On the topology of the groups of type \(\mathfrak{Z}\) (Q2072115)

A group \(G\) admits a cyclic presentation \(P\) if the relators are mapped to each other by certain automorphisms of the free group of a generating set of \(G\). If this generating set consists of \(n\) generators then there is a semi-direct product \(E=G\rtimes C_n\) called the shift extension of \(G\). A spherical picture over a presentation of \(E\) lifts to one over \(P\). In this paper it is shown how this picture determines a Heegaard diagram for a 3-manifold inducing \(P\). The method is demonstrated with the groups of type \(\mathfrak Z\), 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.