Tropicalization of group representations

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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 semialgebraic subsets of the variety of characters of representations of {big square intersection}1(M) i n S L 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 {big square intersection}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 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.

Original languageEnglish
Pages (from-to)279-307
Number of pages29
JournalAlgebraic and Geometric Topology
Volume8
Issue number1
DOIs
StatePublished - 2008

Keywords

  • Bruhat-tits building
  • Character
  • Projective structure
  • Representation
  • Tropical geometry

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