On a question of Etnyre and Van HornMorris
Abstract.
We answer Question 6.12 in [1] asked by Etnyre and Van HornMorris.
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 HornMorris: 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 rightveering mapping classes.
Corollary 1.2.
We have
2. Basic study of quasimorphisms
As shown in [4, Corollary 4.17] the FTDC map is not a homomorphism but a quasimorphism if the surface has negative Euler characteristic. In order to prove Theorem 1.1 we first study general quasimorphisms and obtain a monoid criterion (Theorem 2.2).
Let be a group. A map is called a quasimorphism if
is finite. The value is called the defect of the quasimorphism. A quasimorphism is homogeneous if for all and . Every quasimorphism can be modified to a homogeneous quasimorphism by taking the limit:
A typical example of homogeneous quasimorphism is the translation number
given by:
Here is the group of orientationpreserving homeomorphisms of that are lifts of orientationpreserving homeomorphisms of . The limit does not depend on the choice of . An important property of we will use is that

if then for all .
Given a quasimorphism 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 quasimorphism and prove Theorem 1.1.
Theorem 2.2.
Let be a homogeneous quasimorphism which is a pullback of the translation number quasimorphism , 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) 
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 nonseparating 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 nonseparating 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 nonseparating
By the lantern relation, for any positive integer with we have
thus . This shows that is not a monoid for all and .
(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 nonseparating curves. Thus this case is subsumed into Case 1.
(Case 5) If and applying the 3chain 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.
Acknowledgements
The authors thank John Etnyre for pointing out an error in an early draft of the paper. TI was partially supported by JSPS GrantinAid for Young Scientists (B) 15K17540. KK was partially supported by NSF grant DMS1206770.
References
 [1] John B. Etnyre and Jeremy Van HornMorris, Monoids in the mapping class group. Geom. Topol. Monographs 19 (2015) 319365.
 [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, Rightveering diffeomorphisms of compact surfaces with boundary. Invent. Math. 169 (2007), no. 2, 427449.
 [4] Tetsuya Ito and Keiko Kawamuro, Essential open book foliations and fractional Dehn twist coefficient