Cyclic actions on compression bodies (Q1360187)

Motivated by studying the symmetries of compact orientable surfaces \(\Sigma(g)\) of genus \(g\), the author classifies the cyclic actions on compression bodies and finds necessary and sufficient conditions for a compression body to admit a \(\mathbb{Z}_p\)-action \((p\) is a prime number). Here a compression body is a compact 3-manifold which is homeomorphic to a boundary connected sum of a 3-ball, a handlebody and trivial \(I\)-bundles over closed orientable surfaces \(\Sigma_1,\dots,\Sigma_q\). The fundamental group \(\pi_1M\) of a compression body \(M\) is a free product of surface groups and free groups, that is \(\pi_1M= \pi_1(\Sigma_1) *\cdots* \pi_1 (\Sigma_q)*_rZ\). For the theory see e.g. [\textit{D. McCullough}, \textit{A. Miller} and \textit{B. Zimmermann}, Proc. Lond. Math. Soc., III. Ser., 59, No. 2, 373-416 (1989; Zbl 0638.57017)].