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 1.M / in SLnC1.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 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 SLnC1 to nonarchimedean fields with real surjective valuation. In the case n D 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. 57M60, 57M50, 51E24, 57N16