Dehn twist presentations of hyperelliptic periodic diffeomorphisms on closed surfaces

TL;DR

This paper classifies hyperelliptic periodic diffeomorphisms on closed surfaces and provides Dehn twist presentations for their mapping classes, refining previous classifications and enhancing understanding of their structure.

Contribution

It offers a detailed classification of hyperelliptic periodic diffeomorphisms and explicitly constructs their Dehn twist presentations, advancing the understanding of their algebraic and geometric properties.

Findings

Classification of hyperelliptic periodic diffeomorphisms up to conjugacy

Explicit Dehn twist presentations for hyperelliptic periodic mapping classes

Refinement of Ishizaka's classification results

Abstract

We classify up to conjugacy the group generated by a commuting pair of a periodic diffeomorphism and a hyperelliptic involution on an oriented closed surface. This result can be viewed as a refinement of Ishizaka's result on classification of the mapping classes of hyperelliptic periodic diffeomorphisms. As an application, we obtain the Dehn twist presentations of hyperelliptic periodic mapping classes, which are closely related to the ones obtained by Ishizaka.