Brieskorn manifolds, generalised Sieradski groups, and coverings of lens spaces

TL;DR

This paper explores the relationship between generalized Sieradski groups and Brieskorn manifolds, showing that certain groups correspond to cyclic branched coverings of lens spaces, extending previous results to new cases.

Contribution

It establishes that groups S(2n,5,2) correspond to n-fold cyclic branched coverings of lens spaces, generalizing earlier findings for S(2n,3,2).

Findings

S(2n,3,2) groups are geometric and correspond to 3-manifolds.

S(2n,5,2) groups also correspond to cyclic branched coverings of lens spaces.

Manifolds are classified using Matveev's Recognizer program.

Abstract

The Brieskorn manifolds $B(p,q,r)$ are the $r$-fold cyclic coverings of the 3-sphere $S^{3}$ branched over the torus knot $T(p,q)$. The generalised Sieradski groups $S(m,p,q)$ are groups with $m$-cyclic pre\-sen\-tation $G_{m}(w)$, where defining word $w$ has a special form, depending of $p$ and $q$. In particular, $S(m,3,2) = G_{m}(w)$ is the group with $m$ generators $x_{1}, \ldots, x_{m}$ and $m$ defining relations $w(x_{i}, x_{i+1}, x_{i+2})=1$, where $w(x_{i}, x_{i+1}, x_{i+2}) = x_{i} x_{i+2} x_{i+1}^{-1}$. Presentations of $S(2n,3,2)$ in a certain form $G_{n}(w)$ were investigated by Howie and Williams. They proved that the $n$-cyclic presentations are geometric, i.e., correspond to the spines of closed orientable 3-manifolds. We establish an analogous result for the groups $S(2n,5,2)$. It is shown that in both cases the manifolds are $n$-fold cyclic branched coverings of lens spaces. To classify some of constructed manifolds we used the Matveev's computer program "Recognizer".