On the topology of the groups of type [formula omitted

Abstract

A group G admitting a cyclic presentation P=Pn(w) determines E=G⋊θCn called the shift extension of G. A spherical picture over a presentation for E lifts to one over P, and here it is shown how this picture determines a Heegaard diagram for a 3-manifold inducing P analogous to how spherical diagrams are associated with face pairings. The method is demonstrated with the groups of type Z, an infinite family of cyclically presented groups whose shift extensions split over centrally extended triangle groups. The resulting manifolds are examples of Dunwoody manifolds and break down into two subfamilies, one of which includes and extends earlier results of Cavicchioli, Repovs, and Spaggiari [12], for example containing certain Brieskorn [26] and all Sieradski manifolds [31]. Topological consequences of the solution to the finiteness and fixed point problems for groups of type Z are also considered.