Degenerating slopes with respect to Heegaard distance

Abstract

Let $M=H_{+}\cup_{S} H_{-}$ be a genus $g$ Heegaard splitting with Heegaard distance $n\geq \kappa+2$: (1) Let $c_{1}$, $c_{2}$ be two slopes in the same component of $\partial_{-}H_{-}$, such that the natural Heegaard splitting $M^{i}=H_{+}\cup_{S} (H_{-}\cup_{c_{i}} 2-handle)$ has distance less than $n$, then the distance of $c_{1}$ and $c_{2}$ in the curve complex of $\partial_{-}H_{-}$ is at most $3\mathfrak{M}+2$, where $\kappa$ and $\mathfrak{M}$ are constants due to Masur-Minsky. (2) Let $M^{*}$ be the manifold obtained by attaching a collection of handlebodies $\mathscr{H}$ to $\partial_{-} H_{-}$ along a map $f$ from $\partial \mathscr{H}$ to $\partial_{-} H_{-}$. If $f$ is a sufficiently large power of a generic pseudo-Anosov map, then the distance of the Heegaard splitting $M^{*}=H_{+}\cup (H_{-}\cup_{f} \mathscr{H})$ is still $n$. The proofs rely essentially on Masur-Minsky's theory of curve complex.