On roots of Dehn twists.

On roots of Dehn twists.

Naoyuki Monden
Abstract

Margalit and Schleimer constructed nontrivial roots of the Dehn twist about a nonseparating curve. We prove that the conjugacy classes of roots of the Dehn twist about a nonseparating curve correspond to the conjugacy classes of periodic maps with certain conditions. Furthermore, we give data set which determine the conjugacy class of a root. As a consequence, we can find the minimum degree and the maximum degree, and show that the degree must be odd. Also, we give Dehn twist expression of the root of degree 3.

1 Introduction

Let denote a compact, oriented surface of genus with boundary components, and let denote its mapping class group, the group of isotopy class of orientation-preserving homeomorphisms which are allowed to permute the points on the boundary. The isotopies are also required to permute the points on the boundary. If , we omit it from the notation. Let be the (left handed) Dehn twist about a simple closed curve on .

It is a natural question whether has a root. In other words, does there exist such that for integer degree ? If is a separating curve, it is easy to find roots. For example, the half twist about is a root of degree . However, when is a nonseparating curve, roots were not discovered until recently. In 2009, Margalit and Schleimer [MS] discovered roots of the Dehn twist about a nonseparating curve of degree .

Matsumoto and Montesinos [MM1] introduced a certain type of mapping class of which were called by them pseudo-periodic maps of negative twist. Furthermore, they gave the complete set of conjugacy invariant for a pseudo-periodic map of negative twist. In Section 2 we introduce the definition and the complete set of conjugacy invariant of pseudo-periodic maps of negative twist . In Section 3 we show that inverses of roots of Dehn twists are pseudo-periodic maps of negative twist, and prove Main Theorem A by using the work of [MM1].

Main Theorem A.

Let be a nonseparating curve. Let be the boundary of , which we write to emphasize its orientation.

The conjugacy classes of roots of of degree correspond to the conjugacy classes of periodic maps of order which satisfy following condition:

The action on is the rotation angle of such that .

Moreover, by combining the Riemann-Hurwitz formula, Nielsen’s theorem [Ni] and Main Theorem A we give the following data set which determines the conjugacy class of a root.

Main Theorem B.

The conjugacy class of a root of of degree is determined by the data set which satisfies the following conditions:

(1)

,

(2)

,

(3)

.

As a corollary of Theorem 3.3 we obtain the following result.

Corollary C.

the range of degree is and is odd.

Thus the roots constructed by Margalit and Schleimer have the maximum degree .

In Section 4 we give Dehn twist expression of the root of degree 3. In Section 5 we consider a root of image of the Dehn twist about a nonseparating curve to . We show that the degree must be odd. In Section 6 we consider a root of the Dehn twist about a nonseparating curve in the mapping class group of a surface with boundary components and punctures.

Recently, McCullough and Rajeevsarathy [MR] has independently proved the result similar to Main Theorem A, B and Corollary C. They gave the data set similar to Theorem B in Theorem 1.1 and Main Theorem A was contained in the proof of Theorem 1.1. They proved these results without work of [MM1]. As a corollary of Theorem 1.1 they showed the same results as Corollary C. Moreover, they showed the follow results: In Corollary 3.1 they showed that has a root of degree for all . In Corollary 3.2 they showed that when is prime and , does not have a root of degree  (Corollary 3.2). In Theorem 4.2 they proved that if then the root is one of 2 types of roots  (Theorem 4.2).

The author thanks Darryl McCullough and Kashyap Rajeevsarathy for their comments on the content of this paper.

2 Preliminary

Hereafter, all surfaces will be oriented, and all homeomorphisms between them will be orientation-preserving. For us, a Dehn twist means a left-handed Dehn twist. Let be a closed connected surface of genus .

2.1 Pseudo-periodic map and screw number

Definition 1.

A homeomorphism is a pseudo-periodic map isotopic to a homeomorphism which satisfies the following conditions:

(1)

There exists a disjoint union of simple closed curves on such that .

(2)

The restriction is isotopic to a periodic map.

We call an admissible system of cut curves if each connected component of has negative Euler number. If has negative Euler number, such a system always exists in the isotopy class of .

Nielsen [Ni] introduced the ”screw number” for each component of admissible system of cut curves . The definition is as follows:

We may assume . Let be a fixed cut curve in the system.

Let be the smallest positive integer such that where denotes with an orientation assigned, so preserves the (arbitrarily fixed) direction of .

Let and be the connected components of such that belongs to their adherence in . Let (resp. ) be the smallest positive integer such that (resp. ). Clearly is a common multiple of and .

Since is isotopic to a periodic map, there exists a positive integer such that . We choose the smallest number among such integers and denote it by again. Likewise we choose for .

Let be the least common multiple of and . Then is isotopic to the identity on and . Thus, on the union , is isotopic to the result of a number full Dehn twists performed about . Let () be this number of full twists.

Definition 2.

The rational number is called the screw number of about and is denoted by .

Definition 3.

A pseudo-periodic map is said to be of negative twist if for each component of admissible system of cut curves .

Definition 4.

The curve is said to be amphidrome if is even and , and non-amphidrome otherwise.

Theorem 5 ([MM1], Theorem 2.2).

The conjugacy class of a negative twist is determined by following data:

  • An admissible system of cut curves on .

For all ,

  • : the smallest positive integer such that ,

  • the screw number of , and

  • ’s character of being amphidrome or not.

For each connected component of ,

  • : the smallest positive integer such that ,

  • : the smallest positive integer such that , and

  • the conjugacy class of a periodic map .

  • The action of on the oriented graph whose vertices and edges correspond to connected components of and .

2.2 Valency

Let be an oriented connected surface and let be a periodic map of order . Let be a point on . There is a positive integer such that the points are mutually distinct and . If , we call the point a simple point of , while if , we call a multiple point of . Note that a multiple point is an isolated and interior point of

Let be a set of oriented and disjoint simple closed curves in a surface , and let be a map such that and is periodic. Let be the smallest positive integer such that . The restriction is a periodic map of of order, say, . Then (= the order of ). Let be any point on , and suppose that the images of under the iteration of are ordered as viewed in the direction of , where is an integer with and . So iff . Let be the integer which satisfies

(NB. iff ). Then the action of on is the rotation of angle with a suitable parametrization of as an oriented circle.

Definition 6 ([Ni]).

The triple and are called the valency and the second valency of with respect to .

Nielsen also defined the valency of a boundary curve as its valency with respect to assuming it has the orientation induced by the surface . The valency of a multiple point is defined to be the valency of the boundary curve , oriented from the outside of a disk neighborhood of .

Let be an admissible system of cut curves subordinate to a pseudo-periodic map . We assume that of is non-zero

Thus Matsumoto and Montesinos proved the following proposition :

Proposition 7 ([MM1], Corollary 3.3.1, Corollary 3.7.1).

Let be an annular neighborhood of with . Let be the second valency of with respect to . The second valency of oriented by the orientation induced from is defined as the valency of with respect to the .

Then, If is non-amphidrome,

(1)

,

(2)

is an integer.

If is amphidrome,

(1)

= an even number,

(2)

and ,

(3)

is an integer,

denotes , and denotes .

3 The conjugacy classes of roots of the Dehn twist about a nonseparating curve

3.1 Proof of Main Theorem A

In section 3.1 we do not distinguish a homeomorphism/curve and its isotopy class.

Let be the Dehn twist about a simple closed curve in . Suppose that is a root of of degree . Since we have

we see that . By , must be a periodic map of order . Therefore is a pseudo-periodic map, and an admissible system of cut curves is .

Let be an annular neighborhood of . Let and be the second valencies of and with respect to .

Claim 8.

The simple closed curve is non-amphidrome with respect to .

Proof.

We assume that is amphidrome with respect to .

We will determine the screw number of . Let be . Let , and be the smallest positive integers such that , and . Since and is amphidrome, we have . Moreover, we have , . Thus, by , we can denote . By definition of , is a divisor of (). Then, we have . Therefore, we see that and . From the above arguments, we have

By proposition 7, we have

(Here denotes , and denotes .) However, since and the action of is the rotation of angle in circle, must be equal to . This is a contradiction. ∎

Lemma 9.

Let be a nonseparating simple closed curve in . Then,

Proof.

We use Proposition 7. We will determine the screw number of . Since and is non-amphidrome (by Claim 8), we have . Thus, we can find by the similar argument of Claim 8. Furthermore, we can find that =order of (by =order of for ). By Proposition 7, we have .

Let be the boundary components which corresponds to . Then, is periodic map of order . Therefore, we can get periodic map of period by pasting two disks , to , . Since is non-amphidrome, we can see that fixes the center points , . Consequently, the period of is equal to . We note that is not equal to 0. Because, if then is equal to by the definition of . This is in contradiction to . ∎

Remark 10.

If is separating, then we may allow that is not equal to , that , and that either or is equal to . In other words, , satisfy any one of the following conditions :

(1)

   and ,

(2)

, , and ,

(3)

, , and .

Hereafter, will be a nonseparating curve on . Let be a compact oriented surface of genus with two boundary components , .

Main Theorem A.

The conjugacy classes of the roots of of degree correspond to the conjugacy classes of periodic maps of order which satisfy following condition:

The action on is the rotation angle of such that .

Proof.

We will prove Main Theorem A by using Theorem 5.

Let be a root of Dehn twist of degree . Let be the oriented graph whose vertices and edges correspond to connected components of and . Since is non-amphidrome, we can find that the action of on the oriented graph is identity. From the above arguments, we have , , , , that is non-amphidrome, and that the action of on is identity. These are the same data as .

Since the screw number of is equal to , we see that of is equal to . Therefore, is negative twist. If negative twists are restricted to roots of , the conjugacy class of a root of is determined by and the conjugacy class of a periodic maps by using Theorem 5. It means that the conjugacy class of a root of is determined by and the conjugacy class of a periodic map . By Lemma 9 satisfies the condition.

In Section 3.2 we show that a root is constructed from a periodic map on which satisfies the condition. ∎

3.2 Construction of a root of Dehn twist from a periodic map

Given a periodic map on with the condition of Main Theorem A, we construct a root of Dehn twist as follow:

Let be a periodic map on which satisfies the condition. Figure 1 shows the periodic map .

Figure 1: The periodic map

We will extend to . Let us give an orientation to and as the figure indicates:

Figure 2:

Since is periodic, there are homeomorphisms and such that , . Thus, we can get by gluing to by . Let be a homeomorphism . We define on as follow:

Figure 3 shows a homeomorphism .

Figure 3: the homeomorphism

Figure 4 shows the homeomorphism

Figure 4:

Thus, we can see that . Therefore, is isotopic to . So we may change by isotopy so that . By , . we have

We have a root of with degree . We complete the proof of Main Theorem A.

3.3 The data set which determines the conjugacy class of a root

Let be a multiple point of a periodic map of Main Theorem A and let be the valency of with respect to . We note that . Let be the valency of respect to . Note that and (by ). By Nielsen’s theorem [Ni], it suffices to classify the order of the map, the valencies of multiple points and the valencies of boundary curves.

When we combine the Riemann-Hurwitz formula, Nielsen’s theorem and Main Theorem A, we have the following theorem:

Main Theorem B.

The conjugacy class of a root of of degree is determined by the data which satisfies the following conditions:

(1)

,

(2)

,

(3)

.

where is the genus of .

McCullough and Rajeevsarathy also got the similar data set in [MR]. The notation follows that of [MR].

Corollary C.

the range of degree is and is odd.

Proof.

If , by we have . This is in contradiction to the condition (3). Therefore, we see that is odd.

For , if , and , we can select which satisfy the condition (2). It means that there always exists the root of degree 3 for . In next section, we will give Dehn twist expression of the root of degree 3.

We assume that . By the condition (1) we have so and . From the Condition (2) and (3) we have . Therefore, we see that . It means that is an integer (we note that ). Since and , we see that must be equal to . It means that . Thus we have so . This is in contradiction to . Since Margalit and Schleimer constructed the root of degree , we have . ∎

McCullough and Rajeevsarathy [MR] also proved by the similar arguments.

4 Dehn twist expression of the root of degree 3

We will give Dehn twist expression of the root of degree 3. The key points are to use the star relation which is given by Gervais [Ge].

Consider the torus with 3 boundary components , , and let , , and be simple closed curves in Figure 5

Figure 5: The curves of star relation

The star relation is as follow:

If , then is trivial, and the relation becomes

Let , , and be nonseparating simple closed curves and let be separating simple closed curves in Figure 6.

Figure 6: The curves , , , ,

We define

and

We note . Then, by star relation we have . When we define

is the root of of degree 3.

5 A root of an image of the Dehn twist about a nonseparating curve to

The action of on preserves the algebraic intersection forms, so it induces a representation , which is well-known to be surjective. In this section we will prove that if is nonseparating curve, then an image of Dehn twist has no roots of even degree. We prove the case of . An element matrix satisfies that , where is transpose of and is

Let be a nonseparating simple closed curve in Figure 6 and let be

We assume that , where is

Since , we have . By

and

, we have

(1)

Since , we have

We have . By , we have

(2)

By equations (1) and (2) we have that . This is in contradiction to . For , we can prove by the same arguments.

Since has no roots of degree 2, we can find that a root of cannot have even degree.

6 Remarks

We will consider a root of the Dehn twist about a nonseparating curve in the mapping class group of a surface with boundary components and punctures.

Let be a compact oriented surface with boundary components , and punctures. We denote the mapping class group of , the group of isotopy classes of orientation-preserving homeomorphisms which are allowed to permute punctures and the points on , and restrict to the identity on . The isotopies are also required to permute punctures and the points on and to fix the points on . Let be the Dehn twist about a nonseparating curve in . In this section we do not distinguish a homeomorphism/curve and its isotopy class.

We assume that . Since