Equivalence of Cyclic p-squared Actions on Handlebodies

  • Received : 2017.04.26
  • Accepted : 2018.06.28
  • Published : 2018.09.23


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.


  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.
  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.