Uniform fellow traveling between surgery paths in the sphere graph

TL;DR

This paper proves that the Hausdorff distance between certain surgery paths in the sphere graph is bounded, using properties of the Guirardel core and Rips moves, revealing uniform fellow traveling behavior.

Contribution

It establishes uniform bounds on the Hausdorff distance between surgery paths in the sphere graph, connecting surgeries to Rips moves on the Guirardel core.

Findings

Hausdorff distance between forward and backward surgery paths is at most 2

Hausdorff distance between any two surgery paths with same endpoints is at most 4

Surgery corresponds to Rips moves on the Guirardel core

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.