Moduli of spherical tori with one conical point

In this paper we determine the topology of the moduli space $\mathcal{MS}_{1,1}(\vartheta)$ of surfaces of genus one with a Riemannian metric of constant curvature $1$ and one conical point of angle $2\pi\vartheta$. In particular, for $\vartheta\in (2m-1,2m+1)$ non-odd, $\mathcal{MS}_{1,1}(\vartheta)$ is connected, has orbifold Euler characteristic $-m^2/12$, and its topology depends on the integer $m>0$ only. For $\vartheta=2m+1$ odd, $\mathcal{MS}_{1,1}(2m+1)$ has $\lceil{m(m+1)/6}\rceil$ connected components. For $\vartheta=2m$ even, $\mathcal{MS}_{1,1}(2m)$ has a natural complex structure and it is biholomorphic to $\mathbb{H}^2/G_m$ for a certain subgroup $G_m$ of $\mathrm{SL}(2,\mathbb{Z})$ of index $m^2$, which is non-normal for $m>1$.


Introduction and main results
The subject of this paper is the moduli space of spherical tori with one conical point. We recall that a spherical metric on a surface S with conical points at the points x = {x 1 , . . . , x n } ∈ S is a Riemannian metric of curvature 1 onṠ := S \ x, such that a neighbourhood of x j is isometric to a cone with a conical angle 2πϑ j > 0.
Let us immediately specify what we mean by the moduli space MS g,n (ϑ) of spherical surfaces in this paper. As a set, MS g,n (ϑ) parametrizes compact, connected, oriented surfaces of genus g with a spherical metric that has conical angles (2πϑ 1 , . . . 2πϑ n ) at marked points x 1 , . . . , x n . Two surfaces correspond to the same point of the space if there is a marked isometry from one to the other. In order to define a topology on MS g,n (ϑ), we consider the bi-Lipschitz distance between marked surfaces, as in [15]. Such a distance defines a metric, and the corresponding topology on MS g,n (ϑ) is called the Lipschitz topology; its properties are discussed in Section 6.
As a spherical metric defines a conformal structure on the surface, we have the forgetful map F : MS g,n (θ) → M g,n , where M g,n is the moduli space of conformal structures on (S, x).
Since a neighbourhood of a smooth point on S is isometric to an open set on the sphere equipped with the standard spherical metric, by an analytic continuation we obtain an orientationpreserving locally isometric developing map f :Ṡ → S 2 . Strictly speaking, the developing map is defined on the universal cover ofṠ but it is sometimes convenient to think of it as a multivalued function onṠ.
The developing map defines a representation of the fundamental group ofṠ to the group SO(3) of rotations of the unit sphere S 2 . The image of this representation is called the monodromy group.
The goal of this article is to provide an explicit description of the moduli space MS 1,1 (ϑ) of spherical tori with one conical point.

Main results
Our main results consist of Theorems A-F and they are stated in the following three subsections.

ϑ not an odd integer
Theorem A (Topology of MS 1,1 (ϑ) for ϑ not odd). Take ϑ ∈ (1, ∞) that is not an odd integer and set m = ϑ+1 2 . The moduli space MS 1,1 (ϑ) of spherical tori with a conical point of angle 2πϑ is a connected orientable two-dimensional orbifold of finite type with the following properties.
(i) As a surface, MS 1,1 (ϑ) has genus m 2 −6m+12 12 and m punctures. (ii) The moduli space MS 1,1 (ϑ) has orbifold Euler characteristic χ(MS 1,1 (ϑ)) = − m 2 12 . Moreover, it has at most one orbifold point of order 4 and at most one orbifold point of order 6. All the other points are orbifold points of order 2. Note that for ϑ = 2m this theorem gives a positive answer to the question of Chai, Lin, and Wang [2, Question 4.6.6, a], whether MS 1,1 (2m) is connected.
We refer to [6] for a general treatment of orbifolds. In fact we adopt a slightly more general definition of orbifolds that includes the case in which all points can have orbifold order greater than 1. The definition of orbifold Euler characteristic is given at page 29 of [6]. This is coherent with the definition used, for example, in [16]. A few properties of the orbifold Euler characteristic are listed in Remark 4.7. Note that in [13] we used a different convention and we endowed our moduli spaces with an orbifold structure for which the order of each point is half the number of automorphisms of the corresponding object. Thus, the orbifold Euler characteristics computed in [13] are twice the ones that would be obtained following the convention of the present paper. Remark 1.1 (Orbifold structure and isometric involution). For ϑ not odd, spherical metrics in MS 1,1 (ϑ) are invariant under the unique conformal involution σ of tori (see Proposition 2.17). Thus every such spherical torus is a double cover of a spherical surface of genus 0 with conical points of angles (πϑ, π, π, π), and so the moduli space MS 1,1 (ϑ) is homeomorphic to MS 0,4 ϑ 2 , 1 2 , 1 2 , 1 2 /S 3 as a topological space. On the other hand, the orbifold order of a point in MS 1,1 (ϑ) exactly corresponds to the number of (orientation-preserving) self-isometries of the corresponding spherical torus. This explains why every point of MS 1,1 (ϑ) has even orbifold order, as stated in Theorem A. Thus MS 1,1 (ϑ) is not isomorphic to the orbifold quotient MS 0,4 ϑ 2 , 1 2 , 1 2 , 1 2 /S 3 . An important geometric input on which Theorem A hinges is the notion of balanced spherical triangles and Theorem B, describing the relation between spherical tori and balanced triangles. Definition 1.2 (Spherical polygons). A spherical polygon P with angles π · (ϑ 1 , . . . , ϑ n ) is a closed disk equipped with a Riemannian metric of constant curvature 1, with n distinguished boundary points x 1 , . . . , x n which are called vertices, and such that the arcs between the adjacent vertices are geodesics forming an interior angle πϑ i at the i-th vertex. Two polygons are isometric if there is an isometry between them that preserves the labelling.
Spherical polygons with two or three vertices are called digons or triangles 1 correspondingly. Definition 1.3 (Balanced triangles). A spherical triangle ∆ with angles π · (ϑ 1 , ϑ 2 , ϑ 3 ) is called balanced if the numbers ϑ 1 , ϑ 2 , ϑ 3 satisfy the three triangle inequalities. If the triangle inequalities are satisfied strictly, we call the triangle strictly balanced. If for some permutation (i, j, k) of (1, 2, 3) we have ϑ i = ϑ j + ϑ k we call the triangle semi-balanced. If ϑ i > ϑ j + ϑ k for some i, we call the triangle unbalanced.
We mention that semi-balanced triangles are called marginal in [13] and [11]. Whenever a spherical triangle is realised as a subset of a surface we will induce on it the orientation of the surface. We will say that two oriented spherical surfaces (or polygons) are conformally isometric (or congruent) if there is an orientation preserving isometry from one surface (or polygon) to the other.
Terminology (Integral angles). Throughout the paper, angles will be measured in radiants. Nevertheless, an angle 2πϑ at a conical point of a spherical surface is called integral if ϑ ∈ Z >0 ; similarly, an angle πϑ at a vertex of a spherical polygon is called integral if ϑ ∈ Z >0 . Now we describe a construction that will be omnipresent in this paper. Construction 1.4. To each spherical triangle ∆ with vertices x 1 , x 2 , x 3 one can associate a spherical torus T (∆) with one conical point by taking a conformally isometric triangle ∆ with vertices x 1 , x 2 , x 3 and isometrically identifying each side x i x j with the side x j x i (in such a way that x i is identified to x j and x j is identified to x i ) for i, j ∈ {1, 2, 3}. The angle at the conical point of T (∆), that corresponds to the vertices of the triangles, is twice the sum of the angles of ∆. If ∆ is endowed with an orientation, then T (∆) canonically inherits an orientation.
To state the next result we need two more notions. Let T be a spherical torus with one conical point. An isometric orientation-reversing involution on T will be called a rectangular involution if its set of fixed points consists of two connected components. By a geodesic loop γ based at a conical point x we mean a loop based at x, which is geodesic inṪ = T \ {x} and which passes through x only at its endpoints.
(i) If T does not have a rectangular involution, then there exists a unique (up to a re-ordering) triple of geodesic loops γ 1 , γ 2 , γ 3 based at x that cut T into two congruent strictly balanced spherical triangles. (ii) If T has a rectangular involution, there exist exactly two (unordered) triples of geodesic loops such that each of them cuts T into two congruent balanced triangles. Moreover, such triangles are semi-balanced. These two triples are exchanged by the rectangular involution.
By Theorem B to each spherical torus T one can associate an essentially unique balanced spherical triangle ∆(T ). Such uniqueness will permit us to reduce the description of the moduli space MS 1,1 (ϑ) to that of the moduli space of balanced triangles of area π(ϑ − 1).

ϑ odd integer
The case when ϑ is an odd integer is quite different, as not all spherical metrics are invariant under the unique (nontrivial) conformal involution σ of the tori. We begin by stating our result for metrics that are σ-invariant.
(a) As a topological space, MS 1,1 (2m+1) σ is homeomorphic to the disjoint union of m(m+1) 6 two-dimensional open disks. (b) MS 1,1 (2m + 1) σ is naturally endowed with the structure of a 2-dimensional orbifold with m(m+1) 6 connected components, which can be described as follows. (b-i) If m ≡ 1 (mod 3), then all components are isomorphic to the quotient D of∆ 2 = {y ∈ R 3 + | y 1 +y 2 +y 3 = 2π} by the trivial Z 2 -action. Hence, every point of MS 1,1 (2m+1) σ has orbifold order 2. (b-ii) If m ≡ 1 (mod 3), then one component is isomorphic to the quotient D of∆ 2 by Z 2 × A 3 , where Z 2 acts trivially and A 3 acts by cyclically permuting the coordinates of∆ 2 , and all the other components are isomorphic to D. Hence, one point of MS 1,1 (2m + 1) σ has orbifold order 6 and all the other points have order 2.
The following description of the moduli space of tori with metrics that are not necessarily σ-invariant will be deduce from Theorem C.
(ii) If m ≡ 1 (mod 3), then one component of MS 1,1 (2m+1) is isomorphic to the quotient M of∆ 2 × R by Z 2 × A 3 , where Z 2 acts via the involution (y, t) → (y, −t) and the alternating group A 3 acts by cyclically permuting the coordinates of∆ 2 . All the other components are isomorphic to M. The locus MS 1,1 (2m + 1) σ of σ-invariant metrics correspond to t = 0.
In order to understand what happens for spherical metrics that are not necessarily σinvariant, recall that a spherical surface is called coaxial if its monodromy belongs to a oneparameter subgroup SO (3, R). By [2,Theorem 5.2], every spherical metric on T with one conical point of angle 2π(2m + 1) is coaxial. For this reason the moduli spaces MS 1,1 (2m + 1) are three-dimensional. Indeed, every spherical metric belongs to a 1-parameter family of metrics that induce the same projective structure (see also Corollary A.2): we will say that metrics in the same 1-parameter family are projectively equivalent. Hence MS 1,1 (2m + 1) σ is also isomorphic to the moduli space MS 1,1 (2m + 1)/proj of projective classes of spherical tori of area 4mπ.
Theorems C-D will rely on the following result that links moduli spaces of tori to moduli spaces of balanced triangles with integral angles.
Theorem E (Canonical decomposition of a spherical torus with odd ϑ). Fix a spherical torus with one conical point of angle 2π(2m + 1). In the same projective class there exists a unique spherical torus (T, x) that admits an isometric orientation-preserving involution. Moreover, there exists a unique collection of three geodesic loops γ 1 , γ 2 , γ 3 based at x that cut T into two congruent balanced spherical triangles ∆ and ∆ with integral angles π · (m 1 , m 2 , m 3 ).

ϑ even integral
Our final main result concerns the moduli spaces MS 1,1 (2m), where m is a positive integer. It is known [2,14] that such moduli spaces have a natural holomorphic structure with respect to which they are compact Riemann surfaces with punctures. This is the unique conformal structure which makes the forgetful map to M 1,1 holomorphic. With this structure MS 1,1 (2m) is an algebraic curve.
Theorem F (MS 1,1 (2m) is a Belyi curve). For each integer m > 0 there exists a subgroup G m < SL(2, Z) of index m 2 such that the orbifold MS 1,1 (2m) is biholomorphic to the quotient H 2 /G m . Such G m is non-normal for m > 1. Moreover, the points in H 2 /G m that project to the geodesic ray [i, ∞) in the modular curve H 2 /SL(2, Z) correspond to tori T such that the triangle ∆(T ) has one integral angle.

Analytic representation of spherical metrics
Let (T, x) be a spherical torus with a conical singularity at x of angle 2πϑ. The pull-back of the spherical metric via the universal cover C =T → T has area element e u |dz| 2 . Then function u satisfies the non-linear PDE where δ Λ is the sum of delta-functions over the lattice Λ and T is biholomorphic to C/Λ. So our results describe the moduli spaces of pairs (Λ, u), where u is a Λ-periodic solutions of (1). Equation (1) is the simplest representative of the class of "mean field equations" which have important applications in physics [24].
The general solution of (1) can be expressed in terms of the developing map f : C → CP 1 related to the conformal factor u by and the developing map f = w 1 /w 2 is the ratio of two linearly independent solutions w 1 , w 2 of the Lamé equation: where ℘ is the Weierstrass function of the lattice Λ and c ∈ C is an accessory parameter. So our results can be also interpreted as a description of the moduli space of projective structures on tori whose monodromies are subgroups of SO(3, R).
Most of the known results on spherical tori are formulated in terms of equations (1) and (2). For example, it is proved in [3] that when ϑ is not an odd integer, then the Leray-Schauder degree of the non-linear operator in (1) equals (ϑ + 1)/2 . An especially well-studied case is the classical Lamé equation (2) where ϑ is an integer, see [2,13] and references there. Solutions of (2) with odd integer ϑ are special functions of mathematical physics, [26,20].

The idea of the proof of Theorem A
Here we give a brief summary of the proof of Theorem A, since various parts of it stretch through the whole paper. Fix ϑ > 1 not odd and consider spherical tori with a conical point of angle 2πϑ, and area 2π(ϑ − 1). The proof of Theorem A develops through the following steps.
• On every torus the unique non-trivial conformal involution is an isometry (Proposition 2.17 (i)). • Every spherical torus is obtained by gluing two isometric copies of a spherical balanced triangle with labelled vertices in an essentially unique way (Theorem B, proven in Section 2.4). Such result has a clear refinement for tori with a 2-marking (namely, a labelling of its 2-torsion points), see Construction 4.5. • The doubled space MT ± bal (ϑ) of balanced triangles of area π(ϑ − 1) is the double of the space MT bal (ϑ) of balanced triangles of area π(ϑ − 1) and it describes oriented balanced triangles up to some identifications that only involve semi-balanced triangles (Definition 3.21).
• The space MT bal (ϑ) is an orientable connected surface with boundary and its topology is completely determined (Proposition 3.20) and so is the topology of MT ± bal (ϑ) (Proposition 3.22).

Content of the paper
The relation between spherical tori with one conical point and balanced spherical triangles is established in Section 2, which culminates in the proof of Theorem B. The section contains a careful analysis of the Voronoi graph of a torus and of the action of the unique non-trivial conformal involution σ on its spherical metric.
In Section 3 we describe the topology of the space of balanced triangles of area π(ϑ − 1) and of its double, separately considering the case ϑ non-odd and ϑ odd. Here we visualize the space of spherical triangles with assigned area, which is a manifold, by looking at its image (which we call carpet) through the angle map Θ. The balanced carpet will turn out to be a useful tool in computing the topological invariants of the space of balanced triangles.
In Section 4 we describe the topology of the moduli spaces of spherical tori with one conical point, endowed with the Lipschitz metric (which we study in Section 6). For ϑ non-odd, we first establish a homeomorphism between the doubled space of balanced triangles and the topological space of 2-marked tori using tools from Section 6. Then we prove Theorem A. For ϑ odd, we first prove Theorem E using results from Section 2 and Section 3, which immediately allows us to prove part (a) of Theorem C. Then we endow our moduli space of σ-invariant spherical tori with a 2-dimensional orbifold structure and we prove part (b) of Theorem C. Finally, using one-parameter projective deformations of σ-invariant spherical metrics, we put a 3-dimensional orbifold structure on the moduli space of (not necessarily σ-invariant) tori and we prove Theorem D.
In Section 5 we analyse the moduli space of tori with ϑ even and we prove Theorem F by identifying such moduli space to a Hurwitz space of covers of CP 1 branched at three points. This permits us to exhibit such moduli space as a Belyi curve and to characterize tori that sit on the 1-dimensional skeleton of its dessin. Section 6 deals with properties of the Lipschitz metric on moduli spaces of spherical surfaces with conical points with area bounded from above. The main result of the section is Theorem 6.3 on properness of the inverse of the systole function. Then the treatment is specialized to tori with one conical point of angle 2πϑ with ϑ non-odd (or with ϑ odd and a σ-invariant metric). The section culminates with establishing the homeomorphism between the space 2-marked tori and the doubled space of balanced triangles, needed in Section 4. A last remark explains how to use such result to endow our moduli spaces with an orbifold structure.
In the short Appendix A we prove a general SU(2)-lifting theorem for the monodromy of a spherical surface, and we apply to the case of ϑ odd and ϑ even to explain their special features.

Acknowledgement
We would like to thank Andrei Gabrielov, Alexander Lytchak, Anton Petrunin, and Dimitri Zvonkine for useful discussions.
A. E. was supported by NSF grant DMS-1665115. G. M. was partially supported by INdAM and by grant PRIN 2017 "Moduli and Lie theory". D. P. was supported by EPSRC grant EP/S035788/1.

Voronoi diagram and proof of Theorem B
In this section we will study the Voronoi graphs of spherical tori (T, x, ϑ) with one conical point and prove Theorem B.

Properties of Voronoi graphs, functions and domains
In this subsection we remind the definition of Voronoi graph [22, Section 4] and apply it to spherical tori with one conical point.
Definition 2.1 (Voronoi function and Voronoi graph). Let S be a surface with a spherical metric and conical points x. The Voronoi function The Voronoi graph Γ(S) is the locus of points p ∈Ṡ at which the distance d(p, x) is realized by two or more geodesic arcs joining p to x. We will simply write Γ = Γ(S) when no ambiguity is possible. The Voronoi domains of S are connected components of the complement S \ Γ(S). Each Voronoi domain D i contains a unique conical point x i and this point is the closest conical point to all the points in the domain.
Various properties of Voronoi functions, graphs, and domains of spherical surfaces were proven in [22, Section 4], and the following lemma lists some of the facts needed here. To formulate the last two properties we need one more definition. Definition 2.2 (Convex star-shaped polygons). Let D be a disk with a spherical metric, containing a unique conical point x ∈ D and such that its boundary is composed of a collection of geodesic segments. We say that D is a convex and star-shaped polygon if any two neighbouring sides of D meet under an interior angle smaller than π and for any point p ∈ D there is a unique geodesic segment that joins x with p. Proposition 2.3 (Basic properties of the Voronoi function and graph). Let S be a spherical surface of genus g with conical points x 1 , . . . , x n .
(i) The Voronoi function is bounded from above by π, namely V S < π.
(ii) The Voronoi graph Γ(S) is a graph with geodesic edges embedded in S and contains at most −3χ(Ṡ) = 6g − 6 + 3n edges. (iii) The valence of each vertex of Γ(S) is at least three. For any point p ∈ Γ(S) its valence coincides with the multiplicity µ p , i.e., there exist exactly µ p geodesic segments in S of length V S (p) that join p with conical points of S. (iv) The metric completion of each Voronoi domain 2 is a convex and star-shaped polygon with a unique conical point in its interior. (v) Let γ be an open edge of Γ(S). Let D i and D j be two Voronoi domains adjacent to γ. Let ∆ ⊂ D i and ∆ ⊂ D j be the two triangles with one vertex x i or x j correspondingly, and the opposite side γ. Then ∆ and ∆ are anti-conformally isometric by an isometry fixing γ.
(iv) The convexity is proven in [22, Lemma 4.8]. The fact that each domain is star-shaped follows from the fact that each point p in it can be joined by a unique geodesic segment of length V S (p) with the conical point. Such a segment varies continuously with p, since V S (p) < π.
(v) To find the isometry between ∆ and ∆ just notice that by definition each point p ∈ γ can be joined by two geodesics of the same length with x i and x j . Also these two geodesics intersect γ under the same angle. The isometry between the triangles is obtained by the map exchanging each pair of such geodesics. Example 2.4 (Voronoi graph in a sphere with 3 conical points). Let S be a sphere with three conical points. It follows from Proposition 2.3 (ii) that the Voronoi graph Γ(S) is either a trefoil graph or an eight graph, or an eyeglasses graph, see Figure 1. Indeed, Γ(S) splits S into three disks, and it has at most three edges.
The next definition and remark explain how to define Voronoi functions and graphs for spherical polygons, mimicking Definition 2.1.

Eight graph
Eyeglasses graph Figure 1: Voronoi graphs on a sphere with three conical points Definition 2.5 (Voronoi function and graph of a polygon). Let P be a spherical polygon with vertices x. The Voronoi function V P : P → R is defined as V P (p) := d(p, x). The Voronoi graph Γ(P ) of P consists of points p of two types: first, the points for which there exists at least two geodesic segments of length d(p, x) that join p with x; second, the points p on ∂P for which the closest vertex of P does not lie on the edge to which p belongs.
Remark 2.6 (Doubling a polygon: Voronoi function and graph). To each spherical polygon P one can associate a sphere S(P ) with conical singularities by doubling 3 P across its boundary. Such a sphere has an anti-conformal isometry that exchanges P and its isometric copy P , and fixes their boundary. It is easy to see that the function V S(P ) restricts to V P on P ⊂ S and to V P on P ⊂ S. One can also check that the Voronoi graph Γ(S(P )) is the union Γ(P ) ∪ Γ(P ). As a result, the statements of Proposition 2.3 have their analogues for spherical polygons.
The following lemma gives an efficient criterion permitting one to verify whether a given geodesic graph on a spherical surface is in fact its Voronoi graph. assume by contradiction that there is a point p ∈ D i that is not contained in D i . By definition of D i there is a unique geodesic segment γ(p) of length V S (p) that joins p with x i . Denote by γ (p) the connected component of the intersection γ(p) ∩ D i that contains x i and let p / ∈ D i be the point in its closure. Clearly p belongs to Γ (S). By (a) each component of S \ Γ (S) is star-shaped, so using (b) we get a second (different from γ (p)) geodesic segment of length V S (p ) that joins p with a conical point. Hence p ∈ Γ(S), which contradicts the fact that p is in D i .
We proved that D i ⊂ D i for each i. It follows that D i = D i , hence Γ (S) = Γ(S).  Proof. It is enough to prove the "only if" parts of claims (i), (ii), (iii) the cases are mutually exclusive and so the "if" part will follow as well. For the proof of the "only if" part all three cases are treated in a similar way. Let us consider, for example, the case when Γ(S) is a trefoil graph. Let's show that in this case ϑ i satisfy the triangle inequality strictly. Denote the two vertices of Γ(S) as A and B. The three edges if Γ(S) cut S into three Voronoi disks, each of which contains one conical point. Let us denote these three segments of Γ(S) by γ 1 , γ 2 , γ 3 , as it is shown on the leftmost picture in Figure 2. Let us join each of the x i with the vertices A and B by geodesics x i A, x i B of lengths V S (A) and V S (B) correspondingly. These geodesic segments are depicted in gray.
Consider now the spherical quadrilaterals Ax 3 Bx 1 , Ax 1 Bx 2 and Ax 2 Bx 3 into which the gray geodesics cut S. It follows from Proposition 2.3 (v) for i, j ∈ {1, 2, 3} that the angles of Ax i Bx j at x i and x j are equal. This implies that ϑ 1 , ϑ 2 , ϑ 3 satisfy the triangle inequality strictly.
(ii, iii) In a similar way one treats the cases when Γ(S) is an eight graph or an eyeglasses graph, the corresponding two pictures are shown in Figure 2.
(iv) This is clear from the way Γ(S) is embedded in S, see Figure 2. In particular, if Γ(S) is an eyeglasses graph, d(A,

The circumcenters of balanced triangles
It is well-known that the circumcenter of a Euclidean triangle ∆ is contained in ∆ if and only ∆ is not obtuse. Moreover, in the case when ∆ is right-angled, the circumcenter is the mid-point of the hypotenuse. It is also a classical fact that the circumcenter of a Euclidean triangle is the point of intersection of the axes 4 of its sides. The next theorem is a generalisation of the above statements to spherical triangles. By an involution triangle we mean a triangle that admits an anti-conformal isometric involution that fixes one vertex and exchanges the other two 5 . To prove this theorem we need the following lemma.
Lemma 2.10 (Some isosceles triangles are involution triangles). Let ∆ be a spherical triangle with vertices q 1 , q 2 , q 3 and denote by |q i q j | the length of the side q i q j . Suppose that |q 1 q 2 | = |q 1 q 3 | < π and ∠q 1 < 2π. Then there is an isometric reflection τ of ∆ that fixes q 1 and exchanges q 2 with q 3 . In particular ∠q 2 = ∠q 3 . Moreover, τ pointwise fixes a geodesic segment that joins q 1 with the midpoint of q 2 q 3 and splits ∆ into two isometric triangles. Furthermore, |q 2 q 3 | < 2π.
Proof. Consider first the case when ∠q 1 = π. In this case ∆ can be isometrically identified with a digon so that q 1 is identified with the midpoint of one of its sides. Since each digon has an isometric reflection fixing the midpoints of both sides, the lemma holds. 4 The axis of a segment is the perpendicular through the midpoint of such segment. 5 Note that every Euclidean or hyperbolic isosceles triangle admits an isometric involution exchanging the equal sides. This is not the case for spherical triangles, for example the triangle with angles 5π/2, 13π/2, 9π/2 is equilateral but clearly has no symmetries.
From now on we assume that ∠q 1 = π. Consider the unique spherical triangle ∆ ⊂ S 2 with vertices q 1 , q 2 , q 3 such that |q 1 q 2 | = |q 1 q 3 | = |q 1 q 2 |, ∠q 1 = ∠q 1 , and Area(∆ ) < 2π. We will show that ∆ admits an isometric embedding into ∆ that sends q i to q i . This will prove the lemma since this implies that ∆ is isometric to a triangle obtained by gluing a digon to the side q 2 q 3 of ∆ . And such a triangle clearly has an isometric reflection τ . This will also prove that |q 2 q 3 | < 2π, since |q 2 q 3 | < 2π and either |q 2 q 3 | = |q 2 q 3 | or |q 2 q 3 | + |q 2 q 3 | = 2π.
To prove the existence of the embedding, denote by ι : ∆ → S 2 the developing map of triangle ∆. We may assume that ι(q i ) = q i , ι(q 1 q 2 ) = q 1 q 2 , and ι(q 1 q 3 ) = q 1 q 3 . Note that ι sends q 2 q 3 to the unique 6 geodesic circle that contains ι(q 2 ) and ι(q 3 ). Hence, it is not hard to see that the preimages of ∆ in ∆ form a union of some number of isometric copies of ∆ . One of them, that contains sides q 1 q 2 and q 1 q 3 of ∆, is the embedding we are looking for.
Remark 2.11. We note that this lemma is sharp in the sense that none of the two conditions |q 1 q 2 | = |q 1 q 3 | < π and ∠q 1 < 2π can be dropped.
Proof of Theorem 2.9. (i) Let S(∆) be the sphere obtained by doubling ∆ across its boundary, i.e., by gluing ∆ with the triangle ∆ that is anti-conformally isometric to ∆. Then by Remark 2.6 the graph Γ(S(∆)) is the union of Γ(∆) with Γ(∆ ).
Suppose first that ∆ contains a point O equidistant from all x i 's. Then, since the restriction of V S(∆) to ∆ equals V ∆ , we see that O is equidistant from x i on S as well. So by Proposition 2.3 (iii) the point O corresponds to a vertex of Γ(S(∆)) of multiplicity at least 3. Furthermore, by Lemma 2.8 (iv) we conclude that Γ(S) is either a trefoil or an eight graph. Hence again by Lemma 2.8 the triangle ∆ is balanced.
Suppose now that ∆ is balanced, i.e., ϑ 1 , ϑ 2 , ϑ 3 satisfy the triangle inequality. Then by Lemma 2.8 (i), (ii) the graph Γ(S(∆)) is a trefoil or a eight graph, and so by Lemma 2.8 (iv) there is a point O in S equidistant from all x i . It follows that ∆ contains such a point as well.
(ii) We first prove the "only if" direction. Suppose that O is in the interior of ∆. Then Γ(S(∆)) has two vertices of valence 3. So according to (i), Γ(S(∆)) is a trefoil. Hence, ∆ is strictly balanced by Lemma 2.8 (i).
Suppose that O is on the boundary of ∆. Without loss of generality assume that O is on the side of ∆ opposite to x 1 . For i = 1, 2, 3 let γ i be the geodesic segment of length V ∆ (O) that joins O with x i . Let γ i be the image of γ i in ∆ ⊂ S(∆) under the anti-conformal involution. Since the multiplicity of O in Γ(S) is at most 4 we conclude that γ 2 = γ 2 , γ 3 = γ 3 . Hence, O is the midpoint of the side x 2 x 3 .
To prove the "if" direction one needs to apply Lemma 2.8 (iv). Indeed, if ∆ is strictly balanced, Γ(S(∆)) has two vertices of multiplicity 3 and one of them lies in ∆. If ∆ is semibalanced, Γ(S(∆)) has one vertex and it has to lie on the boundary of ∆.
(iii) Since ∆ is strictly balanced, by (ii) there is a point O in the interior of ∆ equidistant from points x 1 , x 2 , x 3 . Since V ∆ (O) < π, we have |Ox 1 | = |Ox 1 | = |Ox 3 | < π. Hence all three isosceles triangles x i Ox j are involution triangles by Lemma 2.10. (iv) This proof is identical to the proof of (iii) and we omit it.
Remark 2.12. Theorem 2.9 can be used to construct the Voronoi graph Γ(∆) of a balanced triangle ∆ with vertices x 1 , x 2 , x 3 . Indeed, according to this theorem, the geodesic segments Ox i cut ∆ into three or two involution triangles, and using a variation of Lemma 2.7 one can show that Γ(∆) is the union of symmetry axes of these triangles. See Figure 3.
We will see that some results we are interested in about balanced triangles indeed concern the following class of triangles. Definition 2.13 (Short-sided triangles). A spherical triangle short-sided if all its sides have length l i < 2π. In this case, we setl i := min(l i , 2π − l i ). Theorem 2.9 has the following two simple corollaries.
Proof. Let us treat the case when ∆ is strictly balanced. The semi-balanced case is similar. By Theorem 2.9 (iii) the triangle ∆ can be cut into 3 involution triangles x i Ox j where ∠O < 2π and |Ox i | = |Ox j | < π. Applying Lemma 2.10 to the triangle x i Ox j we conclude that |x i x j | < 2π.
Corollary 2.15 (Short geodesic in a balanced triangle). Let ∆ be a balanced triangle with vertices x 1 , x 2 , x 3 . Suppose that {i, j, k} = {1, 2, 3} and such that the valuel k = min(|x i x j |, 2π− |x i x j |) is minimal. Then there is a geodesic segment γ ∆ in ∆ that joins x i with x j and such that (γ ∆ ) =l k ≤ 2π/3, which in fact realizes the minimum distance between distinct vertices.
Proof. Let us again treat the case when ∆ is strictly balanced. Let x i Ox j be three involution triangles in which ∆ is cut. Consider the developing map ι : ∆ → S 2 . Then for each {i, j, k} = {1, 2, 3} the valuel k is equal to the distance between ι(x i ) and ι(x j ) on S 2 , and so d(x i , x j ) ≥ d(ι(x i ), ι(x j )) =l k . For this reason, it is not hard to see, that the minimum of the valuel k is attained for the triangle x i Ox j for which the angle at O is the minimal one. In particular in such a triangle the angle at O is at most 2π/3. It follows that there is a geodesic segment γ ∆ in such a triangle x i Ox j of length less than 2π/3 that joins x i and x j . Since it cuts out of x i Ox j a digon with one side x i x j we conclude that (γ ∆ ) =l k = d(x i , x j ).

Isometric conformal involutions on tori
In this short section we prove the following useful proposition. Lemma 2.16 (Invariance of projective structures on one-pointed tori). Let (T, x) be a flat one-pointed torus and let σ be its unique nontrivial conformal involution. Then every projective structure on T whose Schwarzian derivative has at worst a double pole at x is invariant under σ.
Proof. We represent our torus T as C/Λ where Λ is a lattice in C, and suppose that x corresponds to the lattice points. We also endow T with the corresponding projective structure.
The involution σ pulls back to the map z → −z onT = C. The Schwarzian derivative (see, for example [7]) of a projective structure is a quadratic differential on the torus T . By hypothesis, it has at worst a double pole at x. The vector space of such quadratic differentials is 2-dimensional, generated by the constants and the Weierstrass elliptic function. Hence, all its elements are invariant under the involution σ, and so are all solutions of the associated Schwarz equations. As a consequence, all such projective structures are σ-invariant. Proposition 2.17 (Spherical metrics and conformal involution). Let σ be the unique conformal involution of a spherical torus T that fixes the unique conical point x.
(i) If ϑ / ∈ 2Z + 1, then σ is an isometry. (ii) If ϑ ∈ 2Z + 1, then each projective equivalence class of spherical metrics is parametrized by a copy of R, on which σ acts as an orientation-reversing diffeomorphism. Thus, σ is an isometry for a unique spherical metric in its projective equivalence class.
Proof. Consider the projective structure associated to a spherical metric on (T, x). By Lemma 2.16, such projective structure is σ-invariant. (i) Every spherical metric is non-coaxial by Lemma A.2, and so in each projective equivalence class there is at most one spherical metric. Hence, such metric must be invariant under σ.
(ii) By Lemma A.2, the monodromy ρ of a spherical metric h in MS 1,1 (2m + 1) is coaxial. It cannot be trivial, since this would imply that T covers S 2 with x as unique ramification point. Fix an element α of π 1 (T ) such that ρ(α) = e X = I with X ∈ su 2 . If ι is the developing map associated to h, all the spherical metrics h t projectively equivalent to h have developing maps It can be observed that ι is uniquely determined by requiring that it maps the conical point to the maximal circle fixed by ρ.

Proof of Theorem B
The goal of this section is to prove Theorem B and to make preparations for the proof of Theorem C. Throughout the whole section we will mainly consider the class of tori that have a conformal isometric involution. By Proposition 2.17 we know that such an involution exists automatically in the case when the conical angle is not 2π(2m + 1). We start with the following simple lemma.
Lemma 2.18 (Points of Γ fixed by a conformal isometric involution). Let S be a spherical surface with conical points x that admits an isometric conformal involution σ. Let p be a point inṠ = S \ x fixed by σ. Then p belongs to Γ(S), its multiplicity µ p is even, and there exist exactly µp 2 geodesic segments or loops 7 of lengths 2V S (p) < 2π based at x and passing through p. The point p cuts each such geodesic segment into two halves of equal length.
Proof. Consider any geodesic segment γ of length V S (p) that joins p with one of the conical points. Since σ(γ) = γ we see that p belongs to Γ(S). If p is not a vertex of Γ(S) then γ and σ(γ) are the only two geodesic segments of length V S (p) that join p with x. Clearly, since σ is a conformal involution the union γ ∪ σ(γ) is a geodesic segment or loop based at x. Its length is less than 2π by Proposition 2.3 (i).
The case when p is a vertex of Γ(S) is similar. Since σ is a conformal involution and it sends Γ(S) to Γ(S) we see that the valence of p in Γ S is even. By Proposition 2.3 (iii) the number µ p of geodesic segments of length V S (p) that join p with x is equal to this valence. Clearly, altogether these µ p segments form µp 2 geodesic segments (or loops) of length 2V S (p) for all of which p is midpoint. Now, we concentrate on the case of spherical tori with one conical point. It will be convenient for us to recall first the construction of hexagonal and square flat tori.
Example 2.19 (Flat hexagonal and square tori). Let T 6 and T 4 be the flat tori obtained by identifying opposite sides of a regular flat hexagon and a square correspondingly. Denote by Γ 6 ⊂ T 6 and Γ 4 ⊂ T 4 the graphs formed by the images of polygons boundaries. Then it is easy to check that Γ 6 and Γ 4 are Voronoi graphs in T 6 and T 4 with respect to the images of the centres of the polygons. Lemma 2.20 (Voronoi graph of a spherical torus). Let T be a spherical torus with one conical point and let Γ be its Voronoi graph. Then Γ is either a trefoil or an eight graph. In the first case the pair (T, Γ) is homeomorphic to the pair (T 6 , Γ 6 ). In the second case it is homeomorphic to the pair (T 4 , Γ 4 ).
Proof. By [22, Corollary 4.7] the Voronoi graph Γ has at most three edges and two vertices. Since the complement to the Voronoi graph is a disk, the graph has at least two edges.
Suppose first that Γ has three edges. By [22, Corollary 4.7] the vertices of Γ have multiplicity at least 3, so Γ is a trivalent graph with two vertices, i.e., a trefoil or an eyeglasses graph. It is a classical fact that only the trefoil admits an embedding in the torus with a connected complement. Moreover, such an embedding is unique up to a homeomorphism of the torus 8 . The statement of lemma then clearly holds. The case when Γ has two edges is similar.
The following is the main proposition on which the proof of Theorem B relies.
Proposition 2.21 (From tori to balanced triangles). Let (T, x) be a spherical torus with one conical point x and suppose that T has a non-trivial isometric conformal involution σ. Let Γ(T ) be the Voronoi graph of T .
(i) Suppose Γ(T ) is a trefoil. Then σ permutes the two vertices of Γ(T ), and fixes the midpoints p 1 , p 2 , p 3 of the three edges of Γ(T ). Moreover, there exist exactly three σ-invariant simple geodesic loops γ 1 , γ 2 , γ 3 based at x such that γ i intersects Γ(T ) orthogonally at p i . These geodesic loops cut the torus into the union of two congruent strictly balanced triangles that are exchanged by σ. (ii) Suppose Γ(T ) is an eight graph with the vertex A. Then σ fixes the vertex and the midpoints p 1 , p 2 of the two edges of Γ(T ). Moreover there exist four σ-invariant simple geodesic loops γ 1 , γ 2 , η 1 , η 2 based at x and uniquely characterised by the following properties. Each geodesic γ i intersects Γ(T ) orthogonally at p i . Each geodesic η i passes through A and has length 2d(A, x). Moreover, for i = 1, 2 the triple of loops γ 1 , γ 2 , η i cuts T into the union of two congruent semi-balanced triangles that are exchanged by σ. (iii) T has a rectangular involution if and only if its Voronoi graph is an eight graph. For a torus T with a rectangular involution the triangles in which γ 1 , γ 2 , η 1 cut T are reflections of the triangles in which γ 1 , γ 2 , η 2 cut T .
Proof. (i) Since σ is an isometry of T it sends Γ(T ) to itself. Let's denote the vertices of Γ(T ) by A and B. Since their valence is 3 and σ is a conformal isometric involution, σ can fix neither A nor B. Indeed, begin σ of order 2, if σ fixed A, then it would fix at least one half-edge outgoing from A, and so it would be the identity. Hence σ permutes A and B, which implies in particular that A and B are at the same distance from x. Next, since σ is an orientation preserving involution, and Γ(T ) is a trefoil, from simple topological considerations it follows that σ sends each edge γ i of Γ(T ) into itself. It follows that the midpoints of the edges p 1 , p 2 , p 3 are fixed by σ.
Let us now cut T along Γ(T ) and consider the completionD of the obtained open disk. Clearly,D is a spherical hexagon with the conical point x in its interior. Moreover, σ induces an isometric involution onD without fixed points on ∂D. It follows that σ sends each vertex ofD to the opposite one.
Next, let's denote the vertices ofD by A 1 , B 2 , A 3 , B 1 , A 2 , B 3 as is shown in Figure 4. Here all the points A i correspond to A and B i to B when we assemble T back from the disk. In a similar way we mark midpoints of the sides ofD by p i and p i .

Figure 4: Trefoil case
According to Lemma 2.18, for each i there is a geodesic loop γ i of length 2d(p i , x) based at x for which p i is the midpoint. Let us show that γ 1 , γ 2 , γ 3 cut T into two equal strictly balanced triangles whose vertices are identified to the point x.
Indeed, the first triangle, which we will call ∆ A , is assembled from three quadrilaterals The second triangle ∆ B is assembled from the remaining three quadrilaterals. Clearly, σ(∆ A ) = ∆ B , so these two triangles are congruent.
Finally, ∆ A is strictly balanced according to Theorem 2.9 (i), indeed the point A lies in the interior of ∆ A and is at distance d(A, x) from all the vertices of ∆ A .
(ii) Let us now consider the case when Γ(T ) is an eight graph with a vertex labelled by A. Clearly, A is fixed by σ since this is the unique point of Γ(T ) of valence 4.

Figure 5: Eight graph case
As before we see that the midpoints p 1 , p 2 of the two edges of Γ(T ) are fixed by σ and this gives us two σ-invariant geodesic loops γ 1 and γ 2 . To construct η 1 and η 2 we apply Lemma 2.18 to the point A. Now let us cut T along the Voronoi graph Γ(T ) and consider the completionD of the obtained open disk. Clearly, this disk is a quadrilateral with one conical point in the interior. Let us mark the vertices of this quadrilateral and the midpoints of its edges as it is shown in Figure 5.
As before, the loops γ 1 , γ 2 , η 1 cut T into two congruent triangles exchanged by σ. To show that these triangles are semi-balanced consider one of these triangles obtained as a union of two triangles A 1 xp 2 , A 3 xp 1 and the quadrilateral xp 1 A 2 p 2 . To assemble this triangle one has to identify the pairs of sides (A 1 p 2 , A 2 p 2 ) and (A 2 p 1 , A 3 p 1 ). The resulting triangle is semi-balanced by Theorem 2.9 (ii).
(iii) Suppose first that Γ(T ) is an eight graph. Then we are in the setting of the case 2 of this proposition. Let us construct an involution τ 1 ofD that fixes pointwise γ 1 . We define τ 1 so that Then in order show that τ 1 extends toD it is enough to show that the triangle A 1 xA 4 is isometric to A 2 xA 3 and that the geodesic γ 1 is the axis of symmetry of both triangles A 1 xA 2 and A 3 xA 4 . The former statement follows from Proposition 2.3 (v). To prove the latter statement, note again that A 1 xA 2 is isometric to A 4 xA 3 by Proposition 2.3 (v) and then compose this isometry with σ. This induces desired reflections on both triangles A 1 xA 2 and A 4 xA 3 . The involution τ 2 fixing γ 2 is constructed in the same way.
Suppose now that T has a rectangular involution τ . Let us show that Γ(T ) is an eight graph. Since τ is a rectangular involution, its fixed locus is a union of two disjoint geodesic loops. One of these loops passes through x while the other one, say ξ, is a simple smooth closed geodesic. For any point p ∈ ξ there exist at least two length minimizing geodesic segments that join it with x (they are exchanged by τ ). It follows that ξ lies in Γ(T ). And since a trefoil graph can't contain a smooth simple closed geodesic, we conclude that Γ(T ) is an eight graph.
Later we will need the following statement, which is a part of the proof of Proposition 2.21.
Remark 2.22. Suppose we are in the case (ii) of Proposition 2.21. Consider the four sectors in which geodesic loops η 1 and η 2 cut a neighbourhood of x. Then for each i = 1, 2 the geodesic loop γ i bisects two of these sectors.
The final preparatory proposition of this subsection is the converse to Proposition 2.21.
Proposition 2.23 (From balanced triangles to tori). Let ∆ be a balanced triangle and let ∆ be a triangle congruent to it. Let T (∆) be the torus obtained by identifying the sides of ∆ and ∆ through orientation-reversing isometries.
(iii) If ∆ is semi-balanced then Γ(T (∆)) has one vertex. Moreover the images of the three sides of ∆ in T (∆) coincide with three canonical geodesic loops γ 1 , γ 2 , η i on T (∆) constructed in Proposition 2. 21 2). Here the side of ∆ opposite to the largest angle of ∆ corresponds to η i .

Figure 6: Two isomorphic triangles ∆ and ∆
Proof. (i) Assume first that ∆ is strictly balanced. LetΓ be the graph obtained as the union Γ(∆) ∪ Γ(∆ ). In order to prove thatΓ = Γ(T (∆)), it is enough to show thatΓ satisfies the properties (a) and (b) of Lemma 2.7.
Recall that by Theorem 2.9 (ii) there is a point O in the interior of ∆ that is equidistant from points x i . Denote by p i and p i the midpoints of sides opposite to x i and x i as in Figure 6. Then by Remark 2.12, Γ(∆) is the union of the segments Op i and Γ(∆ ) is the union of the segments Op i . It follows that T (∆) \Γ is a convex and star-shape with respect to x, which means that property (a) of Lemma 2.7 holds. As for property (b), it holds since Γ(∆) and Γ(∆ ) are Vornoi graphs of ∆ and ∆ .
The case when ∆ is semi-balanced case is treated in the same way, so we omit it.
(ii) Since ∆ is strictly balanced, it follows from (i) that Γ(T (∆)) has two vertices. Now, it follows from (i) that for any permutation {i, j, k} the side x i x j ⊂ T (∆) intersects an edge of Γ(T (∆)) at its midpoint and it is orthogonal to it at this point. Hence, by Proposition 2.21 (ii) each geodesic x i x j coincides with the geodesic loop γ k .
(iii) The proof of this result is similar the case (ii) and we omit it.
Remark 2.24. We note that the statement of Proposition 2.23 does not hold for any unbalanced triangle. Indeed, if ∆ is unbalanced one can still construct a torus T (∆) from ∆ and its copy of ∆ . However, the union of the Voronoi graphs of ∆ and ∆ will be an eyeglasses graph in T (∆). Such a graph can never be the Voronoi graph of a torus with one conical point.
Now we are ready to prove Theorem B.
Proof of Theorem B. Let T be a spherical torus with one conical point of angle 2πϑ with ϑ / ∈ 2Z + 1. According to Proposition 2.17, there exists a conformal isometric involution σ on T .
Hence we can apply Proposition 2.21. In particular, by Proposition 2.21 (iii) the torus T has a rectangular involution if and only if Γ(T ) is an eight graph.
(i) The Voronoi graph Γ(T ) of T is a trefoil and we get a collection of three geodesics γ 1 , γ 2 , γ 3 that cut T into two congruent strictly balanced triangles. Such a collection of geodesics is unique on T by Proposition 2.23.
(ii) The Voronoi graph Γ(T ) is an eight graph, and by Proposition 2.21 we get two triples of geodesics γ 1 , γ 2 , η 1 and γ 1 , γ 2 , η 2 both cutting T into two congruent semi-balanced triangles. Again, it follows from Proposition 2.23 that these two triples are the only ones that cut T into two isometric balanced triangle, and they are exchanged by the rectangular involution.

Balanced spherical triangles
The main goal of this section is to describe the space of balanced spherical triangles with assigned area. To do this, we recall in Section 3.1 several theorems describing the inequalities satisfied by the angles of spherical triangles. We also give explicit constructions of such triangles. Section 3.2 is mainly expository. It recalls the results from [11] that the space MT of all (unoriented) spherical triangles has a structure of a three-dimensional real-analytic manifold. From this we deduce that the space of balanced triangles of a fixed non-even area is a smooth bordered surface. In Section 3.3 we describe a natural cell decomposition of the space MT bal (ϑ) of all balanced triangles of fixed area π(ϑ − 1) with ϑ / ∈ 2Z + 1.

The shape of spherical triangles
We start this section by recalling the classifications [9] of spherical triangles. In fact, such triangles are in one-to-one correspondence with spheres with a spherical metric with three conical points, provided we exclude spheres and triangles with all integral angles. Indeed, for each S 2 with a spherical metric and three conical points, that are not all integral, there is a unique isometric anti-conformal involution τ , such that S 2 /τ is a spherical triangle. Conversely, for each spherical triangle ∆ we can take the sphere S(∆) glued from two copies of it. It will be useful to introduce the following notation.

Figure 7: Angle vectors of spherical triangles
We collect the results into three subsections, depending on the number of integral angles, and we remind that there cannot be a triangle with exactly two integral angles.

Triangle with no integral angle
The first result we want to recall from [9] is the following.
Remark 3.2. Let us decipher Inequality (3). Note first, that the subset d 1 ((ϑ 1 , ϑ 2 , ϑ 3 ), Z 3 e ) ≤ 1 ⊂ R 3 is a union of octahedra of diameter 2 centred at points of Z 3 e . The complement to this set is a disjoint union of open tetrahedra. Each such tetrahedron is contained in a unit cube with integer vertices. This collection of tetrahedra is invariant under translations of R 3 by elements of Z 3 e . Theorem 3.1 states that if a point (ϑ 1 , ϑ 2 , ϑ 3 ) ∈ R 3 >0 lies in one of such tetrahedra, the corresponding spherical triangle exists and it is unique. Figure 7 depicts the union of six such tetrahedra in the octant R 3 >0 .
An explicit construction of balanced spherical triangles can be found in [21, Section 3.1.2]. In fact, it was already used by Klein [17].

Triangles with one integral angle
The second result we wish to recall from [9] is the following. Theorem 3.3 (Triangles with one integral angle [9]). If ϑ 1 is an integer and ϑ 2 , ϑ 3 are not integers, then a spherical triangle with angles π · (ϑ 1 , ϑ 2 , ϑ 3 ) exists if and only if at least one of the following conditions is satisfied.
It is obvious that triangles satisfying the hypotheses of Theorem 3.3 (b) are never balanced.
Remark 3.4. It is easy to see that in the case when a triple (ϑ 1 , ϑ 2 , ϑ 3 ) of positive numbers satisfies the triangle inequality and the integrality constraints of Theorem 3.3 (a), there are integers n 1 , n 2 , n 3 ≥ 0 and a number θ ∈ (0, 1) such that ϑ 1 = n 2 + n 3 + 1, ϑ 2 = n 1 + n 3 + θ, Finally, we present a full description of balanced triangles with exactly one integral angle.
Proof. (i) Since ∆ is balanced, by Corollary 2.14 we have At the same time, since the angle ϑ 2 is non-integer, the image ι(x 2 x 3 ) does not belong to C. This means that ι(x 2 ) and ι(x 3 ) are opposite points on S 2 and so |x 2 x 3 | = π.
(ii) Since |x 2 x 3 | = π by part (i), there exists a maximal digon embedded in ∆, with one edge equal to x 2 x 3 . The other edge of such digon must pass through x 1 by maximality, and so it is the concatenation of two geodesics γ 12 from x 1 to x 2 and γ 13 from x 1 to x 3 , that form an angle π at x 1 . It is easy to see that these are the geodesics we are looking for. The uniqueness of γ 12 , γ 13 follows, because n 1 and θ are uniquely determined.
The next lemma is a partial converse to Proposition 3.5 (i).
Proof. Consider the developing map ι : ∆ → S 2 . Since |x i x j | < 2π by Corollary 2.14, we see that ι( In order to show that ϑ 1 is integer it is enough to prove that both images ι(x 1 x 2 ) and ι(x 1 x 3 ) lie on the same great circle. But this is clear, since the points ι(x 2 ) and ι(x 3 ) are opposite on S 2 , while ι(x 1 ) is different from both points.
Last lemma concerns semi-balanced triangles.
(ii) Our hypotheses imply that ϑ 1 = m is an integer. By (i) we obtain that ϑ 2 , ϑ 3 are half-integers.

Triangles with three integral angles
We begin by giving a description of all triangles with integral angles. Proposition 3.8 (Triangles with three integral angles). For any spherical triangle ∆ with integral angles π · (m 1 , m 2 , m 3 ) the following holds.
(i) There exists a unique triple (n 1 , n 2 , n 3 ) of non-negative integers such that m 1 = n 2 +n 3 +1, m 2 = n 3 + n 1 + 1, m 3 = n 1 + n 2 + 1. Moreover, there exist a unique triple of geodesic segments γ 12 , γ 23 , γ 13 ⊂ ∆ with |γ 12 | + |γ 23 | + |γ 13 | = 2π, that join points x i and cut ∆ into the following four domains: the central disk ∆ 0 isometric to a half-sphere and bounded by segments γ 12 , γ 23 , γ 13 ; where each B i is bounded by segments γ jk and x j x k and has angles πn i . (ii) The space of triangles with angles π · (n 1 , n 2 , n 3 ) can be identified with the set of triples of positive numbers l 12 , l 13 , l 23 satisfying l 12 + l 23 + l 13 = 2π (where l ij are interpreted as the lengths of the sides of ∆ 0 ). (iii) All sides of ∆ are shorter than 2π. Moreover, there is at most one side of length π.
Proof. (i) Consider the developing map: ι : ∆ → S 2 . Since all the angles of ∆ are integral, all its sides are sent to one great circle on S 2 . The full preimages of this circle cuts ∆ into a collection of hemispheres. It is easy to see that only one of these hemisphere contains all three conical points, this is the disk ∆ 0 in ∆. The conical points cut the boundary of the disk into three geodesic segments γ 12 , γ 23 , γ 13 . The complement to ∆ 0 in ∆ is the union of the three digons B 1 , B 2 , B 3 .
(ii) It is clear from (i) that ∆ is uniquely defined by the three lengths l ij = |γ ij | and n 1 , n 2 , n 3 . Conversely, for each positive triple l ij with l 12 + l 23 + l 13 = 2π, and each integer triple n 1 , n 2 , n 3 , one constructs a unique spherical triangle.
Assume by contradiction that ϑ 2 and ϑ 3 are not integer, and so we are in the setting of Theorem 3.3. The possibility (b) can't hold because ∆ is balanced. Assume that possibility (a) holds, in which case ϑ 2 − ϑ 3 is an integer, and ϑ 1 + ϑ 2 − ϑ 3 is odd. But then, since ϑ 1 + ϑ 2 + ϑ 3 is also odd, we see that ϑ 3 is integer. This is a contradiction.
We conclude that all ϑ i 's are integer.

Final considerations
The last statement of the section can be derived in many ways. Here we obtain it as a consequence of Theorem 3.1, Theorem 3.3 and Proposition 3.8.
Corollary 3.11 (Triangles are determined the side lengths and angles). Let ∆ be a spherical triangle with angles π · (ϑ 1 , ϑ 2 , ϑ 3 ), and let l i be the length of the side opposite to the vertex x i . Then ∆ is uniquely determined by ϑ i 's and l i 's.
If ϑ 1 is integer, while ϑ 2 and ϑ 3 are not integer, then the triangle ∆ is uniquely determined by the angles ϑ i and the length l 3 by Theorem 3.3.
If ϑ 1 , ϑ 2 ϑ 3 are integer, then it follows from Proposition 3.8 that all triangles with angles ϑ i are uniquely determined by the lengths of their sides.

The space of spherical triangles and its coordinates
Let us denote by MT be space of all (unoriented) spherical triangles with vertices labelled by x 1 , x 2 , x 3 , up to isometries that preserve the labelling. This space has a natural topology induced by the Lipschitz distance (see Section 6). We will denote by ϑ 1 , ϑ 2 , ϑ 3 , l 1 , l 2 , l 3 the functions on MT , defined by requiring that πϑ i (∆) is the angle of the spherical triangle ∆ at x i and l i (∆) is the length of the side of ∆ opposite to x i . By Corollary 3.11 the map Ψ : MT → R 6 , that associates to each triangle its angles and side lengths, is one-to-one onto its image. Moreover, we have the following result. This theorem says that the space MT has a structure of a smooth, connected, analytic manifold and moreover at each point ∆ ∈ MT one can choose three functions among ϑ i and l i as local analytic coordinates. It also follows from Theorem 3.12 that formulas of spherical trigonometry, that are usually stated for convex spherical triangles, hold for all spherical triangles. In particular, for any permutation (i, j, k) of (1, 2, 3) and any ∆ ∈ MT the cosine formula for lengths holds 9 : cos l i sin(πϑ j ) sin(πϑ k ) = cos(πϑ i ) + cos(πϑ j ) cos(πϑ k ). integer. Then the function ϑ 1 + ϑ 2 + ϑ 3 has non-zero differential at ∆.
Proof. (i) Consider the projection map from Ψ(MT ) to the angle space R 3 . According to Theorem 3.1 this map is one-to-one over the subset of (ϑ 1 , ϑ 2 , ϑ 3 ) in R 3 >0 , that satisfy Inequality (3). We need to show that this projection is in fact a diffeomorphism over this set. However, using the cosine formula (4) and the fact that none of ϑ i is integer, we see that the lengths l i depend analytically on the ϑ i 's.
(ii) In case ϑ i are all not integer the statement follows immediately from (i). Suppose that one of ϑ i , say ϑ 1 , is integer. Then, since ∆ is short-sided, using exactly the same reasoning as in the proof of Proposition 3.5 (i), we deduce that l i = π. Now, for any θ > 0 we can glue to the side x 2 x 3 of ∆ the digon with two sides of length π and the angles πθ. The family of triangles thus constructed, that depends on θ, determines a straight segment in Ψ(MT ) starting from Ψ(∆) and the linear function ϑ 1 + ϑ 2 + ϑ 3 restricted to this segment has non-zero derivative.
The following statement is a corollary of Theorem 3.12 and Lemma 3.13. Assume now that ϑ = ϑ 1 + ϑ 2 + ϑ 3 is not an odd integer. Clearly, MT sh is an open subset of MT , and so we deduce from Lemma 3.13 (ii) that MT sh (ϑ) is an open smooth 2dimensional submanifold of MT . The set MT bal (ϑ) is contained in MT sh (ϑ) and its boundary is composed of semi-balanced triangles. We need to show that such triangles form a smooth curve in MT sh (ϑ).
Let ∆ ∈ MT sh (ϑ) be a semi-balanced triangle, say ϑ 1 = ϑ 2 + ϑ 3 . If ϑ 1 , ϑ 2 , ϑ 3 are not integer, from Lemma 3.13 (i) it follows immediately that the curve ϑ 1 − ϑ 2 − ϑ 3 = 0 is smooth in a neighbourhood of ∆. Suppose that one of ϑ i is integer. Then we are in the setting of Lemma 3.7. In particular by Lemma 3.7 (i) we have ϑ 1 + ϑ 2 + ϑ 3 = 2m. But then, applying Lemma 3.7 (ii) we see that all semi-balanced triangles in MT bal (2m) have one integral and two half-integral angles. Such triangles are governed by Proposition 3.5 and their image under the map Ψ forms a collection of straight segments in R 6 . It follows that semi-balanced triangles form a smooth curve in MT sh (2m).
Finally, let's show that MT bal (ϑ) is orientable. This is clear if ϑ is an odd integer, because a disjoint union of open triangles is orientable. In case ϑ is not an odd integer, it suffices to show that MT bal (ϑ) can be co-oriented, since MT is orientable. A co-orientation can indeed be chosen since the function ϑ 1 + ϑ 2 + ϑ 3 = ϑ has non zero differential along the surface MT bal (ϑ) by Lemma 3.13 (ii).

Balanced spherical triangles of fixed area
The goal of this section is to describe the topology of the moduli space MT bal (ϑ) of balanced triangles with marked vertices of fixed area π(ϑ − 1), where ϑ > 1. To better visualize the structure of such space, we introduce the following object.

Case ϑ not odd
Throughout the section, assume ϑ / ∈ 2Z + 1. We will denote by MT Z bal (ϑ) its subset consisting of triangles with at least one integral angle. By Proposition 3.5 this subset is a disjoint union of smooth open intervals in MT bal (ϑ). We will see that it cuts MT bal (ϑ) in a union of topological disks. This decomposition is very well reflected in the structure of the associated balanced carpet, as we will see below. The carpet Crp(ϑ) is composed of a disjoint union of open triangles with a subset of their vertices (the nodes). In order to better visualize such carpets, we will often identify Crp(ϑ) with its projection to the horizontal (ϑ 1 , ϑ 2 )-plane. Figure 8 shows how the projection of Crp  Figure 9 depicts the projection of balanced angle carpets for five different values of ϑ.
The following lemma is a consequence of Theorems 3.1 and 3.3. Hence There exists a point in Crp bal (ϑ) with non-integral coordinates at which ϑ 2 = ϑ 3 .
Proof. (i) Let us split the carpet into two subsets. The first subset consists of points such that none of coordinates ϑ i is integer, and the second subset is where one of coordinates ϑ i is integer.
Is is clear that the first subset is the union of open triangles given by intersecting the plane ϑ 1 + ϑ 2 + ϑ 3 = ϑ with the open tetrahedra that are given by Inequality (3) of Theorem 3.1. Since such plane does not pass through any vertex of the tetrahedra for ϑ non-odd, it follows that the number of triangle only depends on m and so we can compute it for ϑ = 2m. Look at the projection of Crp(2m) inside the (ϑ 1 , ϑ 2 )-plane and enumerate the open triangles as follows: to points of type (0, l + 1/2) with l ∈ {0, 1, . . . , 2m − 1} we can associate a unique triangle, to points of type (n, l + 1/2) with n ∈ {1, . . . , 2m − 1} and l ∈ {0, . . . , 2m − n − 1} we can associate two triangles. The number of such triangles is thus 4m 2 .
(ii) Again, it is enough to consider the case ϑ = 2m. In the balanced carpet ϑ i ≤ m for all i and so the first claim follows from the above enumeration of the nodes. Hence, Crp bal (ϑ) is connected.
As for E, the enumeration in (i) for ϑ = 2m shows that 4E = 4m 2 and so E = m 2 . For ϑ < 2m, the value of E does not change. For ϑ > 2m, there 3m extra triangles intersected by Bal(ϑ), which is exactly the number of nodes sitting in ∂Bal(2m).
Analysis of the map Θ. By Proposition 3.17 the balanced carpet Crp bal (ϑ) consists of E polygons {P l }, bounded by some semi-balanced edges that sit in ∂Bal(ϑ) and some nodes. Note that we are considering P l as closed subsets of Crp bal (ϑ). The closure of P l inside the plane Π(ϑ) is obtained from P l by adding some ideal edges, that have equations of type ϑ i = a + (c + 1)/2 with i ∈ {1, 2, 3} and a ∈ {0, 1, . . . , m − 1}.
For each polygon P l , the real blow-up P l of P l at its nodes is obtained from P l by replacing each node by an open interval (nodal edge): the natural projection P l → P l contracts each nodal edge to the corresponding node. For every l we can fix a realization of P l inside R 2 as the union of an open convex polygon with some of its open edges (nodal edges and semi-balanced edges); the missing edges (ideal edges) correspond to the ideal edges of P l .
We recall that MT bal (ϑ) is a surface by Corollary 3.15 and its boundary consists of semibalanced triangles, and that the map Θ contracts each open interval in MT Z bal (ϑ) to a node by Proposition 3.5 and it is a homeomorphism elsewhere by Lemma 3.13 (i).
It is easy then to see that Θ −1 (P l ) is homeomorphic to P l . Suppose now that two distinct polygons P l and P h intersect in a nodeθ. The preimage Θ −1 (θ) is an open segment and Θ −1 (P l ∪ P h ) is homeomorphic to the space obtained from P l P h by identifying the nodal edges that correspond toθ.
In order to understand such identification, choose an orientation of MT bal (ϑ) in a neighbourhood of Θ −1 (θ) and an orientation of the plane Π(ϑ), so that P l and P l inherit an orientation from Π(ϑ), and each nodal edge of P l is induced an orientation from P l . Together with Corollary 3.15, the last paragraph of the proof of [11,Proposition 4.7] shows that Θ is orientation-preserving on one of the two polygons P l or P h and orientation-reversing on the other. Hence, the two nodal edges corresponding toθ are identified through a map that preserves their orientation; we can also prescribe that such identification is a homothety in the chosen realizations of P l and P h .
Part of the above analysis can be rephrased as follows.
Lemma 3.18. The space MT bal (ϑ) is homeomorphic to the real blow-up of Crp bal (ϑ) at its nodes.
A further step in describing the topology of MT bal (ϑ) is to study its ends. We are now ready to completely determine the topology of the space MT bal (ϑ). Proof. (i) Thanks to Corollary 3.15 we only need to prove that MT bal (ϑ) is connected and of finite type. Since the balanced carpet Crp bal (ϑ) is connected by Lemma 3.17 (ii) and it consists of finitely many nodes and polygons, both claims follow from Lemma 3.18.
(iv) The internal part of MT bal (ϑ) is an orientable surface without boundary and so the Euler characteristic of its cohomology with compact support coincides with its   By Proposition 3.20 (iii) each end of MT bal (ϑ) is associated to a strip S i a (ϑ) with a ∈ {0, 1, . . . , m} and i ∈ {1, 2, 3}, and it is homeomorphic to [0, 1] × R, and so it doubles to punctured disk S 1 × R inside MT ± bal (ϑ), that will be denoted by E i a (ϑ). Hence, we obtain 3m punctures. The genus of g(MT ± bal (ϑ)) = 1 − 3m 2 − 1 2 χ(MT ± bal (ϑ)) is then easily computed.
Proof. The first claim relies on Proposition 3.24. The remaining ones are straightforward.

Moduli spaces of spherical tori
The goal of this section is to describe the topology of the moduli space MS 1,1 (ϑ) and so to prove Theorem A (case ϑ non-odd) and Theorems C-D (case ϑ odd).
We recall that, by isomorphism between two spherical tori, we mean an orientation-preserving isometry. We refer to Section 6 for the definition of Lipschitz distance and topology on MS 1,1 (ϑ) and MS (2) 1,1 (ϑ) needed below. The object of our interest is the following. In order to prove Theorem A it will be convenient to introduce the notion of 2-marking.

Definition 4.2 (2-marking). A 2-marking of a spherical torus T with one conical point x is a labelling of its nontrivial 2-torsion points or, equivalently, an isomorphism
There is a bijective correspondendence between isomorphisms µ : (Z 2 ) 2 → H 1 (T ; Z 2 ) and orderings of the three non-trivial elements of H 1 (T ; Z 2 ): it just sends µ to the triple (µ(e 1 ), µ(e 2 ), µ(e 1 + e 2 )). In fact, the action of SL(2, Z 2 ) on 2-markings corresponds to the S 3 -action that permutes the orderings. If the torus T has a spherical metric with conical point x, the nontrivial conformal involution σ fixes x and its three non-trivial 2-torsion points: the above ordering is then equivalent to the labelling of such three points. In this case, an isomorphism of between two 2-marked spherical tori is an orientation-preserving isometry compatible with the 2-markings. In Remark 6.28 we show that MS 1,1 (ϑ) and MS (2) 1,1 (ϑ) can be endowed with the structure of orbifolds in such a way that the map MS 1,1 (ϑ) → MS (2) 1,1 (ϑ) that forgets the 2-marking is a Galois cover with group S 3 (which is unramified in the orbifold sense).

The case ϑ not odd integer
Because of the relevance for the orbifold structure of the moduli spaces we are interested in, we first classify all possible automorphisms of spherical tori with one conical point. Proposition 4.4 (Automorphisms group of a spherical torus (ϑ non-odd)). Suppose that ϑ / ∈ 2Z + 1. For any spherical torus (T, x) of area 2π(ϑ − 1) the group of automorphisms G T is isomorphic either to Z 2 , or to Z 4 , or to Z 6 .
(iii) For each ϑ there can be at most one torus with automorphism Z 4 and one torus with automorphism Z 6 . (iv) The subgroup of G T of automorphisms that fix the 2-torsion points of T is isomorphic to Z 2 , generated by the conformal involution.
Proof. Recall that by Proposition 2.17 each torus has an automorphism of order 2, namely the conformal involution. Clearly such involution fixes the 2-torsion points of the torus. This implies (iv) and it proves that |G T | is even.
To bound the automorphism group we note that the action of G T fixes x and preserves the conformal structure on T . Hence, in case |G T | > 2 the torus T is biholomorphic to either T 4 or T 6 , and its automorphisms group is Z 4 or Z 6 correspondingly.
Let us now prove the existence part of (i) and (ii). (i) Suppose that d 1 (ϑ, 6Z) > 1. According to Theorem 3.1, this condition is equivalent to existence of a spherical triangle ∆ with angles πϑ/3. Such a triangle has a rotational Z 3symmetry. It follows that the torus T (∆) has an automorphism of order 6.
(ii) Suppose that d 1 (ϑ, 4Z) > 1. According to Theorem 3.1, this condition is equivalent to existence of a spherical triangle ∆ with angles π(ϑ/2, ϑ/4, ϑ/4). This triangle has a reflection, i.e. an anti-conformal isometry that exchanges two vertices of angles πϑ/4. Gluing two copies of ∆ along the edge that faces the angle πϑ/2, we obtain a quadrilateral with four edges of the same length and four angles πϑ/2. It is easy to see that such quadrilateral has a rotational Z 4 -symmetry, and so that T (∆) has an order 4 automorphism.
Let now (T, x, ϑ) be any spherical torus with |G T | > 2 and let us show that it has to be one of two tori constructed above. Consider two cases.
First, suppose that the Voronoi graph Γ(T ) is a trefoil. In this case by Proposition 2.21 and Theorem B there is a unique collection of three geodesic loops γ 1 , γ 2 , γ 3 based at x that cut T into two isometric strictly balanced triangles ∆ and ∆ . This collection is sent by G T to itself, and so |G T | is divisible by three, hence |G T | = 6. It is easy to see then that the subgroup Z 3 ⊂ G T sends ∆ to itself and permutes its vertices. So ∆ has angles πϑ/3 and so we are in case (i). Since ϑ/3 cannot be integer, this also proves the uniqueness of a torus with automorphism group Z 6 .
Suppose now that the Voronoi graph Γ(T ) is an eight graph. Then again by Proposition 2.21 and Theorem B there is a canonical collection of four geodesic loops γ 1 , γ 2 , η 1 , η 2 . Since G T sends the couple η 1 , η 2 to itself, we see that geodesics η 1 and η 2 cut a neighbourhood of x into four sectors of angles πϑ/2. The same holds for the couple of loops γ 1 and γ 2 . Since by Remark 2.22 each γ i bisects two sectors formed by η 1 and η 2 we see that, taken together, the geodesics γ 1 , γ 2 , η 1 , η 2 cut a neighbourhood of x into four eight sectors of angles πϑ/4. Hence γ 1 , γ 2 , η 1 cut ∆ into two semi-balanced triangles with angles π(ϑ/2, ϑ/4, ϑ/4), and so we are in case (ii). The uniqueness of a torus with automorphism group Z 4 follows from the uniqueness of an isosceles triangle with angles π(ϑ/2, ϑ/4, ϑ/4).
We recall in more detail the construction mentioned in the introduction.
The map T (2) is defined by sending an oriented triangle ∆ to the torus T (∆), where we mark by p i the midpoint of the side opposite to the vertex x i of ∆. As for ∆ (2) , we proceed as follows. Let (T, x, p) be a torus with its order 2 points marked by p 1 , p 2 , p 3 .
Suppose first that T does not have a rectangular involution. By Theorem B there is a unique collection of three geodesics loops γ i that cuts T into two congruent strictly balanced triangles ∆ and ∆ . We enumerate the geodesics so that each p i is the midpoint of γ i . Next, we label the vertices of ∆ by x 1 , x 2 , x 3 so that x i is opposite to γ i . Hence, we associate to T a unique strictly balanced triangle with enumerated vertices. In case the vertices of ∆ go in anti-clockwise order, we associate to ∆ the corresponding point in the interior of MT + bal (ϑ), otherwise we associate to ∆ a point in the interior of MT − bal (ϑ). Suppose now that T has a rectangular involution. Then by Theorem B the torus T can be cut into two isomorphic semi-balanced triangles in two different ways. At the same time the rectangular involution sends one pair to the other by reversing the orientation and fixing the labelling of the vertices. This means that the two points associated to T in the boundaries of MT + bal (ϑ) and MT − bal (ϑ) are identified in MT ± bal (ϑ).
At this point we have the tools to prove the following preliminary fact.
Proof. It is very easy to see that T (2) • ∆ (2) is the identity of MS Remark 4.7 (Orbifold Euler characteristic). We recall from the introduction that we are using the definition of orbifold Euler characteritic given at page 29 of [6]. We are particularly interested in two properties enjoyed by the orbifold Euler characteristic: (a) if Y → Z is an orbifold cover of degree d, then χ(Y) = d · χ(Z); (b) if Y is a connected, orientable, two-dimensional orbifold with underlying topological space Y , then where ord(Y) is the orbifold order of a general point of Y and ord(y) is the orbifold order of y ∈ Y , and the sum is ranging over points y ∈ Y that have orbifold order strictly greater than ord(Y). Since we will only compute χ for two-dimensional, connected, orientable orbifolds, property (b) could even be taken as a definition.
The main ingredient for the proof of Theorem A is to show that the map T (2) is a homeomorphism and so that, as a topological space, MS 1,1 (ϑ) is a surface. As a consequence, we can endow MS (2) 1,1 (ϑ) with an orbifold structure (as done in Remark 6.28) in such a way that every point has orbifold order 2, which is coherent with Proposition 4.4 (iv).
1,1 (ϑ) is isomorphic to the quotient of its underlying topological space by the trivial Z 2 -action and its orbifold Euler characteristic is −m 2 /2.
As above, we can endow MS 1,1 (ϑ) with an orbifold structure as in Remark 6.28, in such a way that the orbifold order of a point in MS 1,1 (ϑ) agrees with the number of automorphisms of the corresponding spherical torus.
Let us finally prove Theorem A.
(ii-iii-iv) Clearly χ(MS 1,1 (ϑ)) = χ(MS Let us finish this subsection with a simple corollary of Theorem 4.8. As a topological space, we denote by MS (2) 1,1 (ϑ) the unique smooth compactification of the surface MS (2) 1,1 (ϑ) obtained by filling in the 3m punctures. As above, we endow MS (2) 1,1 (ϑ) with the orbifold strucure given by taking the quotient of its underlying topological space by the trivial Z 2 -action. Proof. Let us comment on the last claim, since the other claims are rather immediate after Theorem 4.8. Recall, that in the proof of Proposition 3.20 (iv) for c ≤ 0 we constructed a decomposition of MT bal (ϑ) in the union of 3m(m + 1)/2 one-cells and m 2 two-cells. One can check that each of theses m 2 cells has exactly three 1-cells in its boundary. Hence, we get a triangulation of the topological space MS (2) 1,1 (ϑ). Note however, that for c > 0 the total number of 2-cells is 2m 2 + 6m and the additional 6m cells are digons rather than triangles.

The case ϑ odd integer
In this subsection we prove Theorems C-D. Our first step will be to prove Theorem E, from which part (a) of Theorem C will be easily obtained.
Proof of Theorem E. According to Proposition 2.17, there is a unique curvature 1 metric on T with angle 2π(2m + 1) in given projective equivalence class, which is invariant under the conformal involution σ of T . Hence we can apply Proposition 2.21 to T endowed with such σ-invariant metric. According to such proposition, there exist three geodesic loops based at the conical point x that cut T into two isometric balanced triangles ∆ and ∆ . By Gauss-Bonnet formula we have Area(∆) = 2πm and so we can apply Lemma 3.10. According to such lemma, ∆ is a balanced triangle with angles 2π · (m 1 , m 2 , m 3 ) where m 1 + m 2 + m 3 = 2m + 1. This finishes the proof of the theorem.
Such a result already allows us to describe MS 1,1 (2m + 1) σ as a topological space.
Proof of part (a) of Theorem C. As in the proof of Theorem E, we can associate to each torus with a σ-invariant metric a unique oriented balanced spherical triangle with integral angles and unmarked vertices. Clearly, an orientation on a triangle is equivalent to a numbering of its vertices up to cyclic permutations. Such correspondence determines a bijective map T : MT bal (2m + 1)/A 3 −→ MS 1,1 (2m + 1) σ where the alternating group A 3 acts by relabelling the vertices of the triangle. Arguments entirely analogous to the ones used in Theorem 6.5 (ii) show that T is continuous and proper, hence a homeomorphism of topological spaces.

6
. The rest of the subsection is devoted to a careful analysis of the orbifold structures on our moduli spaces and to the proof of part (b) of Theorem C and of Thoerem D.

Voronoi graph and decorations
The orbifold structure on our moduli spaces is defined in Remark 6.28, but a more explicit interpretation of such structure for moduli spaces of tori of area 4mπ relies on the notion of decoration.
We begin with a simple lemma. Proof. Consider first the case m = 1. A spherical triangle ∆ 0 with vertices x 1 , x 2 , x 3 of angles (π, π, π) is isometric to a hemisphere and its circumcenter O is at distance π/2 from the boundary of such hemisphere. So the rotations of the hemisphere that take x i to x j fix O. A torus T 0 with a σ-invariant metric h of area 4π is isometric to T (∆ 0 ) and so it has three edges and two vertices. Since σ fixes the Voronoi graph Γ(T 0 ) and pointwise fixes the conical point and the midpoints of the three edges of Γ(T 0 ), it does not fix any other point. In particular, σ exchanges the two vertices of Γ(T 0 ). Moreover, the vertices of Γ(T 0 ) are at distance π/2 from ∂∆ 0 , and so the edges of Γ(T 0 ) have length π. It follows that a (multi-valued) developing map for T 0 sends the vertices of Γ(T 0 ) to the two fixed points O, O for the monodromy, and the edges of Γ(T 0 ) to meridians running between O and O . Note that another spherical metric on T 0 projectively equivalent to h is obtained by post-composing the developing map of h by a Möbius transformation that fixes O and O . Since such transformations preserve the meridians between O and O , the two metrics have the same Voronoi graph. Suppose now m > 1. By Theorem E and Proposition 3.8, a torus T with σ-invariant metric of area 4mπ is obtained from a torus T 0 = T (∆ 0 ) of area 4π as above by gluing a sphere S i with two conical points of angles 2mπ at distance |x j x k | along the geodesic segment x j x k of T 0 . The conclusion then follows from the analysis of the case m = 1.
In order to make the role of the conformal involution σ in the below Constructions 4.14-4.16 more transparent, we will need the following.  Proof. Being an isometry, an automorphism is in particular biholomorphic. It is a classical fact that the only nontrivial biholomorphisms a 2-marked conformal torus (T, x) is the involution σ. By Lemma 4.10 the Voronoi graph Γ(T ) has two vertices and they are exchanged by σ.

Moduli spaces of σ-invariant spherical metrics of area 4mπ
Similarly to what we did in Section 4.1, we first discuss the space of decorated 2-marked tori. Construction 4.14 (Tori with σ-invariant metrics). If σ is the unique (nontrivial) conformal involution of a conformal torus, denote by MS (2) 1,1 (2m + 1) σ the set of 2-marked decorated tori (T, x, p) with a σ-invariant spherical metric of angle (2m + 1)2π at x. We recall that triangles in MT ± bal (2m + 1) have area 2mπ, integral angles and they are strictly balanced. We then define the maps MT ± bal (2m + 1) 1,1 (2m + 1) σ ∆ (2) l l quite as in Construction 4.5. In particular, T (2) sends an oriented triangle ∆ to the 2-marked torus T (∆) obtained as a union of ∆ and ∆ , with the decoration given by the vertex of Γ(T (∆)) that sits inside ∆.
We easily have the following preliminary result.
(iv) is clear, since the 2-marked decorated spherical tori (T, p, v, h) and (T, p, v, σ * h) are isomorphic via the map σ.
In view of Remark 6.28, the final claim follows from (iv). Now we discuss the moduli space MS 1,1 (2m + 1) σ of σ-invariant spherical tori.
Proof of part (b) of Theorem C. Recall that MS 1,1 (2m + 1) σ is endowed with a 2-dimensional orbifold structure by Remark 6.28. By Theorem 4.15 (i), the space MT ± bal (2m+1) is isomorphic to the moduli space of decorated 2-marked tori with a σ-invariant metric. On a fixed such torus, 2-markings are permuted by S 3 and the decorations are exchanged by σ. Hence, the moduli space MS 1,1 (2m + 1) σ is isomorphic (as an orbifold) to the quotient of MT ± bal (2m + 1) by S 3 × 1, σ * . By Proposition 3.25, such quotient can be identified to MT bal (2m + 1)/A 3 × Z 2 , where the alternating group A 3 acts by cyclically relabelling the vertices of the triangles and Z 2 acts trivially by Theorem 4.15 (iv). By Proposition 3.24 the space MT bal (2m + 1) consists of m(m + 1)/2 connected component and it is diffeomorphic to Crp bal (2m + 1) ×∆ 2 . Consider now two cases.
(b-i) Suppose 2m + 1 not divisible by 3. In this case neither of spherical triangles in MT bal (2m + 1) have all equal angles, so the action of A 3 does not send any component to itself. So the number of components of MS 1,1 (2m + 1) σ is m(m+1)

6
, each one is homeomorphic to the quotient D of∆ 2 by the trivial Z 2 -action and so all points have orbifold order 2.
(b-ii) Suppose 2m + 1 divisible by 3. Then the component corresponding to triangles with angles m 1 = m 2 = m 3 = (2m + 1)/3 is the only one that is sent to itself. It contains a unique point fixed by A 3 , namely the equilateral spherical triangle. This point gives rise to an orbifold point of order 6 on MS 1,1 (2m + 1) σ , which belongs to a component homeomorphic to the quotient D of∆ 2 by Z 2 × A 3 , where Z 2 acts trivially. All the other m(m + 1)/2 − 1 components are non-trivially permuted by A 3 , and they are all homeomorphic to D. Hence, there are m(m+1) 6 connected component, and all points except the equilateral spherical triangle have orbifold order 2.

Moduli spaces of spherical metrics of area 4mπ
In order to treat spherical metrics that are not σ-invariant, we need a further construction. We view the topological space MS 1,1 (2m + 1) as a moduli space of decorated, 2-marked tori and we define the following couple of maps 1,1 (2m + 1) ν l l as follows.
In order to define Ξ, let ∆ be an oriented triangle in MT ± bal (2m + 1), and fix a developing map ι for ∆ that sends its circumcentre v to O ∈ S 2 . Extend ι to the universal cover of the torus T (∆), which has a σ-invariant metric h, and is given a 2-marking as in Construction 4.5. For every t ∈ R the map e itR • ι : T → S 2 has the same equivariance of ι, and so the pull-back of the metric of S 2 via such map descends to a spherical metric h t on T . We then define Ξ(T, x, p, v, h, t) := (T, x, p, v, h t ).
In order to define ν, consider a 2-marked decorated spherical torus (T, x, p, v,ĥ), whose metricĥ is not necessarily invariant under the conformal involution σ. Its developing map ι : T → S 2 has monodromy contained in a 1-parameter subgroup that fixes O = ι(ṽ), whereṽ is a lift of v, and a maximal circle E. Note that points in e itR ·E sit at constant distance arctan(2e −t ) from O and that the distance from O corresponds to the distance function d v : T → [0, π] from the vertex v. Thus we also have the function t = − log tan(d v /2) : T → [−∞, ∞]. We remark that a developing map of σ * (T, x, p, v,ĥ) can be obtained by post-composing ι with an isometry of S 2 that exchanges O with −O. Hence, t • σ * = −t. It follows thatĥ is σinvariant if and only if t(x) = 0, namely ι(x) ∈ E for any liftx of x. It is easy to see that modified developing map e −it(x)R • ι has the same invariance as ι and sendsx to E. Hence, the round metric on S 2 pulls back and descends to a σ-invariant metric h on T . We define ν(T, x, p, v,ĥ) := ∆ (2) (T, x, p, v, h, t(x)).
Before proceeding, we need a very simple lemma. Proof. If O is the origin of C and 2|dz| 1+|z| 2 is the spherical line element, then the transformation e itR can be written as z → e −t z. Through the map e itR the metric decreases the most at E = {|z| = 1}, where the Lipschitz constant is exactly 1/ cosh(t).
The first fact about Construction 4.16 is the following. Proof. It is routine to check that the maps Ξ and ν are set-theoretic inverse of each other. Note that the restriction of Ξ to MT ± bal (2m + 1) × {0} is a homeomorphism by Theorem 4.15 (i). Hence, the continuity of Ξ follows from Lemma 4.17.
To show that Ξ is proper, consider a diverging sequence in MT ± bal (2m+1)×R, which we can assume to be contained in a fixed connected component. By Proposition 3.25 an element of such component can be identified by a quadruple (l 1 , l 2 , l 3 , t) with 0 < l i < 2π and l 1 + l 2 + l 3 = 2π. A sequence of quadruples diverges if and only if somel i → 0 or if |t| → ∞ (up to subsequences). Since the systole of the triangle corresponding to (l 1 , l 2 , l 3 ) is min{l i } by Lemma 6.24, the systole of the torus Ξ(l 1 , l 2 , l 3 , t) is at most min{l i }/ log cosh(t) → 0 by Lemma 4.17. It follows that Ξ sends diverging sequences to diverging sequences by Theorem 6.3.

Since MS
(2) 1,1 (2m + 1) is a manifold by Proposition 4.18, it can be endowed with an orbifold structure as in Remark 6.28. We then have the following preliminary result.  1,1 (2m + 1) of 2-marked tori with spherical metric of area 4mπ has the following properties.
(i) As an orbifold, it is isomorphic to the quotient of MT ± bal (2m + 1) × R by the action of the involution σ * that flips the sign of the R-factor. Hence it consists of m(m + 1) components isomorphic to∆ 2 × (R/{±1}).
(ii) The locus in MS Finally we can prove Theorem D.
Proof of Theorem D. The forgetful map MS 1,1 (2m + 1) → MS 1,1 (2m + 1) is an unramified S 3 -cover of orbifolds. By Proposition 4.19 such quotient can be identified to (MT bal (2m + 1) × R)/(A 3 × Z 2 ), where Z 2 acts by flipping the sign of the R-factor and the alternating group A 3 acts by cyclically relabelling the vertices of the triangles.
The rest of argument is entirely analogous to the one used in the proof of Theorem C.

MS 1,1 (2m) and MS
1,1 (2m) as Belyi curves The goal of this section is to identify the moduli spaces MS 1,1 (2m) and MS 1,1 (2m) with Belyi curves and relate their cell decompositions constructed in Corollary 4.9 with the corresponding dessins. We recall [2, Section 2] and [14] that these two spaces have a canonical complex structure. This structure is the unique one with respect to which the forgetful map to M 1,1 and M 1,1 are holomorphic. We also recall that the compactification MS 1,1 (2m) by filling it the 3m punctures has the orbifold structure that makes it isomorphic to the quotient of its underlying topological space (which is in fact a Riemann surface) by the trivial Z 2 -action. The respective forgetful maps extend to the smooth compactifications of all the four orbifolds.
The following definition slightly differs from the usual definition of dessin d'enfant, though it is very similar in spirit. The dessin of ψ can also be seen as the 1-skeleton of the triangulation of S whose open cells are the preimages through ψ of the two open disks in which RP 1 cuts CP 1 .
The main result of this section is the following theorem, which concerns the underlying Riemann surface MS  (iii) The dessin of ψ Bel is composed of tori T such that the triangle ∆(T ) has one integral angle.
In particular, the triangulation given by this dessin is the one described in Corollary 4.9.
Definition 5.3 (Klein group and Klein sphere). The Klein group K 4 is the subgroup of diagonal matrices in SO(3, R). The Klein sphere S Kl is the sphere with three conical points y 1 , y 2 , y 3 of angles (π, π, π) obtained by taking the quotient of the unit sphere S 2 by the action of K 4 ∼ = Z 2 ⊕ Z 2 . We denote by S Kl (R) the circle in S Kl which is invariant under the unique antiholomorphic isometric involution of S Kl .
Using the conformal structure on S Kl given by the spherical metric, we can view S Kl as CP 1 , where y 1 = 0, y 2 = 1, y 3 = ∞, and S Kl (R) as RP 1 .
Remark 5.4 (Klein sphere as a doubled triangle). We note that S Kl can also be obtained by doubling of the spherical triangle ∆ with angles π, π, π across its boundary. This way ∂∆ corresponds to the circle S Kl (R) in S Kl . Recall that, in the triangle with three angles π, each vertex is at distance exactly π 2 from each point of the opposite side. For this reason, points of S Kl (R) are exactly the points on S Kl that are at distance π 2 from one conical point.
The key result to parametrize spherical tori using a Hurwitz space is the following.
Proposition 5.5 (Tori of area (2m − 1)π cover the Klein sphere). Let (T, x) be a spherical torus with a conical point of angle 4πm and with points of order 2 marked by p 1 , p 2 , p 3 . There exists a unique branched cover map ϕ Kl : T → S Kl of degree 4m − 2, which is a local isometry outside of branching points, and such that ϕ Kl (p i ) = y i . Moreover ϕ Kl (x) = y i for i = 1, 2, 3.
Proof. We will first construct the map and then will count its degree. Recall [2, Proposition 1.5.1], that the image of the monodromy map ρ : π 1 (T, x) → SO(3, R) is the Klein group (see also Corollary A.3). Consider the developing map ι : T → S 2 from the universal cover T of T . This map is equivariant with respect to the action of π 1 (T, x) on T by deck transformation and on S 2 by the monodromy representation. Hence, by taking the quotient we get a map ϕ Kl : T → S Kl ∼ = S 2 /K 4 . We will now prove that the constructed map ϕ Kl sends points p i to the three distinct orbifold points of S Kl . This will permit us to label these three points so that ϕ Kl (p i ) = y i . In order to do this, consider the order two automorphims σ of T and denote by S the quotient T /σ. The surface S is a sphere with three conical points of angle π, that are the images of the points p i , and one conical point of angle 2πm. Let us take a liftx ∈ T of x and let σ be the lift of σ to T that fixesx. Since the conical angle at x is an even multiple of 2π, the maps ι and ι • σ coincide in a neighbourhood ofx. It follows that ι is σ-invariant and the map ϕ Kl descends to a map ϕ Kl : S → S Kl . Now, by construction, the map ϕ Kl is a local isometry outside of ramification points. This implies that all three conical points of angle π on S are sent by ϕ Kl to conical points of angle π on S Kl . Finally, to see that the images of the three conical points are distinct, we use the fact that the monodromy of S is generated by three loops winding simply around these points, and it is isomorphic to K 4 . Hence, we proved that points ϕ Kl (p i ) in S Kl are the three distinct conical points of S Kl , and so we can label each ϕ Kl (p i ) by y i . This finishes the construction of the map. Its uniqueness is clear.
Here is already a first corollary of the above proposition.

Proof. To construct the map MS
(2) 1,1 (2m) → H m , we use Proposition 5.5, that associates to each spherical torus (T, x) with a 2-marking the branched cover ϕ Kl : T → S Kl . Using the conformal structure on S Kl given by the spherical metric, we view it as CP 1 , where y 1 = 0, y 2 = 1, y 3 = ∞. By Proposition 5.5 we know that λ = ϕ Kl (x) = 0, 1, ∞. To find the cyclic type of ramification over points (0, 1, ∞, λ), we recall that the map ϕ Kl is a local isometry outside of branching locus, and so for each preimage of the points 0, 1, ∞ the map has branching of order 2. Finally, there is only one conical point in the preimage of λ, hence the cyclic type over λ is (1, . . . , 1, 2m).
The inverse map H m → MS 1,1 (2m) is the following. For each ramified cover T → CP 1 ∼ = S Kl with the prescribed cyclic type, we pull-back the spherical metric of S Kl to T . By Proposition 5.5 the 2-torsion points of T are mapped to y 1 , y 2 , y 3 and we call p i the unique 2-torsion point of T that is sent to y i . This corollary has the following immediate consequence.  Proof. Let us first construct the cover. As in the proof of Proposition 5.5, to each torus T with conical angle 4πm we associate the ramified cover ϕ Kl : T → S Kl ∼ = CP 1 . We know that ϕ Kl (x) = 0, 1, ∞. So we set ψ Bel (T ) = ϕ Kl (x). It is easy to see that this map is an unramified cover of CP 1 \ {0, 1, ∞}.
To justify the holomorphicity statement, let us pull-back the complex structure on CP 1 \ {0, 1, ∞} to the surface MS 1,1 (2m). We claim that this pulled-back structure coincides with the canonical one. Indeed the canonical structure is the unique structure with respect to which the forgetful map to the Riemann surface M (2) 1,1 is holomorphic. However, it is also clear that the modulus of the torus ψ −1 Bel (λ) is a holomorphic function of the local parameter λ on CP 1 \ {0, 1, ∞}.
We need one last lemma. Proof. Let us prove the "if" direction. Suppose that ∆ has an integral angle. Then it has one side of length π. This means that for some i the distance on T from x to p i is π/2. This means that the distance on S Kl between y i and ϕ Kl (x) is π/2. Using Remark 5.4, we deduce that ϕ Kl (x) belongs to S Kl (R). By the definition of the dessin of ψ Bel we see that T belongs to the dessin.
Let us now prove the "only if" direction. Suppose that ϕ Kl (x) belongs to S Kl (R). For example, assume ϕ Kl (x) ∈ y 1 y 2 . Let γ 3 be the geodesic loop on T based at x, whose midpoint is p 3 . Since half of this geodesic is projected by ϕ Kl to the segment that joins y 3 with the segment y 1 y 2 , we see that |γ 3 | = π. From Lemma 3.6, it follows that the angle of ∆ opposite to γ 3 is integral.
Proof of Theorem 5.2. (i) The ramified cover is the extension of the cover constructed in Corollary 5.7 to the compactified spaces.
(iii) This is proven in Lemma 5.8.
(ii) Recall that the topological space underlying MS 1,1 (2m) is a complex curve with exactly 3m punctures. Let q be one of such punctures. Since S Kl is obtained by gluing two spherical triangles along S Kl (R), each y i is adjacent to two triangles. It follows that the ramification degree of ψ Bel at q is equal to half the number of triangles of the dessin adjacent to q. Since MS (2) 1,1 (2m) is glued from two copies of MT bal (ϑ) it is easy to check that the corresponding number can be any odd number from 1 to 2m − 1.
Finally, we can prove Theorem F.
Proof of Theorem F. To prove this result we will realise MS 1,1 (2m) is an unramified orbifold cover of the modular curve H 2 /SL(2, Z). Recall that in Theorem 5.2 we constructed the unramified covering map ψ Bel of degree m 2 from the topological space MS Note that the quotient of CP 1 \ {0, 1, ∞} by the trivial Z 2 -action is an orbifold isomorphic to H 2 /Γ(2), where Γ(2) = {A ∈ SL(2, Z) | A ≡ I (mod 2)}. Hence, the above cover can be promoted to an unramified cover of orbifolds MS  Since MS 1,1 (2m)/S 3 = MS 1,1 (2m) as orbifolds, the covering map then descends to an unramified orbifold covering MS 1,1 (2m) → H 2 /SL(2, Z) of degree m 2 . Note that the cycle type ramification of such cover at infinity is (1, 3, . . . , 2m − 1) by Theorem 5.2 (ii). It follows that, for m > 1, such cover is not Galois and so G m is not a normal subgroup.
The last claim follows from Theorem 5. 6 Lipschitz topology on MS g,n In this section we define a natural topology on the set of spherical surfaces with conical singularities and establish some of its basic properties. We choose the approach using Lipschitz distance, described, for example in [15,Example,page 71].
Let us first recall the definition of Lipschitz distance between two marked metric spaces.
Definition 6.1. Let (X, x 1 , . . . , x n ; d X ) and (Y, y 1 , . . . , y n ; d Y ) be two metric spaces with distinct marked points x i , y i . The Lipschitz distance between them is defined by where dil(f ) = sup and the infimum runs over bi-Lipschitz homeomorphisms between X and Y that send each x i to y i . The value max{dil(f ), dil(f −1 )} is called the bi-Lipschitz constant of the map f . Furthermore, we say that a map f : X → Y is a bi-Lipschitz embedding with constant c ≥ 1 if for any two points x 1 , x 2 we have We will denote by MS g,n the space of genus g surfaces with n marked conical points up to a marked marked isometry. By MS g,n (≤A) we denote the subspace of surfaces with area bounded by A > 0. To state the main two results of this section we recall the notion of the systole of a spherical surface. Definition 6.2 (Systole). The systole sys(S) of a spherical surface S is the half length of the shortest geodesic segment or geodesic loop on S whose end points are conical points of S.
The systole sys(P ) of a spherical polygon P is the minimum of half-distances between all vertices of P and the distances between a vertex of P with the unions of edges not adjacent to the vertex. Such a systole is clearly equal to the systole of the sphere obtained by doubling P along its boundary.
Let MS ≥s g,n (≤A) be the subspace of MS g,n (≤A) of surfaces with systole at least s. Theorem 6.3. MS g,n is a complete metric space with respect to Lipschitz distance. The function sys(S) −1 is proper on MS g,n (≤A) in Lipschitz topology.
Let us denote by MP n the space of all spherical polygons with n cyclically labelled vertices up to isometries that preserve the labelling. We have the following similar result.
Corollary 6.4. The space MP n of spherical polygons with n vertices is complete with respect to Lipschitz distance. For any positive A > 0 the function sys −1 (P ) is proper on the subset MP n of polygons with area at most A.
To prove Theorem 6.3, we show that surfaces from MS ≥s g,n (≤A) admit triangulations into a finite number of relatively large triangles. This is done in Theorem 6.23 which itself relies on Delaunay triangulations, constructed in Theorem 6.15. The proof of Corollary 6.4 is similar.
As an application of Theorem 6.3 and Corollary 6.4, we will get the following result on topology of the space MS 1,1 (2m + 1) σ , L) is a homeomorphism of surfaces.
Recall, that the bijective map T (2) was defined in Construction 4.5, whereas the Lipschitz distance between two 2-marked tori is measured among maps that preserve 2-marking.

Lipschitz metric and its basic properties
Here we collect basic results concerning Lipschitz metric with an emphasis on spherical surfaces. Lemma 6.6. Lipschitz distance defines a metric on the space MS g,n of spherical surfaces of genus g with n conical points.
Proof. Let S 1 and S 2 be genus g spherical surfaces with n conical points. Let's show that d L (S 1 , S 2 ) < ∞, i.e., that there is a bi-Lipschitz map ϕ : S 1 → S 2 . Take a map ϕ : S 1 → S 2 that is a diffeomorphism fromṠ 1 toṠ 2 , sends each conical point x i ∈ S 1 to the corresponding point x i ∈ S 2 , and is radial 10 in a neighbourhood of each x i . Such a map is clearly bi-Lipschitz.
Next note that d L (S 1 , S 2 ) = 0 if and only if S 1 and S 2 are isometric by [1,Theorem 7.2.4].
All the other properties of the metric are obvious.
Definition 6.7. The Lipschitz topology on the moduli space MS g,n of spherical surfaces is the topology induced by the Lipschitz metric.
The next lemma explains how difference in the values of conical angles of two surfaces affects the Lipschitz distance between them. Lemma 6.8 (Continuity of angle functions). Let U 1 , U 2 be neighbourhoods of conical points x 1 , x 2 with conical angles ϑ 1 , ϑ 2 . Suppose f : U 1 → U 2 is a bi-Lipschitz homeomorphism. Then In particular, functions ϑ i are continuous on MS g,n in Lipschitz topology.
Proof. After scaling by a large constant and passing to the limit, we can assume that the metrics on U 1 and U 2 are flat, moreover both U 1 and U 2 are flat cones with conical angles 2πϑ 1 , 2πϑ 2 correspondingly. Note, that as a result the limit quantity max{dil(f ), dil(f −1 )} can only decrease. Replacing f by f −1 if necessary, we can assume that ϑ 1 ≤ ϑ 2 . Let us now reason by contradiction. Assume that Inequality (5) is not satisfied. Consider the radius 1 circle S 1 centred at This contradicts our assumption.
Lemma 6.9 (Continuity of systole function). Let (S 1 , h 1 ), (S 2 , h 2 ) be two spherical surfaces from MS g,n such that d L (S 1 , S 2 ) = d. Then In particular, the function sys(S, h) is continuous on MS g,n in Lipschitz topology.
Proof. Let S be a spherical surface with conical points x 1 , . . . , x n . According to [22], sys(S) is equal to the minimum of half distances between conical points, and half-lengths of all (rectifiable) simple loops based at some conical point x i , contained inṠ ∪ x i and non-contractible inṠ ∪ x i . Any bi-Lipschitz homeomorphsim f from S 1 to S 2 that sends points x i to points x i also sends rectifiable loops based at x i to rectifiable loops based at x i . By definition there for any ε > 0 exists a homemorphism f ε : S 1 → S 2 with bi-Lipschitz constant e d+ε . This clearly explains the above inequalities.

Injectivity radius
Here we prove Proposition 6.11 that gives an estimate on the injectivity radius of points on spherical surfaces in terms of the value of the Voronoi function and the systole of the surface.
Definition 6.10. Let S be a spherical surface and y ∈Ṡ be a non-conical point. The injectivity radius inj(y) is the supremum of r such that S contains an isometric copy of a spherical disk of radius r, embedded in S and centred at y. For a conical point x i ∈ S the injectivity radius is defined to be the minimum of all distances from x i to other conical points, and half lengths of geodesic loops based at x i . Proposition 6.11. Let S be a spherical surface with conical angles 2π(ϑ 1 , . . . , ϑ n ). Then for any y ∈Ṡ we have inj(y) ≥ min(sys(S), V S (y), min Moreover, the following statements hold. (i) If inj(y) < V S (y), then there exists a closed geodesic loop γ ⊂Ṡ of length 2inj(y), based at y. Moreover l(γ) = 2inj(y) < π. (ii) If V S (y) > π 2 then inj(y) = V S (y). (iii) Suppose inj(y) < V S (y) and so V S (y) ≤ π 2 . Then, at least one of the following holds (a) inj(y) > sys(S). (b) There exists i such that ϑ i < 1 2 and inj(y) > min i ϑ i · V S (y).
We will need one lemma to prove this result.
Lemma 6.12. Let D be a spherical disk with one conical point x in its interior. Suppose that the boundary γ of D satisfies (γ) < 2π, and γ is a geodesic loop with a unique non-smooth point y. Then there exits an orientation reversing, isometric involution τ on D.
Proof. Note first, that the angle at x is non-integer, otherwise the univalent developing map from D to S 2 would send γ onto a great circle. Consider the sphere S obtained from D by doubling along γ, and denote by τ γ the corresponding isometric involution. Since not all conical angles of S are integer, there exists a unique anti-conformal isometry τ of S fixing its conical points. Clearly, τ commutes with τ γ and so τ lives γ ⊂ S invariant. Hence τ induces the desired involution on D ⊂ S .
Proof of Proposition 6.11. Since clearly inj(y) ≤ V S (y), Inequality (6) immediately follows from Claim (iii). So, we need to prove Claims (i-iii). (i) Since inj(y) < V S (y), the existence of a geodesic loop of length 2inj(y), based at y is straightforward. Indeed, the midpoint of such a loop is a point at distance inj(y) from y, where disk centred at y of radius inj(y) touches itself. One can check that l(γ) ≤ π, since otherwise there will be points close to the midpoint of γ that can be joined with y by two distinct geodesic segments of length less than inj(y). To see that inj(y) < π 2 we note that in case inj(y) = π 2 the boundary of the open disk centred at y of radius π/2 is a closed geodesic to which the disk is adjacent twice. This means that S is a standard RP 2 , which is impossible since S is orientable.
(ii) Assume V S (y) > π 2 , and suppose by contradiction that inj(y) < V S (y). Let γ be a geodesic constructed in (i). Let 2πθ and 2π(1 − θ) be the angles in which γ cuts the neighbourhood of y, and assume without loss of generality that θ ≤ 1 2 . Take now a point O ∈ S 2 and consider a spherical kite OP 1 QP 2 in S 2 with ∠O = 2πθ, ∠P 1 = ∠P 2 = π/2 and l([OP 1 ]) = l([OP 2 ]) = l(γ)/2. Since θ ≤ 1 2 and l([OP 1 ]) ≤ π 2 , one can check that l([OQ]) ≤ π 2 . In particular the kite lies in the interior of a disk D r centred at O for any r ∈ (π/2, V S (y)). Since V S (y) > r, there exists a locally isometric immersion ι : D r →Ṡ such that ι(O) = y. By pre-composing ι with rotation, we can arrange so that ι sends the sides OP 1 and OP 2 to γ, and ι(P 1 ) = ι(P 2 ) is the mid-point of γ. It is clear then that the segments P 1 Q and P 2 Q are sent by ι to the same geodesic segment inṠ. It follows that ι is not a locally isometric immersion in any neighbourhood of Q. This is a contraction.
(iii) Since inj(y) < V S (y), by (i) there is a simple geodesic loop γ onṠ based at y of length 2inj(y) < π. We will consider separately two possibilities, depending on whether γ is essential 11 onṠ, or non-essential.
Let's assume now that γ is non-essential onṠ. Then γ encircles on S a disk D with at most one conical point in its interior. Since l(γ) < π by (i), the disk D should contain exactly one conical point, which we denote x i . Denote by 2πθ the angle that γ forms at y in D.
Suppose first that θ ≥ 1 2 . In this case γ forms a convex boundary of the surface S\D. Thanks to this, using exactly the same method as in [22, Corollary 3.11] one proves that l(γ) > 2sys(S), and we are in case (a).
Suppose now θ < 1 2 . Since (γ) < π we can apply Lemma 6.12 to D to get its isometric involution τ . This involution fixed the midpoint p of γ, and fixes two geodesic segments yx i and px i that cut D into two isometric right-angled spherical triangles. Let yp be one of two halves of γ. The segments yx i , px i and yp border a triangle x i yp in D with ∠x i = πϑ i , ∠y = πθ 2 , ∠p = π 2 . Since the side yp of the triangle is shorter than π and two adjacent angles are less than π, the triangle is convex. Since |yx i | > |yp|, we have θ i < 1 2 . Applying the sine rule to the triangle x i yp we get sin(|yp|) = sin(πϑ i ) sin(|x i y|). Hence inj(y) = |yp| > sin(πϑ i ) sin(|x i y|) > 2ϑ i sin(V S (y)) > 4 π ϑ i · V S (y), which proves that we are in case (b).

Equivalence of Lipschitz and analytic topologies on MT
In this section we prove that Lipschitz distance between triangles induces the same topology on MT as the topology induced by the embedding in R 6 , described in Theorem 3.12.
Definition 6.13. The relative Lipschitz distance d L (or L-distance) between two spherical triangles is the infimum of log(max(dil(f ), dil(f −1 ))) over all the marked bi-Lipschitz homeomorphisms f : ∆ 1 → ∆ 2 that restrict to a homothety on each edge of ∆ 1 .
The L-distance defines a metric on the space MT of spherical triangles, which we call the L-metric. We have the following natural statement.
Proposition 6.14. The topologies defined on MT by the L-metric and the L-metric coincide with the analytic topology given by the angle-side-length embedding Ψ : MT → R 6 .
Proof. Note that the side-lengths of ∆ are clearly continuous functions in both L and L topologies. The angles of ∆ are continuous in these topologies thanks to Lemma 6.8, applied to the 11 I.e. it doesn't bound onṠ a disk with at most one puncture. double of ∆. Furthermore the L-distance is greater of equal to the L-distance. Hence, the L-topology is finer than the L-topology, which is finer than the analytic topology. For this reason, we only need to show that for any spherical triangle ∆ and a sequence of triangles ∆ i converging to ∆ in R 6 (i.e., in the analytic topology), we have lim d L (∆ i , ∆) = 0. This claim can be proven by exhibiting explicit bi-Lipschitz maps between spherical triangles. We will only treat the case when ∆ is short-sided, since this is the only case needed for the purposes of the paper.
Following [11,Lemma 4.1] denote by U the open subset of MT consisting of triangles with angles πϑ i , where ϑ i < 2. This subset consists of spherical triangles that admit an isometric embedding into S 2 . In particular U lies in MT sh , the space of all short-sided triangles. Let's first prove that the L-topology coincides with the analytic topology on U .
For two spherical triangles ∆ = x 1 x 2 x 3 and ∆ = x 1 x 2 x 3 embedded into S 2 , with incentres I ∆ and I ∆ respectively, define the incentric map Φ : ∆ → ∆ as the unique map satisfying the following properties.
• Φ is a homothety on each edge x i x j .
• For any point p ∈ ∂∆, Φ sends the geodesic segment pI ∆ to a geodesic segment and restricts to a homothety on it. Suppose now we have a sequence of embedded triangles ∆ i ∈ U whose angles and sidelengths converge to that of ∆ ∈ U . Then it not hard to see that the bi-Lipschitz constant of the incentric maps Φ : ∆ i → ∆ tends to 1. Hence ∆ i converges to ∆ in L-topology as well. This proves the statement for U .
Let us denote by U klm ⊂ MT sh the subspace of triangles which can be obtained from an embedded triangle ∆ by repeated gluing of correspondingly (k − 1), (l − 1) and (m − 1) hemispheres to the sides x 1 x 2 , x 2 x 3 and x 3 x 1 of ∆. From Theorem 4.7 and Lemma 5.2 from [11] it follows that the sets U klm give an open cover of MT sh . At the same time, the incentric map Φ between any two triangles ∆ and ∆ from U can be naturally extended to a map Φ :∆ →∆ between triangles with attached hemispheres. Namely a radius of each hemisphere is sent isometrically to a radius and the restriction ofΦ to both sides of each hemisphere are homotheties. Since the Lipschitz constants of Φ andΦ clearly coincide, the statement about the topologies is proven for each U klm and so for the whole space MT sh .

Delaunay triangulations
We now turn to triangulations of spherical surfaces into convex spherical triangles. We will not require the triangulation to induce on the surface the structure of a simplicial complex. In particular, a triangle can be adjacent to a vertex up to 3 times, and to an edge up to 2 times.
The first result is a variation of the famous Delaunay triangulations of the plane [8] (see also [23,Section 14] for a modern exposition and references within). Proposition 6.15 (Delaunay triangulations). Let S be a spherical surface with conical points x 1 , . . . , x n , some of which might have angle 2π. Suppose that the Voronoi function V S is bounded by π 2 . Then there exists a triangulation of S into convex spherical triangles with the following "empty circle" property: for each triangle x i x j x k of the triangulation there exists a vertex v ∈ Γ(S) at equal distance r from x i , x j , x k , such that d(x l , v) ≥ r for all l ∈ {1, . . . , n}.
The proof will follow the proof by Thurston of a similar result [25, Proposition 3.1] concerning triangulations of surfaces with flat metric and conical singularities. We will need the following elementary lemma. Lemma 6.16. Let D, D ⊂ S 2 be two disks of radius less than π 2 . Let x 1 , x 2 ∈ ∂D and x 1 , x 2 ∈ ∂D be four distinct points. Suppose x 1 , x 2 don't lie in the interior of D and x 1 , x 2 don't lie in the interior of D. Then the geodesic segments x 1 x 2 ⊂ D and x 1 x 2 ⊂ D are disjoint in S 2 .
Proof. If D and D are disjoint, there is nothing to prove. Suppose D and D intersect, and let y 1 , y 2 be the two points of intersection of the boundary circles ∂D, ∂D . Let γ be the unique great circle on S 2 passing through y 1 and y 2 . It is now easy to see that the complements D \ D and D \ D lie in different hemispheres of S 2 with respect to γ. It follows that the segments x 1 x 2 and x 1 x 2 also lie in different hemispheres, and so they can intersect only in their endpoints. However, points x i and x i are distinct, so x 1 x 2 and x 1 x 2 are disjoint.
Proof of Proposition 6.15. The proof follows very closely the proof of [25, Proposition 3.1]. Let Γ(S) be the Voronoi graph of S. Let us first explain how to associate to each edge e ⊂ Γ(S) a dual geodesic segmentě with conical endpoints.
Let p ∈ Γ(S) be a point in the interior of an edge e ⊂ Γ(S), and set r = V S (p). Then there exists a locally isometric immersion ι p : D r → S from a radius r < π 2 spherical disk, that sends the centre of D r to p. Exactly two of the boundary points of D r , say y, z, are sent to two conical points x i , x j of S. Denote byě the image ι p (yz). It is easy to see that the segmentě is independent of the choice of p ∈ e.
Let us now deduce from Lemma 6.16 that for any two edges e, e ⊂ Γ S their dual edgesě,ě do not intersect in their interior points. This is similar the proof of [25, Proposition 3.1]. Let D, D be the disks immersed in S, that correspond to e and e . Assume by contradiction thať e,ě intersect in their interior point p. Consider the (multivalued) developing map ι : S → S 2 . The images of D and D under this map are embedded disks, and the images ofě,ě are cords of these disks, intersecting in ι(p). This contradicts Lemma 6.16. Indeed, the endpoints ofě are conical points that belong to ∂D \ D , and the endpoints ofě are conical points that belong to ∂D \ D. Hence, Lemma 6.16 is applicable to the 4-tuple ι(D,ě, D ,ě ).
Next, we associate to each vertex v of Γ(S) a convex polygon embedded in S, whose edgeš e 1 , . . . ,ě k are dual to the half-edges of Γ(S) adjacent to v. To do so, consider the immersion ι v : D r → S of a disk of radius r = V S (v), that sends the centre of D r to v. There will be exactly k points, say y 1 , . . . , y k , on ∂D r whose images in S are conical points. Let P v we the convex hull of the points y i in D r . Then the map ι v is an embedding on the interiorP v of the polygon P v , it may identify some vertices and it may identify an edge to at most one other edge of P v .
2). Suppose that there is a point x ∈ S at distance more than π 4 from points x 1 , . . . , x n . Let us denote such x by x n+1 and let us show that (S, x 1 , . . . , x n+1 ) satisfies conditions 1) and 2) for m = 1. Note that by [22, Lemma 3.10] we have sys(S) ≤ π 2 , which means π 4 ≥ s 2 , and so when we add x n+1 we don't violate 1). It remains to show that the injectivity radius of x n+1 of S is at least s 4 . Let's apply Inequality (6) of Proposition 6.11. We get inj(x n+1 ) ≥ min s, π 4 , min However, by Lemma [22, Lemma 3.13] we know that sys(S) ≤ min i ϑ i · π. So we get inj(x n+1 ) ≥ s 4 . Hence, condition 2) is satisfied for x 1 , . . . , x n+1 . In this way we can go on adding points x n+i until condition 3) is satisfied. Indeed, the process must terminate since the s 8 -neighbourhoods of points x n+i are disjoint disks on S and the area of S is finite.
(ii) To prove the second part of the theorem, we will work with the double S(P ) of P . We will construct a collection of regular points x n+1 , . . . , x n+m ∈ S(P ), such that the surface (S(P ), x 1 , . . . , x n+m ) has the following three properties.
0) The set of points x i is invariant under the isometric involution τ of S(P ) 1) For any i = j, d(x i , x j ) ≥ s 4 for all i = j ∈ {1, . . . , n + m}. 2) For each i the injectivity radius of x i on S is at least s 8 . 3) For any x ∈ S there is a point x i , such that d(x, x i ) ≤ π 4 . Let us explain how to make the first step. Consider P and ∂P as subsets of S(P ). Suppose there is a point y ∈ S(P ) at distance greater than π 4 from x 1 , . . . , x n . In case its distance from ∂P is more than π 8 , we set x n+1 = y, x n+2 = τ (y). In such case conditions 0)-2) are still satisfied for points x 1 , . . . , x n+2 , since d(x n+1 , x n+2 ) ≥ π 4 . Suppose now that d(y, ∂P ) < π 8 . Let y be a point on ∂P closest to y and set x n+1 = y . Clearly, the distance from x n+1 to x 1 , . . . , x n is at least π 8 . For this reason, as in (i) conditions 2) and 3) are still satisfied. This finishes the first step. Now, we repeat the above step until we get a collection of points x 1 , . . . , x n+m in S(P ), that satisfy all four conditions 0)-3). As in the proof of Proposition 6.15, we get a canonical decomposition of S(P ) into convex spherical polygons, invariant under the action of τ , and such that each polygon has side lengths at least s 4 and can be inscribed in a circle of radius at most π 4 . Those polygons whose interior doesn't intersect ∂P should be further cut into triangles by diagonals. Suppose that the interior of a polygon Q intersects ∂P . Then τ (Q) = Q and using a τ -invariant subset of diagonals of Q, one can cut it into a union of triangles exchanged by τ and either a triangle or a trapezoid Q , satisfying τ (Q ) = Q . If Q is a triangle we take Q ∩ P as one of the triangles of the triangulation of P . If Q is a trapezoid, we subdivide further Q ∩ P into two triangles along a diagonal. It is not hard to see that the resulting triangles are (f (s), π 4 )-bounded for certain positive function f (s). That concludes the decomposition of P into triangles.

Systole of balanced triangles
In this section we calculate the systole of a balanced triangle, show that for a balanced triangle ∆ we have sys(∆) = sys(T (∆)). Lemma 6.24. Let ∆ be a balanced spherical triangle with vertices x 1 , x 2 , x 3 , then 2sys(∆) = min i,j (min(|x i x j |, 2π − |x i x j |)).
Moreover the following two statements hold. (i) For any vertex x i of ∆, the distance to the opposite side x i x j is larger than sys(∆).
(ii) Let p ∈ ∂∆ be a point that is not a vertex of ∆. Suppose that η is a geodesic segment in ∆ that joins p with x i and doesn't belong to ∂∆. Then l(η) > sys(∆). (iii) There exists a geodesic segment γ ∆ ⊂ ∆ of length 2sys(∆), that joins two vertices of ∆.