# Equivalence of cyclic $p$-squared actions on handlebodies

## TL;DR

This paper classifies all orientation-preserving cyclic p-squared actions on 3D handlebodies by using graphs of groups, providing a comprehensive enumeration of such symmetries up to equivalence.

## Contribution

It introduces an algebraic framework using graphs of groups to classify and count all cyclic p-squared actions on handlebodies.

## Key findings

- Complete enumeration of _{p^2}-actions on handlebodies
- Characterization of quotient orbifolds via graphs of groups
- Explicit formulas for counting distinct actions

## Abstract

In this paper we consider all orientation-preserving $\mathbb{Z}_{p^2}$-actions on 3-dimensional handlebodies $V_g$ of genus $g>0$ for $p$ an odd prime. To do so, we examine particular graphs of groups $(\Gamma($v$),\mathbf{G(v)})$ in canonical form for some 5-tuple v $=(r,s,t,m,n)$ with $r+s+t+m>0$. These graphs of groups correspond to the handlebody orbifolds $V(\Gamma($v$),{\mathbf{G(v)}})$ that are homeomorphic to the quotient spaces $V_g/\mathbb{Z}_{p^2}$ of genus less than or equal to $g$. This algebraic characterization is used to enumerate the total number of $\mathbb{Z}_{p^2}$-actions on such handlebodies, up to equivalence.