Generalized Neuwirth groups and Seifert fibered manifolds (Q2711608)

A class of groups \(\Gamma^k_n\) is considered which have cyclic presentations on \(n\) generators \(x_1, \ldots, x_n\) and \(n\) relators obtained from the word \(x_1x_2 \ldots x_{n-1}x_n^{-k}\) by cyclically permuting the generators \(x_i\). These groups are generalized Fibonacci groups, for \(k=1\) they have been considered by Neuwirth and are known to be isomorphic to the fundamental groups of closed Seifert fiber spaces. The main result of the present paper shows that this remains true also for \(k\geq 2\): the groups \(\Gamma^k_n\) are fundamental groups of certain 3-manifolds explicitly obtained from the 3-ball by identifying in pairs the faces of a polyhedral decomposition of the 2-sphere; using fundamental groups these 3-manifolds are then identified with certain Seifert fiber spaces.