We study some algebraic properties of a class of group presentations depending on a finite number of integer parameters. This class contains many well-known groups which are interesting from a topological point of view. We find arithmetic conditions on the parameters under which the considered groups cannot be fundamental groups of hyperbolic 3-manifolds of finite volume. Then we investigate the asphericity for many presentations contained in our family.

Abstract. This is a survey of results and open problems on compact 3-manifolds which admit spines corresponding to cyclic presentations of groups. We also discuss questions concerning spines of knot manifolds and regular neighborhoods of homotopically PL embedded compacta in 3-manifolds.

We study various questions about the generalised Fibonacci groups, a family of cyclically presented groups, which includes as special cases the Fibonacci, Sieradski, and Gilbert-Howie groups.

We introduce a family of cyclic presentations of groups depending on a finite set of integers. This family contains many classes of cyclic presentations of groups, previously considered by several authors. We prove that, under certain conditions on the parameters, the groups defined by our presentations cannot be fundamental groups of closed connected hyperbolic 3–dimensional orbifolds (in particular, manifolds) of finite volume. We also study the split extensions and the natural HNN extensions of these groups, and determine conditions on the parameters for which they are groups of 3–orbifolds and high–dimensional knots, respectively.

We study the homotopy type of closed 4-manifolds (or oriented Poincare spaces) with free fundamental group. This gives a partial solution to problem N. 4.53 of [10]. Then we extend the Whitehead-Novikov-Wall theorem for this class of manifolds by using surgery techniques.