TITLE: What is a geodesic in Outer space?
QUESTION [6 upvotes]: The Culler-Vogtmann Outer space $\text{CV}_n$ is an analogue of Teichmuller space for the group $\text{Out}(F_n)$.
Is there any notion of a geodesic path in $\text{CV}_n$? Are there different competing definitions of geodesic?
If so, what would be a simple example of a geodesic path vs. a non-geodesic one, say on $\text{CV}_2$?

REPLY [8 votes]: Besides the geodesic paths of the asymmetric metric $d(\cdot,\cdot)$ that are mentioned in other answers (namely paths such that $d(\gamma(s),\gamma(t)) = t-s$ if $s \le t$), there is another class of paths with many uses known as Stallings fold paths. You can see some discusions of them in the outer space context, with applications, in these lecture notes of Bestvina, these notes of Kapovich and Myasnikov, and this issue of the AMS Memoirs by Handel and myself.