Genus one Birkhoff sections for geodesic flows
We prove that the geodesic flow on the unit tangent bundle to every hyperbolic 2-orbifold that is a sphere with 3 or 4 singular points admits explicit genus one Birkhoff sections, and we determine the associated first return maps.
Key words and phrases:Birkhoff section, geodesic flow, suspension flow
2000 Mathematics Subject Classification:Primary 37D40, 53D25; Secondary 37C27
In this article we investigate from a topological viewpoint a particular family of 3-dimensional flows, namely the geodesic flows on the unit tangent bundle to a hyperbolic 2-orbifold.
Our main subject of investigation is Birkhoff sections. A Birkhoff section for a 3-dimensional flow is a surface whose boundary is the union of finitely many periodic orbits of , and such that every other orbit of intersects the interior of the surface within a bounded time. Introduced by Poincaré and Birkhoff , Birkhoff sections are useful, when they exist, because they provide a description of the flow, minus the orbits forming the boundary of the section, as the suspension flow constructed from the first return map on the section.
Fried proved  that every transitive 3-dimensional Anosov flow admits Birkhoff sections. (We recall that a 3-dimensional flow associated with a vector field on a Riemannian manifold is said to be of Anosov type  if there exists a decomposition of the tangent bundle as , such that is uniformly contracted by the flow and is uniformly contracted by the inverse of the flow. A flow is called transitive if it admits at least one dense orbit.) Now Fried’s proof gives no indication about the complexity of the Birkhoff sections, in particular their genus, and the following is still open:
Question A (Fried).
Does every transitive 3-dimensional Anosov flow admit a genus one Birkhoff section?
A positive answer, that is, the existence of a genus one Birkhoff section, gives rich information about the considered flow. Indeed, if is a Birkhoff section for an Anosov flow, the stable and unstable foliations induce two transverse one-dimensional foliations on that are preserved by the first return map. Moreover, Fried showed  that the first return map is a pseudo-Anosov diffeomorphism. If has genus one (and some condition on the boundary of is satisfied), the first return map is even an Anosov diffeomorphism, and, therefore, it is conjugated to a linear automorphism of the torus—hence it essentially reduces to an element of .
Here we consider Question A for particular 3-dimensional flows, namely the geodesic flows associated with hyperbolic 2-dimensional objects. If is a cocompact Fuchsian group , the quotient of the hyperbolic plane is a compact hyperbolic 2-orbifold. Then the geodesic flow on projects on . The latter is a 3-dimensional Anosov flow, called the geodesic flow on . If no point of has a nontrivial stabilizor, is a hyperbolic surface, and the answer to Question A for the corresponding geodesic flow is positive: a construction of Birkhoff  exhibits two genus one Birkhoff sections. Otherwise, is a 2-orbifold with singular points, and the only known results about Birkhoff sections for the corresponding flows concern orbifolds which are spheres with three singular points of respective order and , with , for which we proved  that every collection of periodic orbits bounds a Birkhoff section, but with no control of the genus of the sections. The first aim of the current article is to establish the following general positive result:
For every hyperbolic 2-orbifold that is a sphere with 3 or 4 singular points, the geodesic flow on admits some (explicit) genus one Birkhoff sections.
Note that Theorem A essentially covers all spherical orbifolds with at most four singular points. Indeed, there exist 2-orbifolds which are spheres with zero, one or two singular points, but they are either bad orbifolds in the sense of Thurston  or of spherical type. In both cases, the associated geodesic flows do not have the Anosov property we are interested in.
As explained above, when an Anosov flow admits a Birkhoff section of genus one, the associated first return map is particularly simple, namely it is conjugated to an automorphism of the torus, hence it is specified by a conjugacy class in . Then the question naturally arises of determining which conjugacy classes occur in this way. For instance, writing for and for , and denoting by the linear automorphism of the torus associated with a -matrix , Ghys , Hashiguchi  and Brunella  showed that, in the case of a hyperbolic surface of genus , the first return maps associated with the above mentioned Birkhoff sections are conjugated to the automorphisms and .
Question B (Ghys).
Let be a matrix in with . Does there exist a hyperbolic 2-orbifold such that the geodesic flow on has a genus one Birkhoff section whose associated first return map is conjugated to ?
We recall (see for instance ) that every element of that is not of finite order is conjugated to a finite product of the matrices and , and that the product is unique up to cyclically permuting the factors and up to exchanging and . Moreover, the element is hyperbolic if and only if the product contains both and . So Question B may be rephrased as a question about which expressions in and can occur. We prove:
For every matrix that can be expressed with at least one and one and at most four or at most four , there exists a hyperbolic 2-orbifold and a genus one Birkhoff section for the geodesic flow on whose first return map is conjugated to .
The matrix has trace and, for , it is eligible for Theorem B. So we immediately deduce:
For every larger than 2, there exists a hyperbolic 2-orbifold and a genus one Birkhoff section for its geodesic flow such that the first return map has trace .
Theorems A and B strengthen the close connection first described by Fried  and Ghys  between the two main known classes of Anosov flows in dimension 3, namely the geodesic flows on negatively curved orbifolds, and the suspension flows associated with linear automorphisms of the torus. In particular, as they go in the direction of positive answers to Questions A and B, a possible interest of these results is to support the idea that there might exist only one transitive Anosov flow up to virtual almost conjugation: two flows on two 3-manifolds are said to be almost conjugated if there exist two finite collections of periodic orbits of such that the manifolds and are homeomorphic and that the induced flows are topologically conjugated, and “virtually” is added when the property may involve a finite covering. A positive answer to Question A would imply that every transitive Anosov flow is almost conjugated to some suspension flow of a torus automorphism, whereas a positive answer to Question B would imply that any two suspension flows would be virtually almost conjugated.
For all with and , the geodesic flow on the 3-manifold admits an explicit Birkhoff section of genus one; the associated first return map is conjugated to
For all larger than with , the geodesic flow on the 3-manifold admits two explicit Birkhoff sections of genus one; the associated first return map are conjugated to
For all larger than with , the geodesic flow on the 3-manifold admits six explicit Birkhoff sections of genus one; the associated first return map are conjugated to
and to the five other classes obtained by permuting the exponents in the latter expression (up to cyclic permutation the 24 possible permutations give rise to six classes).
Proposition C involves three cases. The strategy of proof is the same in every case, but the difficulty is increasing, so that we shall present each proof in a separate section. Starting from the orbifold (resp. , resp. ), the idea is to explicitly construct two (resp. three, resp. four) Birkhoff sections for the geodesic flow, to compute their Euler characteristics (thus checking that their genus is one), and to find a suitable pair of loops on each of them that form a basis of their first homology group. In the first case, these two Birkhoff sections correspond to an avatar of Birkhoff’s construction , but, in the other two cases, the method is new. Then, the idea is to start from one Birkhoff section, to follow the geodesic flow until one reaches another section, and to look at how the loops on the first section are mapped on the second one. Our particular choice of the sections will guarantee that this application is described by a simple matrix of the form . Iterating this observation twice (resp. three, resp. four times), one obtains the expected form for the first return map.
Let us conclude this introduction with two more remarks about particular cases. First, the case in Proposition C leads to a toric section with first return map . This matrix is known to correspond to the monodromy of the figure-eight knot. Therefore, after removing one periodic orbit, the geodesic flow of the -orbifold is conjugated to the suspension flow on the complement of the figure-eight knot—one of the first flows whose periodic orbits have been studied from the topological point of view . Concerning the topology of the underlying 3-manifolds, this implies that the Seifert fibered space can be obtained from by a surgery on the figure-eight knot. A celebrated theorem by Thurston [13, Theorem 5.8.2] states that, for every hyperbolic knot in , only finitely many surgeries yield a non-hyperbolic 3-manifold. Our construction exhibits such an example for the figure-eight knot.
Finally, when goes down to the limit values or , the corresponding orbifolds with 3 singular points are of Euclidean type. The same situation occurs when reaches the limit value . It turns out that the surfaces obtained by extending the construction of Proposition C still are Birkhoff sections for the corresponding geodesic flows (which are no longer of Anosov type), and that the first return maps are given by the same formulas. But, as can be expected, the associated matrices are of parabolic type, namely they are powers of the matrix .
Acknowledgments: It is a pleasure to thank Étienne Ghys for many discussions on topics related to this article, and Tali Pinski for an invitation and a collaboration which are at the origin of this work.
1. The case : symmetric boundary
Here we establish the first case in Proposition C, namely that of an orbifold with three singular points among which one corresponds to an angle . In the whole section, denote two positive integers satisfying . Then is the unique hyperbolic orbifold with three singular points, called , of respective order . We denote by its unit tangent bundle, and by the geodesic flow on . Abusing notation, we also use to refer to a fixed triangle with respective angles in the hyperbolic plane, and we denote by the image of under the symmetry around . The quadrangle is a fundamental domain for the orbifold .
We first describe two particular Birkhoff sections for , called and . As we shall see, these two sections turn out to be isotopic. Indeed we will show that, starting from and following for some (non-constant) time, we first reach , and then come back to . The second step consists in computing the corresponding homeomorphisms from to , and from to . The benefit of considering two sections instead of one is that the associated matrices are especially simple (these are companion matrices). The first return map on each of the two sections is then obtained by composing the homeomorphisms.
1.1. Two Birkhoff sections
The construction of the surfaces and that we propose is similar to Birkhoff’s original construction  of sections for the geodesic flow (although Birkhoff only dealt with surfaces, not with orbifolds), and to A’Campo’s construction  of fiber surfaces for divide links.
We call (resp. ) the set, in , of all unit tangent vectors to that point into the -part (resp. the -part), plus the whole fiber of the point .
The surfaces and are topological surfaces. They are both made of one rectangle whose vertical edges are identified using a rotation at (see Figure 1). Actually we could smooth them and preserve all their properties at the same time, but we do not need that. The two surfaces intersect along the fiber of . For a unit tangent vector at , we fix the convention that, at the point , the surface is infinitesimally pushed along the direction of if points into the -part, whereas is infinitesimally pushed along the direction of if points into the -part. This convention is used only for ordering the intersection points of the lift of a geodesic distinct from with the two surfaces and .
The surfaces and both have one boundary component, namely the lift of in . They are genus one Birkhoff sections for the geodesic flow . Every orbit of distinct from intersects both surfaces alternatively.
That both surfaces have boundary is clear from the definition.
The lifts of in the universal cover of divide into compact -gons and -gons. Let be any geodesic not in the -orbit of , and let be the corresponding orbit of . Then crosses some copy of within a bounded time. When, at the intersection point, goes from an -gon to a -gon, intersects , and when goes from a -gon to an -gon, intersects . Now could also go through directly from a -gon to another one, or from an -gon to another one. This can only happen above a copy of , in which case intersects both surfaces at the same time. So, in all cases, intersects both surfaces within a bounded time. Therefore both and are Birkhoff sections. Owing to the convention about the fiber of , the curve intersects both surfaces alternatively.
As for the genus, since both surfaces consist of one rectangle whose vertical edges are identified using a rotation at , they are made of one 2-cell, four 1-cells (two horizontal and two vertical) and two 0-cells (in the fiber of ), so their Euler characteristics is . Since the surfaces have one boundary component, they are tori. ∎
The loops and generate , and and generate .
The loops and both lie in , and they intersect each other exactly once, namely in the fiber of the intersection between and , at the vector that points toward . This is enough to ensure that these loops generate the fundamental group of a once-punctured torus. The case of the other pair is similar. ∎
1.2. First return maps
For every tangent vector that lies in the surface and not in the fiber of , we define to be the first intersection between the orbit of starting from and the surface . For a tangent vector at that points into the -part, we define as itself (seen as an element of ). In this way, we obtain a map from the whole surface into . Then extends to a map from to , denoted by .
The map is a homeomorphism from to of Anosov type. It is conjugated to the linear homeomorphism whose matrix with respect to the two bases and is .
The continuity of is clear. Since is obtained by following the orbits of , the injectivity is also clear. Therefore is a homeomorphism, and so is . The weak stable and unstable foliations of the geodesic flow in induce on two foliations which are respectively contracting and expanding. Therefore, by Fried’s argument , is of Anosov type.
For determining the images of and , we unfold around the point (see Figure 2) by gluing copies of the quadrangles . We thus obtain a -gon, denoted by (with ). We denote by the middle of the edges (with and , and for , we call the intersection of with .
The curve is made of those vectors above that point toward . Therefore, under the geodesic flow, enters the -gon . It reaches first the fiber of , and then continues until it reaches (the fibers of the points of) the side(s) opposite to in . At that moment, it points into the -part, and therefore belongs to : this is . Depending on the parity of , this curve lies in the fibers of the segment or in the fibers of and (see Figure 2). In both cases, it projects, after modding out by the rotation at , to the curve , yielding .
For , we see that the vector of lying above (which points to ) is fixed by , by definition. When is applied, that is, when we follow the flow , the rest of goes through the -gon . Therefore is made of all the vectors in the fibers of that do not lie above and whose opposite points toward (see Figure 2 bottom left). Since we are only interested in the class of in the first homology group, we can apply any convenient isotopy to inside . We thus rotate all vectors (without changing the fibers in which they lie) in the following way: the vectors in the fibers of the segment are rotated so that they point toward (note that the vector at does not change), similarly the vectors in the fibers of are rotated so that they point toward , and all other vectors (corresponding to vectors in the fibers of , or , or …, or , or ) are rotated so that their opposite points toward . The obtained curve (see Figure 2 bottom right) is equal, in the orbifold , to the concatenation of the opposite of and times . Indeed the parts above and , with the given orientation, add up to the opposite of , and the parts above add up to times . ∎
Arguing similarly, we define a map from to in a way that is exactly symmetric to what we did for .
Proof of Proposition C (first case).
Consider the two Birkhoff sections and given by Definition 1.1 and Lemma 1.2. By Lemma 1.2, starting from any point of and following for some time (which is bounded, but not the same for all points), we reach the surface , and then reach again. Therefore the first return map on is obtained by applying first and then . When compactifying, we obtain a Anosov diffeomorphism which is the product of and . Since an Anosov diffeomorphism of the torus is always conjugated to its action on homology, by Lemma 1.4, the first return map is conjugated to the product .
Since every matrix of the form is equal to , the previous product is equal to , which is conjugated to . For , the exponent is negative, and therefore the formula can be simplified. An easy computation leads then to . ∎
2. The case : non-symmetric boundary
We now turn to the second case in Proposition C, namely that of an orbifold with three singular points of index larger than . In the whole section, denote three integers larger than 2 and satisfying . Then is the unique hyperbolic orbifold with three singular points, called , of respective order . We denote by its unit tangent bundle, and by the geodesic flow on . As in Section 1, we fix a triangle in the hyperbolic plane with respective angles , and we denote by the image of under the symmetry around . The quadrangle is a fundamental domain for the orbifold .
The idea is to mimic the two steps of the case , using three surfaces instead of two. The surfaces we use here cannot be described using Birkhoff’s construction: when we start from a collection of closed geodesics and try to apply the latter, the obtained surface has genus at least two. On the other hand, the second step is similar to that of Section 1, with three maps instead of two.
We now adapt Birkhoff’s construction and describe a surface whose boundary in is the lift of . First we observe that divides the orbifold into five regions: three regions that contain one of the singular points, hereafter called the -, the -, and the -parts, and two triangles delimited by , hereafter called the orthic triangles. Abusing notation, we use the same names when working in the universal cover of .
The surface is the closure of the set of all unit vectors positively tangent to the family , that is, the set .
Taking the closure of is equivalent to adding to the tangent vectors to , the vectors at that point into the -part, the vectors at that point into the -part, and the vectors at that point into the -part or into the orthic triangles.
Next, choosing two other foliations and of the orthic triangles by convex butterflies centered at and respectively, we similarly define surfaces and as the closures of the set of all unit vectors positively tangent to and .
Now, we introduce to be the set of all vectors of that point directly toward the point and to be the set of all vectors whose opposite point directly toward . Then, as in Section 1, and are loops in . The loops and in and and in are defined similarly (see Figure 3).
The surfaces , and have one boundary component, namely the lift of . All three surfaces are genus one Birkhoff sections for . The two curves (resp. , resp. ) form a basis of (resp. , resp. ).
Let be a geodesic in that is not in the -orbit of . Define the code of to be the bi-infinite word in the alphaber describing the different types of the regions crossed by : we write , or when goes through the interior of a -part, a -part, or a -part, and do not write anything when goes through any of the two types of orthic triangle (this coding is neither injective nor surjective, but this is of no importance). Since is not in the orbit of , it cannot cross more than two consecutive orthic triangles. Therefore, the code is indeed bi-infinite. Now, for every factor or , there exists exactly one point where is tangent to the family and contributes that factor to . Since every bi-infinite word contains infinitely many such factors, the lift of in intersects infinitely many times.
For the genus, we observe that is made of the closures of two discs, corresponding to the lifts of the two orthic triangles. With the considered decomposition, there are six edges corresponding to the different segments of , plus three edges in the fibers of and . There are also six vertices, two in each of the fibers of and . Adding the contributions, we obtain for the Euler characteristics of , and therefore one for its genus. For the basis of , as in Lemma 1.2, we observe that the considered loops intersect each other once.
The proof for the other two surfaces is similar. ∎
2.2. First return maps
We now mimic the construction in Section 1 of the maps and , with one difference: the maps , and to be defined all have fixed points (but none in common), which correspond to the intersection points between the surfaces , and . For every tangent vector that lies in the surface and not in the fiber of , we define to be the first intersection between the orbit of the geodesic flow starting from and the surface . For a tangent vector at that points into the -part, we define as itself.
The map is a homeomorphism from the torus to the torus . It is conjugated to the linear homeomorphism whose matrix with respect to the bases and is .
The argument is similar to the one for Lemma 1.4. The continuity and the injectivity of need no new argument.
We now unfold around the point by gluing copies of the quadrangle . We obtain a -gon, that we denote by (with , and , or simply by . We similarly denote by and the corresponding images of and .
We now determine the first intersections of the orbits of that start on the curves and in with the surface . By definition, these curves are the images of and under .
The curve is made of those unit vectors tangent to the family that point toward . Therefore, when following the flow , the points of enter the -part. They first reach the fiber of , and then continue on the other side of until they reach the orthic triangles opposed to the starting ones in the -part. At that moment, the orbits intersect the surface when the geodesics they are following are tangent to the family . They form then a curve that connects to and then to if is even and to to if is odd (see Figure 4 left). In both cases, the curve we obtain projects, in , to the loop , yielding .
As for , we see that the (unique) vector of lying above (which, by definition, points toward ) is fixed by . When is applied, that is, when we follow the geodesic flow, the two parts of that are close to (one delimited by and the intersection of with in the orthic triangle and one delimited by and the intersection of with in ) stay in their respective orthic triangles and become tangent to the family on two curves that join to and to respectively. When the orientation is taken into account, the union of the latter two curves is isotopic to (see Figure 4 right).
The rest of goes through the -part and become tangent to above curves that respectively connect to , to , …, to in the corresponding orthic triangles. Each of these curves can be deformed by isotopy to the corresponding image of , so that, in the quotient orbifold, their union is isotopic to times . ∎
Using an exactly similar construction, we define a map from to and a map from to . We can now complete the argument.
Proof of Proposition C (second case).
Consider the three Birkhoff sections , and given by Lemma 2.2. We argue as in Section 1. Starting from any point of and following the geodesic flow for some bounded time, we reach the surface . When continuing, we then reach , and then again. Therefore the first return map on is obtained by applying first, then , and then . In terms of matrices, and in the basis of , that map is then conjugated to the product . In terms of the standard generators and of , the latter product is equal to , which is conjugated to , as announced.
The other genus one Birkhoff sections are obtained by reversing the direction of the geodesic . By the same construction as above, we obtain three other Birkhoff sections, and the same arguments lead to a first resturn map conjugated to . This result can also be obtained directly by observing that the new Birkhoff sections are obtained from the old ones by rotating all tangent vectors by an angle , and therefore by following the flow in the reverse direction, so that the new monodromy is the inverse of the old one. ∎
3. The case of four singular points
We now turn to the last case in Proposition C. Let denote four integers larger than 2. The case when some of them is equal to requires some slight modifications that we will describe at the end of the section. The proof follows the same scheme as in the case of three singular points, with some modifications that make it more complicated. The most notable one is that the two orthic triangles are replaced by two quadrangles and that the boundaries of the constructed Birkhoff tori have two components.
In distinction to the case of three singular points, there exist many orbifolds with spherical base and four singular points of respective orders . Indeed, Thurston  showed that the associated Teichmüller space has dimension 2. Nevertheless, the argument of Ghys for surfaces  still applies, so that the associated geodesic flows all are conjugated. Therefore, it is enough to consider here one orbifold for every choice of .
From now on, we fix a Fuchsian group such that the quotient orbifold has four singular points of respective orders . We call the geodesic flow on . We also choose a fundamental domain for in , obtained by cutting the orbifold along the shortest geodesic connecting to , to , and to . The fundamental domain is therefore an hexagon, that we denote by . The total angles are at and at , while the sum of the angles at and is and the sum at and is (see Figure 5 bottom).
3.1. Four Birkhoff sections
Let be the oriented closed geodesic whose projection connects the segment to , and be the oriented closed geodesic whose projections connects to , then to , then to , then to , and finally to (see Figure 5). The geodesics and intersect in four points, that we denote by in such a way that both and go through them in this order. The union plays the role of the billiard trajectory in Section 2. It divides into six regions: the -part which has two edges and vertices and , the -part with two edges and vertices and , the -part with vertices and , the -part with vertices and , and two quadrangles, called and in such a way that the two geodesics go clockwise around , and counter-clockwise around .