Equivalence of cyclic $p$-squared actions on handlebodies

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.