Complete invariant geodesic metrics on outer spaces and Jacobian varieties of tropical curves

TL;DR

This paper constructs complete, invariant geodesic and Riemannian metrics on the outer space of marked metric graphs, using tropical geometry and Jacobian maps to study the structure of $ ext{Out}(F_n)$ actions.

Contribution

It introduces new complete invariant metrics on outer space by leveraging tropical curves and their Jacobians, bridging geometric group theory and tropical geometry.

Findings

Constructed complete geodesic metrics on outer space

Established invariance under $ ext{Out}(F_n)$ actions

Connected metric graphs with tropical Jacobian varieties

Abstract

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.