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.
"Sphere paths in outer space." Algebr. Geom. Topol. 12 (4) 2493 - 2517, 2012. https://doi.org/10.2140/agt.2012.12.2493