Complete invariant geodesic metrics on outer spaces and Jacobian   varieties of tropical curves
  Let $\mathrm{Out}(F_n)$ be the outer automorphism group of the free group $F_n$. It acts properly on the outer space $X_n$ of marked metric graphs, which is a finite-dimensional infinite simplicial complex with some simplicial faces missing. In this paper, we construct complete geodesic metrics and complete piecewise smooth Riemannian metrics on $X_n$ which are invariant under $\mathrm{Out}(F_n)$. One key ingredient is the identification of metric graphs with tropical curves and the use of the tropical Jacobian map from the moduli space of tropical curves to the moduli space of principally polarized tropical abelian varieties. 
Lizhen Ji
math.GT