Tropicalization of group representations
Daniele Alessandrini · Algebraic & Geometric Topology · 2008
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 SL nC1 .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 SL nC1 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.