We show that the Hausdorff distance between any forward and any backward surgery paths in the sphere graph is at most . 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 . 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.
"Uniform fellow traveling between surgery paths in the sphere graph." Algebr. Geom. Topol. 17 (6) 3751 - 3778, 2017. https://doi.org/10.2140/agt.2017.17.3751