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 semi-algebraic subsets of the variety of characters of representations of pi(1) (M) in SLn+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 pi(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 SLn+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 Teichmuller spaces. In the general case they are contractible metric spaces with a structure of tropical projective spaces.