An upper bound on distance degenerate handle additions

TL;DR

This paper establishes an upper bound on the set of distance degenerate curves in the curve complex associated with a Heegaard splitting of a 3-manifold, providing a finite diameter region containing all such curves.

Contribution

It introduces a bound on the diameter of the subset of the curve complex that contains all distance degenerate curves for high-distance Heegaard splittings.

Findings

Existence of a finite diameter ball in the curve complex containing all distance degenerate curves.

Applicable to Heegaard splittings with distance at least 3.

Provides a structural understanding of degeneracy in the curve complex.

Abstract

We prove that for any distance at least 3 Heegaard splitting and a boundary component $F$, there is a diameter finite ball in the curve complex $\mathcal {C}(F)$ so that it contains all distance degenerate curves or slopes in $F$.