Signatures of surface bundles and stable commutator lengths of Dehn twists

TL;DR

This paper constructs surface bundles with non-zero signatures, providing new bounds on base genus and fiber genus, and improves upper bounds for stable commutator lengths of Dehn twists through explicit factorizations.

Contribution

It introduces four types of surface bundles with non-zero signatures and refines bounds on stable commutator lengths of Dehn twists via explicit factorizations.

Findings

New upper bounds on minimal base genus for fixed signature and fiber genus.

Asymptotic upper bounds for base genus when fiber genus is odd.

Existence of non-holomorphic surface bundle examples.

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 one gives a new asymptotic upper bound for this number in the case that fiber genus is odd. The third one 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 nonseparating 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.