Mikhail Bogdanov

Monique Teillaud

Delaunay triangulations and cycles on closed hyperbolic surfaces

Keywords: hyperbolic plane, hyperbolic translation, Delaunay triangulation, hyperbolic surface, Fuchsian group, covering space, algebraic software plan hyperbolique, translation hyperbolique, triangulation de Delaunay, surface hyperbolique, groupe fuchsien, revêtement, logiciel algébrique

This work is motivated by applications of periodic Delaunay triangulations in the Poincaré disk conformal model of the hyperbolic plane H 2 . A periodic triangulation is defined by an infinite point set that is the image of a finite point set by a (non commutative) discrete group Γ generated by hyperbolic translations, such that the hyperbolic area of a Dirichlet region is finite (i.e., a cocompact Fuchsian group acting on H 2 without fixed points). We consider the projection of such a Delaunay triangulation onto the closed orientable hyperbolic surface M = H 2 /Γ. The graph of its edges may have cycles of length one or two. We prove that there always exists a finite-sheeted covering space of M in which there is no cycle of length ≤ 2. We then focus on the group defining the Bolza surface (homeomorphic to a torus having two handles), and we explicitly construct a sequence of subgroups of finite index allowing us to exhibit a covering space of the Bolza surface in which, for any input point set, there is no cycle of length one, and another covering space in which there is no cycle of length two. We also exhibit a small point set such that the projection of the Delaunay triangulation on the Bolza surface for any superset has no cycle of length ≤ 2. The work uses mathematical proofs, algorithmic constructions, and implementation.

Introduction

Periodic hyperbolic Delaunay triangulations, which can also be seen as triangulations on hyperbolic surfaces, are used in very diverse fields [START_REF] Sausset | Tuning the fragility of a glassforming liquid by curving space[END_REF][START_REF] Ron | Hyperbolic centroidal Voronoi tessellation[END_REF][START_REF] Bachelot-Motet | Wave computation on the hyperbolic double doughnut[END_REF][START_REF] Chossat | Bifurcation of hyperbolic planforms[END_REF].

The case of closed Euclidean manifolds was already addressed [START_REF] Mazón | The computation of Voronoi diagrams on the Euclidean and spherical 2-orbifolds[END_REF][START_REF] Dolbilin | Periodic Delone tilings[END_REF][START_REF] Caroli | Computing 3D periodic triangulations[END_REF][START_REF] Caroli | Triangulating Point Sets in Orbit Spaces[END_REF][START_REF] Caroli | Delaunay triangulations of point sets in closed Euclidean d-manifolds[END_REF], leading to software packages for the flat torus in 2D and 3D [START_REF] Kruithof | 2D periodic triangulations[END_REF][START_REF] Caroli | 3D periodic triangulations[END_REF]. The algorithm follows the classic incremental algorithm in E d : for each new point p, the set of conflicting simplices (i.e., simplices whose balls contain p) is first computed, and the region that they form is then triangulated with new simplices with apex p. That algorithm relies on the fact that the set of conflicting simplices is always a topological ball. This subsumes that the Delaunay triangulation is a simplicial complex, which is the case when the graph of its edges has no cycle of length ≤ 2.

We study the case of a closed orientable hyperbolic surface M , isometric to a quotient H 2 /Γ, where Γ is a cocompact Fuchsian group acting on H 2 without fixed points; π denotes the natural projection of H 2 onto M that maps each point to its orbit under Γ. For a finite point set P in H 2 , we consider the Delaunay triangulation DT H (ΓP) of the infinite point set ΓP and its projection onto M . The graph of edges of π(DT H (ΓP)) can have non-trivial cycles of length one (1-cycles) or two (2-cycles): for instance, in the extreme case when P consists of only one point p, then the set ΓP is the orbit of p under Γ, so, all vertices of its Delaunay triangulation in H 2 project to the same point of M , i.e., each of its edges projects to a 1-cycle.

In Section 3, we give general conditions under which π(DT H (ΓP)) has no 1-or 2-cycles. We prove that there is always a finite-sheeted covering space M 1 (resp. M 2 ) of M in which, for any set P, there is no 1-cycle (resp. no 2-cycle). Then we focus on the specific case of the Bolza surface M, homeomorphic to the double torus. We give properties of fundamental domains and of closed geodesics in Section 4. In Section 5, we propose a way to construct some finite-sheeted covering spaces, and we exhibit a 4-sheeted covering space in which the projection of a Delaunay triangulation never has 1-cycles, and a 128-sheeted covering space in which it never has 2-cycles. While this has theoretical interest, we turn back to a more practical point of view in Section 6.

Background

For general algebraic and topologic notions, we refer the reader to textbooks such as [START_REF] Shafarevich | Basic notions of algebra[END_REF][START_REF] Armstrong | Basic Topology[END_REF].

Isometries in the Poincaré disk model of the hyperbolic plane

In the Poincaré disk model ( [START_REF] Cannon | Hyperbolic geometry[END_REF], [START_REF] Berger | Geometry[END_REF]Chapter 19]) the hyperbolic plane H 2 is represented as the open unit disk B of the Euclidean plane E 2 . The points on the boundary of B are the points at infinity, their set is denoted as H ∞ = {x ∈ E 2 : x = 1}. Hyperbolic lines, or geodesics, are represented either as arcs of Euclidean circles orthogonal to H ∞ or as diameters of B. See Figure 1(left). As the Poincaré disk is a conformal model of the hyperbolic plane (i.e. it preserves angles), it is widely used in applications [START_REF] Hurdal | Cortical cartography using the discrete conformal approach of circle packings[END_REF][START_REF] Sausset | Tuning the fragility of a glassforming liquid by curving space[END_REF][START_REF] Ron | Hyperbolic centroidal Voronoi tessellation[END_REF][START_REF] Bachelot-Motet | Wave computation on the hyperbolic double doughnut[END_REF][START_REF] Chossat | Bifurcation of hyperbolic planforms[END_REF].

The group of isometries of H 2 is denoted as Isom(H 2 ), and the group of orientation-preserving isome-tries is denoted as Isom + (H 2 ). Each g ∈ Isom + (H 2 ) is of the form g(z) = αz + β βz + α , identified with matrix α β β α , where α, β ∈ C, |α| 2 -|β| 2 = 1.

(1)

There are three types of orientation-preserving isometries of H 2 :

• Elliptic. An elliptic isometry, also called rotation, fixes one point of B, the center of rotation.

• Parabolic. A parabolic isometry, also called limit rotation, fixes no point of B and fixes a unique point of the circle at infinity H ∞ . It permutes the lines ending at the fixed point and preserves the Euclidean circles tangential to the boundary of B at the fixed point.

• Hyperbolic. A hyperbolic isometry, also called translation, fixes no point of B and fixes two points on H ∞ . The axis X g of a translation g is the hyperbolic line ending at the two fixed points. The translation g translates any point along a curve at constant distance to the axis. All such curves have the fixed points of g as infinite points. The distance by which the translation g moves all points of the axis is the same, it is called the translation length l(g) = 2 • acosh 1 2 T r(g) of g, where T r(g) is the trace of the matrix representing g. If a point p is not on the axis of g, then the distance by which g moves p is strictly greater than l(g). See Figure 1(right).

Note that two translations do not commute in general.

Fuchsian groups, hyperbolic surfaces, Dirichlet regions

For more details, we refer to the literature [START_REF] Katok | Fuchsian Groups[END_REF][START_REF] Stillwell | Geometry of surfaces[END_REF][START_REF] Thurston | Three dimensional manifolds, Kleinian groups, and hyperbolic geometry[END_REF][START_REF] Wilson | Curved Spaces[END_REF][START_REF] Thurston | The Geometry and Topology of Three-Manifolds[END_REF].

A Fuchsian group is a discrete subgroup of Isom + (H 2 ). A hyperbolic surface is a connected 2dimensional manifold, such that every point has a neighborhood isometric to a disk of H 2 . A compact hyperbolic surface is called closed. As a consequence of the Killing-Hopf theorem [START_REF] Ratcliffe | Foundations of Hyperbolic Manifolds[END_REF]Theorem 8.5.9], any closed orientable hyperbolic surface M is isometric to a quotient H 2 /Γ, where Γ is a Fuchsian group acting on H 2 without fixed points. Since Γ acts on H 2 without fixed points, it contains no elliptic element; the compactness of M implies that Γ contains no parabolic element [START_REF] Ratcliffe | Foundations of Hyperbolic Manifolds[END_REF]Theorem 6.6.6]. To summarize, a closed orientable hyperbolic surface M is of the form H 2 /Γ, where Γ is a Fuchsian group only containing hyperbolic translations (other than the identity).

The projection map π : H 2 → M = H 2 /Γ is a local isometry and a covering projection. The Dirichlet region D p (Γ) for Γ centered at p is defined as the closure Cl(

V p (Γp)) of the open cell V p (Γp) of p in the Voronoi diagram VD H (Γp) of Γp in H 2 .
It is a compact convex polygon with finitely many sides (Siegel's theorem). The area of M is finite: area(M ) = 4π(genus(M ) -1). Each Dirichlet region D p (Γ) is a fundamental domain for the action of Γ on H 2 , i.e., (i) g∈Γ g(D p (Γ)) = H 2 , and (ii)

∀g 1 , g 2 ∈ Γ, g 1 = g 2 , g 1 (Int(D p (Γ))) ∩ g 2 (Int(D p (Γ))) = ∅.

Closed geodesics, systole, length spectrum

The systole sys(M ) is the least length of a non-contractible loop on M [START_REF] Gromov | Systoles and intersystolic inequalities[END_REF]. There are upper bounds depending on the area and the genus of M [START_REF] Gromov | Filling Riemannian manifolds[END_REF][START_REF] Buser | On the period matrix of a Riemann surface of large genus (with an Appendix by J.H. Conway and N.J.A. Sloane[END_REF]. Clearly, the shortest closed curves on M are simple curves. For a given closed non-contractible loop γ on M , there is a unique closed geodesic that is the shortest closed curve that can be continuously deformed into γ [START_REF] Stillwell | Geometry of surfaces[END_REF][START_REF] Ratcliffe | Foundations of Hyperbolic Manifolds[END_REF]; as a consequence, the shortest closed curves on M are closed geodesics. The closed geodesics of a hyperbolic surface M = H 2 /Γ are in one-to-one correspondence with the conjugacy classes of hyperbolic elements of Γ; The length of the closed geodesic γ corresponding to a conjugacy class [h], h ∈ Γ, is the translation length l(h) [START_REF] Ratcliffe | Foundations of Hyperbolic Manifolds[END_REF]. The axis X h projects to γ by π; it can be thought of as "winding" X h on γ infinitely many times. Clearly, any gX h , where g ∈ Γ, projects onto the same closed geodesic γ by π. The hyperbolic line gX h is the axis of the translation ghg -1 .

Note that the conjugacy classes [h] and [h 2 ] are different; the closed geodesic corresponding to [h 2 ] traverses twice the closed geodesic corresponding to

[h]. An element h ∈ Γ is called primitive in Γ if there is no element g ∈ Γ, such that h = g m , m ∈ Z \ {0}.
The closed geodesics corresponding to primitive elements are also called primitive.

The length spectrum L(Γ) of Γ is the ordered sequence of translation lengths of elements of Γ, i.e., the ordered sequence of lengths of the closed geodesics of M . The multiplicity of l ∈ L(Γ) is the number of distinct conjugacy classes in Γ whose elements have the same length l, i.e., the number of different closed geodesics of the same length (the conjugacy classes [h] and [h -1 ] are different, the corresponding closed geodesics are counted as different geodesics). The systole of M is the first element in L(Γ).

Delaunay triangulation

A triangulation defined by a point set P in H 2 is a Delaunay triangulation iff each triangle satisfies the Delaunay property: its circumscribing disk does not intersect H ∞ and it is empty, i.e. it does not contain any point of P in its interior [START_REF] Bogdanov | Hyperbolic Delaunay complexes and Voronoi diagrams made practical[END_REF]. If a point set P ⊂ H 2 shows degeneracies, then the Delaunay triangulation DT H (P) actually is not uniquely defined. More precisely, let P be a polygon defined by more than 3 points of P, all lying on the same circle C, and such that C does not enclose any other point of P; then each partition of P into triangles is a valid Delaunay triangulation of P . Allowing a misuse of language, unless specified otherwise, the term the Delaunay triangulation actually refers to any Delaunay triangulation.

Projecting DT H (ΓP) on a closed hyperbolic surface

Let M = H 2 /Γ be a closed hyperbolic surface, with projection π : H 2 → M , and P a finite set of points in H 2 . Without loss of generality we assume that no two points of P lie on the same Γ-orbit, otherwise we can always choose a proper subset P ′ ⊂ P, so that ΓP ′ = ΓP. By definition, for any given point p, all elements of the orbit Γp = {gp, g ∈ Γ} project by π to the same point π(p) of M .

DT H (ΓP) is a partition of H 2 by vertices, edges, and triangular (2-)faces. Each g ∈ Γ is an isometry, so, a Delaunay disk circumscribing a triangle (p, q, r), p, q, r ∈ H 2 , is transformed by g into the disk circumscribing (g(p), g(q), g(r)), which is also empty. If the set ΓP shows degeneracies, the Delaunay triangulation of some polygon formed by cocircular points P = (p 0 , p 1 , . . . , p k = p 0 ), k ≥ 4 is not uniquely defined. For any g ∈ Γ, the polygon g(P ) = (g(p 0 ), g(p 1 ), . . . , g(p k ) = g(p 0 )) is also formed by cocircular points. Once a triangulation of P is chosen, the polygon g(P ) can be triangulated by the image by g of the triangulation of P . Then these two triangulations project to the same set of triangles on M by π. From now on, we will consider that DT H (ΓP) is a partition of H 2 by vertices, edges, and triangular (2-)faces.

Proposition 1. Let σ be a face of DT H (ΓP) and let σ be the face σ from which vertices are removed. The restriction of π to σ is injective.

Proof. DT H (ΓP) is a simplicial complex. Assume that the restriction of π to σ is not injective. Then there exist p, q ∈ σ, p = q, such that π(p) = π(q), or in other words q = g(p) for some non-identity g ∈ Γ. Since p ∈ σ, then g(p) ∈ g(σ). On the other hand g(p) = q ∈ σ. Thus q ∈ σ ∩ g(σ) = e q . Since DT H (ΓP) is a simplicial complex, e q is an edge. The point q lies in the interior of that edge. Similarly, p lies in the interior of the edge e p = σ ∩ g -1 (σ). See Figure 2. Since q = g(p) ∈ g(e p ) in DT H (ΓP), q ∈ e q ∩ g(e p ).

p q ep eq σ u v = g(u) w = g(v) g(σ) g(w)
Figure 2: Proof of Proposition 1.

The edges e q and g(e p ) must coincide, since they intersect in the interior point. Since e q = g(e p ) and g is a translation, e q = e p . Let u, v, w be the vertices of σ as shown on the figure. Since g(e p ) = e q , then either g(v) = v or g(v) = w. Since g acts without fixed points on H 2 , g(v) = w. So, the image g(u) is v.

The faces σ and g(σ) are adjacent through the edge e q , so, the image g(w) must be the vertex of g(σ)

opposite to e q . The vertices u, v, w are oriented clockwise. Their images g(u), g(v), g(w) are the vertices v, w, g(w) respectively, which are oriented counterclockwise. This is a contradiction with the fact that g preserves orientation.

If the restriction of π to σ is not injective, the image π(σ) is an embedded geodesic triangle for which at least two vertices coincide, i.e., at least one of the edges of π(σ) is in fact a 1-cycle.

Since π is surjective, the projection π(DT H (ΓP)) forms a partition of M by vertices, edges, and triangular faces. (2) If a 2-cycle in the graph of edges of π(DT H (ΓP)) is a union of two 1-cycles, then it is either freely homotopic to a closed geodesic with at most one transversal self-intersection or freely homotopic to a non-primitive closed geodesic traversing twice a simple closed geodesic. Otherwise, a 2-cycle is freely homotopic to a simple closed geodesic.

A 1-cycle e with a vertex v is a geodesic loop. Note that e is not necessary a closed geodesic: it may form an angle that is not π at vertex v.

Proof. Both results use the following [START_REF] Buser | Geometry and Spectra of Compact Riemann Surfaces[END_REF]: the number of self-intersections of a closed non-contractible curve on M is at least the number of self-intersections of the closed geodesic in its free homotopy class. Then (1) follows from the previous paragraph: a 1-cycle is a geodesic loop without self-intersection. Now let ς be a 2-cycle formed by two edges ς 1 and ς 2 of π(DT H (ΓP)). Consider the case when ς 1 and ς 2 are both 1-cycles. If they coincide, then clearly ς 2 is freely homotopic to a non-primitive closed geodesic traversing twice a simple closed geodesic. If the edges ς 1 and ς 2 do not coincide, then they are internally disjoint. The 2-cycle ς can be parametrized as a path with a transversal self-intersection; this path is freely homotopic to a closed geodesic with at most one transversal self-intersection. Consider the case when ς 1 is not a 1-cycle, then ς 2 is not a 1-cycle either. The edges ς 1 and ς 2 are internally disjoint, so, the 2-cycle ς does not have a self-intersection. When two geodesic paths with the same endpoints in M are homotopic, then they coincide, thus the distinct edges ς 1 , ς 2 are not homotopic. So, the 2-cycle ς 1 ∪ ς 2 is non-contractible. It does not have self-intersections, so must be freely homotopic to a simple closed geodesic.

Largest empty disks and criteria.

Let us first consider disks that do not contain any element of the Γ-orbit of a given point p. The Voronoi diagram of Γp is the partition of H 2 by the translated images of Dirichlet regions D p (Γ) by the elements of Γ. Largest empty disks are centered at images of the vertices of D p (Γ) that are furthest to p; δ p denotes their diameter.

Let us define δ M as δ M = sup{δ z | z ∈ D O (Γ)}, where O denotes the origin. δ M is completely determined by the group Γ. The following inequality holds:

Proposition 3. δ M ≤ 2δ O .
Proof. Let z be a point in H 2 . Let x be one of the vertices of D z (Γ) furthest to z. We denote the disk centered at x with diameter δ z by B x (δ z ). By construction this disk is one of the largest disks whose interiors contain no point of Γz. The point x is contained in a Dirichlet region

D O ′ (Γ), for some O ′ ∈ ΓO. The distance from O ′ to any point of D O ′ (Γ) is not greater than 1 2 δ O ′ , so, dist H (x, O ′ ) ≤ 1 2 δ O ′ . The Dirichlet region D O ′ (Γ) must contain a point of the orbit Γz, say z ′ . The distance from O ′ to z ′ is not greater than 1 2 δ O ′ . Thus the disk B O ′ (δ O ′ ) centered at O ′ and of diameter δ O ′ contains z ′ . Since dist H (x, O ′ ) ≤ 1 2 δ O ′ , the disk B O ′ (δ O ′ ) contains x. Since the disk B O ′ (δ O ′ ) contains both x and z ′ , then dist H (x, z ′ ) ≤ δ O ′ .
By construction the disk B x (δ z ) contains no point of Γz in its interior, so it does not contain z ′ . Thus the radius 1 2 δ z of B x (δ z ) is not greater than dist H (x, z ′ ). So we have

1 2 δ z ≤ dist H (x, z ′ ) ≤ δ O ′ . Since δ z ≤ 2δ O ′ holds for any z, δ M ≤ 2δ O ′ . It only remains to note that δ O ′ = δ O . Inria z x Bx(δz) O ′ z ′ B O ′ (δ O ′ ) Figure 3: Proof of Proposition 3
For a set of points P, we denote as δ P the largest diameter of disks that do not contain any point of P. Clearly, for p ∈ P:

δ P ≤ δ p ≤ δ M .
Proposition 4.

(1) If sys(M ) > δ P , then the graph of edges of π(DT H (ΓP)) has no 1-cycle.

(2) If sys(M ) > 2δ P , then the graph of edges of π(DT H (ΓP)) has no 2-cycle.

Proof.

(1) Assume that sys(M ) > δ P . For each edge e in DT H (ΓP) there exists an empty disk having the endpoints of e on its boundary. The length of the edge e is at most the diameter of the disk, so, it is not larger than δ P . If the endpoints of π(e) coincide, then π(e) is a non-contractible geodesic loop, so, its length is not smaller than sys(M ) by definition, and we get a contradiction. ( 2) Assume now that sys(M ) > 2δ P and there is a 2-cycle ς 1 ∪ ς 2 in π(DT H (ΓP)). The length of the non-contractible loop ς 1 ∪ ς 2 is the sum of the lengths of ς 1 and ς 2 , and therefore is at most 2δ P , so we get a contradiction.

Section 5 shows a covering of the Bolza surface in which the graph π(DT H (ΓP)) has a 2-cycle formed by edges of length δ P , i.e. sys(M ) = 2δ P . So, the criterion is sharp.

Covering spaces.

Let Γ be a subgroup of finite index k in Γ. The quotient space M = H 2 / Γ is a closed hyperbolic surface. We denote as π : H 2 → M the projection map. Consider the map f : M → M such that π = f • π. The surface M is a k-sheeted covering space of M with f as a covering map and a local isometry. Note that then Area( M ) = k • Area(M ), and genus( M ) = k(genus(M ) -1) + 1.

Proposition 5. If sys( M ) > δ M , then for any P, π(DT H (ΓP)) has no 1-cycle.

If sys( M ) > 2δ M , then for any P, π(DT H (ΓP)) has no 2-cycle.

Proof. Let D O be the Dirichlet region for Γ centered at O. Since Γ is discrete, the set P = D O ∩ ΓP is finite. Note that ΓP = Γ P, and as an obvious consequence π(DT H (ΓP)) = π(DT H ( Γ P)). The largest empty disk diameter δ P for the point set Γ P is equal to δ P , and thus it is at most δ M . The rest is a simple consequence of Proposition 4.

The main result of this section follows.

Proposition 6. There exists a finite-sheeted covering space M 1 of M such that sys( M 1 ) > δ M . There exists a finite-sheeted covering space M 2 of M such that sys( M 2 ) > 2δ M .

Proof. We use the fact that for any l 0 > 0, only finitely many closed geodesics on M have length ≤ l 0 [START_REF] Buser | Geometry and Spectra of Compact Riemann Surfaces[END_REF]. These closed geodesics correspond to some conjugacy classes

[g 1 ], [g 2 ], . . . , [g m(l0) ] of Γ.
The group Γ is residually finite, as all Fuchsian groups are [START_REF] Anatoliı | On the faithful representation of infinite groups by matrices[END_REF][START_REF] Scott | Subgroups of surface groups are almost geometric[END_REF], i.e., for each finite subset of Γ, there is a normal subgroup of finite index not containing the elements of that subset. So, there exists a normal subgroup Γ of finite index containing none of g 1 , g 2 , . . . , g m(l0) . Since Γ is a normal subgroup, it contains no elements of The proof is constructive: there are algorithms to compute subgroups of index up to some fixed k in Γ that do not contain a given subset of elements of Γ. Using Proposition 5, these algorithms should compute subgroups giving covering spaces in which for any P the projection of DT H (ΓP) has no 1-or no 2-cycle. The function LowIndexSubgroupsFpGroup( Gamma, H, index, excluded )1 of gap [START_REF]GAP -Groups, Algorithms, and Programming[END_REF] implements such an algorithm, due to Sims [START_REF] Sims | Computation with Finitely Presented Groups[END_REF]. However, its important memory consumption does not allow us to use it for the Bolza surface in practice, so, we will propose our own construction in Section 5.

[g 1 ], [g 2 ], . . . , [g m(l0) ]. As a consequence, there is no closed geodesic of length ≤ l 0 on the surface H 2 / Γ.

Bolza surface

Dirichlet regions

The first property of Dirichlet regions requires to quickly introduce triangle groups [START_REF] Stillwell | Geometry of surfaces[END_REF][START_REF] Ratcliffe | Foundations of Hyperbolic Manifolds[END_REF].

For integers p, q, r ≥ 2, 1 p + 1 q + 1 r < 1, the triangle group T (p, q, r) is defined as the group generated by reflections in the edges of the triangle ∆(p, q, r) whose angles are π p , π q , π r . The triangles {g(∆(p, q, r)), g ∈ T (p, q, r)} partition the plane H 2 .

The group G is a normal subgroup of index 16 without fixed points of T [START_REF] Artin | Algebra[END_REF][START_REF] Buser | Geometry and Spectra of Compact Riemann Surfaces[END_REF][START_REF] Buser | Geometry and Spectra of Compact Riemann Surfaces[END_REF]. The triangle M N O with angles π/2, π/8, π/8 is a fundamental domain for T (2, 8, 8) (Figure 5(left)). Repeated applications of reflections in its two edges incident to O form the fundamental domain of G.

The group G is a normal subgroup of index 96 without fixed points of the triangle group T (2, 3, 8).

The group T (2, 3, 8) is actually the largest subgroup of Isom(H 2 ) in which G is normal, i.e., T (2, 3, 8) = {t ∈ Isom(H 2 ) | tGt -1 = G}.
The projection map from H 2 to M induces the partition of the surface M into triangles ∆(2, 3, 8). See Figure 5(right). Proposition 7. If two points p and q of H 2 lie in the same orbit under T (2, 3, 8), then the Dirichlet regions D p (G) and D q (G) are isometric.

As will be seen in the sequel, this is not true otherwise, which is a striking difference with the Euclidean case. Proof. Let q = r(p) be the image of p under some isometry r ∈ T [START_REF] Artin | Algebra[END_REF][START_REF] Aurich | Periodic orbits on the regular hyperbolic octagon[END_REF][START_REF] Buser | Geometry and Spectra of Compact Riemann Surfaces[END_REF].

Since G is normal in T (2, 3, 8), rGr -1 = G. Then Gq = rGr -1 q = rGp, i.e., the orbit Gq is the image of the orbit Gp under r. The Voronoi diagram VD H (Gq) is the image of VD H (Gp) under r. Therefore D q (G) = Cl(V q (Gq)) in VD H (Gq) is isometric to D p (G).
Notation. For simplicity, in the sequel we denote as g the image g(O) of the origin by an element of g ∈ G. We also label each side of D p (G) by the translation of G that maps D p (G) to the Dirichlet region adjacent through this side.

Let ρ ∈ T (2, 3, 8) be the rotation around the origin O by π/4, and let us denote as

A = (a, b, c, d, a, b, c, d)
the alphabet formed by the generators of G and their inverses, considered as an ordered sequence (Figure 4). The conjugation by ρ acts as a permutation on A.

The Dirichlet regions for G have already been studied [START_REF] Naatanen | On the stability of identification patterns for Dirichlet regions[END_REF]. The rest of this section is devoted to rephrasing the description in a more intuitive way.

For x ∈ A, we denote as v x the vertex of octagon D O (G) that is equidistant to O, x, and ρ -1 xρ. The eight elements of the alphabet A are the elements of the orbit GO that are closest to O. In the same way, the eight elements xz, z ∈ A are closest to x. Any z ∈ A can be written as ρ k xρ -k , where 0 ≤ k < 8, so, the above xz, z ∈ A can be written as

xz = x • ρ k xρ -k (O).
A careful but straightforward observation of the Dirichlet regions of O, x, x • ρxρ -1 , and so on allows to show that: Observation 8. Each vertex v x is incident to eight isometric Dirichlet regions as shown on Figure 6. 4). The eight Dirichlet regions incident to v are centered at the images of O by the following translations, in clockwise order: F = (1, a, ab, abc, abcd = dcba, dcb, dc, d) (Figure 7).

Without loss of generality, we now examine more closely the vertex

v of D O (G) that is incident to D a (G) and D d (G) (Figure
The eight Dirichlet regions

ρ k D v (G), 0 ≤ k < 8, share the common vertex O. The Dirichlet region D v (G) is isometric to D O (G) since v lies in the T (2, 3, 8)-orbit of O (Figure 5); D v (G) is the regular octagon with vertices in F (O).
For a point p ∈ H 2 , let C(p) be the set

C(p) = {g | g ∈ G, D p (G) ∩ D g(p) (G) = ∅}.
For conveniency, we will use the same notation C(p) for the set of points {g(p) | g ∈ C(p)} (it will be clear from the context). Notation C can be remembered as standing for "crown". It is easy to check that for q = t(p), t ∈ T (2, 3, 8) we have

C(q) = {tgt -1 | g ∈ C(p)}.
Let p be a point in triangle SOQ shown in Figure 8. Let us consider the Dirichlet regions that are adjacent to D p (G). • If p lies in the interior of edge OQ, the sides dc and abc degenerate to a vertex of D p (G). On SO, the sides ab and dcb degenerate to a vertex, and on QS b and c degenerate to a vertex. Such a new vertex has degree 4. The Dirichlet region D p (G) is a polygon with 14 sides.

vx O y = ρ -1 xρ x x • ρxρ -1 ω3(x) = x • ρxρ -1 • ρ 2 xρ -2 y • ρ -1 yρ ω ′ 3 (x) = y • ρ -1 yρ • ρ -2 yρ 2 ω4(x) = x • ρxρ -1 • ρ 2 xρ -2 • ρ 3 xρ -3 = y • ρ -1 yρ • ρ -2 yρ 2 • ρ -3 yρ 3
• When p ∈ {O, Q, S}, D p (G) degenerates to an octagon (Proposition 7). Its adjacent octagons are centered at some points in C(p).

In particular: Let us define

C -(O) = C(O) \ {ρ k (abcd)ρ -k (O) | 0 ≤ k < 8}. C -(O)

Short closed geodesics on the Bolza surface

Let us first mention the following result:

Proposition 11. The largest empty disk diameter for M satisfies inequality 4.89 < δ M < 6.62.

Proof. Since the Voronoi cell D O (G) is a regular octagon, the radius of the largest disk that does not contain any point of GO is the distance from O to any vertex of D O (G). It can also be seen as the length l(abcd). The origin together with two adjacent vertices of D O (G) form a isosceles triangle with angles π 4 , π 8 , π 8 . By applying a hyperbolic law of cosines we get that the lateral side of the triangle, which is the radius of the disk, is acosh(3

+ 2 √ 2). As a result δ M ≥ δ O = 2 • acosh(3 + 2 √ 2) > 4.
89. The triangle with angles π 2 , π 3 , π 8 shown in Figure 10 is a fundamental domain for T (2, 3, 8), denoted as D O (T ). Let p be a point in H 2 . δ p is the diameter of a disk B p centered at a vertex of D p (G) and passing through p. Without loss of generality we can assume that B p is centered in D O (T ): otherwise, B p is centered in some Dirichlet region tD O (T ) for T [START_REF] Artin | Algebra[END_REF][START_REF] Aurich | Periodic orbits on the regular hyperbolic octagon[END_REF][START_REF] Buser | Geometry and Spectra of Compact Riemann Surfaces[END_REF], where t ∈ T (2, 3, 8), and we consider q = t -1 (p) instead of p (by Proposition 7, δ p = δ q for any q ∈ T (2, 3, 8)p). Let c denote the center of B p and d c denote the distance from c to O. Let Bp be the disk centered at c and of radius 1 2 δ O + d c . This disk contains the disk circumscribing the octagon D O (G), whose radius is 1 2 δ O , and as a consequence Bp contains at least one representative of Gp in its interior. Since both Bp and B p are centered at c, and Bp contains a point of Gp while B p does not contain any, then Bp must contain B p . The diameter of Bp is δ O + 2d c , so, δ p ≤ δ O + 2d c . To estimate d c we note that d c is not greater than the length of the side of D O (T ) opposite to the vertex with angle π 2 . Calculations using a hyperbolic law of cosines show that d c < 0.860707. So, δ p < 6.62 for any p ∈ H 2 .

We can now observe that there are closed geodesics on M that are shorter than δ M . The generators a, b, c, d, a, b, c, d of G are associated with matrices for k ∈ {0, . . . , 7} respectively, and each element of G is associated with a product of these matrices. As mentioned in Section 2, the translation length of an element of G is given by the trace of its matrix.

1 + √ 2 exp ikπ/4 √ 2 √ 2 + 1 exp -ikπ/4 √ 2 √ 2 + 1 1 + √ 2 O c dc
We just briefly summarize [START_REF] Aurich | Periodic orbits on the regular hyperbolic octagon[END_REF]. For the n th element l n of the length spectrum L(G), n > 0, the following holds: cosh(l n /2) = m + n √ 2, where for any given n, m is the unique odd natural number minimizing |mn √ 2|. The authors also compute the multiplicity of each entry in L(G). Table 1 shows the first 30 elements of the length spectrum of G. As can be seen in the table, there are closed geodesics that are shorter than 4.89, thus shorter than δ M . The second line in the table actually corresponds to closed geodesics whose length is equal to δ O .

Covering spaces of the Bolza surface

As mentioned at the end of Section 3, we give our own construction of coverings spaces of M in which, for any set P, the projection π(DT H (ΓP)) does not contain any 1-or 2-cycle.

Let us first illustrate the idea of the construction on the case of the flat (or Euclidean) torus, to give some intuition.

Back to the Euclidean case

The results in this sections were already presented [START_REF] Bogdanov | Covering spaces and Delaunay triangulations of the 2d flat torus[END_REF], but the proofs are slightly modified here, so that similarities with the hyperbolic case appear more naturally.

We consider the flat torus

T 1 = E 2 /G E 1 ,
where G E 1 is the (commutative) group generated by two translations t x and t y of the same length l along the axes of E 2 . Since G E 1 is Abelian, any x ∈ G E 1 can be uniquely written as x = t α x t β y , where α, β ∈ Z. The size λ 1 (x) of x can be defined as the sum |α| + |β|. The Dirichlet region for G E 1 centered at any point in E 2 is a square of edge length l. It is easy to see that the systole of T 1 is equal to l, and that the largest empty disk diameter δ G E 1 is √ 2l. We denote the Dirichlet region for G E 1 centered at the origin point O by D 1 . Similarly as done for the hyperbolic case, in the sequel the inverse of a translation x of G E 1 will be denoted as x. For simplicity, x both denotes an element x of G E 1 and the image x(O) of the origin O by x.

The group G E 1 is Abelian, so, all its subgroups are normal. Let us consider the subgroup

G E 2 consisting of elements of G E 1 of even size. It is easy to see that G E 2 = t x t y , t x t y . Lemma 12. G E 2 is a subgroup of index 2 in G E 1 . Proof. Let ϕ : G E 1 → Z 2 be the group homomorphism ϕ : G E 1 → Z 2 x → λ 1 (x) mod 2
The subgroup G E 2 is the kernel of ϕ, and the image of ϕ is Z 2 . According to the First Isomorphism Theorem (see, e.g., [START_REF] Artin | Algebra[END_REF]) ker ϕ is a normal subgroup of G E 1 (which is trivial here since G E 1 is Abelian), and For any k > 0, T

G E 1 / ker ϕ ∼ = ϕ(G E 1 ). Therefore, G E 1 /G E 2 ∼ = Z 2 . For any k > 0, if G E 2 k-1 is a subgroup of index 2 k-1 of G E 1 generated by two elements, i.e., G E 2 k-1 = g, h , g, h ∈ G E 1 , we construct the subgroup G E 2 k of G E 2 k-1 as follows: G E 2 k = gh,
2 k = E 2 /G E 2 k is a flat torus, with the corresponding projection map π 2 k : E 2 → T 2 k . The systole sys(T 2 k ) of T 2 k is equal to √ 2 k l.
The torus T 2 k is a covering space of T 1 with covering map

ρ k := π 1 • π -1 2 k . Since the index of G E 2 k in G E 1 is 2 k
, we get the following result: Proposition 14. For any k > 0, T 2 k is a 2 k -sheeted covering space of T 1 .

Let δ(P) denote the diameter of the largest disk in E 2 that does not contain any point of a set P in its interior. Note that for any

p ∈ E 2 , δ(G E 2 k p) = √ 2 k+1 l.
The following is the analog of Proposition 4:

Proposition 15. For a flat torus T = E 2 /G E and a set of points P, (1) if sys(T ) > δ(G E P), then the graph of edges of π(DT E (G E P)) has no 1-cycle.

(2) if sys(T ) > 2δ(G E P), then the graph of edges of π(DT E (G E P)) has no 2-cycle.

Then, from Propositions 14 and 15, we get: For T 8 , we have exactly sys(T 8 ) = 2δ T , so, the above criteria do not directly allow us to conclude. Still, we show that: Proposition 18. If P ⊂ E 2 contains at least two distinct points that do not lie on the same G E 1 -orbit, then the graph of edges of the projection π 8 (DT (G E 1 P)) onto T 8 has no 2-cycle. Proof. The maximum empty disk diameter δ(G E 1 p), for any p ∈ P, is 11). Let us show that δ(G E 1 P) is strictly less than √ 2l. Let B √ 2 be a disk of diameter √ 2l centered at some point c ∈ E 2 and containing no point of G E 1 P in its interior. The disk B √ 2 contains the Dirichlet region D 1 (c) for G E 1 centered at c. The four vertices of D 1 (c), which are all on the same G E 1 -orbit, lie on the boundary of B √ 2 . For p, q ∈ P, q ∈ G E 1 p, there is a representative of G E 1 p and a representative of G E 1 q in D 1 (c). At least one of these representatives lies in the interior of B √ 2 . So, the interior of B √ 2 is not empty. If we add more points, the diameter of the largest empty disk cannot become larger. By Proposition 16, the graph of edges of π 8 (DT (G E 1 P)) has no 2-cycle.

Proposition 16. For any P, (1) if sys(T 2 k ) > δ(G E 1 P), then π 2 k (DT E (G E 1 P)) has no 1-cycle, (2) if sys(T 2 k ) > 2δ(G E 1 P), then π 2 k (DT E (G E 1 P))
√ 2l = 1 2 δ(G E 8 ) (Figure
In Section 5.3, we construct covering spaces of the Bolza surface in a similar spirit: we look for subgroups of G whose associated Dirichlet regions keep some symmetry. Let us first mention some properties of such subgroups of G.

Subgroups of G and triangle groups

There are three triangle groups T (2, 3, 8), T (2, 4, 8), T (2, 8, 8) for which G is a normal subgroup. They all contain the rotation ρ, already defined above, and the reflection τ with respect to the axis X a of translation a. From now on, G denotes a subgroup of G. Obviously, if G is a normal subgroup of one of these three triangle groups, then G = ρ Gρ -1 , G = τ Gτ

O ( G)(O) as ρN O ( G)ρ -1 (O) (since ρ(O) = O). Thus for each g ∈ N O ( G), there is some h ∈ N O ( G) such that g(O) = ρhρ -1 (O)
, and since G = ρGρ -1 , so ρhρ -1 is in G. Then the translation Inria ρhρ -1 • g is in G, and it fixes O. Since G has no fixed point, it must be the identity, and ρhρ -1 = g. This is true for each

g in N O ( G), so N O ( G) = ρN O ( G)ρ -1 . Since N O ( G) generates G, we have G = ρ Gρ -1 .
The same proof works for τ as well: the only property of ρ that we used is that ρ(O) = O, which is also true for τ .

We get the following result as a corollary of the last observation. The rest of this section sketches the construction of small index subgroups of G.

In the sequel, we use the standard notation [g, h] for the commutator of two isometries g and h, i.e.,

[g, h] = ghg -1 h -1 .

Subgroup G 2

We first define the subgroup G 2 of G generated by words of even size on the alphabet A (Figure 13).

The proof of Lemma 12 is actually general, and it allows to show that G 2 is a subgroup of index 2 in G. Hence, G 2 is normal in G.

We can describe the generators of G 2 as follows.

Proposition 22. G 2 is generated by {x • ρxρ -1 = [x, ρ] | x ∈ A}.
2 see also the treatment of degenerate cases at the beginning of Section 3

Proof. It is sufficient to prove that any word of size 2 on A can be generated by this subset. Let us take word yz ∈ G 2 , where y, z ∈ A. As mentioned in Section 4, ρ acts as a permutation on A, i.e., we can write y = ρ k1 xρ -k1 and z = ρ k2 xρ -k2 for some k 1 , k 2 , 0 ≤ k 1 , k 2 < 8. Then yz can be written as

yz = (ρ k1 xρ -k1 • ρ k1+1 xρ -k1-1 ) • (ρ k1+1 xρ -k1-1 • ρ k1+2 xρ -k1-2 ) • . . . • (ρ k2-1 xρ -k2+1 • ρ k2 xρ -k2 ).
The subgroup G 2 can be written in a finitely presented form: The Dirichlet region of O is clearly invariant under ρ and τ . Indeed, since conjugation by ρ or τ leaves A invariant, it preserves the size of a word in G. As a consequence, by Observation 19,

G 2 = ab,
G 2 = ρG 2 ρ -1 = τ G 2 τ -1 .
We can actually show that:

Proposition 24. G 2 is the only subgroup of index 2 of G for which the Dirichlet region centered at O is invariant by ρ.

Proof. Let G ′ 2 be another subgroup of G satisfying these properties. We first remark that if there is some element of A that is in G ′ 2 , then from the symmetry property with respect to ρ, and the fact that conjugation by ρ acts as a permutation in A, then all elements of A are in G ′ 2 , and G ′ 2 = G. So, G ′ 2 does not contain any element of A.

Let us now assume that there exist x, y ∈ A such that xy ∈ G ′ 2 . Since G ′ 2 is a subgroup of index 2 of G, G can be written as a disjoint union of cosets as

G = G ′ 2 ∪ G ′ 2 xy. Also, since y ∈ G ′ 2 , G = G ′ 2 ∪ G ′ 2 y. Then we would have G 2 = G 2 x, i.e., x ∈ G ′
2 . This is a contradiction, showing that all elements of size 2 on alphabet A belong to

G ′ 2 . So, G ′ 2 = G 2 .
The generators of G 2 all have the same translation length, it is actually equal to the translation length of a, b, c, d (this can easily be computed from their matrices). As a result, the hyperbolic surface

M 2 = H 2 /G 2 still

Subgroup G 4

We now define G 4 to be generated by words of even size in the alphabet of generators of G 2 . For x ∈ A, let ω 4 (x) denote the element 6).

x • ρxρ -1 • ρ 2 xρ -2 • ρ 3 xρ -3 (See Figure
Observation 25. The fundamental region D O (G 4 ) has 24 vertices that form the circular sequence

([x, ρ], x, [x, ρ -1 ] | x ∈ A), in counterclockwise order. The set N O (G 4 ) is the circular sequence (ω 4 (x), x • ρ 2 xρ -2 , x • ρ -2 xρ 2 | x ∈ A), in counterclockwise order.
Proof. Let Π be the polygon with vertices As a corollary, using the fact that N O (G 4 ) generates G 4 , we get:

([x, ρ], x, [x, ρ -1 ] | x ∈ A), in counterclockwise order. Consider a vertex x ∈ A of Π. Recall that the elements of the set {x • ρ k xρ -k | 0 ≤ k < 8} are equidistant from x. In particular, x • ρ 2 xρ -2 and x • ρ -2 xρ 2 are equidistant from x. Note that (ρ • x • ρ -1 ) • (ρ 2 • x • ρ -2 ), (ρ 2 • x • ρ -2 ) • (ρ 3 • x • ρ -3 ), and ρxρ -1 • x are vertices of D O (G 2 ) equidistant to O. Thus [x, ρ] • (ρ • x • ρ -1 ) • (ρ 2 • x • ρ -2 ), [x, ρ] • (ρ 2 • x • ρ -2 ) • (ρ 3 • x • ρ -3 ),
Corollary 26. The set {ω 4 (x), x • ρ 2 xρ -2 , x • ρ -2 xρ 2 | x ∈ A} generates G 4 .
In other words, the generators of G 4 are ac, abcd, db, and their conjugates by ρ k , 1 < k < 8 (Figure 14).

In a similar way as done for G 2 , we can show that G 4 is the only subgroup of index 2 of G 2 for which the Dirichlet region centered at O is invariant by ρ.

Proposition 27. The systole of M 4 = H 2 /G 4 is equal to the translation length of abcd.

Proof. The systole sys(M) is equal to the translation length l(a) of the translation a. The translations of G that have the same translation length as a are conjugates hah -1 , where h ∈ T (2, 3, 8). As a ∈ G 4 and G 4 is normal in T (2, 3, 8), it does not contain hah -1 for any h ∈ T (2, 3, 8). Thus sys(H 2 /G 4 ) is greater than l(a).

The second line in the length spectrum of H 2 /G (Table 1) corresponds to geodesics of length l(abcd)

(showing that l(abcd) = 2 • acos(3 + 2 √ 2
) is only technical work, omitted here). Since abcd ∈ G 4 (it corresponds to a violet point abcd(O) in Figure 14), the result follows.

The systole of M 4 is greater than sys(M), but still smaller than δ M . However, as a consequence of Proposition 21: Proposition 28. For any P, the projection of DT H (GP) on M 4 has no 1-cycle.

Covering space for which the projection of DT H (GP) has no 2-cycle

The following proposition motivates resorting to an automatic procedure: Proposition 29. The index of a subgroup G for which sys(H 2 / G) > 2δ M is greater than 32.

Proof. Let M = H 2 / G be a covering space of M for which sys(H 2 / G) > 2δ M . We construct a lower bound on the area of M . Let D be a maximal disk in M (i.e., a disk that does not "overlap"). It can be thought as the maximal inscribed disk in some Dirichlet region for G, so, the diameter of D is equal to sys( M ). The area of D is Area(D) = 2π(cosh R -1), where R is the radius of D, so, if sys( M ) > 2δ M , then Area(D) > 2π(cosh δ M -1). Moreover, clearly, Area( M ) > Area(D). Thus, Area( M ) > 2π(cosh δ M -1).

The area Area(M) is 4π. So, Area( M )/Area(M) > (cosh δ M -1)/2 > 32.9704, using the lower bound in Proposition 11. Therefore M consists of more than 32 sheets of M.

Following Section 3, we are looking for a normal subgroup G of G that contains • no product of two elements forming a simple closed geodesic on M corresponding to a conjugacy class that appears in the portion of the length spectrum L| 2δ M = {l ∈ L(G), l ≤ 2δ M } -we call L 0 the list of these elements,

• no element of conjugacy classes that correspond to unions of two non intersecting 1-cycles on M sharing their vertex -we call L ∪ the list of these elements.

In the sequel, we first show how to compute L 0 , then L ∪ , and we finally show how to find a group not containing their elements. More precisely, we will find a group that contains one of their elements, but still gives a covering space in which the projection of DT H (GP) has no 2-cycle.

L 0 -2-cycles as simple closed geodesics.

We propose an approach for computing the number of self-intersections of a closed geodesic γ corresponding to a conjugacy class [g], g ∈ G, in G. As a consequence, this allows us to compute a portion of the length spectrum L i (G), i > 0 consisting of closed geodesics that have at most i transversal selfintersections. Due to lack of space, we present a simplified version that only allows us to check whether a closed geodesic is simple, which is actually all we need here. If g is non-primitive, then g = ĝn for some primitive ĝ ∈ G and n > 1, the number of transversal self-intersections of γ is n times the number of transversal self-intersections of γ. So, for our purpose, we can assume that g is primitive.

In [START_REF] Aurich | Periodic orbits on the regular hyperbolic octagon[END_REF] the authors compute a portion of L(G), together with one representative for each conjugacy class (Table 1). They show that the axis of each obtained representative intersects the Dirichlet region D O (G), so, X g ∩ D O (G) = ∅. The closed geodesic γ has a transversal self-intersection p iff there exist distinct g 1 , g 2 ∈ [g] whose axes X g1 , X g2 intersect in a lift p ∈ π -1 (p) [START_REF] Poincaré | Analysis situs[END_REF]. Recall that X hgh = hX g for any h ∈ G. If h sends p to D O (G), then X hg1h and X hg2h intersect in D O (G). Thus we consider only the axes that intersect in D O (G).

The method is as follows: We first compute

[g] O = {g ′ ∈ [g] | X g ′ ∩ D O (G) = ∅}. Proposition 30 shows that [g] O is finite. Then we check for pairwise intersections in D O (G) between these axes. Proposition 30. [g] O is finite.
Proof. We take a segment [p, g(p)] of the axis X g . The segment [p, g(p)] is covered by a finite subset of {D x (G) | x ∈ GO}, since G is discrete and the segment is compact. Denote the Dirichlet regions that intersect [p, g(p)] by D i , i ∈ {1, . . . , m + 1}, in the order in which they are traversed by X g from p to g(p) (See Figure 15). Since p ∈ D 1 , then g(p) ∈ gD 1 . On the other hand g(p) ∈ D m+1 . So, the Dirichlet regions gD 1 and D m+1 are adjacent or coincide. Without loss of generality we assume that gD 1 = D m+1 , since we can always choose a covering of [p, g(p)] satisfying this property. Now let us consider the segments X i = X g ∩ D i , i ∈ {1, . . . , m + 1}. We can cover the whole axis X g by images of these segments under G: any point on the axis X g is in some segment g j [p, g(p)], j ∈ Z, which is covered by the images {g j X i | i ∈ {1, . . . , m + 1}}. Moreover, we can cover the axis of any conjugate element hgh by images of the segments X i , i ∈ {1, . . . , m + 1} under G, since the axis X hgh is the image of X g under h. For each X i , there exists a unique h i ∈ G that sends X i to D O (G), and the set {h i (X i )} is finite. If the axis of an element of [g] intersects D O (G), the intersection segment is the image of some X i under an element of G which can only be h i . Thus the number of axes of the elements of [g] that intersect D O (G) is finite. Note that since D m+1 = gD 1 and X g = g(X g ), we have that X m+1 = gX 1 , i.e., h m+1 = g. We have chosen a representative g ∈ [g] O . We set X 1 = X g ∩ D O (G). When X g leaves D O (G), it enters an adjacent region, i.e. there is some h ∈ {a, b, c, d, a, b, c, d}, for which

X g ∩ h(D O (G)) = ∅. We set X 2 = X g ∩ h(D O (G)). Obviously, h 2 = h sends X 2 to D O (G). We take g 2 = h 2 gh 2 , then X g2 ∩ D O (G) = h 2 (X 2 ).
We pursue the same process inductively. At each step we compute g i+1 from g i in a similar way: h i (X i ) ⊂ D O (G), and the segments h i (X i-1 ) and h i (X i+1 ) are in the two regions adjacent to D O (G) and traversed by X gi : 16). We take h ∈ {a, b, c, d, a, b, c, d}, such that

h i (X i-1 ) ⊂ h i (D O (G)) (Figure
X gi ∩ h(D O (G)) = ∅ and h = h i . By construction, h i (X i+1 ) = X gi ∩ h(D O (G)). The inverse h sends h i (X i+1 ) to D O (G), so, h i+1 = hh i sends X i+1 to D O (G).
We take g i+1 = h i+1 gh i+1 = hg i h. We stop at step j such that g j+1 = g. Then we have found Proof. We first examine the predicate checking whether the axis of a translation g h of G intersects D O (G), i.e. whether X g h ∩ D O (G) = ∅. This predicate is considered in detail in [START_REF] Aurich | Periodic orbits on the regular hyperbolic octagon[END_REF]. We briefly sketch the main steps of the computations here. The axis X g h of g h in the Poincaré disk model can be thought of as a Euclidean circle orthogonal to the boundary of the unit disk. The matrix representing g h in the Poincaré disk model is of the form:

[g] O = {g i , 1 ≤ i ≤ j}. DO(G) h i ( X i ) h i ( X i - 1 ) h i ( X i + 1 ) Xg i h hi
α 1 + iα 2 √ 2 -1(β 1 + iβ 2 ) √ 2 -1(β 1 -iβ 2 ) α 1 -iα 2 ,
where

α 1 , α 2 , β 1 β 2 ∈ Q( √ 2
). The Euclidean circle corresponding to X g h is given in Cartesian coordinates by the following equation:

x 2 + y 2 - 2 √ 2 -1 α 2 (β 1 y -β 2 x) + 1 = 0. (2)
The axis X g h intersects D O (G) iff the Euclidean circle corresponding to X g h encloses or contains on the boundary a vertex of D O (G). The later boils down to checking whether the following equality holds.

|α 2 | ≤ (2 - √ 2)(|β 1 | + ( √ 2 -1)|β 2 |)
, where we assume that |β 1 | > |β 2 | (this always can be achieved by rotation of the circle over π 4 around the origin). Now let us come back to the test whether the axes of two g j , g k ∈ [g] O intersect. The axes X gj and X g k intersect if and only if the corresponding Euclidean circles C j and C k intersect. The equation of the Euclidean circle (Equation ( 2)) can be rewritten as follows:

x + √ 2 -1 α 2 β 2 2 + y - √ 2 -1 α 2 β 1 2 = ( √ 2 -1) (β 2 2 + β 2 1 ) α 2 2 -1. (3)
Let the equation of C j be defined by α j2 , β j1 , β j2 ∈ Q( √ 2) substituting respectively α 2 , β 1 , β 2 in Equation [START_REF] Aurich | Periodic orbits on the regular hyperbolic octagon[END_REF]. Similarly, let the equation of C k be defined by α k2 , β k1 , β k2 ∈ Q( √ 2). The squared radius r 2 j of C j is

r 2 j = ( √ 2 -1) (β 2 j2 + β 2 j1 ) α 2 j2 -1.
The squared radius r 2 k of C k is similar. The coordinates of the center of each circle are explicitly presented in Equation [START_REF] Aurich | Periodic orbits on the regular hyperbolic octagon[END_REF]. The squared distance between the centers of C j , C k , which we denote by d 2 jk , can be deduced using the coordinates of the centers of C j , C k and is equal

d 2 jk = ( √ 2 -1) β j2 α j2 - β k2 α k2 2 + β k1 α k2 - β j1 α j2 2 . Note that both r 2 j , r 2 k and d 2 jk are in Q( √ 2). The circles C j , C k intersect if and only if d 2 jk ≤ (r j + r k ) 2 .
The later is equivalent to d 2 jkr 2 jr 2 k ≤ 2r j r k , and by squaring we come to the condition:

(d 2 jk -r 2 j -r 2 k ) 2 ≤ 4r 2 j r 2 k .
The predicate whether X gj , X g k intersect boils down to the evaluation of this condition that uses only algebraic numbers of degree 2, and thus can be evaluated exactly.

The method was implemented in C++. 3 Actually, it computes more generally the number of selfintersections of closed geodesics that are present in the length spectrum L(G). Table 2 shows L| 13.24 , which we are more specifically interested in here. Our code also computes representatives of the corresponding conjugacy classes in L 0 .4 L ∪ -2-cycles as unions of two 1-cycles.

The following proposition states that considering either [g] or [tgt -1 ], for g ∈ G, t ∈ T (2, 3, 8), has the same geometric meaning. Proof. Let us take S = tP. Since G is normal in T (2, 3, 8), then DT H (GS) = DT H (GtP) = DT H (tGP) = tDT H (GP) because t is an isometry.

Let ς be a lift of ς in DT H (GP), with endpoints p ∈ P and g(p). The image t( ς) is in DT H (GS) and its endpoints t(p) and tg(p) are in the same G-orbit, thus the projection π(t( ς)) is a 2-cycle in π(DT H (GS)). It corresponds to the conjugacy class [tgt -1 ].

Let e 1 , e 2 be two edges of DT H (GP) sharing an endpoint, and such that π(e 1 ) ∪ π(e 2 ) is a 2-cycle consisting of two 1-cycles in π(DT H (GP)). Let p and g i (p) be the endpoints of e i , for i = 1, 2, where p ∈ H 2 . For g 1 , g 2 ∈ G, the translation g 1 g 2 sends the endpoint g 2 (p) of the union e 1 ∪ e 2 to its other endpoint g 1 (p). We are going to figure out for which elements g 1 , g 2 this is possible. Note that both g 1 , g 2 are in C(p), and Proposition 10 implies that both g 1 , g 2 are in C(O ′ ) for some O ′ ∈ T (2, 3, 8)O. Assume for the purpose of illustration that p = O. First consider edge e 1 = [O, g 1 ]. Since g 1 is in C(O), e 1 connects two vertices of some octagon ρ k1 D v (G), where 0 ≤ k 1 < 8. We can assume that k 1 = 0, otherwise by Proposition 32 we can consider ρ -k1 g 1 ρ k1 instead of g 1 . Then g 1 ∈ F \{1}. Then we note that d, dc, dcb, dcba are the respective conjugates of a, ab, abc, abcd by the reflection in the line passing through O and abcd (Figure 17). Thus by Proposition 32 it is sufficient to consider only Note that f maps the endpoint g 2 to the endpoint g 1 . Let λ ′ denote the polygonal chain obtained as the union ∪ k∈Z f k (γ ′ ). The vertices of λ ′ lie on two curves that have constant distance to the axis X f of the translation f .

Similarly let γ ′′ denote the union of segments [h, hg 1 ] and [h, hg 2 ]. The endpoint hg 2 is mapped to the endpoint hg 1 by hf h. Let λ ′′ denote the polygonal chain obtained as the union ∪ k∈Z (hf h) k (γ ′′ ). The chain λ ′′ is the image of λ ′ under h. The vertices of λ ′′ lie on two curves that have constant distance to the axis X hf h of hf h. These curves are images of the curves corresponding to λ ′ under h.

The chains λ ′ and λ ′′ intersect transversally at

[O, g 1 ] ∩ [h, hg 2 ]. This point of λ ′ ∩ λ ′′ is unique,
otherwise their corresponding curves intersect more than in one point. Thus X f ∩ X hf h = ∅. In a similar way, we can construct polygonal chains for [p, g 1 (p)] and [h(p), hg 2 (p)]. These polygonal chains must intersect since they have common points at infinity with X f and X hf h respectively. Their intersection point corresponds to the intersection point of the projections of [p, g 1 (p)] and [h(p), hg 2 (p)] on M. where g i , g j , g k ∈ F . Without loss of generality assume that e ′ 2 is the image of e 2 under the (unique) translation of G that sends O to g j . Then clearly the endpoints of e 2 are O and g j g k (Figure 18(right)). Then e 1 ∪ e 2 projects to a 2-cycle since g i and g j g k are on the same G-orbit.

The function generate_words_union_1_cycles implements that enumeration of elements g i g k g j for which e 1 and e ′ 2 do not intersect. 5 It allows to compute the list L ∪ of representatives of conjugacy classes corresponding to unions of 1-cycles. 6Subgroups not containing elements of L = L 0 ∪ L ∪ . The idea of the method is simplistic: we consider one by one the groups that are available at [START_REF] Conder | Regular orientable maps of genus 2 to 301[END_REF], and check using gap whether they are subgroups of G and whether they contain an element of L.

More precisely, the subgroups listed at [START_REF] Conder | Regular orientable maps of genus 2 to 301[END_REF] are quotient groups T (2, m, n)/G where G is a normal subgroup of T (2, m, n) acting on H 2 witout fixed points. They are ordered by genus of the corresponding surface H 2 /G, up to genus 301 [START_REF] Conder | Determination of all regular maps of small genus[END_REF]. G is one of them. Considering normal subgroups in T (2, m, n) allows us to expect to get covering spaces of M that preserve symmetries of the octagon. The method was implemented using gap. 7 The program actually does not find any suitable subgroup of G.

However, while carefully examining its output, we notice that there is a unique subgroup G 128 of G of index 128 that is normal in T (2, 3, 8), and it does not contain any element of L, except (abcd) 2 .

The group G 128 is defined in the following way. We use the finitely presented form of T (2, 3, 8) [START_REF] Stillwell | Geometry of surfaces[END_REF][START_REF] Ratcliffe | Foundations of Hyperbolic Manifolds[END_REF]:

T (2, 3, 8) =< R, S, T | R 3 , S 8 , T 2 , (R • S) 2 , (R • T ) 2 , (S • T ) 2 > .
With the same notation, V denotes the following set of elements [START_REF] Conder | Regular orientable maps of genus 2 to 301[END_REF]

V = {R 3 , S 8 , T 2 , (R • S) 2 , (R • T ) 2 , (S • T ) 2 , S • R • S -3 • R -1 • S • R -1 • S -4 • R • S -1 • R • S 4 • R • S -1 • R • S -3 • R • S 2 , S -1 • R • S -2 • R -1 • S 2 • R • S -2 • R • S -2 • R -1 • S • R -1 • S -2 • R • S 2 • R -1 • S 2 • R -1 • S -2}.
Let now N be the group N =< R, S, T | V > .

N is finite [START_REF] Conder | Determination of all regular maps of small genus[END_REF]. Then G 128 = T (2, 3, 8)/N . Provided that this result, obtained by running software is considered as "proved", we show the following: Proposition 34. For any set of points P, the graph of edges of the projection of DT H (GP) to H 2 /G 128 has no 2-cycle.

Proof. A potential 2-cycle corresponding to (abcd) 2 would be formed by the projections of two edges: the edge incident to p and abcd(p) and the edge incident to abcd(p) and (abcd) 2 (p), for some p ∈ P .

If p is the origin O, each of these edges is the diameter of a largest disk that contains no point of GO in its interior. The same reasoning as in the Euclidean case (see Proposition 18) shows that, if the point set P contains the origin O and at least one more point, then the projection of DT H (GP) to H 2 /G 128 has no 2-cycle. Actually, even if P = {O}, it is always possible to triangulate the fundamental domain D v (G) in such a way that no pair of opposite vertices form an edge; the triangulation of H 2 /G 128 induced by this triangulation has no 2-cycle.

If p = O, Observation 9 shows that the edge with vertices p and abcd(p) cannot be an edge of DT H (GP). So, if P does not contain O, then the projection of DT H (GP) to H 2 /G 128 again does not contain a 2-cycle.

Triangulating the Bolza surface in practice

As done for Euclidean manifolds [START_REF] Caroli | Computing 3D periodic triangulations[END_REF][START_REF] Caroli | Triangulating Point Sets in Orbit Spaces[END_REF][START_REF] Caroli | Delaunay triangulations of point sets in closed Euclidean d-manifolds[END_REF], the incremental algorithm for computing a Delaunay triangulation can be used in a covering space of M in which the graph of its edges is a simplicial comlex, i.e., it has no 2-cycles. It would obviously be of low practical interest to compute in H 2 /G 128 , since we would have to consider 128 images of P. Instead, we can initialize the triangulation with a well chosen set Q of 14 points for which sys(M) > 2δ Q . By Proposition 4(2), this ensures that the triangulation of Q ∪ P on M itself will have no 2-cycle, for any P, so resorting to a covering space is not necessary.

The rest of this section presents the set Q. vertex with angle π/2. Flipping the edge shared by these two triangles gives us two new triangles ∆ 1 and ∆ 2 (See Figure 19(right)). The circumscribing disks of ∆ 1 and ∆ 2 contain no point of T (2, 4, 8)Q T in their interiors. Note that ∆ 1 ∪ ∆ 2 is a fundamental domain for the subgroup T + (2, 4, 8) of orientationpreserving isometries of T [START_REF] Artin | Algebra[END_REF][START_REF] Bachelot-Motet | Wave computation on the hyperbolic double doughnut[END_REF][START_REF] Buser | Geometry and Spectra of Compact Riemann Surfaces[END_REF], and therefore the hyperbolic plane H 2 is covered by the images of ∆ 1 ∪ ∆ 2 under T + [START_REF] Artin | Algebra[END_REF][START_REF] Bachelot-Motet | Wave computation on the hyperbolic double doughnut[END_REF][START_REF] Buser | Geometry and Spectra of Compact Riemann Surfaces[END_REF]. This partition of H 2 gives us the Delaunay triangulation of H 2 defined by the set T (2, 4, 8)Q T . In fact, we get the triangulation DT H (GQ), since T (2, 4, 8)Q T = GQ.

Let δ Q denote the largest empty disk diameter for the set GQ, and d 1 , d 2 denote the diameters of the disks circumscribing ∆ 1 , ∆ 2 respectively. There is a triangle in DT H (GQ) whose circumscribing disk is of diameter δ Q . Since any triangle in DT H (GQ) is the image of ∆ 1 or ∆ 2 under some isometry of T + (2, 4, 8), either d 1 or d 2 is equal to δ Q . Both diameters d 1 and d 2 are less than 1 2 sys(M), which actually is the radius of the inscribed circle in D O (G). So the largest empty disk diameter δ Q is less than 1 2 sys(M). Inserting new points in the triangulation cannot increase the size of empty disks, i.e., δ P∪Q ≤ δ Q for any set P, so by Proposition 4, π(DT H (GP ∪ Q)) has no 2-cycle.

Figure 1 :

 1 Figure 1: The Poincaré disk model. (left) lines and circles. (right) translation g of axis X g .

Proposition 2 .

 2 (1) Any 1-cycle in the graph of edges of π(DT H (ΓP)) is freely homotopic to a simple closed geodesic.

Figure 4 :

 4 Figure 4: Translations a, b, c, d and their inverses.

  Consider a regular octagon in H 2 centered at the origin O and with angle π/4 at each vertex, and the four hyperbolic translations a, b, c, d that pairwise identify opposite sides of the octagon, as in Figure4. They generate a Fuchsian group G for which the octagon is a fundamental domain. The Voronoi diagram VD H (GO) can be seen on Figure9(right). The inverse of an element g ∈ G is denoted as g. The group G can be written in the form of a finitely presented group G = a, b, c, d | abcdabcd , i.e., the quotient of the group a, b, c, d generated by a, b, c, d, by the normal closure (i.e., the smallest normal subgroup) in a, b, c, d of the element abcdabcd. The Bolza surface is M = H 2 /G. It is a closed orientable hyperbolic surface homeomorphic to a torus having two handles, i.e., a double torus.

Figure 5 :

 5 Figure 5: Partition of a fundamental domain for G into fundamental domains for the triangle groups T (2, 8, 8) (left) and T (2, 3, 8) (right), where small circles show the elements of the orbit T (2, 3, 8)O in the octagon.

Figure 6 :Figure 7 :

 67 Figure 6: Centers of Dirichlet regions incident to a vertex v x of D O (G) O

Figure 8 :

 8 Figure 8: Description of Dirichlet regions.

Figure 9

 9 Figure9was computed using the software described in[START_REF] Bogdanov | Hyperbolic Delaunay complexes and Voronoi diagrams made practical[END_REF].

Figure 9 :

 9 Figure 9: Dirichlet region for G centered at a point getting closer to O.

Proposition 10 .

 10 consists of elements of C(O) except abcd and its rotated images around O by powers of ρ, i.e., all elements of C(O) other than those corresponding to octagons that meet octagon D O (G) opposite a vertex. For O ′ = rO, r ∈ T (2, 3, 8), we define C -(O ′ ) = {rgr -1 | g ∈ C -(O)}. From the description above: For any p ∈ H 2 there exists O ′ ∈ T (2, 3, 8)O, such that C(p) ⊂ C(O ′ ). Moreover, if p ∈ T (2, 3, 8)O, then C(p) ⊂ C -(O ′ ).

Figure 10 :

 10 Figure 10: Proof of Proposition 11

- 1 .

 1 Let us introduce N O ( G) defined as the subset of elements g ∈ G such that the Dirichlet regions D O ( G) and D g ( G) are adjacent (i.e., they share a common edge). For G = G, N O (G) = A. More generally, N O ( G) generates G [23]. Observation 19. The Dirichlet region D O ( G) for G centered at O is invariant by ρ (resp. τ ) if and only if G = ρ Gρ -1 (resp. G = τ Gτ -1 ). Proof. If G = ρ Gρ -1 , then the set GO maps onto itself by ρ: ρ GO = ρ Gρ -1 O = GO. Thus the Dirichlet region D O ( G), as a Voronoi cell in VD H ( GO), remains invariant by ρ. Similarly it remains invariant by τ . Now assume that D O ( G) is invariant by ρ. Then in particular N O ( G)(O) = ρN O ( G)(O), and we can rewrite N

Proposition 20 . 5 . 3 Figure 12 :

 205312 Figure 12: Proof of Proposition 21: avoiding edge with vertices O and abdc.1-cycle. Remark that for any g ∈ C -(O), we can choose DT H (GO) so that the geodesic segment with endpoints O and g is an edge of DT H (GO). The result follows.

Observation 23 .Figure 13 :

 2313 Figure 13: Dirichlet region for G 2 .

Figure 14 :

 14 Figure 14: Dirichlet region for G 4 .

  and [x, ρ]ρxρ -1 • x are equidistant from the vertex [x, ρ] of Π, i.e., x • ρ 2 xρ -2 , ω 4 (x), and O are equidistant from the vertex [x, ρ] of Π. The proof is similar for the vertex [x, ρ -1 ].

Figure 15 :

 15 Figure 15: The regions D 1 , D 2 , . . . , D m+1 cover the segment [p, g(p)].

Figure 16 :

 16 Figure 16: Constructing axes.

Proposition 32 .

 32 Let ς be a 2-cycle in π(DT H (GP)) and [g], g ∈ G, be the corresponding conjugacy class in G. For t ∈ T (2, 3, 8), there exists a point set S ⊂ H 2 , such that π(DT H (GS)) has a 2-cycle corresponding to the conjugacy class [tgt-1 ].

Figure 17 :

 17 Figure 17: Edges from p = O.

g 1 ∈

 1 {a, ab, abc, abcd}. Now let us consider edge e 2 = [O, g 2 ]. The endpoints of e 2 are vertices of ρ k2 D v (G) for some 0 ≤ k 2 < 8 and h be the translation of G that sends e 2 in the Dirichlet region D v (G). By definition of DT H (GP) the image h(e 2 ) = e ′ 2 is an edge of DT H (GP). The edges e ′ 2 and e 1 do not intersect. Proposition 33. Let g 1 , g 2 be in C(O). Let h be the translation in G such that the geodesic segments [O, g 1 ] and [h, hg 2 ] intersect transversally. Then for any p ∈ H 2 the projections of the geodesic segments [p, g 1 (p)] and [h(p), hg 2 (p)] on M intersect transversally. Proof. Let f denote the translation g 1 g 2 . Let γ ′ denote the union of segments [O, g 2 ] and [O, g 1 ].

Proposition 33 allowsFigure 18 :

 3318 Figure 18: Enumerating edges. e ′ 2 do not intersect (Figure 18(left)). Let us consider e 1 = [O, g i ] and e ′ 2 = [g j , g k ],where g i , g j , g k ∈ F . Without loss of generality assume that e ′ 2 is the image of e 2 under the (unique) translation of G that sends O to g j . Then clearly the endpoints of e 2 are O and g j g k (Figure18(right)). Then e 1 ∪ e 2 projects to a 2-cycle since g i and g j g k are on the same G-orbit.The function generate_words_union_1_cycles implements that enumeration of elements g i g k g j for which e 1 and e ′ 2 do not intersect.5 It allows to compute the list L ∪ of representatives of conjugacy classes corresponding to unions of 1-cycles.6

4 ∆Figure 19 :

 419 Figure 19: (left) Partition of D O (G) into fundamental domains for T (2, 4, 8). (right) Triangles ∆ 1 , ∆ 2 obtained by edge-flipping.

Figure 20 :

 20 Figure 20: The set Q.

Figure 20 shows

 20 Figure20shows unique representatives in H 2 of the triangles of π(DT H (GQ)). The point set Q contains 9 points in the interior of D O (origin and red points), 4 points on the boundary of D O (green points) and 1 point at its vertex (blue point). In total Q consists of 14 points.

1 11 5

 111 Dirichlet regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 Short closed geodesics on the Bolza surface . . . . . . . . . . . . . . . . . . . . . . . . . . Covering spaces of the Bolza surface 12 5.1 Back to the Euclidean case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Subgroups of G and triangle groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.3 Covering space for which the projection of DT H (GP) has no 1-cycle . . . . . . . . . . . . . 15 5.4 Covering space for which the projection of DT H (GP) has no 2-cycle . . . . . . . . . . . . . 17

6

  Triangulating the Bolza surface in practice 23

Table 1 :

 1 A portion of the length spectrum L(G)[START_REF] Aurich | Periodic orbits on the regular hyperbolic octagon[END_REF].

	length	multiplicity m	n length	multiplicity m	n
	3.05714183896 24	1	1 9.02707171973 12	23 16
	4.89690489536 24	3	2 9.17142551689 72	25 17
	5.82807077544 48	5	3 9.22829508964 96	25 18
	6.11428367792 24	5	4 9.3592716579	192	27 19
	6.67200576991 96	7	5 9.48219414927 48	29 20
	7.10737587411 48	9	6 9.53097705714 192	29 21
	7.26316347512 48	9	7 9.64406564862 96	31 22
	7.59569183041 8	11	8 9.75109975831 336	33 23
	7.88069228867 96	13	9 9.79380979071 24	33 24
	8.1300755289	48	15 10 9.89331068777 192	35 25
	8.22490362326 192	15 11 9.98809463666 192	37 26
	8.43684964052 48	17 12 10.0785887303 192	39 27
	8.62846365652 96	19 13 10.1149054114 96	39 28
	8.70275055643 48	19 14 10.1999558888 384	41 29
	8.87147981074 288	21 15 10.2815363765 96	43 30

  gh . The proof of the following lemma is similar to that of Lemma 12.Lemma 13. For anyk > 0, G E 2 k is a subgroup of index 2 k in G E 1 .By construction, for any k ≥ 0, the generators of G E 2 k are two translations whose vectors are orthogonal and of equal length √ 2 k l. The orbits of G E 2 k are isomorphic to Z 2 and the Dirichlet region D 2 k for G E

	centered at O is a square of edge length	√	2	2 k
		tx	2 t 2 y			t 2 y	t 2 x t 2 y
				txty	ty D8	txty
								D4
	tx	2				tx		D1 O	D2	tx	t 2 x
							√	2l
				txty	ty	txty
		2	√	2l
		tx	2 ty	2		ty	2	t 2 x ty	2

k l (See Figure

11

). l Figure 11: Euclidean case.

Table 2 :

 2 L| 13.24

	# length	mult.	# length	mult.
	0	3.0571	24	15 10.7004 96
	1	4.8969	24	16 10.8758 48
	2	5.8280	48	17 10.9343 96
	3	6.1143	24	18 11.5566 48
	4	7.2632	48	19 11.6561 48
	5	7.5957	8	20 11.7509 144
	6	7.8807	96	21 11.928	96
	7	8.7028	48	22 12.1543 96
	8	8.8715	96	23 12.2286 24
	9	9.0271	12	24 12.2708 48
	10 9.1714	24	25 12.5888 48
	11 9.7511	48	26 12.7306 48
	12 9.7938	24	27 12.7771 192
	13 10.2	96	28 13.0846 96
	14 10.2815 96	29 13.1428 48
				30 13.2147 96

RR n°8434

Γ is a finitely presented group Inria

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Acknowledgements

The authors warmly thank Gert Vegter for early discussions on the construction of covering spaces of the Bolza surface.