On stable commutator length in hyperelliptic mapping class groups

TL;DR

This paper establishes new bounds and exact values for stable commutator length in hyperelliptic mapping class groups, utilizing quasimorphisms and -signatures to analyze algebraic properties.

Contribution

It provides the first explicit bounds and calculations for stable commutator length in these groups, introducing methods involving quasimorphisms and -signatures.

Findings

New upper bounds on stable commutator length for Dehn twists

Exact stable commutator length for specific elements

Linearly independent quasimorphisms in small pointed 2-sphere groups

Abstract

We give a new upper bound on the stable commutator length of Dehn twists in hyperelliptic mapping class groups, and determine the stable commutator length of some elements. We also calculate values and the defects of homogeneous quasimorphisms derived from \omega-signatures, and show that they are linearly independent in the mapping class groups of pointed 2-spheres when the number of points is small.