# Equivalence of Cyclic p-squared Actions on Handlebodies

• Accepted : 2018.06.28
• Published : 2018.09.23

#### 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 (${\Gamma}(v)$, 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 (${\Gamma}(v)$, 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.

#### References

1. D. I. Fuchs-Rabinovitch, On the automorphism group of free products, I, Mat. Sb., 8(1940), 265-276.
2. J. Kalliongis and A. Miller, Equivalence and strong equivalence of actions on handle-bodies, Trans. Amer. Math. Soc., 308(2)(1988), 721-745. https://doi.org/10.1090/S0002-9947-1988-0951625-5
3. D. McCullough, A. Miller, and B. Zimmerman, Group actions on handlebodies, Proc. London Math. Soc., 59(3)(1989), 373-416.
4. J. Prince-Lubawy, Equivalence of $-\mathbb-Z}}_4$-actions on handlebodies of genus g, Kyungpook Math. J., 56(2)(2016), 577-582. https://doi.org/10.5666/KMJ.2016.56.2.577