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

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 $x_1,\dots ,x_m$ 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".

arXiv comment12 pages, 5 figues