On closed leaves of foliations, multisections and stable commutator lengths

TL;DR

This paper constructs examples of foliations with minimal genus closed leaves on 4-manifolds and explores stable commutator lengths in mapping class groups to address bounds on multisection self-intersections.

Contribution

It provides new examples of foliations with genus-minimizing closed leaves and links stable commutator lengths to bounds on multisection self-intersections in surface bundles.

Findings

Examples of foliations with genus-minimizing closed leaves on 4-manifolds.

Asymptotic bounds on self-intersection numbers of multisections.

Connections between stable commutator lengths and surface bundle properties.

Abstract

We give examples of foliations that answer two questions posed by Mitsumatsu and Vogt about the genus minimising properties of closed leaves of 2-dimensional foliations on 4-manifolds. By studying stable commutator lengths in certain stable mapping class groups, we also answer an asymptotic version of another question of theirs concerning bounds on self-intersection numbers of multisections in surface bundles.