ( Σ, [ q n ], n ) n∈N , the space X is also a limit of a subsequence of ( Σ, [ q n ]) n∈N for the Gromov equivariant topology. If the sequence ( Σ, [ q n ]) n∈N converges in Mix(Σ), since Mix(Σ) is separated (by Corollary 7.7), the space X is its unique limit in Mix(Σ).

Moreover, by the geometric construction of a sequence in Flat(Σ) at the end of the proof of [START_REF] Duchin | Length spectra and degeneration of flat metrics[END_REF]Thm. 6 p. 27], we can obtain any mixed structure as a Γ-equivariant subspace of the ultralimit of a sequence of P Flat(Σ). Let us recall this construction of [DLR]. Let (X, d) be a mixed structure on Σ and let Σ 0 be the associated tight subsurface. We build a sequence in Flat(Σ) piece by piece as follows. Let W be a connected component of Σ -Σ 0 which is

  Lemma 4.7 (see [Mor1, Lem. 4.7]) We have d(c, b(α, c)) length(c).

  {e} preserves a boundary component c of W and γ 2 ∈ Γ W does not preserve any boundary component of W , and if (Ax n (γ 1 )) n∈N and (Ax n (γ 2 )) n∈N are two sequences of translation axes of γ 1 and γ 2 , according to Lemma 6.4, the geodesic [Ax n (γ 1 )] n∈N can intersect [Ax n (γ 2 )] n∈N at most at c . Since the sets [Ax n (γ 1 )] n∈N and [Ax n (γ 2 )] n∈N are the sets of fixed points of γ 1 and γ 2 in T W , the unique common point of γ 1 and γ 2 in T W is c . Hence, for every boundary component c of W , the point c is equal to W .

  Let γ ∈ Γ W -{e}. Then γ has a fixed point in T min if and only if γ preserves a boundary component c of W . This fixed point is unique, equal to c . Proof. For every α ∈ C (W ), we have lim ω [qn] (α) = α > 0, and if α is the homotopy class of a boundary component of W , then lim ω [qn] (α) = 0. Hence, an element γ ∈ Γ W -{e} has a fixed point in [ W n ] n∈N if and only if γ preserves a boundary component c of W . For

  Lemma 6.25 The ultralimit [ W n ] n∈N is an R-tree and the action of Γ W on [ W n ] n∈N has a global fixed point W which is equal to c for every boundary component c of W (see the lines following the statement of Lemma 6.2 for the definition). Proof. Let c 1 , c 2 and c 3 be the homotopy classes of the boundary components of W . Let n ∈ N. Assume for a contradiction that the boundary components of W n are pairwise disjoint.

  T at t, w and z , we should have 2π 3π -(θ 1 + θ 2 + θ 3 ) -π < 2π, a contradiction.

			Hence,
	we have d(w , t)	(α) and d(z , t)	(β), and
	d(z, w) d(z, z ) + d(w, w ) d(w, x) -d(w , x) + d(z, x) -d(z , x)
	d(w, x) -( (α) + max{ (α), (β)}) + d(z, x) -( (β) + max{ (α), (β)}),
	which proves the result.
	Assume next that x = y and that [w, z] meets [x, y]. Let t 1 and t 2 be the (possibly equal)
	endpoints of [w, z] ∩ [x,

  and only if [z n ] n∈N = [x n ] n∈N . Similarly, for all n ∈ N, let y n be the first intersection point between Ax n (γ) and F n ( c), and letz n belongs to [y n , Ax n (γ)(-∞)[, then [z n ] n∈N belongs to [F n ( c)] n∈N if and only if [z n ] n∈N = [y n ] n∈N .Moreover, since the distance between I n and J n ω-converges to zero, the intersection between [Ax n (γ)] n∈N and [F n ( c)] n∈N is a segment orthogonal to the boundary components of [F n ( c)] n∈N , possibly reduced to a point if [F n ( c)] n∈N is reduced to a single geodesic.

	zn
	Axn(γ)

We now show that the action of Γ on (X, d) determines Σ 0 , up to isotopy. Let Z be a non trivial π 1 -injective closed connected subsurface of Σ and let Z be a connected component of the preimage of Z in Σ. Assume that the stabilizer Γ Z of Z in Γ is neither trivial nor cyclic, and preserves a unique point Z in X. We can always assume that Z is not a proper subsurface of a bigger subsurface having this property. Then, all the elements of Γ Z preserve a connected components of Σ 0 or of Σ -Σ 0 whose image in Σ is a pair of pants. Hence Z is either a connected component of Σ 0 or the closure of a connected component of Σ -Σ 0 which is a pair of pants. Since no connected component of Σ 0 is a pair of pants, in the first case, the group Γ Z is either a free group of rank at least 3, or if Z is a torus minus a disk, Γ Z is a free group of rank 2. In that case, all the boundary components of Z belong to the same Γ Z -orbit. Moreover, if c is a boundary component of Σ 0 and A is a connected component of Σ -c, then the image of the union of the pieces corresponding to the connected components of Σ 0 or of Σ -Σ 0 , contained in A, by γ ∈ Γ Z , is the union of the pieces corresponding to the connected components of Σ 0 or of Σ -Σ 0 , contained in γ(A). Hence, in that case, the action of Γ Z on the connected components of X -{ Z } has a unique orbit. In the case where Z is a pair of pants, the group Γ Z is a free group of rank 2 and there are three Γ Z -orbits of boundary components of Z. Hence, the action of Γ Z on the connected components of X -{ Z } has three orbits.

strip, union of all the translation axes of γ in [ W n ] n∈N . We will see that the convex hull of γ∈H W F (γ) contains all the points c , with c a boundary component of W (see Lemma 6.2 for the definition). Let W ω be the convex hull of γ∈H W F (γ) minus the points c , with c a boundary component of W (if any).

Lemma 6. 19 The set W ω is a surface without boundary, endowed with a Γ W -invariant halftranslation structure.

Proof of Lemma 6.19. We will need the following lemmas.

Lemma 6.20 The set

n∈N is a surface endowed with a Γ W -invariant Euclidean metric with conical singularities of angles kπ with k ∈ N, k 3.

Lemma 6.21 There exist I ∈ ω and δ > 0 such that for all n ∈ I, the open ball B n (x n , δ) of radius δ contains at most one point of any Γ W -orbit.

Proof. Since [x n ] n∈N does not belong to [∂ W n ] n∈N , there exist I ∈ ω and η > 0 such that for all n ∈ I, the open ball B n (x n , η) is contained in the interior of W n . Let δ = min{ ε 2 , η 3 }. Let (z n ) n∈N be a sequence in Σ such that z n belongs to B n (x n , δ) for all n ∈ N, and let γ ∈ Γ W -{e}. Either γ does not stabilize any boundary component of W n , then d n (z n , γz n ) ε 2δ and γz n does not belong to B n (x n , δ), or γ preserves a boundary component c n of W n , and since d n (z n , c n ) η -δ 2δ, according to Lemma 3.4, we have d n (z n , γz n ) 2δ and γz n does not belong to B n (x n , δ). Hence, for all n ∈ I, the ball B n (x n , δ) contains at most one point of any Γ W -orbit in Σ.

We fix δ as in Lemma 6.21. According to the Gauss-Bonnet formula, and since the angles at the singular points are at least 3π, the number of singular points in W n , and their total angles are uniformely bounded. Consequently, the surface W n has a constant number of singular points of constant angles, for ω-almost all n ∈ N. Hence, there exists k ∈ N such that, for ω-almost all n ∈ N, the ball B n (x n , δ) contains k singular points, denoted by y 1,n , . . . , y k,n with angles θ 1 , . . . , θ k respectively.

Let

). Assume first that there exists η > 0 such that B n (z n , η) does not contain any singular point for ω-almost all n ∈ N. Then, the ball

2 ) is a Euclidean disk. If there does not exist such a η, there exists a sequence of singular points (y n ) n∈N with lim ω d n (z n , y n ) = 0, and according to Lemma 6.21, there exists I ⊆ {1, . . . , k} such that for any sequence of singular points (y n ) n∈N , we have lim ω d n (y n , z n ) = 0 if and only if there exists i ∈ I and I ∈ ω such that y n = y i,n for all n ∈ I. Lemma 6.22 The point [z n ] n∈N is then a singular point of angle 2π + i∈I (θ i -2π).

Definition 7.1 A tree-graded metric space as defined above is called a mixed structure.

Let γ ∈ Γ -{e} and let α γ be the free homotopy class of closed curves defined by γ in Σ. Let W be a non trivial π 1 -injective connected (open) subsurface of Σ. We identify α γ and W with their geodesic representative and geodesic realization, for any hyperbolic metric on Σ, and we denote by α γ,W the closed curve or the union of essential arcs between two (possibly equal) punctures of W , which is the intersection of α γ and W . We still denote by α γ,W the isotopy class (relative to the punctures) of α γ,W in W (that does not depend on the choice of the hyperbolic metric). If W is endowed with a half-translation structure [q W ] as above, let [q W ] (α γ,W ) be the length of a [q W ]-geodesic representative of α W .

Let (X, P) be a mixed structure on Σ, and let m be a hyperbolic metric on Σ. Let (Λ, µ) be the union of the measured hyperbolic laminations on Σ which are dual to the Rtree pieces (Λ has no closed leaf), let c 1 , . . . , c n be the boundary components of Σ 0 , and let t 1 , . . . , t n ∈ [0; +∞[ be the lengths of the corresponding edges in (X, d). For every γ ∈ Γ-{e}, let [q] (α γ ) be the sum of the lengths [q W ] (α γ,W ) of the geodesic representatives of the arcs α γ,W , associated with the different pieces as above.

and let X W be the piece preserved by Γ W . If X W is a point, then X (γ) = 0, and since W is a connected component of Σ 0 or a pair of pants in Σ -Σ 0 , we have i((Λ, µ), α γ ) + [q] (α γ ) + n i=1 t i i(c i , α γ ) = 0. If X W is the dual tree to a measured hyperbolic lamination (Λ , µ ) on W , then

Assume next that γ does not preserve any piece. We replace (X, d) by the space (X , d ) obtained by replacing the edges of length t i (possibly equal to 0), with i ∈ {1, . . . , n}, by edges of length t i + 1.

Then, according to the picture of the introduction, the quotient of (X , d ) by the equivalence relation x ∼ y if and only if x, y belong to the same piece which is not an edge, is the (simplicial) tree T dual to the multicurve ((t i + 1)c i ) i∈{1,...,n} . Let Ax m (γ) be the translation axis of γ in ( Σ, m) and let W 0 , W 1 , . . . , W k (k ∈ N), be the connected components of Σ 0 and of Σ -Σ 0 , that are successively intersected by a fundamental domain of Ax m (γ), for the action of γ Z , starting at a boundary component c of Σ 0 . Then, the translation axis of γ in T has a fundamental domain (for the action of γ Z ) starting at an endpoint of the edge pointwise preserved by the stabilizer Γ c of c in Γ, that successively meets the vertices preserved by Γ W 0 , Γ W 1 , . . . , Γ W k . By the definition of T , the translation axis of γ in X (which is unique since it is not contained in a piece) has a fundamental domain that starts at an endpoint of the edge pointwise preserved by Γ c , and successively meets the pieces

Let Mix(Σ) be the set of Γ-equivariant isometry classes of mixed structures on Σ. We endow Mix(Σ) with the equivariant Gromov topology defined as follows. Let E be a set of metric spaces endowed with an isometric action of Γ. For any X ∈ E , for any finite subset K of X, for any finite subset P of Γ and for any ε > 0, let V (X, K, P, ε) be the set of elements X ∈ E such that there exist a finite subset K ⊆ X and a relation R ⊆ K × K , whose projections on K and K are surjective, such that ∀x, y ∈ K ∀x , y ∈ K ∀γ ∈ P, if xRx and yRy , then |d(x , γy ) -d(x, γy)| < ε.

The sets V (X, K, P, ε) span a topology on E called the equivariant Gromov topology (see for instance [START_REF] Paulin | Topologie de Gromov équivariante, structures hyperboliques et arbres réels[END_REF][START_REF] Paulin | Sur la compactification de Thurston de l'espace de Teichmüller. dans "Géométries à courbure négative ou nulle, groupes discrets et rigidités[END_REF]). The equivariant Gromov topology naturally defines a topology on the set of Γ-equivariant isometry classes of metric spaces endowed with an isometric action of Γ, still called the equivariant Gromov topology, and we endow Mix(Σ) with this topology.

Lemma 7.5 The map X → ( X (γ)) γ∈Γ from Mix(Σ) to R Γ + is continuous and injective.

Proof. First, we prove that the map is injective. Let (X, P) and (X , P ) be two mixed structures on Σ, let Σ 0 and Σ 0 be the associated tight subsurfaces of Σ and let [q] , (Λ, µ) and (t i c i ) i∈{1,...,n} (resp.

[q ] , (Λ , µ ) and (t i c i ) i∈{1,...,n } ) be the length functions, measured hyperbolic laminations and multicurves defined by X and X as in Lemma 7.4. Assume that for every γ ∈ Γ, we have X (γ) = X (γ). Let W 0 be a connected component of Σ 0 , which is not a strip. The stabilizer Γ W 0 of W 0 fixes a point W 0 in X. Since X (γ) = 0 if and only if X (γ) = 0, all the elements of Γ W 0 have a fixed point in X . Let us show that they have a common fixed point. As in the proof of Lemma 7.4, let (X 2 , d 2 ) be the space obtained by replacing the edges of length t i (possibly equal to 0), with i ∈ {1, . . . , n }, by edges of length t i + 1, and let T 2 be the quotient space (X 2 , d 2 )/ ∼, by the equivalence relation generated by x ∼ y if x and y belong to a common piece which is not an edge. The quotient metric space T 2 is the simplicial tree dual to {(t i + 1)c i } i∈{1,...,n } , and it is endowed by the action of Γ defined by the action of Γ on X 2 . All the elements of Γ W 0 have a fixed point in T 2 . According to a Lemma of Serre (see [START_REF] Shalen | Dendrology of groups : an introduction[END_REF]p. 271]), the subgroup Γ W 0 has a global fixed point in T 2 , and by definition of the action of Γ on X 2 , and hence on T 2 , this point is a vertex of T 2 . Hence, Γ W 0 preserve a piece of X 2 , and hence it preserves a piece of X . Since all the elements of Γ W 0 have a fixed point in X , by the definition of the action of Γ on X , this piece is a point, and Γ W 0 preserve a connected component W 0 of Σ 0 or of Σ -Σ 0 whose image W 0 in Σ is a pair of pants, and W 0 is contained in W 0 , up to isotopy. However, W 0 is either a torus minus a disk or we have χ(W 0 ) -4, hence W 0 cannot be contained in a pair of pants. Hence, up to isotopy, W 0 is contained in a connected component W 0 of Σ 0 . Similarly, up to isotopy, any cylinder component of Σ 0 is contained in a connected component of Σ 0 or of Σ -Σ 0 which is a pair of pants. In both cases, it is isotopic to a cylinder contained in a connected component of Σ 0 . Hence, up to isotopy, any connected component W 0 of Σ 0 is contained in a connected component W 0 of Σ 0 .

Conversely, there exists a connected component Z 0 of Σ 0 such that W 0 is contained in Z 0 , up to isotopy. Since the connected components of Σ 0 are pairwise disjoint, we have W 0 = W 0 . Hence, the connected components of Σ 0 and of Σ 0 are equal, up to isotopy, and Σ 0 = Σ 0 , up to isotopy. Notably, we have n = n and {c 1 , . . . , c n } = {c 1 , . . . , c n }.

Let W be a connected component of Σ -Σ 0 . Let X W (resp. X W ) be the piece of X (resp. X ) preserved by Γ W , and let H W be the set of non trivial and non peripheral elements of Γ W . The pieces X W and X W are reduced to a point if and only if W = p( W ) is a pair of pants.

If X W is a tree dual to a measured hyperbolic lamination, according to Lemma 6.13, for every ε > 0, there exists γ ∈ H W such that X (γ) < ε. If X W is the completion of a half-translation structure on a surface, then there exists ε > 0 such that X (γ) ε for every γ ∈ H W . And the same properties hold for X W . Hence, the piece X W is the completion of a surface if and only if the piece X W is also the completion of a surface. Assume that X W and X

W are the completions of ( W , [ q W ]) and ( W , [ q W ]). Since ( X (γ)) γ∈Γ W = ( X (γ)) γ∈Γ W , according to [START_REF] Duchin | Length spectra and degeneration of flat metrics[END_REF]Thm. 1], the quotients ( W , [ q W ])/Γ W and ( W , [ q W ])/Γ W are isometric. Notably, the corresponding length functions X W and X W are equal, and by addition, we have

and by soustraction i((Λ, µ), α γ ) + n i=1 t i i(c i , α γ ) = i((Λ , µ ), α γ ) + n i=1 t i i(c i , α γ ). Since (Λ, µ) ∪ (t i c i ) i∈{1,...,n} and (Λ , µ ) ∪ (t i c i ) i∈{1,...,n} are some measured hyperbolic laminations and since the boundary of Σ is empty, according to [START_REF] Fathi | Travaux de Thurston sur les surfaces[END_REF]Exp. 7], we have (Λ, µ) ∪ (t i c i ) i∈{1,...,n} = (Λ , µ ) ∪ (t i c i ) i∈{1,...,n} , and since the laminations Λ and Λ have no closed leaf, we have (Λ, µ) = (Λ , µ ) and (t i c i ) i∈{1,...,n} = (t i c i ) i∈{1,...,n} . Finally, according to Lemma 7.3, there exists a Γ-equivariant isometry between (X, d) and (X , d ). Hence, the map is injective.

Next, we prove that the map is continuous. It is sufficient to prove that for every γ ∈ Γ -{e}, the map X → X (γ) from Mix(Σ) to R + is continuous. Let γ ∈ Γ, let X ∈ Mix(Σ) and let (X n ) n∈N be a sequence of Mix(Σ) that converges to X.

Case 1. Assume that γ is elliptic in X. Let x ∈ X be a fixed point of γ. Let P = {γ}, K = {x} and ε > 0. Then, for n large enough there exists

Case 2. Assume that γ is hyperbolic in X. Let Ax X (γ) be a translation axis of γ in X and let x ∈ Ax X (γ). Then, we have

neither a cylinder nor a pair of pants and let W be a connected component of the preimage of W in Σ. Then, the piece X W is non empty.

If the piece X W is the completion of the universal cover of a half-translation structure [q W ] on W , let q W ∈ Q(W ) be a representative of [q W ] and let (W n , q W,n ) be the half-translation surface obtained by cutting vertical slits at the punctures of (W, q W ) of length 1 n 2 . If X W is the dual tree to a filling (and minimal) measured hyperbolic lamination (Λ W , µ W ) on W , let q W be a quadratic differential on W whose vertical measured foliation is associated with (Λ W , µ W ) (by the map of [Lev]), and let (W n , q W,n ) be the surface obtained from the half-translation surface (W, 1 0 0 1 n 2

• q W ) by cutting vertical slits at the punctures of length

Finally, let q 0 be a quadratic differential on the interior of the connected components of Σ 0 which are not cylinders, that can be extended at the punctures (with possibly singularities of angle π), and whose vertical foliation is minimal (on every connected component, which is always possible) and let us cut vertical slits of lengths 1 n at the punctures. Thus, we get a surface (Σ 0 , q 0 ). We set (Σ 0,n , q 0,n ) = (Σ 0 , 1 n q 0 ). For every free homotopy class c of boundary components of Σ 0 , let C c,n be the flat cylinder of height X (X c ), where X c is the edge associated to a lift c of c, and of girth 1 n 2 . Any free homotopy class of boundary components c of Σ 0 bounds two connected components W and W of Σ 0 or of Σ -Σ 0 which are not cylinders. If W and W are not pair of pants, for all n ∈ N, we glue isometrically the boundary components of C c,n on the corresponding slits of W n and of W n .

Finally, if W is a pair of pants, it is bounded by three boundary components of Σ 0 . We cover it by gluing the three corresponding cylinders, and we glue their other boundary components on the corresponding slits of Σ 0,n .

We thus obtain a locally Euclidean metric on Σ with conical singular points of angle kπ, with k ∈ N and k 3. Moreover, all the slits on which are glued the flat cylinders are vertical. Hence, the surface has a singular foliation, whose leaves are vertical (with respect to the quadratic differentials on the pieces). Hence, this metric comes from a quadratic differential q n on Σ. Let ( Σ, [ q n ]) n∈N be the corresponding sequence of Flat(Σ). Then, the sequence converges to X ∈ Mix(Σ). Hence the closure of P Flat(Σ) for the equivariant Gromov topology is P Mix(Σ).

Let d be a distance on P Mix(Σ) that induces the topology. Let (X n ) n∈N be a sequence in P Mix(Σ). For all n ∈ N, there exists a sequence (X n,k ) k∈N in P Flat(Σ) such that d(X n,k , X n ) < 1 k . We have seen that there exists X ∈ P Mix(Σ) such that the sequence (X n,n ) n∈N ω-converges to X in P Mix(Σ). Then, the sequence (X n ) n∈N ω-converges to X. Hence, the space P Mix(Σ) is sequencially compact, and since it is Hausdorff, it is compact.

Finally, let us prove that P Mix(Σ) -P Flat(Σ) is closed. By the definition of the mixed structures, and according to Lemma 6.13, the set Mix(Σ) -Flat(Σ) is the set of the elements X ∈ Mix(Σ) such that for all ε > 0, there exists γ ∈ Γ -{e} such that X (γ) < ε. And a sequence (Y k ) k∈N of Flat(Σ) ω-converges to an element of Flat(Σ) if and only if the ω-limits of the translation distances of the non trivial elements of Γ are finite and uniformely bounded below by the injectivity radius of the limit of the sequence in Flat(Σ). According to Lemma 6.12, this happens if and only if there exists ε > 0 such that for all γ ∈ Γ -{e} and ω-almost all n ∈ N, we have Y k (γ) ε (and lim ω Y k (γ) < +∞).

Let (X n ) n∈N be a sequence of Mix(Σ) -Flat(Σ) that converges to X ∈ Mix(Σ), and for all n ∈ N let (X n,k ) k∈N be a sequence in Flat(Σ) that converges to X n . For all n ∈ N, there exists

n . Since the sequence (X n,k(n) ) n∈N converges to X in Mix(Σ), X does not belong to Flat(Σ). Hence Mix(Σ) -Flat(Σ) is closed, and so is P Mix(Σ) -P Flat(Σ).

Theorem 7.9 The map [X] → [ X (γ)] γ∈Γ from P Mix(Σ) to PR Γ is a homeomorphism onto its image, equivariant under the action of the mapping class group of Σ.

Proof. It is a consequence of Lemma 7.5 and Theorem 7.8.

In [START_REF] Duchin | Length spectra and degeneration of flat metrics[END_REF]Thm. 4], Duchin-Leininger-Rafi define an embedding [q] → L [q] of Flat(Σ) into the space C(Σ) of geodesic currents on Σ (see for instance [START_REF] Bonahon | The geometry of Teichmüller space via geodesic currents[END_REF] for the definition of the geodesic currents), uniquely defined such that for every α ∈ C (Σ), i(L [q] , α) = [q] (α). Moreover, in [START_REF] Duchin | Length spectra and degeneration of flat metrics[END_REF]§. 5], they define the space M (Σ) of mixed structures on Σ as the subset of C(Σ) of geodesic currents which are the sum of a geodesic current L [q ] , defined by i(L [q ] , α γ ) = [q ] (α γ ) for every γ ∈ Γ where [q ] is a half-translation structure on a subsurface Σ of Σ (possibly equal to Σ), whose connected components are π 1 -injective and have a negative Euler characteristic, and of a measured hyperbolic laminations (for any hyperbolic metric on Σ) whose support is disjoint from Σ (up to isotopy). Let PC(Σ) = (C(Σ)-{0})/R + * and PM (Σ) be the image of M (Σ) -{0} in PC(Σ). They prove at [START_REF] Duchin | Length spectra and degeneration of flat metrics[END_REF]Thm. 6] that the closure of the image of P Flat(Σ) in PC(Σ) is exactly PM (Σ). Let Φ be the map from Mix(Σ) to C(Σ) that associates to the lengths function [q] , the measured hyperbolic lamination (Λ, µ) and the multicurve (t i c i ) i∈{1,...,n} associated with (X, P) ∈ Mix(Σ), the geodesic current L [q] + (Λ, µ) + (t i c i ) i∈{1,...,n} (where the measured hyperbolic laminations are seen as geodesic currents).

Lemma 7.10 The map Φ is a homeomorphism which is an extension to Mix(Σ) of the map

Proof. It is a consequence of the fact that the map µ → (i(µ, α)) α∈C (Σ) is an embedding of C(Σ) into its image in (R + ) C (Σ) (see for instance [START_REF] Duchin | Length spectra and degeneration of flat metrics[END_REF]Thm. 10,11] and of Theorem 7.9.