Fenchel-Nielsen coordinates

The behaviour of Fenchel-Nielsen distance under a change of pants decomposition

D. Alessandrini Daniele Alessandrini, Max-Plank-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany daniele.alessandrini@gmail.com L. Liu Lixin Liu, Department of Mathematics, Sun Yat-Sen University, 510275, Guangzhou, P. R. China, and Hausdorff Research Institute for Mathematics, Poppelsdorfer Allee 45 D-53115 Bonn Germany mcsllx@mail.sysu.edu.cn A. Papadopoulos Athanase Papadopoulos, Université de Strasbourg and CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France, and Hausdorff Research Institute for Mathematics, Poppelsdorfer Allee 45 D-53115 Bonn Germany athanase.papadopoulos@math.unistra.fr  and  W. Su Weixu Su, Department of Mathematics, Sun Yat-Sen University, 510275, Guangzhou, P. R. China, and Hausdorff Research Institute for Mathematics, Poppelsdorfer Allee 45 D-53115 Bonn Germany su023411040@163.com
July 13, 2019
Abstract.

Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition and given a base complex structure on , there is an associated deformation space of complex structures on , which we call the Fenchel-Nielsen Teichmüller space associated to the pair . This space carries a metric, which we call the Fenchel-Nielsen metric, defined using Fenchel-Nielsen coordinates. We studied this metric in the papers [1], [2] and [3], and we compared it to the classical Teichmüller metric (defined using quasi-conformal mappings) and to another metric, namely, the length spectrum, defined using ratios of hyperbolic lengths of simple closed curves metric. In the present paper, we show that under a change of pair of pants decomposition, the identity map between the corresponding Fenchel-Nielsen metrics is not necessarily bi-Lipschitz. The results complement results obtained in the previous papers and they show that these previous results are optimal.


AMS Mathematics Subject Classification: 32G15 ; 30F30 ; 30F60.

Keywords: Teichmüller space, Fenchel-Nielsen coordinates, Fenchel-Nielsen metric.

L. Liu and W. Su are partially supported by NSFC grant No. 10871211.

1. Introduction

This paper is in the lineage of the papers [1], [2], [3] and [6], in which we studied and compared various metrics on Teichmüller spaces of surfaces of finite or of infinite topological type. The first important thing to know in that respect is that some definitions that are equivalent to each other in the Teichmüller theory of surfaces of finite type are no more equivalent in the setting of surfaces of infinite type. Indeed, there are Teichmüller spaces that are associated to a surface of infinite type that are distinct (in the set-theoretic sense) and that would be equal if the same definitions were made in the case of a surface of finite type. Furtermore, each such space associated to a surface of finite or of infinite type carries a natural distance function, and it is an interesting problem to study the relations between the various spaces, their distance functions and their topologies.

It is necessary to have different names to the various spaces that arise, and we briefly recall the terminology.

We use the name quasi-conformal Teichmüller space for the “classical” Teichmüller space defined using quasi-conformal mappings and equipped with its Teichmüller metric.

In the paper [1], we introduced the Fenchel-Nielsen Teichmüller space, a certain space of equivalence classes of complex structures on a surface, equipped with a distance, called the Fenchel-Nielsen distance, defined using Fenchel-Nielsen coordinates. This Teichmüller space and its metric depend on the choice of a pair of pants decomposition of the surface. The Fenchel-Nielsen Teichmüller space was a fundamental tool in our work, because it has explicit coordinates and an explicit distance function, and we used it in the other papers mentioned to describe and understand the other Teichmüller spaces. One of the results obtained was that if the lengths of all the boundary curves of the pair of pants decomposition are bounded above by a uniform constant then there is a set-theoretic equality between the Fenchel-Nielsen Teichmüller space and the quasiconformal Teichmüller space. Furthermore, the identity map between the two Teichmüller spaces, equipped with their respective metrics, is a locally bi-Lipschitz homeomorphism. This gives an explicit description of the global topology and the local metric properties of the quasiconformal Teichmüller space.

In the paper [3], we obtained similar local comparison results between the Fenchel-Nielsen Teichmüller space and the so-called length spectrum Teichmüller space, another deformation space of complex structures, whose definition and metric are based on the comparison of lengths of simple closed curves between surfaces.

In the cases of surfaces of finite type, the various Teichmüller spaces coincide set-theretically, but there are still interesting questions on the local and global comparison of the metrics that are defined on these spaces.

In the present paper, we prove that under a change of the pair of pants decomposition, the identity map between the corresponding Fenchel-Nielsen Teichmüller spaces is not bi-Lipschitz in general. This result holds for surfaces of finite and for those of infinite type. We first prove this result in the case where the surface is a torus with one hole or a sphere with four holes. In this case, the two pair of pants decompositions are obtained from each other by a single elementary move. The proof is based on explicit computations that use formulae obtained by Okai in [8]. We then deduce an analogous result for arbitrary surfaces of finite or of infinite type.

To state the theorems precisely, we now introduce some minimal amount of notation. We refer the reader to Section 2 for more details.

If is a surface equipped with a complex structure, we denote by the quasi-conformal Teichmüller space of , equipped with the quasi-conformal distance , and by the Fenchel-Nielsen Teichmüller space of with reference to the pair of pants decomposition , equipped with its associated Fenchel-Nielsen distance .

In Section 6 we prove the following:

Theorem 1.1.

Let be an orientable surface which is either of finite topological type of negative Euler characteristic and which is not a pair of pants, or of infinite topological type. Let be a pair of pants decomposition of . Then we have the following:

  1. There exists another pair of pants decomposition such that for every base complex structure on we have as sets, but the identity map between the two spaces equipped with their Fenchel-Nielsen distances and respectively is not Lipschitz.

  2. For every base complex structure on , consider the space . Then the identity map from to is not Lipschitz.

  3. If is of infinite topological type, there exists a base complex structure on and another pair of pants decomposition such that if is the space , then the identity map from to is not continuous.

  4. If is of infinite topological type, there exists a base complex structure on such that if is the space , then the identity map from to is not continuous.

From this result, we can see that the local comparison results we obtained in the papers [1] and [3] are optimal in the sense that they cannot be extended to global comparison results, since the global metric geometry of the Fenchel-Nielsen distance depends on the choice of the pair of pants decomposition.

2. The Fenchel-Nielsen metric

In this section, we recall a few facts from our previous papers, which will help making the present paper self-contained.

Let be an orientable connected surface of finite or of infinite topological type, that can have punctures and/or compact boundary components. The complex structures that we consider on are such that each boundary component has a regular neighborhood that is bi-holomorphically equivalent to a bounded cylinder and each puncture has a neighborhood that is bi-holomorphically equivalent to a punctured disc.

We start by reviewing the definition of the Fenchel-Nielsen metric on the Teichmüller space of . The definition depends on the choice of a pair of pants decomposition of .

A pair of pants decomposition of is a decomposition into generalized pairs of pants glued along their boundary components, where a generalized pair of pants is a sphere with three holes, a hole being either a point removed (leaving a puncture on the pair of pants) or an open disc removed (leaving a boundary component on the pair of pants). The curves in the above definition are the closed curves on (including the boundary components) that define the decomposition. It is well-known that every surface of finite topological type with negative Euler characteristic admits a pair of pants decomposition. It also follows from the classification of surfaces of infinite type that every such surface admits a pair of pants decomposition [1].

To every complex structure on we can associate a hyperbolic metric, called the intrinsic metric, which was defined by Bers in [4]. This metric is conformally equivalent to the given complex structure, and every boundary curve is a geodesic for that metric. The definition of the intrinsic metric is recalled in [1]. In the sequel, when we talk about geometric objects (geodesics, length, angles, etc.) associated to a complex structure on , it is understood that these are associated to the intrinsic metric on . A pair of pants decomposition of the surface equipped with its intrinsic metric is said to be geodesic if each is a geodesic closed curve in with respect to this metric. In [1] we proved that every topological pair of pants decomposition of is homotopic to a unique geodesic pair of pants decomposition. (Note that this is not true for general hyperbolic metrics on . For example, the Poincaré metric of may not satisfy this property if this metric is different from the intrinsic metric, and this may happen; we discussed this fact in [1].)

Given a closed curve on the surface , we make the convention that we shall call also the unique geodesic representative of this closed curve with reference to the intrinsic metric.

Given a complex structure on and a geodesic pair of pants decomposition of this surface, then for any closed geodesic there is a well-defined length parameter , which is the length of this closed geodesic, and a twist parameter , which is defined only if is not the homotopy class of a boundary component of . The quantity is a measure of the relative twist amount along the geodesic between the two generalized hyperbolic pair of pants (which might be the same) having this geodesic in common. In this paper, the value is a signed distance-parameter, that is, it represents an amount of twisting in terms of a distance measured along the curve, as opposed to an angle of twisting parameter (whose absolute value would vary by after a complete Dehn twist).

For any complex structure on , its Fenchel-Nielsen parameter relative to is a collection of pairs

where it is understood that if is a boundary geodesic, then it has no associated twist parameter, and instead of a pair , we have a single parameter .

Given two complex structures and on , their Fenchel-Nielsen distance (with respect to ) is defined as

(1)

again with the convention that if is a boundary component of and therefore has no associated twist parameter, we consider only the first factor.

If is a surface of finite topological type, then is always finite, and the function defines a distance on the Teichmüller space of , which we will denote simply by . If, instead, is of infinite topological type, can assume the value infinity. In this case we fix a base complex structure on , and we define the Fenchel-Nielsen Teichmüller space of as the set of homotopy classes of complex structures on such that is finite. We denote this space by . The function is a distance function in the usual sense on this space, and it makes it isometric to the sequence space .

When it is important to stress on the dependence on a given pair of pants decomposition , we shall denote the Teichmüller space by (this dependence of the space on may happen only for surfaces of infinite type), and the Fenchel-Nielsen distance by (the distance function depends on also for a surface of finite type).

If a pair of pants decomposition is obtained from another pair of pants decomposition by a finite number of elementary moves (represented in Figures 1 and 2 below), then for any basepoint we have the set-theoretic equality .

In the last section of this paper we will prove that for every surface , there exist two pair of pants decompositions such that for every base complex structure , the Fenchel-Nielsen Teichmüller spaces are the same (that is, we have a set-theoretical equality ), but the identity map between the two spaces is not Lipschitz with reference to the two Fenchel-Nielsen distances and .

3. The effect of an elementary move on the torus with one hole

In this section, the surface is a torus with one hole, where the hole can be either a boundary component or a puncture. A pair of pants decomposition of is determined by a unique simple closed curve on which is not homotopic to a point or to the hole.

We consider two distinct pair of pants decomposition and of defined by two essential simple closed curves and satisfying , as represented in Figure 1.

Figure 1. The two curves and intersect at one point. An elementary move replaces one of these curves by the other one.

Let be a complex structure on , equipped with its intrinsic hyperbolic metric. At the hole, can have geodesic boundary, and in this case we denote by its length, or it can have a cusp, and in this case we write . We denote by the length and twist parameters of the curve , for the decomposition , and by the length and twist parameters of the curve , for the decomposition . We need a formula relating these values. This was done by Okai ([8]) in the case where the hole is a boundary component, but we also need similar formulae for the case of a cusp. In the following proposition we obtain this with a continuity argument. In these formulae, the case where means that at the hole the surface has a cusp.

Proposition 3.1.

With the above notation we have, for all ,

Proof.

When , these formulae are proved in [8]. To see that they also hold when (the case of a cusp) we use the shear coordinates relative to an ideal triangulation. We recall that in the case of a torus with one hole equipped with a complex structure (and the corresponding intrinsic metric), an ideal triangulation has three edges, say , and associated to every edge there is a real number, called the shear parameter. Thus, we have three numbers, . The boundary length can be read from the shear coordinates: (see [10, Prop. 3.4.21]), and the cusp corresponds to the case where . Here we need the property that the parameters can also be written as continuous functions of the shear coordinates. To see this one can show that for every element of the fundamental group of the surface, the coefficients of the matrix corresponding to in the holonomy representation can be written as continuous functions of the shear coordinates. The length of the corresponding closed geodesic in the surface is a continuous function of the trace of this matrix, hence all simple closed curve lengths are continuous functions of the shear coordinates. For the twist parameters, we note that their absolute values can be written as a continuous function of the lengths of some simple closed curves.

To show that the above formulae are valid for , we just need to note that both the left hand side and the right hand side of the equations are continuous functions of the shear coordinates. It is known that shear coordinates extend to the case of surfaces with cusps (see [10, Chapter 3]), see also [9]). Then as these functions are equal when , i.e. when , and as this subset is dense in the space of shear coordinates, these functions are equal everywhere, that is, including the case where . ∎

For simplicity, we choose a complex structure on the torus with one hole such that its intrinsic metric has the property that the two closed geodesics in the homotopy classes of and meet perpendicularly. To see that such a choice is possible, we can start with an arbitrary complex structure on (equipped with its intrinsic metric) and we cut this surface along the closed geodesic ; in this way is cut into an essential geodesic arc, which is homotopic to a unique geodesic arc that is perpendicular to the two boundary components of that arise from cutting the surface along . Then we glue back the two components in such a way that in the resulting surface the two endpoints of match. We obtain a complex structure with the desired property. Such a complex structure corresponds to the case in the notation of Okai [8].

Performing a Fenchel-Nielsen twist of magnitude along , we obtain from a new complex structure that has Fenchel-Nielsen coordinates denoted by in the coordinate system associated to and in the coordinate system associated to . The coordinates of the complex structure that we have chosen are and in the bases and respectively.

Since is obtained by a time- twist along the curve , we have for all .

We need to estimate and .

By the formula for length in Proposition 3.1, we have

and

Thus, we have

Using the fact that , we obtain

which implies

As a result, and using (which follows from our hypothesis that and meet perpendicularly for ), we have

Now we estimate the twist parameter, using the formula for the twist in Proposition 3.1. For the purpose of the computations, we use the following notation:

and

The quantities and are functions of and , and we have

We shall focus on the case where is small and are bounded.

Consider the second order expansions near .

Then we have

and

This gives

Note that the term in the previous formula depends on and only via continuous functions of . In particular, there exists an such that for we have

where is a constant that depends only on and the upper bound of .

Note that . From this and the properties of the logarithm function we obtain

Thus, we have

From the continuity of the function, if is bounded, then there is a constant that depends only on the upper bound for such that . Therefore,

where is a constant that depends only on the upper bound of .

We record the above results for the length and twist parameters in the following proposition.

Proposition 3.2.

Let be a complex structure on with the following properties:

  1. has either geodesic boundary of length , or a cusp (and in the latter case we write );

  2. there exist a pair of perpendicular simple closed geodesics on with .

Let be the complex structure obtained from by performing a Fenchel-Nielsen twist of magnitude along . Let and be the Fenchel-Nielsen coordinates of in the coordinate systems associated to and respectively. (Note that .)

Assume and are bounded above by some constant . Then there exist constants and , both depending only on , such that for all , we have

(2)

and

Note that the second inequality in (2) follows from one version of the Collar Lemma, which says that there exists such that for , we have .

4. The effect of an elementary move on the sphere with four holes

In this section, the surface is the sphere with four holes, where each hole can be either a boundary component or a cusp. We equip with two pair of pants decompositions , defined by two essential simple closed geodesics satisfying .

Let be a complex structure on equipped with its intrinsic metric. Near each of the four holes, can have a geodesic boundary, or a cusp. We denote by the lengths of the boundary components (as before with the convention that is zero if the corresponding hole is a cusp); see Figure 2. We denote by the length and twist parameter respectively of the curve , for the decomposition , and by the length and twist parameter respectively of the curve , for the decomposition . We need a formula relating these values. Like in the case of the torus with one hole, Okai wrote such formulae in [8] in the case where all the holes are boundary components, and we need to see that the formulae also hold in the case where some of the holes are cusps. In the following proposition we deduce this by a continuity argument, as we did in the previous section for the case of the one-holed torus.

Proposition 4.1.

With the above notation we have

and

Proof.

When all the boundary lengths are positive () these formulae are proved in [8]. To see that they also hold when some is zero, we proceed as in the proof of Proposition 3.1. We fix an ideal triangulation, now having edges, so we have shear coordinates. For the length can be expressed as the absolute value of the sum of the shear coordinates relative to the edges adjacent to the hole involved (see [10, Prop. 3.4.21]), and the cusp corresponds to the case where this sum is zero. With exactly the same argument as in Proposition 3.1 we can see that the parameters can be written as continuous functions of the shear coordinates. Hence both the left hand side and the right hand side of the equations are continuous functions of the shear coordinates. As these functions are equal when , and as this subset is dense in the space of shear coordinates, the functions are equal everywhere, including on the set where the values are zero. ∎

Like in the case of the torus with one hole, we choose such that and intersect perpendicularly, in order to simplify the computations. Performing a Fenchel-Nielsen twist of magnitude along , we obtain from a new complex structure and we denote its Fenchel-Nielsen coordinates in the coordinate system associated to and in the coordinate system associated to . As before we have . Then we also have the following proposition.

Figure 2. An elementary move on the sphere with four holes replaces one of the interior curves drawn by the other one.
Proposition 4.2.

Let be a complex structure on with the following properties:

  1. every hole of is a geodesic boundary component or a cusp, with lengths denoted by (with the convention that is zero if the corresponding hole is a cusp);

  2. there exist a pair of perpendicular simple closed geodesics on with .

Let be the complex structure obtained from by performing a Fenchel-Nielsen twist of magnitude along . Let and be the Fenchel-Nielsen coordinates of in the coordinate systems associated to and respectively. (Note that .)

Then, if and are bounded from above by some constant , there exist constants , all of them depending only on , such that for we have

(3)

and

Proof.

We note as in the case studied before that the second inequality in (3) follows from the Collar Lemma.

To prove the proposition, we use the formulae in Proposition 4.1. Setting

and

the formula for the length parameter becomes

For , this formula becomes

Thus, we have

where is a constant that depends only on . Using the same estimates as in Section 3, we obtain

Now we estimate the twist . We assume that the twist is positive, to avoid taking absolute values. The formula in Proposition 4.1 gives

(4)

where

Note that for all . Let , and .

As , we have , and . As a result, we can assume that is sufficiently small, so that and are not less than . Then

(We used the fact that for .)

Now we need to estimate . To do this, note that by the first formula of Proposition 3.1, we have

Using the fact that we conclude that if is small, there exists a constant depending only on such that

First we estimate :

where is a constant depending only on and .

Then we estimate :

where and are constants depending only on and .

Then, finally:

Now if is small, then there is a constant such that . (We use the formula .) Therefore,

This proves Proposition 4.2. ∎

5. Comparing Fenchel-Nielsen distances

We reformulate Proposition 3.2 and Proposition 4.2 in the following.

Proposition 5.1.

Let be a complex structure on or that satisfies the assumptions in Proposition 3.2 or Proposition 4.2 respectively. Assume that , (in the case where ) and (in the case where ) are bounded above by some constant . Then there exist constants , and , all depending only on such that for , we have

and

For or , let and denote the Fenchel-Nielsen coordinates on associated to and respectively. As above, for any complex structure on , we denote by the complex structure obtained by the time- Fenchel-Nielsen twist of along . We have the following:

Proposition 5.2.

Suppose that is either the torus with one hole or the sphere with four holes , with and being homotopy classes of closed curves in satisfying in the case and in the case . Then, there exist a sequence of points such that

Proof.

Let , be a sequence of complex structures on satisfying the following properties:

  1. for any the geodesics in the classes of and intersect perpendicularly;

  2. the hyperbolic length tends to as ;

  3. the hyperbolic length of the holes of the torus with one hole or of the sphere with four holes is bounded by some fixed constant .

It is clear that . By Proposition 5.1, we have

and

As a result,

as . ∎

We conclude with the following

Corollary 5.3.

With the notation of Proposition 5.2, the identity map between the metrics and is not Lipschitz. More precisely, there does not exist any constant satisfying for all and in .

6. General surfaces

The aim of this section is to prove Theorem 1.1, which is the analogue of Corollary 5.3 for an arbitrary surface of finite or infinite type.

The idea is to start with an arbitrary pair of pants decomposition of , to take in it an embedded subsurface of type or with boundary curves belonging to the system , and to modify the pair of pants decomposition into a pair of pants decomposition by an elementary move performed inside this subsurface, according to the scheme used in Sections 3 and 4.

There is one complication in doing this. Even though in the new pair of pants decomposition the length parameters of all the non-modified curves for the complex structure obtained by twisting along the curve is the same in the systems and , the situation is not the same for the twist parameters. Performing the Fenchel-Nielsen twist along the curve does not modify the twist parameters of the closed curves in the system that are different from , but in the system , the twist parameters of the curves that are on the boundary of the subsurface or do not remain constant. This is the main question that we deal with now.

Let be a complex structure on equipped with a geodesic pair of pants decomposition , and let be a closed geodesic in the interior of .

The closed geodesic is either in the interior of a torus with one hole or of a sphere with four holes that is defined by the pair of pants decomposition . We denote such a one-holed torus or a four-holed sphere by .

Let be the pair of pants decomposition of obtained from by an elementary move on , replacing this curve with a curve which is contained in and which has minimal intersection number with .

Figure 3.

We take a sequence of complex structures on satisfying . By the collar lemma, we then have .

We assume that the curve is adjacent to two distinct pairs of pants in the decomposition . The other case can be dealt with in the same way. We shall use the notation of Figure 3.

We shall apply Proposition 4.2, and for this we assume that is a complex structure such that