An upper bound on distance degenerate handle additions

Let $V \cup_S W$ be a Heegaard splitting with a boundary component $F$. If $r$ is an essential simple closed curve or a slope in $F$, then there is a Heegaard splitting $V (r) \cup_S W$ obtained by attaching a $2$-handle along $r$ on $V$. It was conjectured by Ma and Qiu that for almost all choices of $r$, the Heegaard distance $d(V(r),W)$ is the same to $d(V,W)$.

By studying handle additions and local properties of the curve complex, we prove that if the distance of $V \cup_S W$ is at least $3$, then there is a finite diameter ball in the curve complex $\mathcal{C}(F)$ so that it contains all distance degenerate curves or slopes in $F$. Together with a result proved by Lustig and Moriah, it gives an affirmative answer to Ma and Qiu’ conjecture.

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This work was partially supported by NSFC No. 11601065, 12131009 andScience and Technology Commission of Shanghai Municipality (STCSM), grantNo. 22DZ2229014.

Received 16 April 2021

Accepted 13 October 2021

Published 12 August 2024