Heegaard Splittings and Pseudo-anosov Maps

LetM andM− be oriented 3-dimensional handlebodies whose boundary is identified with an oriented surface S of genus g > 1 in such a way that the orientation of S agrees with the orientation of ∂M and does not with the one of ∂M−. Given a mapping class f ∈ MCG(S) we consider the closed, oriented 3-manifold Nf =M + ∪f M− obtained by identifying the boundaries of M and M− via f . In this note we study geometric and topological properties of Nfn where f n is a sufficiently large power of a generic pseudo-Anosov mapping class; here, a pseudo-Anosov mapping class f ∈ MCG(S) is generic if its stable lamination λ is not a limit, in the space PML of projective classes of measured laminations on S, of meridians of M and its unstable lamination λ− is not a limit of meridians of M−. Recall that a meridian in M± is an essential simple closed curve in ∂M± which is homotopically trivial in M±. The term generic is appropriate because Kerckhoff [Ker90] proved that the closure in PML of the set of meridians of M and M− have zero measures with respect to the canonical measure class of PML. We should point out that the above construction is due to Feng Luo by using an idea of Kobayashi (cf. Hempel [Hem01]). These were constructed as examples of Heegaard splittings where the minimum distance in the complex of curves of the Heegaard surfaces between a pair of meridians induced by the two handlebodies is sufficiently large. Our first result is that the manifold Nfn admits, for n large enough, a negatively curved metric: