Volume 17, issue 6 (2017)

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Uniform fellow traveling between surgery paths in the sphere graph

Matt Clay, Yulan Qing and Kasra Rafi

Algebraic & Geometric Topology 17 (2017) 3751–3778
Abstract

We show that the Hausdorff distance between any forward and any backward surgery paths in the sphere graph is at most $2$. From this it follows that the Hausdorff distance between any two surgery paths with the same initial sphere system and same target sphere system is at most $4$. Our proof relies on understanding how surgeries affect the Guirardel core associated to sphere systems. We show that applying a surgery is equivalent to performing a Rips move on the Guirardel core.

Keywords
sphere graph, Guirardel core, surgery path
Primary: 20E36