On a question of Etnyre and Van Horn-Morris

On a question of Etnyre and Van Horn-Morris

Tetsuya Ito Department of Mathematics, Graduate School of Science, Osaka University
1-1 Machikaneyama Toyonaka, Osaka 560-0043, JAPAN
tetito@math.sci.osaka-u.ac.jp http://www.math.sci.osaka-u.ac.jp/ tetito/
 and  Keiko Kawamuro Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA keiko-kawamuro@uiowa.edu
July 26, 2019
Abstract.

We answer Question 6.12 in [1] asked by Etnyre and Van Horn-Morris.

1. Introduction

Let be a compact oriented surface with boundary. Let denote the mapping class group of , namely the group of isotopy classes of orientation preserving homeomorphisms of that fix the boundary pointwise. Let be a boundary component of and let

denote the fractional Dehn twist coefficient FDTC) of with respect . See Honda, Kazez and Matić’s paper [3] for the definition of the FDTC. For we define the following sets ([1, p.344]):

The following theorem answers [1, Question 6.12] of Etnyre and Van Horn-Morris: For which the set forms a monoid?

Theorem 1.1.

Let be a surface that is not a pair of pants and has negative Euler characteristic. Let be a boundary component of . The set and hence is a monoid if and only if .

Remark 1.

If is a pair of pants then is a monoid if and only if .

Theorem 1.1 states that is not a monoid. But contains , the monoid of right-veering mapping classes.

Corollary 1.2.

We have

2. Basic study of quasi-morphisms

As shown in [4, Corollary 4.17] the FTDC map is not a homomorphism but a quasi-morphism if the surface has negative Euler characteristic. In order to prove Theorem 1.1 we first study general quasi-morphisms and obtain a monoid criterion (Theorem 2.2).

Let be a group. A map is called a quasi-morphism if

is finite. The value is called the defect of the quasi-morphism. A quasi-morphism is homogeneous if for all and . Every quasi-morphism can be modified to a homogeneous quasi-morphism by taking the limit:

A typical example of homogeneous quasi-morphism is the translation number

given by:

Here is the group of orientation-preserving homeomorphisms of that are lifts of orientation-preserving homeomorphisms of . The limit does not depend on the choice of . An important property of we will use is that

  1. if then for all .

Given a quasi-morphism and let

It is easy to see that:

Proposition 2.1.

The set forms a monoid if .

The following Theorem 2.2 gives another monoid criterion for . We will later apply Theorem 2.2 to the quasi-morphism and prove Theorem 1.1.

Theorem 2.2.

Let be a homogeneous quasi-morphism which is a pull-back of the translation number quasi-morphism , namely, there is a homomorphism such that . Then forms a monoid for .

Proof.

Let . Assume to the contrary that id not a monoid. There exist such that . That is . Take an integer so that

(2.1)

By the definition of the defect we have

(2.2)

By (2.1) and (2.2) we get

Let and . By the property we have .

On the other hand, since by the property again we have for all . Therefore and

which is a contradiction. ∎

3. Proofs of Theorem 1.1 and Corollary 1.2

Proof of Theorem 1.1.

According to [4, Theorem 4.16], if the FDTC has for some homomorphism . This fact along with Theorem 2.2 shows that is a monoid if and . Since the set is also a monoid if and .

Next we show that is not a monoid for . For any non-separating simple closed curve and any boundary component of we have . Therefore we have i.e.,

(3.1)

(Case 1) Recall that for any surface of genus the group is generated by Dehn twists along non-separating curves (see p.114 of [2]). If were a monoid then this fact and (3.1) imply that which is clearly absurd. Thus is not a monoid if and .

(Case 2) If and let be the boundary components and be the simple closed curves as shown in Figure 1-(1). Let and . Since are non-separating

By the lantern relation, for any positive integer with we have

thus . This shows that is not a monoid for all and .

Figure 1. (1) Case 2. (2) Case 3. (3) Case 5.

(Case 3) If and add additional boundary components in the place of as shown in Figure 1-(2). By a similar argument using the lantern relation we can show that is not a monoid for all and any . By the symmetry of the surface we can further show that is not a monoid for all and .

(Case 4) If and the group is generated by Dehn twists about non-separating curves. Thus this case is subsumed into Case 1.

(Case 5) If and applying the 3-chain relation [2, Proposition 4.12] to the curves in Figure 1-(3) we get . By the same argument as in Case 2 we can show that is not a monoid for all .

Parallel arguments show that does not form a monoid for . ∎

We close the paper by proving Corollary 1.2.

Proof of Corollary 1.2.

Let be a non-separating simple closed curve. By (3.1) we observe that

Acknowledgements

The authors thank John Etnyre for pointing out an error in an early draft of the paper. TI was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 15K17540. KK was partially supported by NSF grant DMS-1206770.

References

  • [1] John B. Etnyre and Jeremy Van Horn-Morris, Monoids in the mapping class group. Geom. Topol. Monographs 19 (2015) 319-365.
  • [2] Benson Farb and Dan Margalit, A primer on mapping class groups. Princeton Mathematical Series, 49. Princeton University Press, Princeton, NJ, 2012. xiv+472 pp.
  • [3] Ko Honda, William H. Kazez and Gordana Matić, Right-veering diffeomorphisms of compact surfaces with boundary. Invent. Math. 169 (2007), no. 2, 427-449.
  • [4] Tetsuya Ito and Keiko Kawamuro, Essential open book foliations and fractional Dehn twist coefficient
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