Simple Connectivity of Spheres in the Curve Complex

Richard Cao; Rishibh Prakash

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arXiv · 2510.23890

Simple Connectivity of Spheres in the Curve Complex

Richard Cao·Rishibh Prakash

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Abstract

For a fixed radius $r$ and a point $o$ in the curve complex of a surface, we define the sphere of radius $r$ to be the induced subgraph on the set of vertices of distance $r$ from $o$. We show that these spheres are almost simply connected for surfaces of high enough complexity, in the sense that loops in the sphere bound an embedded disk contained in a small neighborhood of the sphere.

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Richard Cao, Rishibh Prakash. 2025-10-27. Simple Connectivity of Spheres in the Curve Complex. https://arxiv.org/abs/2510.23890

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