On upper bounds on stable commutator lengths in mapping class groups

TL;DR

This paper establishes new bounds on the stable commutator lengths of Dehn twists in mapping class groups and hyperelliptic mapping class groups, revealing differences based on curve types.

Contribution

It provides the first known bounds for stable commutator lengths of Dehn twists in specific mapping class groups, highlighting distinctions between separating and nonseparating curves.

Findings

New upper bounds for Dehn twists in mapping class groups

New lower bounds for Dehn twists in hyperelliptic mapping class groups

Stable commutator lengths differ for separating and nonseparating curves in genus 2

Abstract

We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating curve on an oriented closed surface of genus 2 are not equal to each other.