Uniform fellow traveling between surgery paths in the sphere graph

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.