Construction of Fully Faithful Tropicalizations for Curves in Ambient Dimension 3

In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the tropicalization has nice properties. We construct, for any Mumford curve $X$, a map to a three-dimensional torus, such that the tropicalization is isometric to a subgraph of the Berkovich space $X^{\rm an}$, called the extended skeleton. In this case, we say the tropicalization is "fully faithful." Additionally, given a map $X$ to a toric variety $Y$, which induces a fully faithful tropicalization, we show that we can extend the map to $X \to Y \times (\mathbf{P}^1)^n$ such that the new tropicalization is smooth and fully faithful.