TY - JOUR
T1 - Tropicalization of group representations
AU - Alessandrini, Daniele
PY - 2008
Y1 - 2008
N2 - 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.
AB - 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.
KW - Bruhat-tits building
KW - Character
KW - Projective structure
KW - Representation
KW - Tropical geometry
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U2 - 10.2140/agt.2008.8.279
DO - 10.2140/agt.2008.8.279
M3 - Article
AN - SCOPUS:72949095145
SN - 1472-2747
VL - 8
SP - 279
EP - 307
JO - Algebraic and Geometric Topology
JF - Algebraic and Geometric Topology
IS - 1
ER -