The Tropical Moduli Space of Degree-3 Rational Maps

We construct and study the tropical moduli space ${\mathcal{M}}_3^{\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}}$ of degree-$3$ tropical rational maps $\mathbb{T}{\text{PP}}^1\to \mathbb{T}{\text{PP}}^1$ up to post-composition. Using a combinatorial description in terms of slope sequences, we classify all such maps and show that there are exactly ten combinatorial types. This yields a polyhedral model of ${\mathcal{M}}_3^{\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}}$ parametrized by gap lengths between break points. We determine the automorphism groups and obtain a stratification by explicit linear conditions. We also relate the construction to tropical Hurwitz theory and describe a natural compactification via degenerations of the parameters.