Algebraic & Geometric Topology 8 (2008) 279–307

DOI: 10.2140/agt.2008.8.279

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 π₁(M) in SL_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 π₁(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.

Keywords

projective structure, Bruhat–Tits building, tropical geometry, character, representation

Mathematical Subject Classification

Primary: 51E24, 57M50, 57M60, 57N16

References

Publication

Received: 26 July 2007
Accepted: 20 November 2007
Published: 12 March 2008

Authors