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 [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

 c(−,C):Mod(S)→Q

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]):

 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 [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

 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

• [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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