ON HYPERBOLIC 3-MANIFOLDS WITH SYMMETRIC HEEGAARD SPLITTINGS

Abstract

We construct a family of hyperbolic 3-manifolds by pairwise identifications of faces in the boundary of certain polyhedral 3-balls and prove that all these manifolds are cyclic branched coverings of the 3- sphere over certain family of links with two components. These extend some results from (5) and (10) concerning with the branched coverings of the whitehead link. There are two well known results about the realization of closed 3-manifolds. One is that any closed orientable 3-manifold can be obtained by Dehn surgeries on the components of an oriented link in the 3-sphere. The other one says that any closed 3-manifold can be represented as a branched covering of some link in the 3-sphere. So if we consider a link in the 3-sphere, we can construct many classes of closed orientable 3-manifolds by considering its branched coverings or Dehn surgeries along it. The description of closed 3-manifolds as polyhedral 3-balls, whose finitely many boundary faces are glued together in pairs, is a further standard way to construct 3-manifolds (see (3), (4), (10), (11), and (12)). If the polyhedral 3-ball admits a geometric structure and the face identifica- tion is performed by means of geometric isometries, then the same geometric structure is inherited by the quotient manifold (see (10), (12), and (15)). Many authors have studied the connections between the face identification procedure and the representation of closed 3-manifolds as branched coverings of the 3- sphere. In (10) Helling, Kim and Mennicke considered a family of polyhedral 3-balls Pn depending on a positive integer n, and for any coprime positive in- tegers n and k, they defined a pairwise gluing of faces in the boundary of Pn yielding a closed orientable 3-manifold Mn,k. In the sequel, they proved that Mn,k is an n-fold strongly cyclic covering of the 3-sphere branched over the Whitehead link and classified, up to isometry, those coverings. More general