SSPINES OF MINIMAL LENGTH
BRUNO MARTELLI, MATTEO NOVAGA, ALESSANDRA PLUDA, AND STEFANO RIOLO
Abstract.
In this paper we raise the question whether every closed Riemannian manifoldhas a spine of minimal area, and we answer it affirmatively in the surface case. On constantcurvature surfaces we introduce the spine systole , a continuous real function on moduli spacethat measures the minimal length of a spine in each surface. We show that the spine systoleis a proper function and has its global minima precisely on the extremal surfaces (thosecontaining the biggest possible discs).We also study minimal spines , which are critical points for the length functional. Wecompletely classify minimal spines on flat tori, proving that the number of them is a properfunction on moduli space. We also show that the number of minimal spines of uniformlybounded length is finite on hyperbolic surfaces. Introduction
In this paper we study the spines of closed Riemannian manifolds that have minimal areawith respect to the codimension-one Hausdorff measure. We describe the general setting, weprove that such spines exist on surfaces, and then we study the constant curvature case indetail.In differential topology, a spine of a closed smooth n -manifold M is a smooth finitesimplicial complex P ⊂ M such that M minus a small open ball collapses onto P . Inparticular M \ P is an open ball.In all cases, we suppose that dim P < dim M , and this is the only restriction we make ondimensions: for instance, any point is a spine of the sphere S n for all n ≥
1. The polyhedron P may also have strata of varying dimensions: for instance a natural spine of S × S is theunion of the sphere S × q and the circle p × S .A compact manifold M has many different spines: as an example, one may give M aRiemannian structure and construct P as the cut locus of a point [8]. The manifold M has typically infinitely many pairwise non-homeomorphic spines, with portions of varyingdimension.The notion of spine is widely employed in topology: for instance, it may be used todefine a complexity on manifolds [19, 20, 17], to study group actions [6] and properties ofRiemannian manifolds [3]. In dimensions 2 and 3 spines (with generic singularities) arisenaturally and frequently as the dual of 1-vertex triangulations. Topologists usually consider Mathematics Subject Classification.
Key words and phrases. spine; length functional; constant curvature surfaces. a r X i v : . [ m a t h . G T ] N ov pines only up to isotopy, and relate different spines (or triangulations) via some moves like“flips” on surfaces (see for instance [5, 11, 15, 23]) and Matveev-Piergallini moves [18, 21]on 3-manifolds.However, it seems that spines have not been much studied from a geometric measuretheory point of view, and this is the main purpose of this paper.If M is a closed Riemannian manifold of dimension n ≥
2, every spine P ⊂ M has awell-defined finite ( n − H n − ( P ) called area . For instance,a point in S n has zero area; the spine of S × S described above has the same area 4 π of S . We are interested here in the following problem. Question 1.1.
Does every closed Riemannian manifold M of dimension n ≥ have a spineof minimal area? As an example, the answer is yes for all spheres S n equipped with any Riemannianmetric, since S n has a spine of zero area (a point). The reader may notice here that we needto allow smaller-dimensional spines like points to get a positive answer to Question 1.1 onspheres (and also on other manifolds, see Section 2.1).This paper is essentially devoted to surfaces: their spines have dimension ≤ length to indicate their area. Theorem 1.2.
Every closed Riemannian surface S has a spine Γ of minimal length. Thespine Γ is: (1) a point if S is diffeomorphic to a sphere, (2) a closed geodesic if S is diffeomorphic to a projective plane, (3) finitely many geodesic arcs meeting at trivalent points with angle π otherwise. The theorem says everything about spheres, so we restrict our attention to the othersurfaces S . In analogy with minimal surfaces, we say that a spine Γ ⊂ S is minimal if it isas prescribed in points (2) or (3) of the theorem: either a closed geodesic, or finitely manygeodesic arcs meeting with angle π at trivalent points. These conditions are similar to thatof having zero mean curvature for hypersurfaces. A spine of minimal length is minimal, butthe converse may not hold: a minimal spine is a critical point of the length functional, whilea spine of minimal length is a global minimum.It is now natural to study these geometric objects on closed surfaces of constant curva-ture. Recall that all constant curvature metrics on a closed oriented surface S , consideredup to orientation-preserving isometries and global rescalings, form the moduli space M (S)of S . The moduli space is not compact: on the torus T , the space M (T) is the (2 , , ∞ )orbifold and is homeomorphic to a plane.We completely classify all minimal spines in flat tori. √ √ √ √ √ √
111 23 345 56
Figure 1.
The number of minimal spines on each oriented flat torus in modulispace M ( T ), considered up to orientation-preserving isometries of T . Themoduli space M ( T ) is drawn via the usual fundamental domain in H . Ateach point z ∈ M ( T ), the number c ( z ) is the smallest among all numberswritten on the adjacent strata (the function c is lower semi-continuous). Theorem 1.3.
Every oriented flat torus T contains finitely many minimal spines up toorientation-preserving isometries. The number of such minimal spines is the proper function c : M (T) → N shown in Fig. 1. We see in particular that the square and the hexagonal torus at z = i and z = e π i are theonly flat tori that contains a unique minimal spine up to orientation-preserving isometries.We also discover that every flat torus T contains finitely many minimal spines up to isometry,but this number increases (maybe unexpectedly?) and tends to ∞ as the flat metric tendsto infinity in moduli space.By looking only at spines of minimal length we find the following. Proposition 1.4.
Every oriented flat torus T has a unique spine of minimal length up toisometries of T . It also has a unique one up to orientation-preserving isometries, unless itis a rectangular non square torus and in this case it has two. Let S : M ( T ) → R be a function that assigns to a (unit-area) flat torus z ∈ M ( T )the minimal length S( z ) of a spine in T . The function S may be called the spine systole because it is analogous to the (geodesic) systole that measures the length of the shortest losed geodesic. We will prove that S has the following expression:S( z ) = (cid:115) | z | − |(cid:60) ( z ) | + √ (cid:61) ( z ) (cid:61) ( z ) . and has a unique global minimum at the hexagonal torus.Now that the picture on the flat tori is perfectly understood, it is time to turn to highergenus hyperbolic surfaces and see if minimal spines have the same qualitative behavior there.Let S g be a closed orientable surface of genus g ≥ M ( S g ) be the moduli space of S g . Every hyperbolic surface in M ( S g ) has a spine of minimum length by Theorem 1.2 andits length defines a spine systole S : M ( S g ) → R .An extremal surface is a hyperbolic surface that contains a disc of the maximum possibleradius in genus g . Such surfaces were defined and studied by Bavard [7] and various otherauthors, see [13] and the references therein. We can prove the following. Theorem 1.5.
The function
S : M ( S g ) → R is continuous and proper. Its global minimaare precisely the extremal surfaces. A full classification of all minimal spines on all hyperbolic surfaces would of course bedesirable; for the moment, we content ourselves with the following.
Proposition 1.6.
Every closed hyperbolic surface S has finitely many minimal spines ofbounded length. In particular, we do not know if a closed hyperbolic surface has finitely many minimalspines overall (counting spines only up to isometry does not modify the problem, since theisometry group of S is finite).In higher dimensions, Question 1.1 has already been addressed by Choe [9] who hasprovided a positive answer for all closed irreducible 3-manifolds. The techniques used inthat paper are much more elaborate than the ones we use here.2. Preliminary definitions
Spines.
We recall some well-known notions that apply both in the piecewise-linear andin the smooth category of manifolds.A smooth finite simplicial complex (or a finite polyhedron ) in a smooth n -manifold M with (possibly empty) boundary is a subset P ⊂ int( M ) homeomorphic to a simplicialtriangulation, such that every simplex is diffeomorphic through some chart to a standardsimplex in R n .The subset P has a well-defined regular neighborhood , unique up to isotopy: this is apiecewise-linear notion that also applies in the smooth category [14]. A regular neighborhood N of P is a compact smooth n -dimensional sub-manifold N ⊂ M containing P in its interiorwhich collapses simplicially onto P for some smooth triangulation of M ; as a consequence \ P is an open collar of ∂N . If the manifold M itself is a regular neighborhood of P wesay that P is a spine of M .This definition of course may apply only when ∂M (cid:54) = ∅ , so to define a spine P of a closedmanifold M we priorly remove a small open ball from M . In particular we get that M \ P is an open ball in this case.In this paper we only consider spines P of closed manifolds M . We also consider byassumption only spines P with dim P < dim M , so that if M is endowed with a Riemannianmetric P has a well-defined and finite ( n − H n − ( P ) called area .The area can be zero in some cases because we allow P to have dimension strictly smallerthan n −
1. For instance, any point is a spine of S n and any complex hyperplane is a spineof complex projective space CP n , because their complements are open balls: in both casesthe spine has zero area because it has codimension bigger than 1.A spine may have strata of mixed dimensions: for instance a natural spine for S m × S n is the transverse union of two spheres S m × q and p × S n (the reader may verify that thecomplement is an open ball).In the following we consider only the case in which the dimension of the manifold M istwo. Remark 2.1.
The dimension of a spine Γ of a closed surface S is less or equal one and itshomotopy type is completely determined by that of S . Indeed, • Γ is homotopically equivalent to a point if and only if S is diffeomorphic to a sphere.Indeed, a regular neighbourhood of a point must be a disc. • Γ is homotopically equivalent to a circle if and only if S is diffeomorphic to a projectiveplane. Indeed, a regular neighbourhood of a circle could be nothing but a M¨obiusstrip, because an annulus has too many boundary components. • Finally, assume that S is nor diffeomorphic to a sphere, neither to a projective plane.A spine of S is an embedded graph Γ ⊂ S . Denoting with e the number of edges,with v the number of vertices of Γ and by χ the Euler characteristic, a necessarycondition for Γ to be a spine of S is χ ( S ) − χ ( S − B ) = χ (Γ) = v − e . (2.1)2.2. Networks in Riemannian surfaces.
Let S be a Riemannian surface. An embeddedgraph Γ ⊂ S is a network : a union of a finite number of supports of simple smooth curves γ i : [0 , → S , intersecting only at their endpoints. A point in which two or more curvesconcur is called multipoint . Each curve γ i of the network has length H (Im ( γ i )) and thelength of Γ is the sum of the lengths of all the curves, that is, L (Γ) = H (cid:32) n (cid:91) i =1 (cid:0) Im (cid:0) γ i (cid:1)(cid:1)(cid:33) = n (cid:88) i =1 H (cid:0) Im (cid:0) γ i (cid:1)(cid:1) . (2.2) n the following, we will search the minima of this functional, restricted to the set ofspines of a closed Riemannian surface S , endowed with the Hausdorff topology.2.3. First variation of the length functional.
Let Γ be a network in a closed Riemanniansurface S . Let Φ t with t ∈ [0 , T ] be a smooth family of diffeomorphisms of S , with Φ ( x ) = x for all x ∈ Γ and let Γ t = Φ t (Γ). Consider the vector field X on S defined as X ( x ) = ddt Φ t ( x ) (cid:12)(cid:12)(cid:12) t =0 . The first variation formula for the length functional L for a network Γ is (see for in-stance [16]): ddt L (Γ t ) (cid:12)(cid:12)(cid:12) t =0 = (cid:90) Γ div τ X d H , (2.3)where with div τ we denote the tangential divergence.Let x , . . . , x m be the multipoints of Γ. At every multipoint x j , l j curves concur. For k = 1 , . . . l j denote with τ kj the unit tangent vector to each of that curves at x j . In particular, ddt L (Γ t ) (cid:12)(cid:12)(cid:12) t =0 = − (cid:90) Γ H · X d H + m (cid:88) j =1 l j (cid:88) k =1 τ kj · X , (2.4)where H is the curvature of the curve. A network Γ is stationary if there holds (cid:90) Γ div τ X d H = 0 for every vector field X on S , (2.5)Thanks to (2.4), condition (2.5) is equivalent to require that each curve of the network Γis a geodesic arc and the sum of the unit tangent vectors of concurring curves at a commonendpoint is equal to zero.We will actually consider only spines, and reserve the word minimal for a more restrictiveconfiguration where only three edges concur at each multipoint, see Definition 3.2. Thereason for that is that spines of minimal length will always be of this type.2.4.
Rectifiable sets and stationary varifolds.
Let S be a Riemannian surface. Werecall that a set Γ ⊂ S is countably -rectifiable if it can be covered by countably manyimages of Lipschitz maps from R to S , except a subset H − negligible.Let Γ be a countably 1-rectifiable, H -measurable subset of S and let θ be a positivelocally H -integrable function on Γ. Following [1, 2] we define the rectifiable -varifold (Γ , θ ) to be the equivalence class of all pairs ( (cid:101) Γ , (cid:101) θ ), where (cid:101) Γ is countably 1-rectifiable with H (cid:16) (Γ \ (cid:101) Γ) ∪ ( (cid:101) Γ \ Γ) (cid:17) = 0 and where (cid:101) θ = θ H -a.e. on Γ ∩ (cid:101) Γ. The function θ is called multiplicity of the varifold.A varifold (Γ , θ ) is stationary if there holds (cid:90) Γ θ div τ X d H = 0 , (2.6) or any vector field X on S . Notice that, if θ is constant and Γ is a network, condition (2.6)is consistent with (2.5). Recalling (2.3) this means that the network Γ is a critical point ofthe length functional. 3. Riemannian surfaces
In this section we prove an existence result for the spines of minimal length of a closedRiemannian surface (Theorem 1.2).
Proposition 3.1.
Consider a closed Riemannian surface S not diffeomorphic to a sphere.Then there is a constant K > such that L (Γ) ≥ K for every spine Γ of S .Proof. Let r > S . Every homotopically non-trivial closed curvein S has length at least 2 r . Every spine Γ contains at least one homotopically non-trivialembedded closed curve and hence has length at least K = 2 r .To verify the last fact, recall that S \ Γ is an open 2-disc and hence the inclusion map i : Γ (cid:44) → S induces a surjection i ∗ : π (Γ) → π ( S ). Since π ( S ) (cid:54) = { e } the spine Γ containssome homotopically non-trivial loop, and this easily implies that it also contains an embeddedone. (cid:3) Proof of Theorem 1.2.
Thanks to the topological observations made in Subsection 2.1,the case of the sphere is trivial and that of the projective plane is well known. Therefore,we will henceforth suppose that S is neither diffeomorphic to a sphere nor to a projectiveplane. We want to prove that (cid:96) = inf (cid:8) L (Γ) | Γ is a spine of S (cid:9) . (3.1)is a minimum. We subdivide the proof into four steps. Step 1: Γ n minimizing sequence of spines converges to a closed connected and rectifiableset Γ ∞ . We have just defined (cid:96) = inf Γ ⊂ S H (Γ)where Γ varies among all spines of S . Let Γ n be a minimizing sequence of spines, that is asequence such that H (Γ n ) → (cid:96) . In particular, the sets Γ n are closed, connected, rectifiable,and S \ Γ n is homeomorphic to an open disc for every n .Thanks to Blaschke Theorem [4, Theorem 4.4.15], up to passing to a subsequence, thesequence converges Γ n → Γ ∞ to a compact set Γ ∞ in the Hausdorff distance, and by GolabTheorem [4, Theorem 4.4.17] we get H (Γ ∞ ) ≤ lim inf n H (Γ n ) = (cid:96) (3.2)and Γ ∞ is connected. Moreover by the Rectifiability Theorem [4, Theorems 4.4.7], the limitset Γ ∞ is rectifiable and connected by injective rectifiable curves. Step 2:
Structure of Γ ∞ . e have proved that Γ ∞ has minimal length (cid:96) . We can associate to Γ ∞ a rectifiablevarifold with multiplicity 1.Since Γ ∞ is of minimal length, it follows that the corresponding varifold is stationary,hence is composed by a finite number of geodesic segments, joining finitely many nodes (see[2]). The minimality of Γ ∞ also implies that each node is the meeting point of exactly threecurves, forming angles of π by standard arguments: suppose by contradiction that more thanthree curves concur at meeting point O . Consider a sufficiently small neighbourhood of thismultiple point O where there are not other nodes. Take two segments concurring in O formingan angle < π and two points P and Q of these segments in the considered neighbourhood.Lift P and Q to the tangent plane of the surface at O through the exponential map. Weknow that in the tangent plane the Steiner configuration is the minimal length configurationjoining O , P and Q . Replace the part of two curves on the surface joining P and Q with O inthe neighbourhood with the image through the exponential map of the three segments of theSteiner configuration. Repeating iteratively this procedure, we obtain only triple junctions. Step 3:
The set Γ ∞ intersects every homotopically non-trivial closed curve γ . If γ is a homotopically non-trivial closed curve, then each Γ n intersects γ , because γ cannot be contained in the open disc S \ Γ n . Hence, the Hausdorff limit Γ ∞ also intersects γ because γ is compact. Step 4:
The set Γ ∞ is a spine. Let S be the set of all closed subsets of Γ ∞ that intersect every homotopically non-trivial closed curve in S . By Zorn’s lemma there is a Γ ∈ S which is minimal with respectto inclusion. We now prove that Γ is a spine.The open set S \ Γ contains no non-trivial closed curve. If S \ Γ is an open disc we aredone: we prove that this is the case. If one component U of S \ Γ is not an open disc, itis not simply connected: hence it contains a simple closed curve γ which is homotopicallynon-trivial in U , but is necessarily trivial in S since it does not intersect Γ. Therefore γ bounds a disc D ⊂ S . Since D (cid:54)⊂ U , the intersection D ∩ Γ is non-empty: if we remove D ∩ Γfrom Γ we obtain another element of S strictly contained in Γ, a contradiction.Therefore S \ Γ consists of open discs only. If they are at least two, there is an arc inΓ adjacent to two of them: by removing from Γ an open sub-arc in this arc we get againanother element in S strictly contained in Γ.We know that Γ ⊂ Γ ∞ is a spine, and hence Γ = Γ ∞ , for if not the length of Γ would bestrictly smaller than that (cid:96) of Γ ∞ , a contradiction by (3.1). (cid:3) Definition 3.2.
A spine of a Riemannian surface S is minimal if it is a point, a closedgeodesic, or if it is composed by finitely many geodesic arcs, meeting with angle π attrivalent points.We have shown that a spine of minimal length is minimal. Of course, the converse maynot hold. However, a minimal spine is a stationary point of the length functional thanksto (2.4). emark 3.3. Notice that the extension of the existence result in higher dimension presentseveral difficulties: there is no higher dimensional version of Golab Theorem, because of thelack of semicontinuity of the Hausdorff measure. Also, it is not clear if a limit of spines isstill a spine. However, as we already observed, the existence of a spine of minimal area in aclosed irreducible 3-manifold has been proved in [9].
Remark 3.4.
If the surface S is neither diffeomorphic to a sphere nor to a projectiveplane, we have shown that minimal spines are trivalent graphs. Hence, adding the equation3 v = 2 e to (2.1), we get that the number of edges and that of vertices of a minimal spineare completely determined by the topology of S .3.1. Non positive constant curvature surfaces.
We restrict the attention to the caseof non positive constant curvature surfaces. The goal is to show that minimal spines arelocal minimizers for the length functional, justifying our choice for the adjective “minimal”.Let us begin with a definition and a well-known lemma about the convexity of the distancefunction in H that we take from [10]. Definition 3.5.
Let x , x be points in H , R or S and λ ∈ [0 , convex combination x = λx + (1 − λ ) x is defined as follows:in R : x = λx + (1 − λ ) x in H , S : x = λx + (1 − λ ) x (cid:107) λx + (1 − λ ) x (cid:107) where in the H case we are considering the hyperboloid model in R with the Lorentzianscalar product (cid:104)· , ·(cid:105) and (cid:107) v (cid:107) = (cid:112) − (cid:104) v, v (cid:105) . Lemma 3.6.
Let x , x , y , y be points of the hyperbolic plane H . For λ ∈ (0 , , considerthe convex combinations x λ = λx + (1 − λ ) x and y λ = λy + (1 − λ ) y . Then, we have d ( x λ , y λ ) ≤ λd ( x , y ) + (1 − λ ) d ( x , y ) , with equality only if x , x , y , y belong to the same line. Here d denotes the distance in H .Proof. Without loss of generality, for simplicity, we prove only the case λ = , therefore x λ (resp. y λ ) is the midpoint of x and x (resp. y and y ). If x λ = y λ the theorem is trivial,hence we suppose x λ (cid:54) = y λ .Let σ p be the reflection at the point p ∈ H . The map τ = σ y λ ◦ σ x λ translates theline r containing the segment x λ y λ by the quantity 2 d ( x λ , y λ ): hence it is a hyperbolictransformation with axis r . We call z i = τ ( x i ) and note that z = σ y λ ( x ), hence d ( x , y ) = d ( z , y ).The triangular inequality implies that d ( x , z ) ≤ d ( x , y ) + d ( y , z ) = d ( x , y ) + d ( x , y ) . (3.3)We notice that the equality holds only if x , y and z belong all to the same line. z x x z λ x λ y y λ y Figure 2.
A hyperbolic transformation has minimum displacement on its axis r , hence2 d ( x λ , y λ ) = d ( x λ , z λ ) = d ( x λ , τ ( x λ )) ≤ d ( x , τ ( x )) = d ( x , z ) , (3.4)and the equality holds only if x (and hence x ) is in r . Finally we get d ( x λ , y λ ) ≤ ( d ( x , y )+ d ( x , y )) and hence d is convex.Notice that the equality holds both in (3.3) and in (3.4) only if x , x , y , y belong tothe same line r . (cid:3) Theorem 3.7.
Minimal spines of closed surfaces of non positive constant curvature are localminima for the length functional among spines, with respect to the Hausdorff distance.If the curvature is negative, these are strict local minima.Proof.
We prove the first statement by contradiction.Consider a sequence of spines Γ n of the surface S converging in the Hausdorff distance toa minimal spine Γ, such that L (Γ n ) < L (Γ). The minimal spine Γ of S is a network composedby geodesic arcs joining k triple junctions x , ··· , x k , where the number k depends only onthe topology of S (see Remark 3.4). For n big enough, also Γ n have k triple junctions x ,n , ··· , x k,n . Moreover, we can suppose that for n big enough Γ n are composed only bygeodesic segments. Indeed if Γ n are not composed by geodesic segments, we can replace Γ n with (cid:101) Γ n , union of geodesic arcs, with the same triple junctions of Γ n , and the value of thelength functional decreases L ( (cid:101) Γ n ) ≤ L (Γ n ) < L (Γ).For n big enough, and for every λ ∈ [0 ,
1] and i ∈ { , . . . k } , take the convex combination x λi,n = (1 − λ ) x i + λx i,n and define Γ λn as the spine obtained by joining the points x λi,n withgeodesic segments in the same pattern of Γ. We get a continuous family of spines { Γ λn } λ ∈ [0 , such that Γ n = Γ n and Γ n = Γ. By Lemma 3.6, the continuous function F n ( λ ) = L (Γ λn ) isconvex (convexity of the distance function is easily proved also in the Euclidean case) and F n (1) ≤ F n (0). We also have F (cid:48) n (0) = 0 because the minimal spine Γ is a stationary pointof the length functional. This implies that L (Γ n ) = F n (1) ≥ F n (0) = L (Γ) and we have acontradiction.In the hyperbolic case, Lemma 3.6 provides strict convexity and hence Γ is a strict localminimum. (cid:3) emark 3.8. It is not restrictive to consider local minima of the length functional onlyamong spines and not in the larger class of networks. Indeed, if we take a smooth family Φ t of diffeomorphism of S with t ∈ [0 , T ] and a spine Γ and we consider a small perturbation ofΓ via these diffeomorphisms, Γ t = Φ t (Γ) is still a spine for t small enough. In particular, inthe proof of Theorem 3.7, we show that L (Γ) ≤ L (Φ t (Γ)), for all Φ t and for t small enough.4. Flat tori
In this section, we analyse minimal spines of the closed surface of genus 1: the torus T = S × S . In particular, we will fully determine all the minimal spines on T , endowedwith any Euclidean metric.4.1. Minimal spines of Riemannian tori.
From Remark 3.4, for any Riemannian metricon the torus T , minimal spines have exactly 2 vertices and 3 edges. There are only two kindsof graph satisfying these properties: the θ -graph and the eyeglasses (see Figure 3). a ) θ -graph b ) eyeglass Figure 3.
The two trivalent graphs with three edges.Both graphs can be embedded in a torus, but only the first in a way to obtain a spine.Indeed, it is easy to find a θ -spine in the torus (see Figure 4- a ), and actually infinitely manynon isotopic ones. Consider, instead, the eyeglasses as an abstract graph. Thickening theinteriors of its edges, we get three bands. To have a spine on T , it remains to attach them,getting a surface homeomorphic to T − B . This is impossible. The band corresponding toone of the two “lenses” of the eyeglasses has to be glued to itself. The are only two ways todo this: one would give an unorientable surface (see Figure 4- b ), the other would have toomany boundary components (see Figure 4- c ).We have shown the following a ) b ) c ) Figure 4. a ) A regular neighbourhood of a θ -spine on T . b ) − c ) An eyeglassesspine on T does not exist. roposition 4.1. Trivalent spines on the torus are θ -graphs. Combined with Theorem 1.2, we get the following
Theorem 4.2.
On every Riemannian torus, minimal spines (exist and) are embedded θ -graphs with geodesic arcs, forming angles of π at their two meeting points. Basics on flat tori.
From now, we consider only constant curvature Riemannian met-rics on T . These correspond to flat (i.e. Euclidean) structures on T . We make a very roughintroduction about the basic concepts of the topic to fix the notations, referring the readerto [10] for details.Let us define • the Teichm¨uller space T of the torus as the space of isotopy classes of unit-area flatstructures on T , • the mapping class group Mod of the torus as the group of isotopy classes of orientationpreserving self-diffeomorphisms of T . • the moduli space M of the torus as the space of oriented isometry classes of unit-areaflat structures on T .The group Mod acts on the set T and M is the quotient by this action. The set T is endowed with a natural metric (the Teichm¨uller metric), which makes it isometric tohyperbolic plane H . The action of the mapping class group can be identified with a discreteaction by isometries.Visualizing the hyperbolic plane with the upper-half space model {(cid:61) ( z ) > } ⊂ C , theaction is given by integer M¨obius transformations and the mapping class group is isomorphicto the group SL ( Z ) of unit-determinant 2 × {± I } , so that the moduli space M has the structure of a hyperbolic orbifold, with π M (cid:39)
PSL ( Z ). A fundamental domain for the action is the hyperbolic semi-ideal triangle D = (cid:8) | z | ≥ , |(cid:60) ( z ) | ≤ (cid:9) , with angles π , π and 0 (in grey in Figure 5). The quotient M is the complete (2 , , ∞ )-hyperbolic orbifold of finite area, with one cusp and two conical singularities of angles π and π . We see M as usual as D with the boundary curves appropriately identified.We will also be interested in the following space: • the non-oriented moduli space M no is the space of all isometry classes of (unoriented)unit-area flat structures on T .We have M no = M / ι where ι is the isometric involution that sends an oriented flat torusto the same torus with opposite orientation: the lift of ι to the fundamental domain D is thereflection with respect to the geodesic line i R + and M no is the hyperbolic triangle orbifold M no = D ∩ {(cid:60) ( z ) ≥ } with angles π , π and 0. We do not need to fix an orientation on T to talk about spines,hence we will work mainly with M no . e i π − zD H ∼ = T Figure 5.
Every z ∈ H in upper half-plane represents a flat torus obtainedby identifying the opposite edges of the parallelogram with vertices 0, 1, z , z + 1. A fundamental domain D for the action of PSL ( Z ) is colored in grey.Every flat torus T has a continuum of isometries: the translations by any vector in R and the reflections with respect to any point x ∈ T . The tori lying in the mirror sides of M no have special names and enjoy some additional isometries: • the rectangular tori are those lying in i R + , • the rhombic tori are those in the other sides of M no , namely {| z | = 1 } ∪ {(cid:60) ( z ) = } .These flat tori are obtained by identifying the opposite sides of a rectangle and a rhombus,respectively. The tori in the cone points z = i and e πi are the square torus and the hexagonaltorus . On the hexagonal torus, the length d of the shortest diagonal of the rhombus equalsthe length l of any of its sides, while we have d ≥ l and d ≤ l on the sides | z | = 1 and (cid:60) = respectively (we can call these rhombi fat and thin , repsectively).The rectangular and rhombic tori are precisely the flat tori that admit orientation-reversing isometries.Teichm¨uller and moduli spaces have natural Thurston and Mumford-Deligne compacti-fications. In the torus case, these are obtained respectively by adding the circle “at infinity” ∂ H = R ∪ {∞} to T = H and a single point to M or M no . We denote the lattercompactifications by M and M no .4.3. Hexagons.
The study of minimal spines on a flat torus is intimately related to that ofa particular class of Euclidean hexagons.
Definition 4.3. A semi-regular hexagon is a Euclidean hexagon with all internal angles π and with congruent opposite sides.We define the moduli space H as the space of all oriented semi-regular hexagons con-sidered up to homotheties and orientation-preserving isometries. Similarly H no is definedby considering non-oriented hexagons and by quotienting by homotheties and all isometries.We get a map H → H no that is at most 2-to-1. wo opposite sides of a semi-regular hexagon are parallel and congruent. A semi-regularhexagon is determined up to isometry by the lengths a, b, c > H = (cid:8) ( a, b, c ) | a, b, c > (cid:9) / R > × A H no = (cid:8) ( a, b, c ) | a, b, c > (cid:9) / R > × S where the multiplicative group of positive real numbers R > acts on the triples by rescaling,while A and S act by permuting the components.We can visualize the space H no in the positive orthant of R by normalizing ( a, b, c ) suchthat a + b + c = 1 and a ≥ b ≥ c , and in this way H no is a triangle with one side removed(corresponding to c = 0) as in Fig. 6. The other two sides parametrize the hexagons with a = b ≥ c and a ≥ b = c . The regular hexagon is of course ( , , ). The space H no isnaturally an orbifold with mirror boundary made by two half-lines and one corner reflectorof angle π .The space H of oriented semi-regular hexagons is obtained by doubling the triangle H no along its two edges. Therefore H is topologically an open disc, and can be seen as an orbifoldwhere the regular hexagon is a cone point of angle π . The map H → H no may be interpretedas an orbifold cover of degree two.The orbifold universal covering ˜ H → H is homeomorphic to the map z (cid:55)→ z from thecomplex plane to itself. The orbifold fundamental group π H is isomorphic to the cyclicgroup Z /
3, acting on the plane by rotations of angle π . ca b a ≥ b = c c = a = b ≥ c (1 , ,
0) ( , , )( , , D Z / Z / H no D Z / Z / H no Z / Z / H H
Figure 6.
On the left, the orbifold structures on the moduli spaces of semi-regular hexagons, and their compactifications. On the right, a parametrizationof H no by a triangle with one side removed in R . oth H and H no have natural compactifications, obtained by adding the side with c = 0,which consists of points ( a, − a,
0) with a ∈ [ , a < π and π , while (1 , ,
0) should be interpreted as a segment – a doubly degenerate hexagon. The underlying spaces of the resulting compactifications H and H no are bothhomeomorphic to closed discs.4.4. Spines.
We now introduce two more moduli spaces S and S no which will turn out tobe isomorphic to H and H no .Let the moduli space S be the set of all pairs ( T, Γ), where T is a flat oriented torus andΓ ⊂ T a minimal spine, considered up to orientation-preserving isometries (that is, ( T, Γ) =( T (cid:48) , Γ (cid:48) ) if there is an orientation-preserving isometry ψ : T → T (cid:48) such that ψ (Γ) = Γ (cid:48) ). Wedefine analogously S no as the set of all pairs ( T, Γ) where T is unoriented, quotiented by allisometries. Again we get a degree two orbifold covering S → S no .The opposite edges of a semi-regular hexagon H are congruent, and by identifying themwe get a flat torus T . The boundary ∂H of the hexagon transforms into a minimal spineΓ ⊂ T in the gluing process. This simple operation define two maps H −→ S , H no −→ S no . Proposition 4.4.
Both maps are bijections.Proof.
The inverse map is the following: given ( T, Γ), we cut T along Γ and get the originalsemi-regular hexagon H . (cid:3) We will therefore henceforth identify these moduli spaces and use the symbols H and H no to denote the moduli spaces of both semi-regular hexagons and pairs ( T, Γ). Of courseour aim is to use the first (hexagons) to study the second (spines in flat tori).4.5.
The forgetful maps.
The main object of this chapter is the characterization of theforgetful maps p : S −→ M , p : S no −→ M no that send ( T, Γ) to T forgetting the minimal spine Γ. The fiber p − ( T ) over an orientedflat torus T ∈ M can be interpreted as the set of all minimal spines in T , considered up toorientation-preserving isometries of T . Likewise the fiber over an unoriented torus T ∈ M no is the set of minimal spines in T , considered up to all isometries.As we said above, we identify H , H no with S , S no and consider the compositions (whichwe still name by p ) p : H ∼ −→ S −→ M , p : H no ∼ −→ S no −→ M no . The map p is described geometrically in Fig. 7. b c abc Figure 7.
The composition p : H → S → M . z = ˜ p ( a, b, c )0 1 c (cid:48) b (cid:48) a (cid:48) Figure 8.
How to construct z from ( a (cid:48) , b (cid:48) , c (cid:48) ): we pick the tri-pod with angles π and lengths a (cid:48) , b (cid:48) , c (cid:48) , and place it in the up-per half-plane so that the end-points of the edges c (cid:48) and b (cid:48) liein 0 and 1. The point z is theendpoint of the edge a (cid:48) . Proposition 4.5.
The map ˜ p : H no −→ H ( a, b, c ) (cid:55)−→ c − ab + ac + bc b + c + bc ) + i √ ab + bc + acb + c + bc is a lift of p : H no → M no . The map ˜ p is injective and sends homeomorphically the triangle H no onto the light grey domain drawn in Fig. 9.Proof. The map ˜ p is defined in Fig. 8. We first rescale the triple ( a, b, c ), which by hypothesissatisfies a ≥ b ≥ c and a + b + c = 1, to ( a (cid:48) , b (cid:48) , c (cid:48) ), by multiplying each term by 1 / √ b + c + bc .Now we pick the tripod with one vertex and three edges of length a (cid:48) , b (cid:48) , c (cid:48) and with angles π , and we place it in the half-space with two vertices in 0 and 1 as shown in the figure(we can do this thanks to the rescaling). The third vertex goes to some z ∈ H and we set˜ p ( a, b, c ) = z . he map ˜ p is clearly a lift of p and it only remains to determine an explicit expressionfor ˜ p . Applying repeatedly the Carnot Theorem, we get:Arg( z ) = cos − c − ab + ac + bc (cid:112) ( b + c + bc ) ( a + c + ac ) , | z | = (cid:114) a + c + acb + c + bc . Finally, ˜ p ( a, b, c ) = z = 2 c − ab + ac + bc b + c + bc ) + i √ ab + bc + acb + c + bc . The proof is complete. (cid:3) ˜ p ( H no ) ˜ p ( H ) a ≥ b = c c = a = b ≥ c Figure 9.
The lift ˜ p sends H no homeomorphically to the light grey domain onthe left. It sends the two sides a ≥ b = c and a = b ≥ c of H no to two geodesiclines in H , and sends the line at infinity c = 0 to the constant-curvature linedashed in the figure. In the oriented setting, the lift ˜ p sends H to the wholebigger grey domain but is discontinuous on the segment s of hexagons of type a = b ≥ c .Fig. 9 shows that the lift ˜ p sends the two sides a = b ≥ c and a ≥ b = c of the triangle H no to geodesic arcs in H . Recall that the compactification H no is obtained by addingthe singly-degenerate parallelograms ( a, − a,
0) with a ∈ [ ,
1) and the doubly-degenerate P = (1 , , p also extends to the parallelograms, and sends them to the dashedline in Fig. 8, but it does not extend continuously to P , not even as a map from H no to H .However, the map p from moduli spaces does extend. Proposition 4.6.
The map p : H no → M no extends continuously to a map p : H no → M no . roof. Send the doubly degenerate point P to the point at infinity in M . (cid:3) The oriented picture is easily deduced from the non-oriented one. We can lift the map p : H → M to a map ˜ p : H → H in Teichm¨uller space whose image is the bigger (bothlight and dark) grey domain in Fig. 9, however the map ˜ p is discontinuous at the segmentconsisting of all hexagons with a = b ≥ c , which is sent to one of the two curved geodesicarcs in the picture. The map p : H → M extends continuously to a map p : H → M .The M¨obius transformation z (cid:55)→ iz + √ − i z − √ i gives a more symmetric picture of the image˜ p ( H ) inside the hyperbolic plane in the Poincar´e disc model, as can be seen in Figure 10. ie i π e i π ∞− Figure 10.
A picture of ˜ p ( H ) in the Poincar´e disc model. Remark 4.7.
Now, it is possible to quantify the “distance” between two spines on differentflat tori, in a natural way. Indeed, thanks to the map p and the Teichm¨uller metric, themoduli space of spines can be endowed with a structure of hyperbolic orbifold.For, identify − by a representation ρ − the action of the orbifold fundamental group π H on the universal covering ˜ H with the action on the hyperbolic plane of the group Γ,generated by an order-three elliptic rotation about the point e i π . To get a developing map d : ˜ H → H for the hyperbolic structure on H , lift ρ -equivariantly the map p .The developed image d ( ˜ H ) of H is the interior in H of the regular non-geodesic idealtriangle joining the points ∞ , 0, 1 in Figure 10. The set ˜ p ( H ) is a fundamental domain forthe action of Γ and the space H is identified with the quotient d ( ˜ H ) / Γ .Hence, the moduli space of spines is an infinite-area incomplete hyperbolic orbifold. Itscompletion, supported on the complement of the doubly degenerate hexagon
H \ { P } , isan orbifold with non-geodesic boundary. Its boundary is an infinite constant-curvature linewhose points are (all the) four-valent geodesic spines with alternating angles π and π .Similarly, H no \ { P } is a complete infinite-area hyperbolic orbifold with (unbounded,non-geodesic) boundary and two mirror edges. .6. The number of minimal spines.
We are ready to determine the fiber p − ( T ) forevery flat torus T in M and M no , which may be interpreted as the set of all minimal spinesin T , considered up to (orientation-preservingly or all) isometries of T . √ √ √ √ √ √
11 22 233 34 56 √ √ √ √ √ √
111 23 345 56
Figure 11.
The number of minimal spines on each flat torus, in the unori-ented and oriented setting. At each point z in M no or M , the number c no ( z )or c ( z ) is the smallest number among all numbers written on the adjacentstrata (both functions are lower semi-continuous). Theorem 4.8.
Every unoriented flat torus T has a finite number c no ( T ) of minimal spinesup to isometries of T . Analogously, every oriented flat torus T has a finite number c ( T ) ofminimal spines up to orientation-preserving isometries. The proper functions c no : M no → N and c : M → N are shown in Fig. 11.Proof. We can identify H no with its image H = ˜ p ( H no ) and note that the restriction of theprojection π : H → M no to H is finite-to-one. Indeed, for z, z (cid:48) ∈ H , π ( z ) = π ( z (cid:48) ) if and onlyif (cid:61) z (cid:48) = (cid:61) z and (cid:60) z (cid:48) = ±(cid:60) z + n for some integer n . The number c no ( z ) is the cardinality ofthe fiber p − ( z ) and is easily shown to be as in Fig. 11-(left).The oriented case is treated analogously: in that case ˜ p ( H ) is the bigger grey zone in H and p ( z ) = p ( z (cid:48) ) if and only if z (cid:48) = z + n for some n ∈ Z . We get the picture inFig. 11-(right). (cid:3) The subtler cases are those where T has additional symmetries: in Fig. 12 we show theminimal spines of T as the metric varies from rectangular to square, fat rhombic, hexagonal,and finally thin rhombic. igure 12. The minimal spines in some oriented tori having additional sym-metries. As we pass from rectangular to thin rhombic, we find 2, 1, 2, 1, andthen 3 spines up to orientation-preserving isometries. They reduce to 1, 1, 2,1, 2 up to all isometries, including orientation-reversing ones. Rectangles andrhombi that are thinner (ie longer) than the ones shown here have additionalminimal spines that wind around the thin part.4.7.
Length.
Let us define the length of a union of curves on a flat torus in the moduli space M as its normalized length, that is its one dimensional Hausdorff measure on a unit-arearepresentative torus. For example, if you consider the explicit representative given by theparallelogram R -generated by 1 and z ∈ C , you have to divide lengths by the square root ofthe area of that parallelogram, that is (cid:112) (cid:61) ( z ).Generalizing the well known concept for closed geodesics on surfaces (see [10]), we definethe length spectrum of minimal spines of a flat torus T as the set of lengths of its minimalspines. Actually, we can define a more accurate spectrum, a set of triples corresponding tothe lengths of the sides of any spine on T , in decreasing order, that is (cid:40) ( a, b, c ) (cid:112) ( b + c + bc ) (cid:61) (˜ p ( a, b, c )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( a, b, c ) ∈ p − ( T ) (cid:41) . We have shown that both sets are finite, for any flat torus T .In the following, we determine the spines of minimal length. Theorem 4.9.
Every unoriented flat torus has a unique spine of minimal length up toisometry. In the oriented setting, instead, the same holds with the exception of the rectangularnon square tori, for which there are exactly two.Proof.
Fix a torus T ∈ M no . In the proof of Theorem 4.8, we observed that all points of π − ( T ) ∩ H ⊂ H have the same imaginary part, so the lengths of the tripods we found withthe map ˜ p can be compared without need of normalization. Clearly, only one is the shortest:that corresponding to z ∈ D ∩ {(cid:60) ( z ) ≥ } .In the oriented case, consider the representative z of T ∈ M with z ∈ D \ { z ∈ ∂D | (cid:60) ( z ) < } . There are three cases: • if (cid:60) ( z ) >
0, the unique shortest spine is that associated to z itself; if (cid:60) ( z ) <
0, it is associated to z + 1; • if (cid:60) ( z ) = 0, that is T is a rectangular torus, both spines associated to z and z + 1are of minimal length. These two spines are the same only for the square torus,indeed a rotation of π sends one to the other.These evident assertions can be verified by trigonometry, or by computing the lengthsthrough the formula showed in the following . (cid:3) The explicit expression on ˜ p ( H ) ⊂ H for the length function L : H → (0 , + ∞ ), whichto a spine assigns its length, can be simply computed by the law of sines: L ( z ) = (cid:115) | z | − (cid:60) ( z ) + √ (cid:61) ( z ) (cid:61) ( z ) . The formula for the spine systole
S :
M → (0 , + ∞ ), a generalized systole which assignsto a flat torus the length of its shortest spines isS( z ) = (cid:115) | z | − |(cid:60) ( z ) | + √ (cid:61) ( z ) (cid:61) ( z ) . Both functions are proper and almost everywhere smooth. They extend to continuousfunctions L : H → (0 , + ∞ ] and S : M → (0 , + ∞ ].For every continuous function f : X → R on a topological 2-manifold X , recall that apoint p ∈ X is said to be regular if there is a (topological) chart around p = 0 in which f ( x, y ) = const + x . Otherwise, it is critical . A critical point p ∈ X is said to be nondegenerate if in a local (topological) chart around p = 0 we have either f ( x, y ) = const − x + y , or f ( x, y ) = const ± ( x + y ). The non degenerate critical points are necessarilyisolated. The function f is topologically Morse if it is proper and all critical points are nondegenerate. The classical Morse theory works also in the topological category. Remark 4.10.
The functions L : H → (0 , + ∞ ) and S : M → (0 , + ∞ ) are topologicallyMorse. For both functions, the set of sublevel k is empty if k < √ √
2, a point if k = √ √ √ √ ≈ . Hyperbolic surfaces
Let now S g be a closed orientable surface of genus g ≥
2. The oriented hyperbolic metricson S g form the moduli space M ( S g ) and the minimum length of a spine furnishes the spinesystole S : M ( S g ) −→ R . We now prove some facts on the function S. heorem 5.1. The function S is continuous and proper. Its global minima are precisely theextremal surfaces.Proof. The function S is clearly continuous because the length of spines varies continuouslyin the metric. We now prove properness as an easy consequence of the Collar Lemma [10].By Mumford’s compactness theorem the subset M ε ( S g ) ⊂ M ( S g ) of all hyperbolicmetrics with (closed geodesic) systole ≥ ε is compact for all ε >
0, and the Collar Lemmasays that for sufficiently small ε > S ∈ M ( S g ) \ M ε ( S g ) has a simpleclosed curve γ of length < ε with a collar of diameter C ( ε ), for some function C such that C ( ε ) → ∞ as ε →
0. Every spine Γ of S g must intersect γ and cross the collar, hence L (Γ) ≥ C ( ε ) and therefore S is proper.We now determine the global mimima of S. Let Γ be a spine in S ∈ M ( S g ) of minimallength. The spine has 6 g − S along Γ we get ahyperbolic polygon P with 12 g − π . The length of Γ is halfthe perimeter of P .Porti has shown [22] that, among all hyperbolic n -gons with fixed interior angles, theone with smaller perimeter is the unique one that has an inscribed circle. Therefore amongall polygons P with angles π the one that minimizes the perimeter is precisely the regularone R , that is the one whose sides all have the same length. We deduce that the globalminima for S are the hyperbolic surfaces that have R as a fundamental domain, and theseare precisely the extremal surfaces, as proved by Bavard [7]. (cid:3) It would be interesting to investigate the function S and check for instance whether it isa topological Morse function, see [12].In the flat case we have shown that the number of minimal spines is finite. In thehyperbolic setting, we do not know if the same is true. To conclude the section, we prove apartial result:
Theorem 5.2.
The number of minimal spines with bounded length of a closed hyperbolicsurface S is finite.Proof. In the proof of Theorem 1.2, it results clear that every set of minimal spines ofequibounded length is compact. We now prove that for hyperbolic surfaces every such setis discrete, hence finite. By contradiction, let Γ n be a sequence of distinct minimal spinesof S of equibounded length, converging in the Hausdorff distance to the minimal spine Γ.Moreover L (Γ n ) → L (Γ). For every λ ∈ [0 ,
1] and every n big enough, we construct, exactlyas in the proof of Theorem 3.7, the (not necessarily minimal) spine Γ λn and continuousfunction F n ( λ ) = L (Γ λn ). The surface S is hyperbolic, therefore, by Lemma 3.6, F n ( λ ) isstrictly convex. Both Γ and Γ n are minimal spines, that is stationary points of the lengthfunctional, hence, F (cid:48) n (0) = F (cid:48) n (1) = 0 and F n is constant in λ : a contradiction. (cid:3) References [1]
W. K. Allard , On the first variation of a varifold , Annals of Math. (1972), 417–491. W. K. Allard – F. J. Almgren , The Structure of Stationary One Dimensional Varifolds . Invent.Math. (1976), 83–97.[3] S. B. Alexander – R. L. Bishop , Spines and Homology of Thin Riemannian Manifolds with Bound-ary , Adv. Math. (2000), 23–48.[4]
L. Ambrosio – P. Tilli . “Topics on Analysis in Metric Spaces,” Oxford University Press, 2004.[5]
J. Aramayona – T Koberda – H. Parlier , Injective maps between flip graphs , arXiv:1409.7046 [6] A. Ash , Small-dimensional classifying spaces for arithmetic subgroups of general linear groups , DukeMath. J. (1984), 459–468.[7] C. Bavard , Disques extr´emaux et surfaces modulaires , Ann. Fac. Sci. Toulouse Math. (1996), 191–202.[8] M. A. Buchner , Simplicial structure of the real analytic cut locus , Proc. Amer. Math. Soc. (1977),118–121.[9] J. Choe , On the existence and regularity of fundamental domains with least boundary area , J. Differ-ential Geom. (1989), 623–663.[10] B. Farb and D. Margalit “A primer on mapping class groups,” Princeton University Press, Prince-ton, NJ (2012).[11] S. Fomin – M. Shapiro – D. Thurston , Cluster algebras and triangulated surfaces. I. Clustercomplexes , Acta Math. (2008), 83–146.[12]
M. Gendulphe , The injectivity radius of hyperbolic surfaces and some Morse functions over modulispaces , arXiv:1510.02581 [13] E. Girondo – G. Gonzalez-Diez , On extremal Riemann surfaces and their uniformizing Fuchsiangroups , Glasgow Math. J. (2002), 149–157.[14] M. H. Hirsch , Smooth regular neighborhoods , Ann. Math. (1962), 524–530.[15] M. Korkmaz – A. Papadopoulos , On the ideal triangulation graph of a punctured surface , Ann.Inst. Fourier (2012), 1367–1382.[16] F. Maggi
Sets of Finite Perimeter and Geometric Variational Problems. An introduction to geometricmeasure theory.
Cambridge University Press, Cambridge (2012).[17]
B. Martelli , Complexity of PL manifolds , Alg. & Geom. Top. (2010), 1107–1164.[18] S. Matveev , Transformations of special spines and the Zeeman conjecture , Math. USSR-Izv. (1988), 423–434.[19] , Complexity theory of three-dimensional manifolds , Acta Appl. Math. (1990), 101–130.[20] , “Algorithmic Topology and Classification of 3-Manifolds”, second edition, Algorithms andComputation in Math. , Springer, Berlin (2007).[21] R. Piergallini , Standard moves for standard polyhedra and spines , Rendiconti Circ. Mat. Palermo (1988), 391–414.[22] J. Porti , Hyperbolic polygons of minimal perimeter with given angles , Geom. Dedicata (2012),165–170.[23]
D. Sleator – D. D. Tarjan – W. P. Thurston , Rotation distance, triangulations, and hyperbolicgeometry , J. Amer. Math. Soc. (1988), 647–681. ipartimento di Matematica “Tonelli”, Largo Pontecorvo 5, 56127 Pisa, Italy E-mail address : martelli at dm dot unipi dot it Dipartimento di Matematica “Tonelli”, Largo Pontecorvo 5, 56127 Pisa, Italy
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