Fix a complete non-Archimedean valued field $K$. Any subscheme $X$ of $(K*)^n$ can be tropicalized by taking the closure of the coordinate-wise valuation. This was the foundational setup of tropical geometry. In recent years the picture has been completed to a commutative diagram including the analytification of $X$ in the sense of Berkovich. The corresponding tropicalization map is continuous and surjective and is also coordinate-dependent. Work of Payne shows that the Berkovich space $X^{an}$ is homeomorphic to the projective limit of all tropicalizations. A natural question arises: given a concrete $X$, can we find a split torus containing it and a continuous section to the tropicalization map? If the answer is yes, we say that the tropicalization is faithful.
The curve case was worked out by Baker, Payne and Rabinoff. The underlying space of an analytic curve can be endowed with a polyhedral structure locally modeled on an R-tree with a canonical metric on the complement of its set of leaves. In this case, the tropicalization map is piecewise linear on the skeleton of the curve (modeled on a semistable model of the algebraic curve). In higher dimensions, no such structures are available in general, so the question of faithful tropicalization becomes more challenging.
In this talk, we show that the tropical projective Grassmannian of planes is homeomorphic to a closed subset of the analytic Grassmannian in Berkovich sense. Our proof is constructive and it relies on the combinatorial description of the tropical Grassmannian (inside the split torus) as a space of phylogenetic trees by Speyer-Sturmfels. We also show that both sets have piecewiselinear structures that are compatible with our homeomorphism and characterize the fibers of the tropicalization map as affinoid domains with a unique Shilov boundary point. Time permitted, we will discuss the combinatorics of the aforementioned space of trees inside tropical projective space