Horizon saddle connections, quasi-Hopf surfaces and Veech groups of dilation surfaces
HHORIZON SADDLE CONNECTIONS, QUASI-HOPF SURFACES ANDVEECH GROUPS OF DILATION SURFACES
GUILLAUME TAHAR
Abstract.
Dilation surfaces are generalizations of translation surfaces where the geo-metric structure is modelled on the complex plane up to affine maps whose linear partis real. They are the geometric framework to study suspensions of affine interval ex-change maps. However, though the SL (2 , R ) -action is ergodic in connected componentsof strata of translation surfaces, there may be mutually disjoint SL (2 , R ) -invariant opensubsets in components of dilation surfaces. A first distinction is between triangulable andnon-triangulable dilation surfaces. For non-triangulable surfaces, the action of SL (2 , R ) is somewhat trivial so the study can be focused on the space of triangulable dilationsurfaces.In this paper, we introduce the notion of horizon saddle connections in order to refinethe distinction between triangulable and non-triangulable dilation surfaces. We also in-troduce the family of quasi-Hopf surfaces that can be triangulable but display the sametrivial behavior as non-triangulable surfaces. We prove that existence of horizon saddleconnections drastically restricts the Veech group a dilation surface can have. Contents
1. Introduction 12. Dilation surfaces 33. Horizon saddle connections 54. Veech groups 11References 141.
Introduction
A dilation structure on a surface is a geometric structure modelled on the complex planewith structural group the set of maps x (cid:55)→ ax + b with a ∈ R ∗ + and b ∈ C . In most cases,a dilation structure is defined on the complement of a finite set of singularities, just liketranslation structures, see [10].Dilation surfaces have many common features with translation surfaces: notions of straightlines, angles and direction are well-defined (they are defined on C and transported by pull-back to the dilation surface). In particular, for every angle θ ∈ S , we can define a directional foliation . whose leaves are straight lines in direction θ in every chart. Theseleaves are the trajectories of the dynamics defined by the directional foliation. Just liketranslation surfaces are suspensions of interval exchange maps. Dilation surfaces are sus-pensions of affine interval exchange maps. These foliations have been studied in [1]. Justlike moduli spaces of translation surfaces, moduli spaces of dilation surfaces decomposeinto strata that are analytic orbifolds on which there is an action of SL (2 , R ) that encom-passes a renormalization process. Date : May 12, 2020.
Key words and phrases.
Dilation Surface, Veech group, Saddle connection. a r X i v : . [ m a t h . G T ] J un n strata of translation surfaces, most interesting dynamic properties depend on the clo-sure of the SL (2 , R ) -orbit. Besides, the orbit of a generic translation surface is dense inits connected component of the stratum. We look for an analogous framework for dilationsurfaces.However, there are no SL (2 , R ) -invariant dense open sets in strata of dilation surfaces. Anidea exposed in [2] is that we should focus on nice subsets of strata. A first criterion is toconsider dilation surfaces that are triangulable in a geometric sense. We require that theedges of the triangulation are saddle connections, that is geodesic segments between sin-gularities (and without singularities in their interior). These triangulable dilation surfacesform a SL (2 , R ) -invariant open set of the moduli space.In this paper, we generalize this criterion by introducing the notion of horizon saddle con-nections . If a dilation surface displays such objects, then the action of SL (2 , R ) on itsimplifies drastically. Definition 1.1.
In a dilation surface, a k -horizon saddle connection γ is a saddle connec-tion such that no trajectory crosses γ strictly more than k times.In particular, non-triangulable dilation surfaces admit -horizon saddle connections, seeProposition 3.1 for details.Inclusion of the moduli space T of triangulable dilation surfaces in the moduli space D of dilation surfaces can be generalized into an infinite sequence of inclusions of SL (2 , R ) -invariant open sets. The example of quasi-Hopf surfaces (see Subsection 3.2 for details)shows that there are triangulable surfaces with some horizon saddle connections. Theorem 1.2.
In the moduli space of dilation surfaces D , for any k ≥ the set H k ofsurfaces without k -horizon saddle connections is a SL (2 , R ) -invariant open set. We have · · · ⊂ H k ⊂ · · · ⊂ H ⊂ T ⊂ D Theorem 1.2 is proved in Section 3.Horizon saddle connections constrain the action of SL (2 , R ) . In the following, the Veechgroup V ( X ) of a dilation surface X is the stabilizer of the action of SL (2 , R ) . There isa remarkable similarity with the three types of Veech groups for translation surfaces withpoles: continuous, cyclic parabolic or finite, see [5, 6] for details. Theorem 1.3.
In a dilation surface X such that there are at least three distinct directionsof horizon saddle connections, then the Veech group of X is finite. Theorem 1.4.
In a dilation surface X such that there are exactly two distinct directionsof horizon saddle connections, then there are two cases:(i) V ( X ) is finite (of order , or );(ii) the Veech group of X is conjugated to (cid:26)(cid:18) a k a − k (cid:19) | k ∈ Z (cid:27) with a ∈ R ∗ + or its productwith the subgroup generated by (cid:18) − (cid:19) .In the latter case, the surface is a rational quasi-Hopf surface. Theorem 1.5.
In a dilation surface X such that there is exactly one direction of horizonsaddle connections, then there are three cases:(i) Hopf surfaces: the Veech group of X is conjugated to (cid:26)(cid:18) a b a − (cid:19) | a ∈ R ∗ , b ∈ R (cid:27) ;(ii) the Veech group of X is cyclic parabolic: conjugated to (cid:26)(cid:18) k (cid:19) | k ∈ Z (cid:27) or its product ith {± Id } .(iii) the Veech group of X is trivial or {± Id } . Theorems 1.3 to 1.5 are proved in Section 4.The structure of the paper is the following:- In Section 2, we recall the background about dilation surfaces: moduli space, linearholonomy, triangulations.- In Section 3, we introduce the notion of horizon saddle connections and prove the finitenesslemma. We also discuss some examples: dilation tori, surfaces with chambers, quasi-Hopfsurfaces, dilation surfaces with finitely many saddle connections.- In Section 4, we present some preliminary results about Veech groups and prove the maintheorems about restriction of the Veech group in presence of horizon saddle connections.2.
Dilation surfaces
Dilation structure.
A dilation structure on a Riemann surface X is a kind of affinestructure. We follow the definitions of [2]. Definition 2.1.
A dilation surface is a Riemann surface X with a finite set Λ ⊂ X ofsingularities and an atlas of charts on X \ Λ with values in C and such that:(i) transition maps are of the form x (cid:55)→ ax + b with a ∈ R ∗ + .(ii) the affine structure extends around every element of Λ to a "conical singularity" char-acterized by its topological index and its dilation ratio (see Subsection 2.2).One difficulty in the study of dilation surfaces is that there is no notion of distance sincetwo segment with different lengths and the same direction are equivalent up to affine maps.However, the ratio of lengths of two saddle connections sometimes makes sense.Let α and β be two saddle connections of a dilation surface X that intersect (possibly atthe ends of the segment) each other. Then, we consider a chart covering a disk D (possiblywith a slit) centered on the intersection point. In D , the length of the two intersectingbranches is well-defined. The portion these branches represent in their saddle connectionis also well-defined. Therefore, the local length ratio of α and β is well-defined. It onlydepends on the choice of the intersection point and possibly the order in the couple ( α, β ) if the intersection point is a singularity with a nontrivial dilation ratio. This constructionwill help in some crucial results about Veech groups.2.2. Linear holonomy.
In a dilation surface X , for every closed path γ of X \ Λ , wecan cover γ with charts of the atlas. The transition map between the first chart and thelast chart is an affine map. Its linear part is well-defined up to conjugacy. This numberis obviously a topological invariant. Therefore, we have a representation of the pointedfundamental group of X \ Λ into R ∗ + . Since the latter is Abelian, the representationfactorizes through H ( X \ Λ , Z ) (Hurewicz theorem). Thus, we get a group morphism ρ : H ( X \ Λ , Z ) −→ R ∗ + we denote by linear holonomy of loops.The local geometry of a conical singularity is determined by two topological numbersassociated to a simple loop γ around it:(i) the linear holonomy ρ ( γ ) ;(ii) the topological index i ( γ ) .The neighborhood of such a singularity is constructed starting from an infinite cone ofangle i ( γ )2 π and a ray starting from the origin of the cone. Then, we identify the rightpart of the ray with the left part of the ray with a homothety ratio of ρ ( γ ) . igure 1. Hyperbolic cylinder of angle π and dilation ratio λ . Thissurface formed by a unique hyperbolic cylinder is a Hopf torus .The conical singularities satisfy a Gauss-Bonnet formula. It is remarkable that thedilation ratio appears as something like an "imaginary curvature". The following formulahas been proved as Proposition 1 in [3].
Proposition 2.2.
In a surface of dilation of genus g ≥ with conical singularities s , . . . , s n of angle k i π and dilation ratio λ i , we have:(i) (cid:80) ni =1 ( k i −
1) = 2 g − ;(ii) (cid:80) ni =1 ln ( λ i ) = 0 . In particular, there is no dilation surface in genus zero. It would be necessary to intro-duce a notion of pole among the singularities of the affine structure.2.3.
Moduli space and strata.
For any g, n ≥ , we consider the set X of dilation struc-tures on a given compact topological surface of genus g with n marked points. We definethe moduli space D g,n of dilation surfaces of genus g with n singularities as the quotientof X by the group of orientation-preserving diffeomorphisms. This space is an analyticorbifold of real dimension g −
1) + 3 n , see [8] for details.For any sequence of integers a = ( a , . . . , a n ) of integers such that (cid:80) ni =1 a i = 2 g − and any sequence of positive real numbers λ = ( λ , . . . , λ n ) such that (cid:81) ni =1 λ i = 1 , there isa (possibly empty) stratum D g,n ( a, λ ) of D g,n . These strata are analytic orbifolds of realdimension g −
1) + 2 n + 1 . SL (2 , R ) acts on the moduli space of dilation surface by composition with the coordinatesmaps with values in C . The action preserves the linear holonomy ρ so it also preservesstrata. The Veech group V ( X ) of a dilation surface X is the stabilizer of this group action.It is a subgroup of SL (2 , R ) .2.4. Cylinders.
For any closed geodesic, the first return of a small segment orthogonal tothe geodesic is a map of the form x (cid:55)→ λx with λ ∈ R ∗ + . We say that the closed geodesicis flat if λ = 1 . Otherwise, it is hyperbolic.Flat closed geodesics belong to families that describe flat cylinders (just like in trans-lation surfaces). Hyperbolic closed geodesics also describe cylinders. We refer to them as affine cylinders . They are obtained from an angular portion of an annulus (or a cycliccover of an annulus) by gluing the two sides on each other, see Figure 1. The isomorphismclass of an affine cylinder is completely determined by two numbers:(i) The affine factor λ (dilation ratio along hyperbolic closed geodesics).(ii) The angle θ (determined by the angular portion of the annulus considered).In translation surfaces, the horocyclic flow and its conjugates (action of unipotent ele-ments of SL (2 , R ) preserving the direction of the closed geodesics of the cylinder) modify he twist of a flat cylinder. There is a similar phenomenon for affine cylinders.We consider an affine cylinder of dilation factor λ whose boundary is formed by asaddle connection with a marked point in direction α and another saddle connection witha marked point in direction β . Then we consider the conjugate of the Teichmüller flow A t that contracts direction α with a factor e − t and expands direction β with a factor e t . Theflow of this subgroup on this surface is periodic and the affine cylinder is invariant by anyelement A t of the flow such that t = ln ( λ )2 . This will be useful in Subection 3.3 when wewill study quasi-Hopf surfaces.2.5. Triangulable dilation surfaces.
In the framework of dilation surfaces, a (geomet-ric) triangulation is a topological triangulation where the edges are saddle connectionsand where every conical singularity is a vertex. Having a triangulation of dilation surfaceprovides a nice parametrization of its neighborhood in the moduli space (by deformingthe triangles). However, not every dilation surface admits a triangulation. Indeed, affinecylinders of angle at least π are not triangulable.A geodesic trajectory entering in an affine cylinder cannot intersect a hyperbolic geodesicthat shares the same direction. Let I ⊂ S be the closure of the interval of the directionsof closed geodesics of the affine cylinder. If a trajectory γ whose direction do not belongto I enters the cylinder, then it will cross every closed geodesic and finally leave by theopposite boundary. On the contrary, if its direction θ belongs to I , then γ will accumulateon the hyperbolic geodesic of direction θ the closest to the entry point (in the foliation ofthe cylinder by closed geodesics).In particular, if the angle of the affine cylinder is at least π , then there is no saddle con-nection joining the two boundaries of the cylinder. If there were such a segment, it wouldbelong to a direction that is a direction of a hyperbolic geodesic of the cylinder and itwould be forced to cross it. Thus, there is no geometric triangulation of the affine cylinder.Veech proved in some unpublished course notes the converse result (see [9]). Theorem 2.3 (Veech) . A dilation surface X admits a geometric triangulation if and onlyif there is no affine cylinders of angle at least π in X . The idea of the proof is that without affine cylinders of angle at least π , every affineimmersion of an open disk extends to an immersion of its closure. This geometric lemmaallows to build a Delaunay triangulation whose dual is a geometric triangulation. Theproof clearly generalizes to the case of dilation surfaces with geodesic boundary. Besides,a different proof is given in [7] by proving that any dilation surface with boundary that isnot a triangle or an affine cylinder of angle at least π has an interior saddle connection.Even if the surface is not triangulable, a maximal system of non-intersecting saddleconnections cuts out the surface into a union of affine cylinders of angle at least π (theyare clearly disjoint from each other) and a triangulable locus (without affine cylinders ofangle at least π , the construction of a triangulation in the surface with boundary works).It will be used as a substitute of triangulation.3. Horizon saddle connections
A horizon saddle connection is a segment such that for some number k , there is no geo-desic trajectory crossing it strictly more than k times. In particular, if k < k (cid:48) , any k -horizonsaddle connection is a k (cid:48) -saddle connection. There is no such horizon saddle connectionsin translation surfaces. Indeed, the foliation in the generic direction is minimal so everysaddle connection is crossed infinitely many times by some trajectory. Therefore, absence f horizon saddle connections indicates that we are not too far from the case of translationsurfaces.The following proposition is the starting point of the generalization of our triangulabilitycondition. Proposition 3.1.
The saddle connections that belong to the boundary of an affine cylinderof angle at least π are -horizon saddle connections.Proof. A trajectory that enters in such an affine cylinder cannot leave it (they accumulateon an hyperbolic geodesic of the cylinder, see Subsection 2.1 for details). Therefore, nosaddle connection of the boundary can be crossed two times. (cid:3)
The key property of horizon saddle connections is that for any k , there are finitely many k -horizon saddle connections. This property drastically rigidifies the action of SL (2 , R ) .In a similar way, in translation surfaces with poles, most the geometry of a surface isencompassed in a polygon called the core of the surface. The fact that there are finitelymany saddle connections in the boundary of the core makes more rigid the action of SL (2 , R ) on strata of such surfaces, see [6] for details. Lemma 3.2.
In a dilation surface, for any k ≥ , there is at most a finite number of k -horizon saddle connections.Proof. We choose a maximal geodesic arc system of X (that is a decomposition into disjointtriangles and affine cylinders of angle at least π ). For any k , there is a finite number offree homotopy classes of loops that cross each edge of the system at most k times. Thereis a unique geodesic representative in every homotopy class and it minimizes the geometricintersection number. Therefore, there cannot be infinitely many distinct k -horizon saddleconnections. (cid:3) Remark . We actually do not know if there is dilation surface with k -horizon saddleconnections for arbitrary high k . We expect that there is a topological bound on themaximal number of the maximal crossing number of a horizon saddle connection.Just like triangulable dilation surfaces define a SL (2 , R ) -invariant open set, absence of k -horizon saddle connections defines for every k an invariant open set. Proof of Theorem 1.2.
Let X be a surface without k -horizon saddle connections. Wechoose a geometric triangulation of X and a neighborhood of X in the moduli space suchthat the same geodesic triangulation holds in the neighborhood. For any topological arcthat crosses the edges more than k times (there is a finite number of such arcs), its geodesicrepresentative in X is formed by several saddle connections each of which is crossed bytrajectories with an arbitrary high number of intersections. Such a trajectory still existsin a neighborhood of X . Thus, for each arc, there is neighborhood where this arc cannotbe a horizon saddle connection (at least a compact part of the trajectory that crosses morethan k times persists). Since there is a finite number of such arcs, there is a neighborhoodof X where there is no horizon saddle connections. (cid:3) Genus one.
Even in the simplified situation of dilation surfaces of genus one, tri-angulated surfaces and surfaces free from horizon saddle connections define distinct opensubsets of the strata.In [4], Ghazouani proved that any dilation torus with n singularities can be decomposedinto at most n flat and affine cylinders (see Proposition 9). The sketch of the proof is thefollowing: Gauss-Bonnet formula implies that every singularity has a conical angle of π .For a given transverse curve, the homeomorphism of the circle induced by the directional ow changes continuously and monotonically with the direction. Therefore, in some direc-tions, the rotation number is rational and there is at least one close geodesic. This closedgeodesic belongs to a cylinder. This cylinder is bounded by a chain of saddle connectionsjoining marked points. Since they have a conical angle of π , the other side of the chainof saddle connections also belongs to a cylinder.A dilation surface belongs to the locus of triangulated surfaces T D ,n in D ,n if and onlyif every affine cylinder of the decomposition has an angle strictly smaller than π .In the following proposition, we introduce a significantly stronger criterium. Proposition 3.4.
For a dilation surface X of genus one that admits a decomposition into c cylinders. Let A , . . . , A c ⊂ S be the closures of the sets of directions of geodesics of thecylinders of the decomposition. Exactly one of the following two statements holds:(i) c (cid:83) i =1 A i = S and every boundary saddle connection of the decomposition is a -horizonsaddle connection. There are no other horizon saddle connections in the surface.(ii) c (cid:83) i =1 A i (cid:54) = S and every saddle connection is crossed infinitely many times by sometrajectory.Proof. In the first case, there is a closed geodesic in any direction. Therefore, for any tra-jectory α , there will be a closed geodesic γ (or equivalently a chain of saddle connectionsseparating two cylinders of the decomposition) in the same direction α cannot cross. Forany saddle connection β that is the boundary of two cylinders, it is clear that if α crosses β two times, then it should cross γ at least one time. Conversely, any saddle connectionthat does not separate cylinders is crossed infinitely many times by the closed geodesics ofthe cylinder.In the second case, we consider a closed geodesic γ whose direction is in c (cid:83) i =1 A i and adirection θ in the complement of c (cid:83) i =1 A i that is not the direction of some saddle connectioneither (since the complement is any open set, it is always realizable). Then we consider thefirst return map φ on γ defined by the directional flow in direction θ (every trajectory insuch a direction eventually leaves every cylinder whose geodesics have directions c (cid:83) i =1 A i ).Map φ is a homeomorphism of the circle. There are two cases. If the rotation numberof φ is irrational, then the directional foliation in direction θ has minimal leaves that aredense in the whole surface and cross every saddle connection infinitely many times. If therotation number is rational, then there is a closed geodesic in direction θ , then it definesa cylinder whose closed geodesics intersect every cylinder of the first decomposition. Thiscylinder can be completed to form another decomposition into cylinders. Closed geodesicsof a given decomposition represent the same free homotopy class. The intersection numberof the loops of the two decompositions is not trivial. Therefore, every saddle connection inthe boundary of a cylinder of the first decomposition is crossed infinitely many times bysome trajectory. (cid:3) In any stratum D ,n ( λ ) , we define H ,n ( λ ) to be the SL (2 , R ) -invariant open set (seeTheorem 1.2) formed by surface without horizon saddle connections. We have the followingstrict inclusions of invariant open sets: H ,n ( λ ) (cid:40) DT ,n ( λ ) (cid:40) D ,n ( λ ) e proved a dichotomy among dilation tori between those with horizon saddle connec-tions (where one cylinder decomposition covers every direction) and horizon-free dilationtori. In the first case, there is only one cylinder decomposition because any other de-composition would contain a closed geodesic that would cross every cylinder of the firstdecomposition. Horizon saddle connections make this situation impossible. Therefore,the shape of the cylinders of the unique cylinder decomposition provides global coordi-nates for this locus in the moduli space. SL (2 , R ) acts separately on each cylinder of thedecomposition. We should not expect any interesting recurrence behaviour.3.2. Surfaces with chambers.
A natural question in the study of moduli spaces of geo-metric structures is about the connected components. In [2], the authors study the space DT , of triangulable dilation surfaces of genus two with only one conical singularity. Thisspace fail to be connected and there is an exceptional connected component formed bysurfaces split into two chambers separated by a closed saddle connection. This cannothappen in the framework of translation surfaces and we can understand this situation us-ing horizon saddle connections.A chamber is a dilation surface with boundary formed by a pentagon with two pairs ofparallel sides glued on each other. The remaining side is the boundary of the chamber.Gluing the boundaries of two chambers provides a dilation surface of genus two with onesingularity.Clearly, the boundary of a chamber is a -horizon saddle connection. Since such a cham-ber cuts out the surface into two connected components, a trajectory that crosses it shouldcross it again in the reverse way.Considering only surfaces without horizon saddle connections, we eliminate the excep-tional component formed by surfaces with two chambers. Thus, we could expect that strataof dilation surfaces without horizon saddle connections have exactly the same connectedcomponents as translation surfaces.3.3. Hopf and quasi-Hopf surfaces.
Hopf surfaces are examples of non-triangulabledilation surfaces. They are the only case of surfaces of genus at least two where the Veechgroup is not discrete (see Section 4 for details). They are also the only surfaces whosesaddle connections all belong to the same direction.
Definition 3.5.
A Hopf surface is a dilation surface covered by a union of disjoint affinecylinders of angle kπ where k is an integer number and such that the saddle connections ofthe boundary of every affine cylinder lie in the same directions. In particular, every saddleconnection of a Hopf surface is a -horizon saddle connection.We introduce a mild generalization of Hopf surfaces. Definition 3.6.
A dilation surface X is quasi-Hopf if there is a pair of directions α and β such that X is covered by a union of disjoint affine cylinders whose boundary saddleconnections belong to directions α or β . We distinguish integer affine cylinders (whoseangle is an integer multiple of π ) from non-integer affine cylinders.Some quasi-Hopf surfaces can be conjugated (using the action of SL (2 , R ) ) to surfaceswhose affine cylinders have an angle of π . They are examples of triangulable surfaces thatadmits nevertheless horizon saddle connections, see Figure 2. Proposition 3.7.
In a quasi-Hopf surface, if the union of the directions of the closedgeodesics of any two consecutive cylinders in the decomposition is the whole circle of di-rections (except perhaps α and β ), then every boundary saddle connection of a cylinder igure 2. A quasi-Hopf surface of genus with four conical singularitiesof angle π . It is covered by six affine cylinders of angle π . The eight saddleconnections drawn are horizon. in the decomposition is a -horizon saddle connection. There is no other horizon saddleconnection.Proof. A horizon saddle connection cannot cross a cylinder because it would be crossedinfinitely many times by a closed geodesic. Therefore, they belong to the boundary of thecylinders of the decomposition. Each of them is a -horizon saddle connection because itis the boundary of two cylinders that together admits a closed geodesic in every direction(excepted α and β ). Therefore, any trajectory crossing such a saddle connection remainsforever in one of the two cylinders. (cid:3) The action of the one-parameter subgroup A of diagonal matrices of SL (2 , R ) contractingdirection α and expanding direction β on a quasi-Hopf surface is interesting. It cruciallydepends on the commensurability of the dilation ratios of affine cylinders. Proposition 3.8.
We consider a quasi-Hopf surface X formed by affine cylinders whoseboundary saddle connections belong to exactly two direction α and β . The A -orbit of X in the stratum is closed if and only the dilation ratios of non-integer affine cylinders arelog-commensurable.Proof. For any quasi-Hopf surface, we consider the set C of non-integer affine cylinders (wealso choose an orientation on them in such a way that it goes from direction α to direction β ). For every cylinder i ∈ C , λ i is the linear holonomy along the hyperbolic geodesics ofthe cylinder. We consider the flow A t that contracts direction α with a factor e − t andexpands direction β with a factor e t .The dilation action preserves any affine cylinder of C but modifies their twist, see Sub-section 2.4. Every non-integer affine cylinder of the decomposition is preserved (with itstwist) by any element A t of the flow such that t = ln ( λ )2 . Besides, there could be additionalsymmetries such that the subgroup that preserves the cylinder and its twist is generatedby t = θ i = ln ( λ )2 d i where d i is an integer. These exponents ( θ , . . . , θ c ) define a charac-teristic ratio . The action of A t on a cylinder i only depends on the class of t in R /θ i Z .Equivalently, we could consider the image of t in R C / Z C by the map t (cid:55)→ ( θ t, . . . , θ c t ) .Clearly, the A -orbit is closed if and only if exponents ( θ i ) i ∈ C are commensurable. In otherwords, exponents ( ln ( λ i )) i ∈ C should be commensurable. (cid:3) The latter condition defines the subclass of rational quasi-Hopf surfaces .3.4.
Dilation surfaces with finitely many saddle connections.
An open questionraised in [1] is about characterization of dilation surfaces with finitely many saddle connec-tions. A related question asks if for any dilation surfaces, every point belongs to a saddleconnection or a closed geodesic. Surfaces with finitely many saddle connections provideseasy examples of this phenomenon. It also exemplifies -horizon saddle connections, see igure 3. A dilation surface where sides h, i, j, k, l, m, n, o are -horizonsaddle connections (boundaries of affine cylinders of angle at least π )whereas saddle connection f is -horizon. Trajectory s is an example oftrajectory cutting twice f .Figure 3. We can generalize this example to get k -horizon saddle connections for an arbi-trary number k .We provide here a characterization of dilation surfaces with finitely many saddle connec-tions in terms of types of trajectories. Trajectories are oriented. We say that a trajectoryis critical if it starts from a conical singularity (and is not a saddle connection). Theorem 3.9.
For a dilation surface X , the following statements are equivalent:(i) There are finitely many saddle connections in X .(ii) Every trajectory either hits a conical singularity or accumulates on a closed geodesicof an affine cylinder of angle at least π .Proof. We first prove that Proposition (i) implies Proposition (ii). For a surface X withfinitely many saddle connections, around every conical singularity, the directions of criticaltrajectories (finite cover of the unit circle) are divided into finitely many sectors separatedby the directions of saddle connections. The critical trajectories of a given sector form theimmersion of an infinite cone in X for the following reason. If the immersion of the singularsector cannot be extended in some point, then, up to zooming close to this point, we canfind an embedding of the disk that does not extend to its boundary. Veech’s theorem (seeAppendix of [2]) then implies that the sector in fact belongs to a hyperbolic cylinder ofangle at least π (and the angle of the sector is at least π ).Thus, a small enough neighborhood of the singularity in such a cone either belongs to thetriangulable locus or to a cylinder of angle at least π (the boundary of the triangulablelocus is a union of saddle connections). In the latter case, the angle of the sector is exactly π and every critical trajectory of the sector belongs to this cylinder (and accumulates onone of its hyperbolic geodesics). The other sectors have an angle strictly smaller than π because there are enough saddle connections to get a geometric triangulation.There is an incidence relation on the set of sectors. The left boundary of the image of theinfinite cone of a sector A is formed by saddle connections and at most one infinite criticaltrajectory. There are finitely many saddle connections so either the boundary contains aninfinite critical trajectory or there is a periodicity in the saddle connections of the bound-ary. In the latter case, these saddle connections are the boundary of a cylinder. Since thereare infinitely many saddle connections in a flat cylinder or a cylinder of angle smaller than π , this cylinder is one of those with an angle of magnitude at least π . If there is no suchperiodicity, the boundary contains an infinite critical trajectory that belongs to anothersector B . The critical trajectories of sector B that belong to the directions of sector A areentirely contained in the infinite cone of sector A . This implies in particular that the totalangle of sector B is strictly bigger than that of sector A , see Figure 4. Since there arefinitely many sectors, the sectors of the triangulable locus of X with the biggest angle are igure 4. The infinite cones defined by sectors A and B are in blue andred. The intersection of these two cones is in purple.thus incident to sectors of angle π (those that belong to affine cylinders of angle at least π ). If every critical trajectory of a sector finally enters into some affine cylinders of angleat least π , then it is clearly the same for every sector that is incident to it (the property toaccumulate in some closed geodesic of a cylinder is an open condition of the direction ofthe trajectory). We prove thus step by step that in every sector, every critical trajectoryends in some affine cylinder of angle at least π .Then, for any direction θ (without loss of generality, we assume it is the horizontal di-rection), we draw the saddle connections and the critical trajectories in this direction. Weget finitely many infinite horizontal strips (no flat cylinders since there are finitely manysaddle connections). The intersection of any critical trajectory with the triangulable locusis compact. Therefore, in each of these strips, each horizontal trajectories goes from thesame affine cylinder to the same affine cylinder.Then, we prove that that Proposition (ii) implies Proposition (i). Proposition (ii) impliesin particular that every trajectory starting from a conical singularity either is a saddleconnection or accumulates on a closed geodesic of an affine cylinder. If moreover thereare infinitely many saddle connections in the surface, then there is a conical singularity A starting from which a critical trajectory γ is approached in direction by a sequence ofsaddle connections ( γ n ) n ∈ N starting from A . Without loss of generality, we assume thesequence is monotonically converging in the counterclockwise direction. If γ finally entersinto an affine cylinder of angle at least π and accumulates on a closed geodesic, then thisis the same for every critical trajectory starting from A with a direction in some openneighborhood of θ . This contradicts accumulation of saddle connections in this direction.On the contrary, if γ is a saddle connection from A to another singularity B , then sequence ( γ n ) n ∈ N also accumulates on the trajectory δ starting from B in the direction obtained bya counterclockwise rotation of angle π from the ending direction of γ . The infinite sequenceof saddle connections just passes γ and B from the right, see Figure 5. The same reasoningholds for δ if it is also a saddle connection. Finally, we automatically get a contradictionwith the existence of an accumulation direction of saddle connections. Consequently, thereare finitely many of them. (cid:3) Veech groups
General results.
The first structure theorem for Veech groups of dilation surfaceshas been proved in [2]. igure 5. Saddle connection γ continued by critical trajectory δ on whichaccumulates a sequence of saddle connections. The infinite open cone start-ing from sector B formed by trajectories entering in some affine cylinder isin yellow. Theorem 4.1 (Dichotomy theorem) . Let X be a dilation surface of genus g ≥ , thenthere are two possible cases:(i) X is a Hopf surface and V ( X ) is conjugated to the subgroup of upper triangular elementsof SL (2 , R ) ;(ii) V ( X ) is a discrete subgroup of SL (2 , R ) . The condition on the genus is necessary because there are some affine tori whose Veechgroup is SL (2 , R ) . They are constructed as the exponential of some flat tori (see Subsec-tion 4.4 of [2] for details).Parabolic directions in translations surfaces correspond to cylinder decompositions. Thereis a similar result for dilation surfaces. Proposition 4.2.
Let X be a dilation surface such that V ( X ) contains a parabolic ele-ment. Then the decomposition of X into invariant components of the parabolic directionis formed by affine and flat cylinders whose boundary saddle connections belong to theparabolic direction. Moduli of the flat cylinders should be commensurable.Proof. We first follow a part of the proof of Proposition 4 in [2] to prove that in parabolicdirections, critical trajectories are saddle connections. By contradiction, we assume thereis an infinite critical trajectory γ in a parabolic direction. Without loss of generality, weassume γ is vertical. Since it is infinite, γ has an accumulation point x . Any neighborhood U of x is crossed infinitely many times by γ . Accumulation point x is approached in U by an infinite sequence of vertical lines. Any element of GL (2 , R ) that acts as a scalingon γ also acts as a scaling on U ∩ γ . Consequently, it acts as the same scaling on U .Therefore, any element of GL (2 , R ) that preserves the vertical direction in this dilationsurface is the identity up to a scalar factor. It is thus the identity in the Veech group ofthe dilation structure (elements are considered up to a scalar factor since lengths are notglobally preserved). We proved that there cannot be parabolic elements that preserve thedirection of an infinite critical trajectory. Every critical trajectory in a parabolic directionis a saddle connection.Cutting along saddle connections in the parabolic direction provides a decomposition ofthe dilation surface into invariants components. Since conical singularities are cut outinto sectors of angle equal to π , discrete Gauss-Bonnet implies that the components aretopological annuli. These components are thus either flat cylinders or affine cylinders theangle of which are integer multiples of π . Since the flat cylinders are preserved by theparabolic element, their moduli should be commensurable. (cid:3) yperbolic directions in a dilation surface are directions that are preserved by an hy-perbolic element of the Veech group of the surface. There is no closed geodesics nor saddleconnections in hyperbolic directions of translation surfaces. Such phenomena can appearfor some specific dilation surfaces. Lemma 4.3.
Let X be a dilation surface such that V ( X ) contains a hyperbolic element φ . If a saddle connection γ lies in a hyperbolic direction, then γ belongs to the commonboundary of two cylinders (see Figure 3 for examples of cylinders with only a part of theboundary in common).Proof. Without loss of generality, we can assume γ is vertical and that φ is expanding inthe vertical direction and contracting in the horizontal direction. If γ is an edge of sometriangle A (whose other sides are not in the vertical direction), then φ provides a sequenceof triangles bounding γ and such that the directions of the other edges approach the verticaldirection. There cannot be a horizontal side α in triangle A e because in this case the actionof φ on the triangle would provide another triangle B with γ as a vertical side and a portionof α as a horizontal side. This is impossible because there is no singularity in the interiorof a saddle connection.If none of the sides is horizontal, their local length ratio (see Subsection 2.1 for details)is bounded by below by the ratios of their vertical coordinates. Therefore, either there isa conical singularity inside γ (which is impossible) or there is another saddle connectionthat forms an angle of π with γ . We can iterate this procedure to get a chain of saddleconnections in the vertical direction. Since there is a finite number of them, these saddleconnections are the boundary of a cylinder. If γ is not an edge of a triangle, then it is aboundary saddle connection of an affine cylinder of angle at least π . (cid:3) Proposition 4.4.
Let X be a dilation surface such that V ( X ) contains a hyperbolic element φ . If a saddle connection or a closed geodesic belong to a hyperbolic direction, then X is aquasi-Hopf surface.Proof. If an affine cylinder is preserved by an hyperbolic element of the Veech group, thenits boundary saddle connections belong to invariant directions of the element. Besides,there is no flat cylinder whose closed geodesics belong to an hyperbolic direction. In agiven hyperbolic direction, there are at most finitely many saddle connections or closedgeodesics. Therefore, at least one power of the hyperbolic element preserves the cylinderthey belong to. Consequently, we get invariant cylinders whose boundary is connectedwith other invariant cylinders until all the surface is covered. (cid:3)
Veech groups in presence of horizon saddle connections.
There is a lot of ex-amples of dilation surfaces with many directions of horizon saddle connections. FollowingProposition 3.4, some dilation tori feature cylinder decompositions such that every bound-ary saddle connections is horizon. We can also glue chambers (see Subsection 3.2) on theboundary of a polygon. If there are at least three distinct directions of horizon saddleconnections, then the Veech group of the surface is finite.
Proof of Theorem 1.3.
The set of directions of horizon saddle connections is finite andglobally preserved by elements of the Veech group. Therefore, for any element φ of theVeech group, there is an integer n such that φ n preserves every direction of horizon saddleconnection. Since they are at least three, then φ n is the identity. Thus, any element of theVeech group is elliptic. Following Theorem 1 in [2], the Veech group of a dilation surfacethat is not a Hopf surface is discrete. Therefore, the Veech group of X is conjugated to afinite rotation group. (cid:3) In the case of dilation surfaces with exactly two directions of horizon saddle connections,we have to take into account the specific situation of quasi-Hopf surfaces we introducedpreviously. roof of Theorem 1.4. A parabolic element cannot globally preserve two distinct directionsso the Veech group is discrete (see Theorem 1 of [2]) and contains only elliptic and hyper-bolic elements. Elliptic elements globally preserve the two distinct directions so they canonly preserve each of them or intertwine them. Therefore, elliptic elements form a finitegroup conjugated to the trivial group or the finite group of rotations of order two or four.If the Veech group contains a hyperbolic element, then this element preserves the two di-rections of horizon saddle connections. Every hyperbolic element preserves the same pairof directions. Since the surface is not Hopf (otherwise there would be only one directionof horizon saddle connections), the Veech group is discrete so the hyperbolic elements be-long to the same cyclic group. Following Proposition 4.4, the surface is then quasi-Hopf.Proposition 3.8 provides the condition a quasi-Hopf surfaces should satisfy to admis ahyperbolic element of the Veech group that preserves the two directions of the boundarysaddle connections of the cylinders. (cid:3)
The case of dilation surfaces with only one direction of horizon saddle connections in-cludes that of Hopf surfaces.
Proof of Theorem 1.5.
The elliptic elements preserve the direction of horizon saddle con-nections so they belong to the group {± Id } . Every parabolic element preserves this direc-tion. Therefore, either the Veech group is not discrete (the surface is then a Hopf surfaceand its described by Theorem 4.1) or the parabolic elements all belong to the same cyclicparabolic group. If there is an hyperbolic element in the Veech group of the surface, thenit preserves the horizon saddle connections. These saddle connections thus are boundariesof cylinders whose boundary saddle connections belong to one of the two hyperbolic di-rections. The surface is then quasi-Hopf. If there is only one direction of horizon saddleconnections, then the surface is Hopf. (cid:3) Acknowledgements.
The author is supported by a fellowship of Weizmann Instituteof Science. This research was supported by the Israel Science Foundation (grant No.1167/17). The author thanks Selim Ghazouani for interesting discussions. The authoralso thanks the anonymous referee for their patience and their valuable remarks.
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Faculty of Mathematics and Computer Science, Weizmann Institute ofScience, Rehovot, 7610001, Israel
E-mail address : [email protected]@weizmann.ac.il