Sphere paths in outer space

TL;DR

This paper investigates the geometry of outer space by estimating the lengths of sphere paths using surgery processes, relating metrics to intersection numbers, and analyzing their progress within thick parts of the space.

Contribution

It introduces new estimates for sphere path lengths in outer space and connects the Lipschitz metric to intersection numbers, advancing understanding of outer space geometry.

Findings

Sphere paths can be estimated using surgery processes.

Paths in thick parts of outer space make definite progress.

Lipschitz metric relates to intersection numbers in outer space.

Abstract

We give estimates on the length of paths defined in the sphere model of outer space using a surgery process, and show that they make definite progress in some sense when they remain in some thick part of outer space. To do so, we relate the Lipschitz metric on outer space to a notion of intersection numbers.