A group G admitting a cyclic presentation $\mathcal{P}={\mathcal{P}}_n(w)$ determines $E=G{⋊}_{\theta }{C}_n$ called the shift extension of G. A spherical picture over a presentation for E lifts to one over $\mathcal{P}$, and here it is shown how this picture determines a Heegaard diagram for a 3-manifold inducing $\mathcal{P}$ analogous to how spherical diagrams are associated with face pairings. 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, 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 $\mathfrak{Z}$ are also considered.

承认循环呈现的G组$\mathcal{磷}={\mathcal{磷}}_n(瓦)$ 决定 $乙=G{⋊}_{\theta }{C}_n$称为G的平移扩展。E演示文稿上的球形图片提升到一个$\mathcal{磷}$，这里展示了这张图片如何确定 3 流形诱导的 Heegaard 图 $\mathcal{磷}$类似于球面图如何与人脸配对相关联。该方法用类型的组来演示$\mathfrak{Z}$，一个无限的循环呈现的群，其移位扩展分裂在中心扩展的三角形群上。由此产生的流形是 Dunwoody 流形的例子，并分解为两个子族，其中一个包括并扩展了 Cavicchioli、Repovs 和 Spaggiari [12] 的早期结果，例如包含某些 Brieskorn [26] 和所有 Sieradski 流形 [31]。类型群有限性和不动点问题解的拓扑结果$\mathfrak{Z}$ 也在考虑中。