## Abstract

A Brieskorn manifold B(p, q, r) is the r-fold cyclic covering of the 3-sphere S^{3} branched over the torus knot T(p, q). Generalized Sieradski groups S(m, p, q) are groups with an m-cyclic presentation G_{m}(w), where the word w has a special form depending on p and q. In particular, S(m, 3, 2) = G_{m}(w) is the group with m generators x_{1},…, x_{m} and m defining relations w(x_{i}, x_{i+1}, x_{i+2}) = 1, where w(x_{i}, x_{i+1}, x_{i+2}) = x_{i}, x_{i+2}, x_{i+1}^{−1}. Cyclic presentations of the groups S(2n, 3, 2) in the form G_{n}(w) were investigated by Howie and Williams, who showed that the n-cyclic presentations are geometric, i.e., correspond to spines of closed 3-manifolds. We establish a similar result for the groups S(2n, 5, 2). It is shown that in both cases the manifolds are n-fold branched cyclic coverings of lens spaces. To classify some of the constructed manifolds, we use Matveev’s computer program “Recognizer”.

Original language | English |
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Pages (from-to) | S175-S185 |

Number of pages | 11 |

Journal | Proceedings of the Steklov Institute of Mathematics |

Volume | 304 |

DOIs | |

Publication status | Published - 1 Apr 2019 |

## Keywords

- 3-manifold
- branched covering
- Brieskorn manifold
- cyclically presented group
- lens space
- Sieradski group