On the generalized Dunwoody $3$-manifolds

Abstract

We introduce a family of orientable $3$-manifolds induced by certain cyclically presented groups and show that this family of $3$-manifolds contains all Dunwoody $3$-manifolds by using the planar graphs corresponding to the polyhedral description of the $3$-manifolds. As applications, we consider two families of cyclically presented groups, and show that these are isomorphic to the fundamental groups of the certain Dunwoody $3$-manifolds $D_{n}$ ($n \geq 2$) which are the $n$-fold cyclic coverings of the $3$-sphere branched over the certain two-bridge knots, and that $D_{n}$ is the $(\mathbb{Z}_{n}\oplus \mathbb{Z}_{2})$-fold covering of the $3$-sphere branched over two different $\Theta$-curves.