Tropicalization of group representations

Abstract

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 ${\pi }_1\left(M\right)$ in $S{L}_{n+1}\left(ℝ\right)$. 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 ${\pi }_1\left(M\right)$ 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 $S{L}_{n+1}$ 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.