Combinatorics of a class of groups with cyclic presentation

We study a family of combinatorial closed 3-manifolds obtained from polyhedral 3-balls, whose finitely many boundary faces are glued together in pairs. Then we determine geometric presentations of their fundamental groups, and find conditions under which such groups are infinite and/or aspherical. Moreover, we show that our presentations are a natural generalization of those considered by Prishchepov in [M.I. Prishchepov, Asphericity, atoricity and symmetrically presented groups, Comm. Algebra 23 (13) (1995) 5095–5117]. Finally we illustrate some geometric and topological properties of the constructed manifolds, as, for example, a combinatorial description of them as cyclic coverings of the 3-sphere branched over some specified classes of knots.