A geometric study of generalized Neuwirth groups

We define a family of groups with balanced presentations and prove that these groups correspond to spines (or, equivalently, to Heegaard diagrams) of a certain class of Seifert fibered 3-manifolds. These manifolds are constructed from triangulated 3-balls by identifying pairs of boundary faces via orientation-reversing homeomorphisms. Then we describe the manifolds as cyclic branched coverings of certain lens spaces when the groups are cyclically presented. Finally, we give explicit computations of the Casson-Walker-Lescop invariant and the Rohlin invariant for many manifolds in the above class.