In this paper we give an interpretation to the boundary points of the
compactification of the parameter space of convex projective structures on an
–manifold
. These
spaces are closed semi-algebraic subsets of the variety of characters of representations
of in
. The
boundary was constructed as the “tropicalization” of this semi-algebraic
set. Here we show that the geometric interpretation for the points of the
boundary can be constructed searching for a tropical analogue to an action of
on a
projective space. To do this we need to construct a tropical projective space with many
invertible projective maps. We achieve this using a generalization of the Bruhat–Tits
buildings for
to nonarchimedean fields with real surjective valuation. In the case
these
objects are the real trees used by Morgan and Shalen to describe the boundary points
for the Teichmüller spaces. In the general case they are contractible metric spaces
with a structure of tropical projective spaces.
