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 \(X^{\textrm{an}}\) $X^{\text{an}}$ , called the extended skeleton. In this case, we say the tropicalization is “fully faithful.” Additionally, given a map from X to a toric variety Y, which induces a fully faithful tropicalization, we show that we can extend the map to \(X \rightarrow Y \times (\textbf{P}^1)^n\) $X\to Y\times {({\mathbf{P}}^1)}^n$ such that the new tropicalization is smooth and fully faithful.