Abstract:

Affine Bruhat–Tits buildings are geometric spaces extracting the combinatorics of algebraic groups. The building of PGL parametrizes flags of subspaces or lattices in a finite-dimensional vector space—or, equivalently, norms up to homothety and were first studied by Goldman and Iwahori.
Tropical geometry provides a piecewise-linear shadow of algebraic geometry. In the case of a linear embedding P^r to P^n, its tropicalization is a polyhedral complex that depends only on the associated (valuated) matroid.

In this talk, we explore the interplay between the building of PGL and tropical linear spaces. Motivated by Payne’s result that the analytification of a variety can be viewed as the limit of all its tropicalizations, we show that a suitable compactification of the building is the limit of all tropicalized linear subspaces of rank r, as the embedding and ambient projective dimension vary. This space is, in fact, the tropical linear space associated to the universal realizable valuated matroid, extending a result of Dress and Terhalle.

This is joint work with Luca Battistella, Kevin Kühn, Martin Ulirsch, and Alejandro Vargas.