### Abstract

The first aim of this paper is to give four types of examples of surface bundles over surfaces with non-zero signature. The first example is with base genus 2, a prescribed signature, a 0-section and the fiber genus greater than a certain number which depends on the signature. This provides a new upper bound on the minimal base genus for fixed signature and fiber genus. The second example gives a new asymptotic upper bound for this number in the case that fiber genus is odd. The third example has a small Euler characteristic. The last is a non-holomorphic example. The second aim is to improve upper bounds for stable commutator lengths of Dehn twists by giving factorizations of powers of Dehn twists as products of commutators. One of the factorizations is used to construct the second examples of surface bundles. As a corollary, we see that there is a gap between the stable commutator length of the Dehn twist along a non-separating curve in the mapping class group and that in the hyperelliptic mapping class group if the genus of the surface is greater than or equal to 8.

Original language | English |
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Journal | Journal of the London Mathematical Society |

DOIs | |

Publication status | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- 20F12
- 57M07 (primary)
- 57N05 (secondary)
- 57R22
- 57R55

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Signatures of surface bundles and scl of a Dehn twist.** / Monden, Naoyuki.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Signatures of surface bundles and scl of a Dehn twist

AU - Monden, Naoyuki

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The first aim of this paper is to give four types of examples of surface bundles over surfaces with non-zero signature. The first example is with base genus 2, a prescribed signature, a 0-section and the fiber genus greater than a certain number which depends on the signature. This provides a new upper bound on the minimal base genus for fixed signature and fiber genus. The second example gives a new asymptotic upper bound for this number in the case that fiber genus is odd. The third example has a small Euler characteristic. The last is a non-holomorphic example. The second aim is to improve upper bounds for stable commutator lengths of Dehn twists by giving factorizations of powers of Dehn twists as products of commutators. One of the factorizations is used to construct the second examples of surface bundles. As a corollary, we see that there is a gap between the stable commutator length of the Dehn twist along a non-separating curve in the mapping class group and that in the hyperelliptic mapping class group if the genus of the surface is greater than or equal to 8.

AB - The first aim of this paper is to give four types of examples of surface bundles over surfaces with non-zero signature. The first example is with base genus 2, a prescribed signature, a 0-section and the fiber genus greater than a certain number which depends on the signature. This provides a new upper bound on the minimal base genus for fixed signature and fiber genus. The second example gives a new asymptotic upper bound for this number in the case that fiber genus is odd. The third example has a small Euler characteristic. The last is a non-holomorphic example. The second aim is to improve upper bounds for stable commutator lengths of Dehn twists by giving factorizations of powers of Dehn twists as products of commutators. One of the factorizations is used to construct the second examples of surface bundles. As a corollary, we see that there is a gap between the stable commutator length of the Dehn twist along a non-separating curve in the mapping class group and that in the hyperelliptic mapping class group if the genus of the surface is greater than or equal to 8.

KW - 20F12

KW - 57M07 (primary)

KW - 57N05 (secondary)

KW - 57R22

KW - 57R55

UR - http://www.scopus.com/inward/record.url?scp=85068073416&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068073416&partnerID=8YFLogxK

U2 - 10.1112/jlms.12247

DO - 10.1112/jlms.12247

M3 - Article

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

ER -