Tropicalization of group representations

Details Tropicalization of group representations Tropicalization of group representations

2007-03-20

arxiv.org/abs/math/0703608v3

Mathematics

Geometric Topology

27 pages, 1 figure; Changes in version 2: minor changes, some references added. Changes in version 3: the paper has been updat

Scientific paper

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 the fundamental group of 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 the group 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 non-archimedean 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.

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