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

TL;DR

The paper addresses the problem of classifying irreducible periodic diffeomorphisms on a genus $g$ surface that commute with an involution $\iota_g$ whose quotient is a torus $T^2$. It develops a framework based on total valency (Nielsen data) and leverages Riemann–Hurwitz, Harvey’s criterion for cyclic coverings, and torus-quotient dynamics to constrain possible lifts. The authors obtain a finite list of conjugacy classes for such irreducible actions, realized by specific generators $h_{n,p}$ (e.g., $h_{6,1}$, $h_{8,1}$, $h_{8,5}$, $h_{12,2}$, $h_{12,3}$) up to conjugacy, and prove the nonexistence of irreducible roots for certain constructed examples $F_1$ and $F_2$. This sharpens the understanding of symmetry groups of surfaces with torus quotients and highlights a marked contrast with Ishizaka’s hyperelliptic (sphere-quotient) classification, by yielding a finite, explicit catalog of possibilities.

Abstract

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

Figures (12)

• Figure 1: An involution $\iota_{g}$ such that $\Sigma_g / \langle \iota_g \rangle$ is homeomorphic to $T^2$
• Figure 2: A diffeomorphism $h_{n,p}$
• Figure 3:
• Figure 5: Glue the boundaries of the complements of disk neighborhoods of the fixed points of $(\Sigma_g, h_{4g,2g})$ and $(\Sigma_g, h_{4g,2g}^{4g-1})$ to obtain a diffeomorphism that commutes with an involution $\iota_{2g+1}$
• Figure 6:

Theorems & Definitions (43)

• Example 1.1: Takahashi
• Example 1.2
• Theorem 1.3
• Theorem 2.1: Nielsen Nielsen, see also Ashikaga-Ishizaka, Hirose
• Proposition 2.2: Nielsen Nielsen
• Remark 2.3
• Proposition 2.4: Riemann-Hurwitz formula
• Proposition 2.5: Harvey Harvey
• Theorem 2.6: Kasahara Kasahara
• Theorem 2.7: Wiman, see Hirose-Kasahara