High distance Heegaard splittings of 3-manifolds

J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631–657] used the curve complex associated to the Heegaard surface of a splitting of a 3-manifold to study its complexity. He introduced the distance of a Heegaard splitting as the distance between two subsets of the curve complex associated to the handlebodies. Inspired by a construction of T. Kobayashi [T. Kobayashi, Casson–Gordon's rectangle condition of Heegaard diagrams and incompressible tori in 3-manifolds, Osaka J. Math. 25 (3) (1988) 553–573], J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631–657] proved the existence of arbitrarily high distance Heegaard splittings.In this work we explicitly define an infinite sequence of 3-manifolds {Mn} via their representative Heegaard diagrams by iterating a 2-fold Dehn twist operator. Using purely combinatorial techniques we are able to prove that the distance of the Heegaard splitting of Mn is at least n.Moreover, we show that π1(Mn) surjects onto π1(Mn−1). Hence, if we assume that M0 has nontrivial boundary then it follows that the first Betti number β1(Mn)>0 for all n⩾1. Therefore, the sequence {Mn} consists of Haken 3-manifolds for n⩾1 and hyperbolizable 3-manifolds for n⩾3.