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DEGENERATING SLOPES WITH RESPECT TO HEEGAARD DISTANCE

By Jiming Ma and Ruifeng Qiu

Abstract

Abstract. Let M = 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 − ∪c i 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

Topics: Heegaard distance, Dehn filling, handle addition
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