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
and  Keiko Kawamuro Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
July 26, 2019
###### Abstract.

We answer Question 6.12 in  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

 c(−,C):Mod(S)→Q

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

 FDTCr,C(S):={ϕ∈Mod(S)∣∣ϕ=idS or c(ϕ,C)≥r}

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

 ⋃r>0FDTCr(S)⊊Veer+(S)⊊FDTC0(S).

## 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

 D(q):=supg,h∈G|q(gh)−q(g)−q(h)|

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:

 ¯¯¯q(g):=limn→∞q(gn)n

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

 τ:˜Homeo+(S1)→R

given by:

 τ(g)=limn→∞gn(0)n=limn→∞gn(x)−xn

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

 Gr:={g∈G∣∣g=idG or q(g)≥r}.

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) q(gn)−q((gh)n)=n(q(g)−q(gh))>D(q).

By the definition of the defect we have

 (2.2) ∣∣q(g−n(gh)n)+q(gn)−q((gh)n)∣∣≤D(q).

By (2.1) and (2.2) we get

 q(g−n(gh)n)≤−q(gn)+q((gh)n)+D(q)<−D(q)+D(q)=0.

Let and . By the property we have .

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

 (G−n(GH)n)(0)>(G−nGn)(0)=0,

## 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) T±1γ∈FDTC0,C(S)⊂FDTCr,C(S).

(Case 1) Recall that for any surface of genus the group is generated by Dehn twists along non-separating curves (see p.114 of ). 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

 T±1x,T±1y,T±1z∈FDTC0,C(S)⊂FDTCr,C(S).

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

 c((TxTyTz)−n,C)=c(T−naT−nbT−ncT−nd,C)=−n

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 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

 Tγ∈Veer+(S)∖(⋃r>0FDTCr(S)) and T−1γ∈FDTC0(S)∖Veer+(S).

## 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

•  John B. Etnyre and Jeremy Van Horn-Morris, Monoids in the mapping class group. Geom. Topol. Monographs 19 (2015) 319-365.
•  Benson Farb and Dan Margalit, A primer on mapping class groups. Princeton Mathematical Series, 49. Princeton University Press, Princeton, NJ, 2012. xiv+472 pp.
•  Ko Honda, William H. Kazez and Gordana Matić, Right-veering diffeomorphisms of compact surfaces with boundary. Invent. Math. 169 (2007), no. 2, 427-449.
•  Tetsuya Ito and Keiko Kawamuro, Essential open book foliations and fractional Dehn twist coefficient
You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters   