Limits of tropical linear spaces

The building of PGL parametrizes norms on a finite-dimensional vector space up to homothety and was first studied by Goldman and Iwahori as a piecewise-linear analogue of symmetric spaces. Tropical geometry, on the other hand, provides piecewise-linear shadows of algebraic geometry, and the tropicalization of a linear embedding depends only on the associated (valuated) matroid.

Motivated by Payne's influential result that the Berkovich analytification of a variety is the limit of all its tropicalizations, we show that a natural compactification of the building of PGL is the limit of all tropicalized linear subspaces, as the embedding and ambient projective dimension vary. This space turns out to be the tropical linear space of the universal realizable valuated matroid, extending a result of Dress and Terhalle. Time permitting, we will discuss real tropical analogues of these results. This is based on joint work with Luca Battistella, Kevin Kühn, Martin Ulirsch, and Alejandro Vargas.