On a classification of irreducible periodic diffeomorphisms on surfaces which commute with certain involution

TL;DR

This paper classifies irreducible periodic automorphisms of surfaces that commute with specific involutions, extending previous work on hyperelliptic automorphisms to cases where the quotient surface is a torus.

Contribution

It provides a conjugacy classification for irreducible periodic automorphisms commuting with involutions resulting in a torus quotient, expanding understanding beyond hyperelliptic cases.

Findings

Classification up to conjugacy achieved

Automorphisms commuting with involutions with torus quotient characterized

Extends Ishizaka's hyperelliptic automorphism classification

Abstract

Ishizaka classified up to conjugacy hyperelliptic periodic automorphisms of a surface. Here, an involution $I$ on a surface $\Sigma_{g}$ is hyperelliptic if and only if $\Sigma_{g}/\langle I \rangle$ is homeomorphic to $S^2$. In this article, we give a classification up to conjugacy for irreducible periodic automorphisms of a surface $\Sigma_{g}$ which commute with involutions $\iota$ such that $\Sigma_{g}/\langle \iota \rangle$ is homeomorphic to $T^{2}$.