On certain classes of hyperbolic 3-manifolds
The authors study an interesting class of closed orientable (mostly hyperbolic) 3-manifolds \(M(n,k)\), defined for positive integers \(n\) and \(k\), from various different points of view. These manifolds can be obtained by glueing in pairs the faces of certain 3-dimensional polyhedra, by an identification pattern with a rotational symmetry of order \(n\). Using handle decompositions (Heegaard splittings) and handle cancellations (or alternatively, coloured graphs) it is shown that the manifolds \(M(n,k)\) are exactly the \(n\)-fold cyclic branched coverings of the Whitehead link, with branching orders \(n\) and \(n/d\) associated to its two components where \(d = (n,k)\) is the greatest common divisor (so, if \(d>1\), these coverings are not strongly cyclic, and this is one of the first studies of such a situation). This generalizes constructions in a paper by \textit{H. Helling, A. C. Kim} and \textit{J. L. Mennicke} [Some honey-combs in hyperbolic 3-space, Commun. Algebra 23, No. 14, 5169--5206 (1995; Zbl 1002.57503)] for the case \(d=1\) (where hyperbolic realizations of the polyhedra are constructed). The manifolds \(M(n,k)\) are also \(n/d\)-fold strongly cyclic branched coverings of \((d+1)\)-component links \(L_{d+1}\) in the 3-sphere formed by a chain of \(d\) unknotted circles plus their axis of \(d\)-symmetry (\(L_2\) is just the Whitehead link). Presentations of the fundamental groups and the homology groups of the manifolds are obtained, and the manifolds are classified up to homeomorphism resp. isometry. Combinatorial descriptions of the manifolds by coloured graphs are given. Finally, the manifolds obtained by Dehn surgery on the components of the strongly invertible links \(L_{d+1}\) are studied, obtaining railroad systems and spines associated to presentations of their fundamental groups; using the Montesinos algorithm, they are 2-fold branched coverings of links in the 3-sphere which are explicitly determined.
scientific article; zbMATH DE number 1463440