Equivalence of $\mathbb{Z}_{4}$-actions on handlebodies of genus $g$

TL;DR

This paper classifies all orientation-preserving $Z_4$-actions on 3D handlebodies of genus g, using algebraic graph of groups methods to enumerate distinct actions up to equivalence.

Contribution

It provides a complete algebraic framework to classify and count all $Z_4$-actions on handlebodies of genus g, extending understanding of symmetries in 3D topology.

Findings

Enumeration of $Z_4$-actions up to equivalence

Graph of groups characterization of handlebody orbifolds

Explicit counting method for group actions

Abstract

In this paper we consider all orientation-preserving $\mathbb{Z}_{4}$-actions on $3$-dimensional handlebodies $V_g$ of genus $g>0$. We study the graph of groups $(\Gamma($v$),\mathbf{G(v)})$, which determines a handlebody orbifold $V(\Gamma($v$),{\mathbf{G(v)}})\simeq{V_g}/\mathbb{Z}_{4}$. This algebraic characterization is used to enumerate the total number of $\mathbb{Z}_{4}$ group actions on such handlebodies, up to equivalence.