Tropicalization of group representations

In this paper we give an interpretation to the boundary points of the compactification of the parameter space of convex projective structures on an n-manifold M. These spaces are closed semialgebraic subsets of the variety of characters of representations of {big square intersection}₁(M) i n S L _n+1 (R). 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 {big square intersection}₁(M) 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 SL _n+1 to nonarchimedean fields with real surjective valuation. In the case n=1 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.