Degenerating Slopes with Respect to Heegaard Distance

LetM = H+∪SH− be a genus g Heegaard splitting with Heegaard distance n ≥ κ + 2: (1) Let c1, c2 be two slopes in the same component of ∂−H−, such that the natural Heegaard splitting M i = H+ ∪S (H− ∪ci 2 − handle) has distance less than n, then the distance of c1 and c2 in the curve complex of ∂−H− is at most 3M+ 2, where κ and M are constants due to Masur-Minsky. (2) Let M∗ be the manifold obtained by attaching a collection of handlebodies H to ∂−H− along a map f from ∂H to ∂−H−. If f is a sufficiently large power of a generic pseudo-Anosov map, then the distance of the Heegaard splitting M = H+ ∪ (H− ∪f H ) is still n. The proofs rely essentially on Masur-Minsky’s theory of curve complex.