Brieskorn manifolds, generated Sieradski groups, and coverings of lens space

The Brieskorn manifold $\mathscr B(p,q,r)$ is the $r$-fold cyclic covering of the three-dimensional sphere $S^{3}$ branched over the torus knot $T(p,q)$. The generalised Sieradski groups $S(m,p,q)$ are groups with $m$-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 $x_{1},\ldots,x_{m}$ and $m$ defining relations $w(x_{i}, x_{i+1}, x_{i+2})=1$, where $w(x_{i}, x_{i+1}, x_{i+2}) = x_{i} x_{i+2} x_{i+1}^{-1}$. Cyclic presentations of $S(2n,3,2)$ 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 the spines of closed three-dimensional 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 branched cyclic coverings of lens spaces. For the classification of the constructed manifolds, we use Matveev's computer program “Recognizer.”